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Informe final D03I1039

1. 10 0 650 0 00 0 2200 8 0 ezo 0 ozto 220 0 010 0 s20 0 zv 0 620 0 8 1 0 s20 0 9e 0 500 0 0930 200 0 S0 0 p00 0 030 00 0 161 0 6 0 0 1 2 S s00 0 1021 Z00 0 Si 0 SOV AIS y 1 sagon anoj y pwd eee s 9utdg JU 103 eyeq sonny no WOOL SIJPUUIS JOIOWPILG 3 Y ty zy ly ri ery X 17 ZE 7 ey Cortazar and Naranjo 0 7 0 6 0 5 0 4 Std Dev 0 1 01 1999 07 1999 12 1999 06 2000 12 2000 06 2001 12 2001 FIGURE 2 Time series of the standard deviation for state variable x The figure displays the time evolution from January 1999 to December 2001 of the standard deviation for state variable x in the four factor model which is obtained from the first entry of matrix P the covariance matrix of state variable estimation errors 1 4 1 2 ACA ER 1 0 08 Std Dev 00 0 4 0 2 OL 1999 07 1999 12 1999 06 2000 12 2000 06 2001 12 2001 FIGURE 3 Time series of the standard deviation for state variable x The figure displays the time evolution from January 1999 to December 2001 of the standard deviation for state variable x in the four factor model which is obtained from the last diagonal entry of matrix P the covariance matrix of state variable estimation errors statistically significant The standard deviation measurement error par
2. 18 90 50 14 Nota M N moneda nacional M E moneda extranjera SITUACION SIN PROYECTO Situacion para primer ano de actividades MERCADO SITUACION SIN PROYECTO Para cada Producto o Servicio a evaluar aa A A 0 o Mercado Sin Proyecto Producto o Sevicio 1 Unidades Mercado Sin Proyecto Producto o Servicio 1 Unidades INGRESOS Cantidad anual Cantidad anual Tipo de unidad a Mercado Nacional Mercado Export Ingresos Mercado Ingresos Mercado Total de Ingresos Productos o servicios considerar Precio Unitario Unidades Unidades Nacional MM Exportador MM MM Servicios a Usuarios a ES 297 0 0 TOTAL INGRESOS Ce RE COSTOS MANO DE OBRA CALIFICADA Costo mensual Costo total mensual Costo total anual unitario MM Gerente 4000000 Profesional 1350000 Tecnico 750000 Administrativo 550000 Otros definir 0 AA ASA 8 2 MANO DE OBRA NO CALIFICADA Costo mensual Costo total mensual Costo total anual Ne Cargos unitario Obreros y Jornaleros Otros definir INSUMOS PARA LA PRODUCCION Insumos para la Tipo de unidad a Costo Unitario Cantidad mensual Costo total mensual Costo total anual producci n considerar Unidades MM MM Insumos de Oficina 1 Promedio 500000 2 Si 2 0 0 0 0 3 0 0 0 0 PAS A AO A OOO Y BIENES DE CAPITAL Moneda AMAM Tipo de cambio AMAM Tipo de unidad a Costo total anual Bienes de Capital considerar Cantidad Costo Unitario
3. MM 1 N computadores 8 500000 2 Mantenci n Oficinas 10 1000000 3 0 0 A LA 4 10 0 TOTALES es TE C OTROS COSTOS Tipo de unidad a Costo Unitario Cantidad mensual Costo total mensual Costo total anual Otros costos considerar Unidades MM MM 1 Arriendo Mensual 1000000 1 12 2 0 0 0 3 0 0 0 Pd A 2 TOTAL COSTOS ANUALES 276 RESUMEN DE INVERSIONES ITEM EN MONEDA EN MONEDA TOTAL GENERAL naciona oms Jexrmasena ans cas Otros definir TOTAL po Maquinaria y equipos loa le SITUACION CON PROYECTO Situaci n para primer a o de actividades CURVA DE ADOPCION DE LA TECNOLOGIA Para cada Producto o Servicio Mercado Con Proyecto Producto 1 Unidades Mercado Con Proyecto Producto 1 Unidades INGRESOS Moneda Extranjera EA Tipo de cambio gt antidad anual antidad anua Productos o Tipo de unidad a Mercado Nacional Mercado Export Ingresos Moneda Ingresos Moneda Total de Ingresos servicios considerar Precio Unitario Unidades Unidades Nacional MM Extranjera MM MM EE Le MEN MN E lt A COSTOS MANO DE OBRA CALIFICADA Moneda Extranjera IEA Tipo de cambio Cid Para todos los productos y todas las Unidades de Negocio osto total anual osto total anual Costo mensual Costo total mensual Moneda Nacional Moneda Extranjera N Puestos Cargos unitario MM Gerente 4000000 0 Profesional 1350000 113 4 Tecnico 750000 Administrati
4. 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 IJFE 317 TERM STRUCTURE ESTIMATION 11 Bond Maturity Years Jul 2001 Chronological Time Dec 2001 Figure 4 Graphical description of available Chilean government inflation protected discount and coupon bond daily data for the second semester of 2001 A black cell indicates that data were available for the corresponding maturity at a given day three factors Therefore this important difference in estimation errors suggests that a three factor model is necessary to explain the complex dynamics of the Chilean yield curve To illustrate the ability of the approach to fit observed prices on a day with a large number of transactions Figure 5 shows the yield curve derived from the model for 01 09 1997 We see that the model is able to fit very well observed yields and this is representative of the sample period Recall that in Figure 2 we illustrated the inability of the curve fitting methods to provide for reliable long term rates for a day when only short term bonds were traded Figure 6 shows the yield curve obtained for the same day 10 06 1999 using our proposed methodology We see that the estimated yield curve not only correctly fits observed yields for that day but also is consistent with the previous day observations Note that the yield curve shown has been constructed using
5. Calculation Time Mine LSM Calculation Time Mine 50 LSM Reduced Base 0 T 2 12 22 32 Number of Regressors Fig 10 Relative computer calculation time for solving the extended Brennan and Schwartz mine model as a function of the number of regressors when using the standard and the reduced base implementation of the LSM method 5 Conclusions Real options valuation ROV is an emerging paradigm that provides helpful insights for both valuing and managing real assets It provides more precise quantifications on the value of available strategic and operational flexibilities than traditional discounted cash flow techniques Despite its potential the ROV approach has not yet made a strong inroad in corporate decision making due to several reasons one of which is the requirement to keep models too simple to obtain solutions within a reasonable amount of effort In this paper we show how it is possible to solve complex multidimensional American options using computer based simulation procedures The implementation is validated using the one factor Brennan and Schwartz 23 model with the reported finite difference solution We then extend the Brennan and Schwartz 23 to include a three factor price model and solve it using the proposed methodology Comparative static analyses are provided This paper argues that these new simulation methods have the potential of expanding significantly the use of the ROV appro
6. S00 0 90Z3 0 P00 0 01Z 0 20 0 s 900 0 evt 010 0 6ez t 200 0 1990 oo o tsp o 00 0 Siyo 40100 IUO SLOFIVY ON S404904 23A S4010 anog D puvd 140430 4 I4 100 8SZ 1 92z eS 000 0 00 0 8900 800 6500 621 0 5900 20 0 S90 0 ELO O 0200 900 0 t100 981 o 920 0 20 0 roo o 091 0 00 0 9S1 o Coo o 210 100 944 v00 0 1 0 1 Svo zoL 000 0 2000 0000 090 0 1600 1010 S60 0 6910 t800 z100 9900 6100 Ep00 9 0 0 9900 000 r0 0 0100 0v0 0 Z910 seo 0 ere o Se0 0 co o sz0 0 1 20 0 0 2b3 0 2 0 0 S o 500 0 vie o 0100 9 z 0 600 0 0 2 0 00 0 051 0 9000 9 1 0 290 0 1 8e0 0 ss 2 Z100 esei tero 0 1890 uow anj g Maprny 92444 ou vd STAVL unn sno anong ONG LE2z 9Sz 000 0 S10 0 s0 0 9S0 0 1S0 0 210 0 150 0 900 0 z0 0 sto o Coo O 1610 00 0 2910 200 0 089 0 MOLDY ON 0S2 eze 000 0 500 0 ZS1 0 8910 2 00 S10 0 090 0 S100 090 0 900 0 150 0 890 0 1S0 0 oit o Sz0 0 cer o 800 0 0S 0 v00 0 SZO too o Z61 0 900 0 9 9 1 200 0 s8p 0 seseujuged U UMOYS OJE SIQUO PIEPUBIS OJON 7601 280 ZSE 000 0 00 0 640 0 20 0 6800 110 bs0 0 Z00 0 6S0 0
7. 1979 A continuous time approach to the pricing of bonds Journal of Banking and Finance 3 2 133 155 Brenner RJ Harjes RH Kroner KF 1996 Another look at models of the short term interest rate Journal of Financial and Quantitative Analysis 31 85 107 Broze L Scaillet O Zakoian J M 1995 Testing continuous time models of the short term interest rate Journal of Empirical Finance 2 199 223 Chan KC Karolyi GA Longstaff FA Sanders AB 1992 Comparison models of the short term interest rate Journal of Finance 47 1209 1227 Chen R R Scott L 1993 Maximum likelihood estimation for a multifactor equilibrium model of the term structure of interest rates Journal of Fixed Income 3 1431 Chen R R Scott L 2003 Multi factor Cox Ingersoll Ross models of the term structure estimates and tests from a Kalman filter model Journal of Real Estate Finance and Economics 27 2 Cortazar G Schwartz ES 2003 Implementing a stochastic model for oil futures prices Energy Economics 25 3 215 238 Cox JC Ingersoll J Ross S 1985 A theory of the term structure of interest rates Econometrica 53 385 407 Dai Q Singleton KJ 2000 Specification analysis of affine term structure models Journal of Finance 55 5 1943 1978 De Jong F 2000 Time series and cross section information in affine term structure models Journal of Business amp Economics Statistics 18 3 300 314 De Jong F Santa Clara P 1999 The dynamics of the forward interest ra
8. 2007 A Multicommodity Model of Futures Prices Using Futures Prices of One Commodity to Estimate the Stochastic Process of Another 4th Annual Conference of Asia Pacific Association of Derivatives APAD Gurgaon India June 20 22 2007 3 OTROS INFORMES TECNICOS 4 EVALUACI N CIENTIFICO TECNOLOGICA A continuaci n se resume un an lisis FODA del proyecto Fortalezas del Proyecto Las metodolog as cient ficas desarrolladas para abordar ausencia de transacciones La plataforma tecnol gica computacional incluyendo rutinas computacionales y plataforma WEB Las bases de datos construidas El equipo humano especializado capacitado La reputaci n en el mercado Debilidades del Proyecto La vulnerabilidad financiera que lo expone a ataques de eventuales competidores que inicien una guerra de precios Exigencia de mantener innovaci n permanente como protecci n de mercado Oportunidades del Proyecto Posibilidades de expansi n internacional Amenazas Entrada de competencia nacional e internacional 5 EVALUACI N ECON MICO SOCIAL An lisis comparativo con la evaluaci n ex ante de la Formulaci n del Proyecto A continuaci n se discute y actualiza la evaluaci n econ mico social presentada en la Formulaci n del Proyecto Tal como se plantea en la Formulaci n del proyecto el beneficio social principal de este proyecto se genera al contribuir a la modernizaci n del mercado financiero nacional Un
9. Art culos revista nacional ISI Cap tulos libro nacional ISBN Art culos revista internacional ISI 3 Cap tulos libro nacional Libros publicaci n nacional Cap tulos libro internacional ISBN Seminarios nacionales 1 Cap tulos libro internacional Seminarios internacionales 2 1 Libros publicaci n internacional Congresos nacionales 1 ProyectosI amp D Congresos internacionales 2 Tesisdoctorales Simposios nacionales 1 Tesismagister 4 Simposios internacionales 2 IIC J Postdoctorados Cursos TY Proyectos detitulos 3 Reconocimientos de laboratorio Tallere o Presentaciones en Congresos Internacionales 9 Otro especificar Propuestas de normativa o Coo e Otros especificar Oto especificar realizados por el proyecto con expositores y o ponencias nacionales 2 realizados por el proyecto con expositores y o ponencias de extranjeros 3 CAPACIDADES CIENTIFICO TECNOLOGICAS PRODUCTOS Y SERVICIOS DESARROLLADOS POR EL PROYECTO Durante la realizaci n del proyecto se desarroll la capacidad de desarrollar metodolog as e implementar soluciones en el mbito de la Ingenier a Financiera y Gesti n del Riesgo Los tres principales servicios ya desarrollados se distribuyen v a Internet como m dulos independientes de la plataforma RiskAmericaPlus Estos son Servicio 1 M dulo SVC El SVC o Sistema de Valorizaci n de Carteras consi
10. Computers amp Operations Research 35 2008 113 129 117 return volatility declines with maturity One factor mean reverting models can be found for example in 46 48 With one factor models however all futures returns are assumed to be perfectly correlated which is not consistent with empirical evidence To account for a more realistic price behavior two factor models with mean reversion were introduced Examples are 2 4 Later Cortazar and Schwartz 7 proposed a three factor model for commodity prices and estimated it using oil futures showing that the model exhibits low estimation errors In this paper we calibrate the Cortazar and Schwartz 7 three factor model with copper futures and use it as an extension of the Brennan and Schwartz 23 model of a copper mine The model has three state variables the commodity spot price S the demeaned convenience yield y and the expected long term spot price return vr Commodity spot prices follow a geometric Brownian motion Spot price returns have an instantaneous drift equal to the expected long term return v minus short term deviations from the convenience yield y Both y and v are mean reverting the first one to zero and the second one to a long term average V The authors show that the three factors allow for an increased flexibility of the model which makes it able to match both the shape of the futures price curves and also the volatility term structure two key attributes
11. IJFE 317 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 JFE 317 TERM STRUCTURE ESTIMATION El The Kalman filter may be applied to dynamic models that are in a state space representation which include measurement and transition equations At each point in time the measurement equation relates a vector of observable variables z with a vector of state variables x which in general is not observable z Hx d v v N 0 R 11 where z is a m x 1 vector H is a m x n matrix x is an x 1 vector d is a m x 1 vector and v isa m x 1 vector of serially uncorrelated Gaussian disturbances with mean 0 and covariance matrix R Even though we have implicitly assumed that vector z of observable variables is of a fixed size we will later relax this assumption to allow for missing observations Also note that the measurement equation contains a disturbance term to allow for measurement errors in the observed data Measurement equation 11 also assumes the existence of a linear relation between observed variables and state variables This assumption will also be relaxed later on The transition equation describes the dynamics of the state variables X AXi HC En amp N O Q 12 where A is a n x n matrix e is an n x vector and e is an n x 1 vector of serially uncorrelated Gaussian disturbances with mean 0 and covariance matrix Q Under this
12. Instituci n RISKAMERICA Seleccione el archivo que contiene la informaci n a valorizar Archivo Co Fecha 27 03 2007 Seleccione el tipo de servicio O Servicio TasasMercado O Servicio TasasMercado y Valorizaci n O servicio Valorizaci n Descargar Estructura de Tasas Descargar Archivos Hist ricos 4 1 2 M dulo Indices Este m dulo entrega informaci n referida al comportamiento del mercado financiero Esta descripci n se realiza en t rminos de distintas familias y clases de activos incluy ndose tanto renta fija como variable RiskAmerica RiskAmerica ndices Portfolio mmodity Informacion Publicaciones Contacto Acerca Responsabilidades elo Valores al 23 03 2007 a gt Gr fico Valor a A Bonos de Gobierno Indice Valor Ret MTD YTD Bonos Corporativos Gobierno 1868 76 0 03 069 1 32 oer Letras Hipotecarias Corporativo 2346 58 0 0 82 1 43 Hipotecarias 1999 51 0 03 095 2 43 1800 Intermediaci n Financiera Intermediaci n 1280 63 002 029 131 1380 0 Bonos de Reconocimiento 920 0 Indices Renta Variable Montos Yigentes al 23 03 2007 o Ql gt 460 0 Constructor de Indices Comparador de Indices Gob Global m Corp Global m IF Global LH Global Para cada uno de los m s de 100 ndices existentes se entrega informaci n de su composici n as como de su comportamiento en t rminos de retornos y riesgos RiskAmerica RiskAmerica Indices Portfolio
13. RMSE in sample RMSE out of sample Discount bonds 19 0 1 0 14 0 12 Coupon bonds 1 1 5 0 25 0 33 21 1 5 2 5 0 16 0 23 2 5 3 5 0 17 0 21 23 3 5 4 5 0 13 0 15 4 5 5 5 0 16 0 16 5 5 6 5 0 06 0 06 25 6 5 7 5 0 05 0 06 7 5 8 5 0 06 0 09 27 8 5 9 5 0 06 0 08 9 5 10 5 0 05 0 06 29 10 5 11 5 0 04 0 04 11 5 12 5 0 03 0 03 12 5 13 5 0 03 0 03 31 13 5 14 5 0 03 0 02 14 5 15 5 0 02 0 02 33 15 5 16 5 0 03 0 03 16 5 17 5 0 03 0 03 17 5 18 5 0 03 0 03 35 18 5 19 5 0 03 0 04 19 520 0 03 0 04 37 Total 39 0 20 0 10 0 11 41 43 independent of the state variables is obtained by applying Ito s lemma to equation 8 N N 1 2 45 oR t gt una 29 47 i l j where 49 T exp k T u t 30 30 Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe IJFE 317 Color in Web B W in Print 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 IJFE 317 14 G CORTAZAR ET AL Volatility Structure of Interest Rates 1997 2001 Model Volatility A Empirical Volatility from Bond Yields 2 Volatility 0 1 5 3 5 5 5 7 5 9 5 11 5 13 5 Maturity Years Figure 7 Volatility structure of interest rates 1997 2001 There are two difficulties in computing empirical estimates of the interest rate volatilities First most of the data consist of amortiz
14. Scott L 1993 Maximum likelihood estimation for a multifac tor equilibrium model of the term structure of interest rates Journal of Fixed Income 3 14 31 l Journal of Futures Markets DOI 10 1002 fut gas hike oe P OU Futures Prices l Cortazar G amp Schwartz E S 1994 The valuation of commodity contingent claims Journal of Derivatives 1 27 35 Cortazar G amp Schwartz E S 2003 Implementing a stochastic model for oil futures prices Energy Economics 25 215 238 Cortazar G Schwartz E S amp Naranjo L F 2003 Term structure estima tion in iow frequency transaction markets A Kalman filter approach with incomplete panel data working paper Los Angeles University of California Anderson School of Management Cox J C Ingersoll J amp Ross S 1981 The relation between forward prices and futures prices Journal of Financial Economics 9 321 46 Culp C L amp Miller M H 1994 Hedging a flow of commodity deliveries with futures Lessons from Metallgesellschaft Derivatives Quarterly 1 7 15 Dai Q amp Singleton K J 2000 Specification analysis of affine term structure models Journal of Finance 55 1943 1978 de Jong F 2000 Time series and cross section information in affine term structure models Journal of Business amp Economics Statistics 18 300 314 de Jong F amp Santa Clara P 1999 The dynamics of the forward interest rate curve A
15. mmodity Descripcion de Indices Descripci n del Indice indice indice a a Seleccionar Indice Fecha de Consulta 09 01 2007 E Mis Indices Constructor Indices Renta Fija Indices Renta Variable Constructor de Indices Constructor de Indices rp Composici n 09 01 2007 Caracter sticas 0901 2007 Estadisticas 04 01 2007 Comparador de Indices Nombre indice Valor 1 116 30 a Graficar Moneda y O Bl Ticker indice 1 D a 0 0267 Fecha Inicio 02 01 2006 MTD 0 61 H Papeles 2 YTD 0 61 Monto MMM 0 00 1 A o 9 98 Duraci n a os 9 76 DesvEst 1 a o 4 15 E or o Plazo a os 13 24 TIR 3 53 m s220 Convexidad a os 123 67 Serie de Tiempo Graficar var W oO Grafico a cry Graficar Tipo lv Gy E Asimismo se pueden comparar y realizar an lisis entre los distintos ndices Ri skAmerica SVC ndices Portfolio mmodity Indices Renta Fija Opciones a 2 Gr fico Yalor a cry Indices Renta Variable a Agregar ndices aj 1 0 Constructor de Indices es Graficar DesvEst Comparador de Indices J 66 Fecha Inicio Val Comparador de Indices aor 44 Valor Normalizado Fecha T rmino Indices Seleccionados RiskAmerica Plus Gob Global m Corp Global m IF Global Y Gob Global LH Global BR Global IF Global LH Global BR Global Se entrega ademas la posibilidad de construir y indices personalizados
16. nuevo 1 001 39 1 0695 024 014 014 523 000 1 00 NA ooo 0 00 indice NA NA NA NA NA NA NA NA NA NA NA RF_1 NA NA NA ONA NA NA NA NA NA NA NA RV NA NA NA NA NA NA NA NA NA NA NA indices Cargados 25 4 1 2 M dulo Portfolio Este m dulo entrega informaci n de de riesgo retorno y performance tanto de carteras existentes del mercado como de carteras propias En Informaci n de Mercado entrega informaci n informaci n de riesgo de retornos y de performance de carteras p blicas de AFP y de Fondos Mutuos como se muestra en las siguientes p ginas web Ri eA erica Indices Portfolio Commodity a 3 Retornos Riesgo Performance Fecha Inicio 03 01 2003 ES Indicadores de Performance td Fecha T rmino 23 03 2007 ES Fondo A Riesgo ADSON Fondos Fondo A lv Riesgo Relativo Libre de Riesgo IF Global Iv Asset Allocation p Merca An lisis de Desempe o ercado Ponderado Globa Valor Normalizado Ey i E BansanDER E currum E hssirar O PLANMITAL O PROADA E SANTAMARIA Sis Ponderado E Sis Promedio T 1 Shape 2 Modigliani 3 Jensen 4 Treynon 5 Inf Ratio Matriz de Indicadores R A Sharpe Modigliani JensenC Treynon Inf RatioC BANSANDER 182 _ 70 110 820 16 CUPRUM 18 784 J oss J 826 _ 153 C H na 178 Orra oor 812 151 PLANMITAL 178 778 0 27 809 152 promos 177 771
17. oa sors 15 SANTAMARIA 174 758 158 701 14 1 1 En Riesgo Absoluto se entregan herramientas para calcular el Value at Risk de carteras propias tanto por m todos param tricos como por simulaci n hist rica Risks merica RiskAmerica Ingreso de Proyecto Value at Risk ar mmodity Histograma Yariaci n Cartera YaR royecto FondoCH w Le Cartera ssr_piezo0 v l az Param trico Intervalo de Tiempo Simulaci n Hist ar Fecha Inicio Fecha T rmino Ingreso de Par metros ar Valor inicial Cartera Nivel de Confianza 1000 95 v Horizonte de Evaluaci n 1 d a An lisis de aR B z A Var Var Posici n 0 Incremental En Riesgo Relativo se entregan herramientas para calcular el Tracking Error y el VaR Relativo a entre dos carteras Ri skAmeric a RiskAmerica Portfolio Frecuencia 4 12 10 08 Resultados 0 6 0 4 0 2 00 02 04 06 08 Retomo del Portafolio 1 0 Valor Inicial Portfolio P rdida VaR al 5 00 Ganancia al 95 00 P rdida Ganancia Max Prob Gan Requerida mmodity Informacion de Mercado Tracking Error Gr ficos Param trico Ingreso de Proyectos El 2 Proyectos FondoC_H v ke Cartera HAB_Dic2006 m ke Benchmark SIsT_Dic2006 M Be An lisis Se mpe o Tracking Error El Para
18. parameters to be estimated any measurement error crucially affects the shape of the fitted curve An extreme case is when the number of parameters to be estimated is larger than the number of observed prices in this case there is an infinite number of curves that fit the observed prices Figure 1 illustrates this extreme but not uncommon in emerging markets case of a date with fewer prices than model parameters by plotting two of the infinite term structures that perfectly explain observed prices This example is taken from one of the many dates in the Chilean government bond market with extremely thin trading Curve fitting methods clearly cannot be applied to dates with very low number of transactions A second problem of these static curve fitting methods when used in markets with infrequent trading occurs when the prices for short or for long term bonds are not available even if the number of observed prices is sufficient for the estimation Curve fitting methods provide reasonable estimates within the time range spanned by the available prices but provide much less reliable estimates for extrapolations outside this range In many emerging markets it is common that for some dates long term bonds are not traded but the need for a complete term structure estimation for valuation and hedging purposes remains Bond Yields 12 22 2000 8 6 ES TD 4 o S A Observed Bond Yields 2 N amp S Method
19. resultado exitoso en cuanto a nuevas herramientas de gesti n de carteras como es el caso de lo ocurrido con este proyecto debiera inducir que las administradoras de fondos de pensiones pueden mejorar la gesti n de sus carteras y de esta manera obtener mayores retornos de sus inversiones sin incrementar el riesgo asumido El principal impacto econ mico social del proyecto es el Incremento Marginal de la Rentabilidad aplicado a una Fracci n de los Fondos Potenciales que pudieran verse beneficiados con las herramientas de gesti n de carteras desarrolladas Por ltimo se debe estimar el adelantamiento de los flujos en n mero de a os que representa la realizaci n del proyecto comparado con la situaci n base sin proyecto i e se estima que si no hubiera habido proyecto otro similar se hubiera desarrollado teniendo el mismo impacto despu s Los cuatro par metros anteriores son dif ciles de estimar y de ellos depende el resultado de los indicadores econ micos sociales En la Evaluaci n ex ante presentada en la formulaci n del proyecto se asumieron los siguientes valores para estos par metros Incremento Marginal de la Rentabilidad 0 25 Fracci n 25 de las administradoras 60 de los activos 15 Fondos Potenciales MMUS 40 000 Adelantamiento 1 a o si no se hace el proyecto los flujos se realizan 1 a o despu s Con lo que el beneficio econ mico social por a o se estim en US 15 millones y los flujos netos
20. used as regressors to explain the realized present value in trajectory then the least square regression is equivalent to solving the following optimization problem 2 N M Min Y Cre Y Tal LS ato 17 o 1 j l The optimal coefficients are then used to estimate the expected continuation value Gre At M G Sp_a Y LUST 0 18 j l Fig 2 shows discounted continuation values of our simple copper mine for all N simulated paths and the expected continuation function computed as the solution to the regression of these values on powers of the spot copper price Then the optimal decision for each price path is to choose the maximum between two values the immediate exercise and the expected continuation value Once we have worked ourselves backwards until t 0 we have a final vector of continuation values for each price path which averaged provides us with an estimation of its expected value which in turn when compared with the immediate exercise value gives the option value at time 0 Option value Max Q S A G Sp 19 3 2 Implementing the extended Brennan and Schwartz model In this section we show how to implement the LSM approach to solve the Brennan and Schwartz 23 model for any price process including the options to abandon a mine to close an open mine and to open a closed mine Fig 3 may be useful to understand the nature of the problem by describing all possible states during the simu
21. 9mmodity Risk merica RiskAmericaPlus busca proveer un conjunto de servicios enfocados a satisfacer las necesidades espec ficas del mercado financiero chileno y apoyar de una manera integral la gesti n y decisiones de inversi n de los distintos actores del mercado Hasta la fecha la investigaci n financiera mundial se ha centrado principalmente en comprender y modelar el comportamiento de los mercados desarrollados donde se concentran los mayores vol menes de inversi n Sin embargo los mercados emergentes debido a su menor tama o han sido relegados a un segundo plano en esta materia gener ndose as una escasez de herramientas que se ajusten correctamente a la realidad espec fica de estos mercados Considerando esta necesidad la Pontificia Universidad Cat lica de Chile a trav s de su Laboratorio de Investigaci n Avanzada en Finanzas FINlabUC ha creado RiskAmericaPlus y desarrollado Ingenier a Financiera orientada especificamente a resolver problemas relevantes para el mercado nacional El desarrollo de RiskAmericaPlus ha sido llevado a cabo por la Universidad Cat lica de Chile y apoyado por CONICYT a trav s de FONDEF Banco Santander AFP Habitat y Fundaci n COPEC Universidad Cat lica logrando de esta manera la combinaci n necesaria entre excelencia acad mica y experiencia de mercado que han permitido desarrollar una completa familia de herramientas acordes a las necesidades reales y con la r
22. Bond Yields 1 N amp S Method Bond Yields 2 0 T 7 0 5 10 15 20 Maturity Years Figure 1 Two different estimations of yield curves from Chilean government inflation protected discount and coupon bond data using the Nelson and Siegel method for 12 22 2000 Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe IJFE 317 11 13 15 17 19 21 23 25 27 29 Color in Web B W in Print 31 33 35 37 39 41 43 45 47 Color in Web B W in Print 49 51 IJFE 317 4 G CORTAZAR ET AL Figure 2 illustrates a 20 year term structure estimate of the coupon bond yield in Chile for 10 06 1999 a date in which there are sufficient bond prices but the maturity of the longest bond traded was only 6 years We use all pure discount and coupon bonds traded on that date to compute the implied pure discount yield curve using the Svensson 1994 method Once this curve is obtained we compute the yields of coupon bonds with maturities from 0 5 to 20 years priced using the implied pure discount yield curve estimated earlier This coupon yield curve is then plotted in Figure 2 together with the yields of all market transactions on 10 06 1999 and on the day before From Figure 2 we can see that prices of traded bonds with similar maturities did not change much between both dates and that long term bonds were traded only on the first da
23. ESTIMATION METHODOLOGY The Kalman filter is an estimation methodology that recursively calcu lates optimal estimates of unobservable state variables with the use of all past information Consistent parameter estimates can be obtained by maximizing the likelihood function of error innovations In the finance lit erature the Kalman filter has been used to estimate and implement sto chastic models of commodities Schwartz 1997 Schwartz amp Smith 2000 Sorensen 2002 interest rates Babbs amp Nowman 1999 Cortazar Schwartz amp Naranjo 2003 de Jong 2000 de Jong amp Santa Clara 1999 Duan amp Simonato 1999 Geyer amp Pichler 1999 Lund 1994 1997 and other relevant economic variables Pennacchi 1991 Although widely used in a complete panel data setting most literature has not focused on using the Kalman filter where there are missing observations in the panel a common feature in many commodity futures markets One of the characteristics of the Kalman filter is that state variables estimates are obtained with the use of a rich information set that includes past information and not only current prices Moreover it can allow for measurement errors in observable variables that may be induced by mar ket imperfections or by the inability of a model with a restricted number of factors to explain the whole structure of contemporaneous observations The Kalman filter may be applied to dynamic models that are in a state space
24. LAMES ITAM FMA Econometric Society Tsinghua University 6 INFORMS Hong Kong International 2006 7 15th annual meeting of the European Financial 1 Management Association China Hong Kong 25 06 2006 28 06 2006 Espafia Madrid 28 06 2006 01 07 2006 Ri INFORMS EFMA 8 2005 FMA Annual Meeting EEUU 12 10 2005 15 10 2005 9 9th Annual International Conference Real Francia Paris 23 06 2005 25 06 2005 Options Theory Meets Practice Rh ROG EDC EFA 2004 Meeting 18 06 2004 21 08 2004 1 EFA a B IDENTIFICACION DEL PERSONAL DEL PROYECTO QUE PARTICIPO EN EVENTOS N del evento Descripci n de la persona vinculada al evento Rol en el evento T tulo de la Exposici n Certificaci n N de la tabla RUT Apellido Apellido Nombres 1 Expositor si fue expositor Si fue asistente anterior paterno materno 2 Asistente 1 De asistencia Otro especificar 2 De aprobaci n 1 6066335 1 Cortazar Sanz Gonzalo 1 Lanzamiento Oficial de RiskAmercaPlus 4940618 5 Majluf Sapag Nicol s 1 Schwartz G Eduardo 1 Hedge Funds Riesgos y Oportunidades Gonzalo 1 A Multicommodity Model of Futures Prices Using Futures Prices of One Commodity to Estimate the Stochastic Process of Another Gonzalo 1 Term Structure Estimation in Markets with Infrequent Trading Gonzalo 1 Term Structure Estimation in Markets with Infrequent Trading Gonzalo Term Structure Estimation in Markets w
25. Piloto Planta 30 06 2006 30 08 2006 01 08 2006 Portfolioriskv1 Dis Evento Programado Programacion Evento De 30 06 2006 30 12 2006 10 01 2007 Difusion Ds Prototipo probado a Nivel Piloto Planta 30 06 2006 02 01 2007 02 01 2007 Portfoliobenchmarksv1 all Plan de Experimentos 31 definido Plan De Experimentos Definido Productofinal M 15 12 2006 30 12 2006 30 12 2006 Experimentos Criticos Ds Efectuados Experimentos Criticos Efectuados Productofinal 25 12 2006 30 12 2006 20 01 2007 Dis Tes s 0 Tesis De Pregrado 30 12 2006 30 12 2006 31 12 2005 30 12 2006 30 12 2006 17 01 2006 Tesis De Magister Dise o de Prototipo Dr emitido Diseno De Productofinalriskportfoliov2 20 01 2007 30 12 2006 30 12 2006 Evento De Difusion 30 12 2006 30 12 2006 29 03 2007 Oy Publicaciones 30 12 2006 30 12 2006 17 04 2007 Prototipo Probado a nivel 31 Laboratorio Prototipo Probado A Nivel Laboratorio Productofinal 30 01 2007 30 01 2007 25 04 2007 Prototipo Probado a nivel Pilota Planta 05 03 2007 05 03 2007 30 03 2007 Prototipo Probado A Nivel Piloto planta Productofinal Reprogramados Este proyecto no ha reprogramado resultados Eliminados Ning n resultado ha sido eliminado No Logrados No hay resultados No Logrados Detalle de Resultados No hay resultados ingresados para este proyecto ANEXO 3 PLANILLA PRESUPUESTARIA INICIAL Y DE EJEC
26. Porcentaje de Rentabilidad Adicional Anual de Fondos de Pensiones Valor Escenario Pesimista 0 10 Descripci n Variable 2 Porcentaje de Cartera Administrada que es impactada por nuevas tecnolog as de Gesti n Optima de Cateras Valor Escenario Optimista 100 Descripci n Variable 3 Porcentaje de Activos en Cartera que es impactada por nuevas tecnolog as de Gesti n Optima de Cateras Valor M s Probable 0 25 A O 1 2 3 4 5 6 7 8 9 10 INGRESOS Moneda nacional 0 0 0 297 535 772 891 1010 1069 1129 Moneda extranjera equivalente 0 0 O 10457 10457 10457 10457 10457 10457 10457 Total ingresos 0 0 O 10754 10991 11229 11348 11466 11526 11585 COSTOS Mano de obra calificada 0 0 0 226 357 487 552 616 646 676 Mano de obra no calificada 0 0 0 0 0 0 0 0 0 0 Insumos M N 0 0 0 24 26 29 32 35 39 43 Bienes de capital M N 0 0 0 14 15 17 19 20 23 25 Otros M N 0 0 0 12 13 15 16 18 19 21 Total costos 0 0 0 276 412 548 618 689 727 765 INVERSIONES En moneda nacional 0 509 509 6 6 6 6 6 6 6 En moneda extranjera 0 0 0 0 0 0 0 0 0 0 Total inversiones 0 509 509 6 6 6 6 6 6 6 BENEFICIOS 0 509 509 10472 10573 10675 10723 10772 10793 10815 CON PROYECTO La informaci n aqu contenida se extrae de la hoja SITUACION CON PYTO En millones de pesos Variables Cr ticas Var 1 Var 2 Var 3 Unidad m3 kg I ton etc Valor Escenario Pesimista Descripci n Variable 1 Porcentaje de Rentabilidad Adicional Anual de Fondos de Pensiones Valor Escenario
27. Usuario G RiskAmerica Halo RiskAmeric SVC Indices Portfolio mmodity 79 x Constructor T Descripci n de Indices IICA Descripci n del Indice indice indice El Indices Renta Variable Seleccionar Indice Fecha de Consulta 0940172007 ES bmposici n 09 01 2007 Caracteristicas 0401 2007 Estadisticas 0401 2007 Hombre indice Valor 1 116 30 Graficar Moneda iv o Ticker indice 1 D a 0 0267 Fecha Inicio 02 01 2006 MTD 0 61 H Papeles 2 YTD 0 61 Monto MMM 0 00 1 A o 9 98 Duraci n a os 9 76 DesvEst 1 a o 4 15 Hour 66 71 Plazo a os 13 24 TIR 3 53 MN 29 204 Convexidad a os 123 67 Serie de Tiempo Grafico 2 cy Graficar Tipo v El E Tambi n se pueden cargar indices generados externamente Usuario Gonzalc RiskAmeric gt RiskAmerica Portfolio mmodity Mis Indices T Familia de ndices Descripci n de ndices Comparador de indices Familia de Indices Indices Renta Fija Agregar a Bl Fecha de Consulta 23nsnm007 ES Mis Indices Constructor de Indices Estadisticas Caracteristicas ic DesvEst H Monto Durac Plazo ES Hombre Valor 1D a MTD YTD 1A o TIR i 3 1 a o Instr MMM a os a os Renta Fija 2 El cob ace 1 263 78 0 000 034 1 98 7 98 107 533 10 201041 254 3 06 El Gob cero 1 928 86 0 014 0 78 144 759 223 264 173 62903 447 447 Drenta Variable 0 Didices Construidos 4
28. build option value and investment decisions Journal of Financial Economics 1987 18 2 7 27 He H Pindyck R Investment in flexible production capacity Journal of Economics Dynamics and Control 1992 16 575 99 Cortazar G Schwartz ES A compound option model of production and intermediate investment Journal of Business 1993 66 4 517 40 Dixit A Entry and exit decisions under uncertainty Journal of Political Economy 1989 97 620 38 Pindyck R Investment of uncertain cost Journal of Financial Economics 1993 34 1 53 76 Ekern S An option pricing approach to evaluating petroleum projects Energy Economics 1988 10 91 9 Trigeorgis L Evaluating leases with complex operating options European Journal of Operations Research 1996 91 69 86 Brennan MJ Trigeorgis L Project flexibility agency and competition Oxford Oxford University Press 2000 Dixit A Pindyck R Investment under uncertainty Princeton NJ Princeton University Press 1994 Decamps J Mariotti T Villeneuve S Investment timing under incomplete information European Economic Association Annual Congress 2003 Lambrecht B Perraudin W Real options and preemption under incomplete information Journal of Economic Dynamics and Control 2003 27 4 619 43 Miltersen KR Schwartz ES R amp D investments with competitive interactions NBER Cambridge MA 2003 Murto P Nasakkala E Keppo J Timing of investments in oligopoly under uncertainty a framework for numerical analysis E
29. convenience yield independently Gibson amp Schwartz 1990 this article models the cost of carry c Schwartz amp Smith 2000 defined as the difference between the instantaneous risk free interest rate r and the convenience yield 6 Transformations of the Model This subsection shows how to rewrite a model through an affine trans formation and this procedure is later applied to the Gibson and Schwartz 1990 model to show that this is a particular case of the N factor model presented in this article Consider the following model logS h Z ult 4 de KZ B di Edw 5 where is a vector of state variables dW dW Q dt Z t is a scalar function h and B are vectors and matrices K and Y need not be diagonal Then the following affine transformation T L LZ t may be applied to the original state vector to obtain a new state vector x T If the matrix L is invertible then there exists a one to one correspondence between the state variables of the two models The new model is log S P x w t 6 dx Kx B dt 2 dw 7 Obviously this assumption could casily be relaxed in order to study stationary models of commodity prices Actually when the model was estimated with this assumption relaxed no significant differences were found Mso the model performs well for other commodities as will be shown later when calibrated with copper futures It is assumed for s
30. erica Servicio de Valorizaci n de Carteras SVC consiste en un servicio diario de valorizaci n y asignaci n de tasas para los distintos instrumentos de renta fija del mercado nacional El mercado chileno de instrumentos de renta fija presenta una baja frecuencia de transacciones lo que dificulta la obtenci n diaria de precios de mercado para los distintos activos que componen las carteras de renta fija nacional Diariamente s lo el 0 5 de los instrumentos del mercado presentan transacciones por lo que se hace necesario contar con una metodolog a que permita estimar las tasas de los nemot cnicos restantes Portfolio 9mmodity Con el fin de solucionar estos problemas y de proporcionar herramientas que apoyen las decisiones de inversi n y gesti n del riesgo financiero hemos desarrollado un modelo de estimaci n de tasas para los instrumentos de renta fija del mercado nacional Este modelo se caracteriza por maximizar el uso de la informaci n de mercado disponible a trav s de un registro de transacciones diario e hist rico Esta informaci n es utilizada a trav s de consultas de spreads as como en la estimaci n de Estructuras de Referencia mediante modelos din micos de no arbitraje aplicados al mercado nacional los cuales permiten obtener valorizaciones que se ajustan a los movimientos del mercado y que poseen volatilidades estables y consistentes para los distintos papeles www riskamerica c
31. fico Valor B z Bonos de Gobierno Indice Valor Ret MTD YTD Bonos Corporativos Gobierno 1868 76 003 069 1 32 2300 0 Letras Hipotecarias Corporativo 2346 58 0 0 82 1 43 sae Hipotecarias 199951 0 03 095 2 3 12500 Intermediaci n Financiera Intermediacion 128063 002 0 29 1 31 1380 0 Bonos de Reconocimiento 920 0 Indices Renta Variable Montos igentes al 23 03 2007 Ql ay 460 0 gt mm ty m nm a e A S gt gt y gt D gt gt 5S gt Constructor de Indices Dogs ees Comparador de Indices RiskAmerica Ph Gob Global Corp Global m IF Global LH Global Para cada uno de los m s de 100 ndices existentes se entrega informaci n de su composici n as como de su comportamiento en t rminos de retornos y riesgos RiskAmerica merica Indices Portfolio mmodity Mis Indices Constructor Descripci n de ndices ESSE Descripci n del Indice indice indice a aj Indices Renta Variable a m Seleccionar Indice Cal Fecha de Consulta Constructor de Indices Ir agit 2007 Constructor de Indices Composici n 09 01 2007 o Comparador de Indices Hombre indice Valor 1 116 30 i a Moneda e ol A Ticker indice 1 Dia 0 0267 Caracter sticas 0901 2007 Estadisticas 09 01 2007 Fecha Inicio 02 01 2006 MTD 0 61 1 Papeles 2 YTD 0 61 Monto MMM 0 00 1 A o 9 98 Duraci n a os 9 76 DesvEst 1 a o 4 15 E our 66 71 Plazo
32. filter is thus a particular type of Bayesian estimation Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe IJFE 317 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 IJFE 317 8 G CORTAZAR ET AL Another useful characteristic of the Kalman filter under the normality assumption is that it provides consistent model parameters estimates y when maximizing the log likelihood function of error innovations 1 1 a log LO 5D log F 5 vF v 20 t t where y represents a vector containing the unknown parameters Moreover the covariance matrix of the estimation errors I may be obtained from the information matrix I 0 log Lap ET 21 Xy 4 2 Kalman filter applied to incomplete panel data As already stated existent literature stresses on the use of the Kalman filter methodology with complete panel data sets However it is not necessary to assume a fixed number of observable variables at each time period in order to apply the Kalman filter Let m be the number of observations available at time which need not be equal to the number of observations available at any other date This means that the number of observations available at any date is time dependent The measurement equation is again z Hx d v v N 0 R 22 but now z is a m x 1 vector H is a m x n mat
33. fixed income instruments with few market transactions Indices Service which provides benchmarks on market behavior of several asset classes and Portfolio Service which provides risk return and performance information for market and private portfolios These services are already been used by many financial institutions in Chile The project also generated journal publications and congress presentations and provided support for master s and undergraduate thesis strengthening the activities of the FINlabUC Laboratorio de Investigaci n Avanzada en Finanzas de la Pontificia Universidad Cat lica de Chile 2 SINTESIS DE RESULTADOS DEL PROYECTO RESULTADO RESULTADO cantidad Importancia econ mica social Productos mejorados Nuevos productos Nuevos procesos Procesos mejorados Nuevos servicios Servicios mejorados Contratos empresas productoras Convenios contratos para proyecto de escalamiento Otros registros de propiedad Otro especificar IIC C Otro_ especificar Importancia cient fica tecnol Capacidades cient fico tecnol gicas obtenidas Patentes Registro de variedades vegetales Capacidad de desarrollar metodolog as e implementar soluciones en el mbito de la Ingenier a Financiera y Gesti n del Riesgo Otros Resultados C amp T Otros Resultados C amp T Articulos revista nacional ISSN Articulos revista nacional Art culos revista internacional ISSN__ 3 Art culos revista internacional
34. for price model selection The dynamics of the state variables are dS vw yz dt 01 dzi 6 St dy Ky dt 02 dz2 7 dv a v v dt 03 dz3 8 with dz1 dz2 pj2 dt dzjdz3 p43dt dz2dz3 p23 dt 9 Defining 4 as the risk premium for each of the three risk factors the risk adjusted processes are dS vr y 41 dt 01 dzj 10 t dy Ky 42 dt o2 dz3 11 dv a v vy 43 dt 03 dz3 12 with dz dz5 pia dt dz3 dz3 p23dt dz dz3 p43 de 13 Following the same estimation procedure used in Cortazar and Schwartz 7 for oil prices we calibrate this model for copper using all futures traded between 1991 and 1998 at NYMEX obtaining the parameter values shown in Table 1 The model allows for all three state variables to be correlated providing a greater flexibility which is in line with empirical evidence It is interesting to note that most parameter values including the factor correlations exhibit a sign and magnitude similar to those reported in 7 for oil Also the model fits the empirical data with a mean absolute error of 0 2 and exhibits similar theoretical and empirical volatilities as shown in Fig 1 Using this three factor price model to extend the Brennan and Schwartz 23 real option model we obtain a much better model specification With this new price process and following the general framework described in the previou
35. formulation with state variables Journal of Financial and Quantitative 34 131 157 Deaton A amp Laroque G 1992 On the behavior of commodity prices Review of Economic Siudies 59 1 23 Duan J C amp Simonato J G 1999 Estimating and testing exponential term structure models hy Kalman filter Review of Quantitative Finance and Accounting 13 111 135 Duffie D amp Singleton K J 1997 An econometric model of the term struc ture of interest rate swap yields Journal of Finance 52 1287 1321 Geyer A L J amp Pichler S 1999 A state space approach to estimate and test multifactor Cox Ingersoll Ross models of the term structure Journal of Financial Rescarch 22 107 130 Gibson R Schwartz E S 1990 Stochastic convenience yield and the pricing of oil contingent claims Journal of Finance 45 959 976 Harvey A C 1989 Forecasting structural time series models and the Kalman filter Cambridge Cambridge University Press Heath D Jarrow R amp Morton A 1992 Bond pricing and the term struc ture of interest rates A new methodology for contingent claims valuation Econometrica 60 77 105 Hilliard J E amp Reis J 1998 Valuation of commodity futures and options under stochastic convenience yields interest rates and jump diffusions in the spot Journal of Financial and Quantitative Analysis 33 61 86 Langetieg T C 1980 A multivariate model of the term
36. hence the dimension of the function range depends on the number of observations available at time In a complete panel data setting this time dependence disappears 16 These instruments are actually issued by the Chilean Central Bank an institution equivalent to the Federal Reserve in the US 17 In practice this is done by expressing payments in another unit the UF Unidad de Fomento which is updated every month using the previous month inflation 18 Curiously the figure resembles a DNA pattern 19 Implementation issues of the model can be found in the Appendix hr Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe IJFE 317 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 JFE 317 TERM STRUCTURE ESTIMATION 17 REFERENCES Andersen TG Lund J 1997 Estimating continuous time stochastic volatility models of the short term interest rate Journal of Econometrics 77 343 377 Babbs SH Nowman KB 1999 Kalman filtering of generalized Vasicek term structure models Journal of Financial and Quantitative Analysis 34 1 115 130 Ball C Torous W 1996 Unit roots and the estimation of interest rate dynamics Journal of Empirical Finance 3 215 238 Bliss RR 1996 Testing term structure estimation methods Advances in Futures and Operations Research 9 197 231 Brennan MJ Schwartz ES
37. it can be seen that most parameters are sta ble across different panels which shows the reliability of the model when applied to the oil market All mean reversion parameters k for all three panels are highly significant and show the existence of strong mean reversion in oil prices Volatility parameters g are also highly sig nificant and stable across panels Correlation parameters p are almost all significant As found in the literature Schwartz 1997 the long term growth rate parameter 4 and most risk premium parameters are not Journal of Futures Markets DOI 10 1002 fut sigh seein ap a o ae SSI y8 ee szt 000 0 0 0 0000 600 0 890 0 921 0 050 0 zo 0 S900 ccoo 050 0 40 0 S90 0 910 0 vt0 0 8Z 0 y00 0 6410 v00 0 1110 8000 zt 0 POZ SO Str srt OSi Zeb S2 961 goa z400 000 0 s oo 000 0 800 0 000 0 rooo gi Ep10 90z 0 08z 0 zero azto sero oz10 6Gai o 690 0 6e0 0 0800 9z0 0 2400 seo o 4800 6800 260 0 yoo 600 se0 0 240 0 pooo 2800 9z0 0 2600 2000 600 0z0 0 1200 vsg0 820 0 ZZS 0 Sz0 0 8100 80 o Zizo 2200 Zr 0 9S0 0 icoo o o 0200 2 0 0 881 0 Ooo zeto S20 0 6220 600 0 SSE 0 1 loo soco 600 0 vero 900 0 zeco 8000 izo 00 0 esz 0 1400 0 6510 S00 0 S61 D
38. main reasons First many real options have a longer maturity which makes risk modeling critical and may force considering many risk factors as opposed to the classic Black and Scholes approach with only one risk factor Second real investments many times exhibit a more complex set of interacting American options which make them more difficult to value In recent years new approaches for solving American options have been proposed which coupled with an increasing availability of computing power have been successfully applied to solving long term financial options In this paper we explore the applicability of one the most promising of these new methods in a multidimensional real option setting O 2006 Elsevier Ltd All rights reserved Keywords Real options Simulation Natural resources Valuation Finance 1 Introduction Even though in the last two decades there has been an increasing literature on the benefits of using the contingent claim approach to value real assets limitations on solving procedures and computing power have often forced aca demics and practitioners to simplify these real option models to a level in which they loose relevance for real world decision making There are two main reasons why real option models may present a higher challenge than their financial option counterparts to be solved First many real options have a longer maturity which makes risk modeling critical and may Corresponding author Tel 56 2 35
39. measure At maturity we assume the switching option has no value thus Vr x j 0 j Hl K 1 The switching option can then be solved recursively as follows Moving backwards in time in t T At the value of the option is maximized among all feasible future stages Vra X J max 1 K CFr a x i CH 4 60 a PO a 2 At times t t t1 tr z r the value of the option can be computed as a function of current cash flows and the conditional expectation of the value in the following period For example in tr 24z j 1 K 6 116 G Cortazar et al Computers amp Operations Research 35 2008 113 129 where r is the risk free rate between time tr zar and tr_a Er 2A1 represents the conditional expectation at time tr 24r under the risk neutral probability measure Consequently the initial value of the switching option Vo x j can be solved by this backward recursion where j represents the initial state To determine the critical vector of state variables x that triggers the transition between different states of production we must find the values that equate the conditional expectations between states of production In the original Brennan and Schwartz 23 the project is a contingent claim on copper price which follows a one factor model thus dS udt o dz 4 Si in which is the instantaneous price return is the return volatility and dz is an increment to a standard Gaus
40. n Cartera royecto YaR FondoC_H v pe B Param trico Intervalo de Tiempo Fecha Inicio Fecha T rmino Ingreso de Par metros Walor inicial Cartera Nivel de Confianza Simulaci n Hi Horizonte de Evaluacion 1 An lisis de aR B A Var Var Posici n 0 Incremental 95 M dia a Frecuencia 1 4 1 2 10 08 0 6 04 0 2 00 02 04 06 08 Retomo del Portafolio 4 Resultados Valor Inicial Portfolio P rdida YaR al 5 00 Ganancia al 95 00 P rdida Ganancia M x Prob Gan Requerida 10 En Riesgo Relativo se entregan herramientas para calcular el Tracking Error y el VaR Relativo a entre dos carteras RiskAmer ica Portfolio mmodity Informacion de Merca Ingreso de Proyectos Tracking Error An lisis empe o Tracking Error Param trico Matriz Yarianza Covarianza a gt Proyectos FondoC_H v pe Cartera HAB_Dic2006 Iw ae Benchmark SIST_Dic2006 vw pe ar Simulaci n Hist a gt Seleccione Fecha Inicio Fecha T rmino 01 01 2 HIST Mat_1a iv ar An lisis de Tracking Error ATE TE Posici n 0 Incremental A TE ATE Posici n 1 Posici n Bench Todos Gr ficos Param trico A TE Posici n 0 TE Incremental ATE Posici n 1 A TE Posici n Bench IPSA IF Global LH Global Gob Global Corp Global YEN USA USD UK_EUR Pac
41. pueden agregar sus instituciones reguladoras A continuaci n se entrega una lista actualizada de los clientes potenciales provenientes de AFP Fondos Mutuos Bancos Cias de Seguros A estas listas habr a que agregar corredoras de bolsa securitizadoras proveedores de diversos servicios financieros e instituciones reguladoras Nombre Bansander S A Cuprum S A Habitat S A Planvital S A Provida S A Santa Mar a S A Tipo Administradoras de Fondos Mutuos Nombre Administradora General de Fondos Security S A Banchile Administradora General de Fondos S A BancoEstado S A Administradora General de Fondos Bandesarrollo Administradora General de Fondos S A BBVA Administradora General de Fondos S A BCI Administradora de Fondos Mutuos S A BICE Administradora General de Fondos S A Boston Administradora General de Fondos S A Celfin Capital S A Administradora General de Fondos Consorcio S A Administradora General de Fondos Corp Banca Administradora General de Fondos S A Cruz del Sur Administradora General de Fondos S A Euroam rica Administradora General de Fondos S A IM Trust S A Administradora General de Fondos Larra n Vial Administradora General de Fondos S A Legg Mason Chile Administradora General de Fondos S A Penta Administradora General de Fondos S A Principal Administradora General de Fond
42. que transparenta los mercados y permite una mejor competencia y asignaci n de recursos financieros Desde marzo de 2006 los precios para todos los instrumentos de renta fija son entregados v a Internet a todos los fondos mutuos nacionales como precio oficial para ser utilizados en el c lculo diario de la cuota de todos los fondos mutuos seg n circular de la Superintendencia de Valores y Seguros y en acuerdo con la Asociaci n de Administradoras de Fondos Mutuos de Chile ESPERADOS Mejores decisiones de inversi n y de gesti n del riesgo para carteras de inversi n de las instituciones financieras del pa s a medida que vayan adoptando las herramientas que actualmente est n en fase de prueba 3 2 IMPACTOS CIENTIFICO TECNOLOGICOS PRODUCIDOS Nuevas metodolog as de valorizaci n y gesti n del riesgo principalmente para mercados con pocas transacciones como son los mercados de econom as emergentes como la chilena Esto se ha traducido en publicaciones tesis de mag ster y presentaciones en conferencias acad micas internacionales ESPERADOS Nuevas publicaciones en preparaci n orientadas a formas de gestionar carteras de inversi n y a modelos multi activos 3 3 IMPACTOS INSTITUCIONALES PRODUCIDOS Fortalecimiento del FINlabUC Laboratorio de Investigaci n Avanzada en Finanzas tanto en actividad reconocimiento y equipamiento Fortalecimiento del programa de Mag ster en Ciencias de la Ingenier a con incremento en el
43. representation The measurement equation relates a vector of observable variables z with a vector of state variables x z H x d y v N O R 19 Joumal of Futures Markets DOI 10 1002 fut Oil Futures Prices where z is a m X 1 vector H is a m X n matrix x is a n X 1 vector d is a m X 1 vector v is a m X 1 vector of serially uncorrelated Gaussian disturbances with mean O and covariance matrix R of dimension m X m and m is the number of observations available at time t Measurement Equation 19 assumes the existence of a linear rela tion between observed variables and state variables As noted above in this model the logarithm of futures prices is a linear function of state variables Nevertheless the Kalman filter could be modified Harvey 1989 to allow for nonlinear measurement equations as would be the case if for example commodity option prices alone or in combination with futures prices were used as observations The transition equation describes the stochastic process followed by the state variables x Ax t e N 0 Q 20 where A is an n X n matrix c is an n X vector and is an n X 1 vec tor of serially uncorrelated Gaussian disturbances with mean 0 and covariance matrix Given this state space representation the Kalman filter calculates optimal estimates x of state variables and the variance covariance matrix P E x amp x R The Kalma
44. representation the state variables have a multivariate normal distribution This assumption can also be relaxed to include non Gaussian models for the state variables Equations 11 and 12 define what is called the state space representation The Kalman filter provides optimal estimates X of the state variables given all the information up to time t Let P be the covariance matrix of the estimation errors P E x X X E 13 Then given X _ and P _ which include all the information up to time 1 1 the estimator of the state variables and the covariance matrix of the estimation errors at time t are Xar1 AX 1 0 14 Pr APA F Q 15 Equations 14 and 15 are usually called the prediction step When new information represented by z becomes available it is used to obtain an optimal estimate of the state variables and of the error covariance matrix X Kae F Po HF ly 16 P Po Py HF HP 17 where F HPH R 18 Vp 2 HX 4 1 d 19 Equations 16 and 17 correspond to what is usually called the update step Intuitively the update step is just the calculation of the conditional expectation of state variables x given all the history of observations and the new information z i e X E _ x z It can be shown that this conditional expectation is in fact an optimal estimation in a mean square error sense and corresponds to equation 16 The Kalman
45. well developed markets where numerous bonds are traded every day for different maturities these static methods generate yield curves that accurately fit current bond transactions Bliss 1996 There are however other features besides goodness of fit to observed prices that are desirable in a term structure model such as the time series stability of the term structure curves obtained This stability can be analysed by observing the sequence of daily term structure estimations implied by the model It might well be the case that the model fits very well the existing bond prices or yields but it implies large daily movements of yields for maturities that are not traded This is not an issue for liquid markets but as we shall see is a serious problem for thin markets One way of assessing the stability of the term structure curves obtained is to compare the volatilities from the model with actual volatility from the data In markets with a complete cross section of prices for each date volatility of interest rates computed from the estimated term structures will closely match historical data and the stability of the model is not an issue However for sparse data sets in which at each date there are only a few different bond maturities traded stability will become an important criterion for judging the reliability of the term structure estimation When the number of observed prices for a particular date is not sufficiently larger than the number of
46. with any number of stochastic factors To show the implementation of the approach we estimate a three factor generalized Vasicek model using Chilean government bond price data The approach however may be used in any market with infrequent trading a common characteristic of many emerging markets Copyright 2007 John Wiley amp Sons Ltd JEL CODE W Ml Al KEY WORDS M MH H 1 INTRODUCTION There are two issues that are of central importance in term structure analysis One is the modelling and estimation of the current term structure of spot rates which is essential for valuing and hedging cash flows that are linearly related to the discount function The second is the modelling and estimation of the dynamics of the term structure which is required for valuing and hedging cash flows that are non linear functions of the term structure all types of options These two issues have been addressed independently in the literature For current term structure estimation most authors have proposed parametric and non parametric methods for fitting curves to current bond prices or yields without regard to past prices McCuloch 1971 1975 Vasicek and Fong 1982 and Fisher et al 1994 among others use spline curve fitting methods to estimate the current term structure Nelson and Siegel 1987 and Svensson 1994 use parsimonious Correspondence to Eduardo S Schwartz University of California at Los Angeles USA T E mail eschwart and
47. y tres millones seiscientos treinta y dos mil setecientos treinta pesos La diferencia de 1 670 564 un mill n seiscientos setenta mil quinientos sesenta y cuatro pesos con respecto a lo girado por FONDEF ha sido reintegrada mediante cheque nominativo cruzado a nombre de CONICYT por el mismo monto Las instituciones beneficiaras declaran haber utilizado el subsidio para financiar los recursos que consulta el proyecto e Aporte de los beneficiarios La instituci n hizo aportes a la ejecuci n del proyecto con recursos valorados en 352 786 millones de pesos Dicho monto lo enteraron con 122 873 millones de pesos en recursos de las propias instituciones beneficiarias y con 229 913 millones de pesos en recursos aportados por las empresas y otras contrapartes del proyecto Los recursos declarados de contraparte satisfacen el porcentaje m nimo exigible por bases del Concurso Las instituciones beneficiarias declaran que los montos detallados de los aportes de las diferentes fuentes se encuentran en el ANEXO 3 de este informe 7 Objetivos y Resultados obtenidos Objetivos Generales El objetivo principal del proyecto es desarrollar herramientas aplicaciones y servicios computacionales que aprovechen en forma efectiva las tecnolog as asociadas a internet para modernizar el sistema financiero nacional apoyando una mejor gesti n de carteras de inversi n en activos transados en el mercado nacional e internacional Los desarrollos
48. 003 25 3 215 38 8 Hsu J Schwartz ES A Model of R amp D Valuation and the Design of Research Incentives UCLA Anderson School 2003 128 10 11 12 13 14 21 22 23 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 43 44 45 46 47 48 50 G Cortazar et al Computers amp Operations Research 35 2008 113 129 Cortazar G Schwartz E Casassus J Optimal exploration investments under price and geological technical uncertainty a real options model R amp D Management 2001 31 2 181 9 Kulatilaka N Operating flexibilities in capital budgeting substitutability and complementarity in real options In Trigeorgis L editor Real options in capital investments new contributions New York Praeger 1995 Bernardo A Chowdry B Resources real options and corporate strategy Journal of Finance 2002 63 211 34 Bossaerts P Simulation estimators of early optimal exercise Graduate School of Industrial Administration Carnegie Melon University 1988 Tilley JA Valuing American options in a path simulation model Transactions of the Society of Actuaries 1993 45 42 56 Barraquand J Martineau D Numerical valuation of high dimensional multivariate American securities Journal of Financial and Quantitative Analysis 1995 30 3 301 20 Raymar S Zwecher M A Monte Carlo valuation of American call options on the maximum of several stocks Journal of Derivatives 1997 5 1 Fall 7 23 Br
49. 44272 fax 56 2 5521608 E mail address gcortaza ing puc cl G Cortazar 0305 0548 see front matter O 2006 Elsevier Ltd All rights reserved doi 10 1016 j cor 2006 02 016 114 G Cortazar et al Computers amp Operations Research 35 2008 113 129 force the use of several risk factors as opposed to only one like in the classic Black and Scholes 1 stock option model Second often real investments exhibit a more complex set of nested and interacting American options which make them more difficult to value In the valuation of natural resource investments for example until only a few years ago most commodity price models considered only one risk factor and constant risk adjusted returns These earlier models have several undesirable implications including that all futures returns should be perfectly correlated and exhibit the same volatility which is not in line with empirical evidence In recent years however many multifactor models of commodity prices have been proposed being much more successful than previous one factor models in capturing the observed behavior of commodity prices like mean reversion and a declining volatility term structure 2 5 7 On the other hand the real options literature has also evolved and models increasingly take into account the different types of flexibilities available to decision makers when managing their projects These flexibilities include the options to abandon a project to shut down produ
50. 6 Riskmatrixv 1 Prototipo Probado a nivel 9 Laboratorio Prototipo Probado A Nivel Laboratorio Riskmatrix 30 12 2005 30 12 2006 30 12 2006 Plan de Experimentos definido Plan De Experimentos 15 04 2006 15 04 2006 28 04 2006 Definido Portfoliobenchmarks Plan de Experimentos 3 definido Plan De Experimentos Definido Portfoliorisk Experimentos Cr ticos Efectuados Experimentos Criticos 25 04 2006 25 04 2006 28 04 2006 Efectuados Portfoliobenchmarks 15 04 2006 15 06 2006 15 06 2006 pil Experimentos Cr ticos 3 Efectuados Experimentos Criticos Efectuados Portfoliorisk hs Dise o de Prototipo emitido 25 04 2006 30 06 2006 30 06 2006 wl 30 04 2006 30 04 2006 28 04 2006 Diseno De Portfoliobenchmarksv 1 Dise o de Prototipo 3 emitido 30 04 2006 30 07 2006 30 06 2006 Diseno De Portfolioriskv 1 Di Tes s Iniciada 30 06 2006 30 06 2006 30 06 2006 Inicio Tesis Pregrado Publicaci n solicitada 17 Documento De Trabajo De 30 06 2006 30 06 2006 30 06 2006 Publicacion Prototipo Probado a nivel Das Laboratorio Prototipo Probado A Nivel 30 06 2006 30 06 2006 30 06 2006 Laboratorio Portfoliobenchmarks ll Prototipo Probado a nivel 3 Laboratorio Prototipo Probado A Mivel 30 06 2006 30 06 2006 30 06 2006 Laboratorio Portfoliorisk is Tes s Iniciada Inicio Tesis De Postgrado 30 06 2006 30 06 2006 30 06 2006 Prototipo probado a Nivel 23
51. 70 to 7 12 Figure 6 displays the RMSE by maturity for each of the models ht can seen that the inclusion of more factors improves the fit for all maturities especially for short and long term contracts Another way to measure model stability is to compare the RMSE of out of sample and in sample estimations For this purpose the RMSE for vears 2002 2003 and 2004 is calculated with the use of Panel A param eters Table IV displays these results and shows that the RMSE for these vears is similar to the RMSE obtained for 2001 It is important to note that these results are obtained under the assumption that the spot price is nonstationary which seems reasonable for oil Schwarz amp Szakmary 1994 Moreover Bessembinder et al 1995 find large and significant mean reversion in agricultural com modities and crude oil futures markets smaller but statistically signifi cant mean reversion in metals futures and weak mean reversion in financial futures markets Thus one may be concerned that the nonsta tionarity of the model combined with the embedded assumption of mean Journal of Futures Markets DOI 10 1002 tut 262 Cortazar and Naranjo TABLE IV In Sample RMSE for the Year 2001 and Out of Sample RMSE for Years 2002 2004 Calculated with Panel A 1992 2001 Parameters for Different Number of Factors 2001 In sample 2002 2003 1F 5 43 4 38 5 93 2F 1 38 1 22 1 88 3F 0 60 0 66 0 71 4F 0 36 0 53 0 46 TABLE V In
52. 990 estimate from government bond data the instantaneous risk free interest rate r p A Futures Prices The price of a futures contract at time t and maturing at T can then be found as the expected value of the spot price under the risk neutral measure Cox Ingersoll amp Ross 1981 F x t T EX S 16 As shown in the Appendix the expected value in Equation 16 can be computed as i l Fx t T exp xin gt eS oy pt u Laa t tea l 5 l BS E 17 i 2 K jel K K Journal of Futures Markets DOI 10 1002 tut 252 Cortazar and Naranjo One important advantage of this model is its tractability with explicit futures price formulas even for an arbitrary number of factors In addition the logarithm of the futures price is a linear function of state variables which is useful when estimating the model with a Kalman filter based procedure Because the state variables have a multivariate normal distribu tion any linear combination of state variables will also distribute normal allowing maximum likelihood techniques to be used Finally the model volatility term structure of futures returns can be obtained from Equations 2 and 17 aj r gt Y 0 0 pe 18 ist j l Given that 0 as the maturity of a futures contract grows the volatility of futures returns converges to a constant given by a which is the volatility of the first state variable
53. CIONES causas o motivo del efectuados en moneda retiro incorporaci n o modificaci n del aporte del a o A F P Habitat 104912 Riskamerica Dictuc SA 125 001 C ENSE ANZA OBTENIDA IA 5 4 RENDICION FINAL DE GASTOS INSTITUCI N BENEFICIARIA MONTO TOTAL MONTO TOTAL MONTO TOTAL MONTO RENDIDO aprobado CONTRATO adjudicado GIRADO POR DEVOLUCI N O 1 mas reajustes 2 FONDEF 3 GIRO 3 1 Pont Universidad Cat lica 153 632 760 158 000 000 155 303 294 1 670 534 de Chile 5 5 ORGANIZACION Y EQUIPO DE TRABAJO El proyecto se organiz considerando Una direcci n General Gonzalo Cortazar Una unidad de Investigaci n Eduardo Schwartz y Gonzalo Cortazar con apoyo de Jaime Casassus Una unidad de Productos y Transferencia a Usuarios Nicol s Majluf y C Mery AFP Habitat Un pool de profesionales e investigadores que eran asignados a distintos proyecto de acuerdo a las necesidades Las tareas se gestionaron considerando Subproyectos a cargo de distintos profesionales con reportes semanales internos reportes mensuales con reuni n directiva ampliada con participaci n de prof AFP y reuniones trimestrales del directorio del proyecto que inclu a al Gte de Inversiones de la AFP Ense anza obtenida 5 6 OBSERVACIONES CONCLUSIONES Y RECOMENDACIONES DE LA DIRECCI N DEL PROYECTO 1 Sobre el proyecto 2 Sobre su Instituci n 3 Sobre otr
54. D 2002 2004 Average number Maximum maturity Year of daily observations years Panel A iS Panel B 1992 22 3 1993 22 3 1994 21 3 1995 25 4 1996 31 4 Panel C 1997 34 7 1998 31 7 1999 31 7 2000 933 7 2001 34 7 Panel D 2002 34 7 2003 34 7 2004 33 7 The data are divided into four different panels in Table I Panel A includes all futures contracts traded between 1992 and 2001 which cor respond to 70 584 observations In order to analyze the effect if any in parameter estimates of the introduction of long term contracts in 1997 the data are divided into Panel B 1992 1996 and Panel C 1997 2001 with 30 424 and 40 160 observations respectively Panel D is used for out of sample testing purposes only and includes futures traded between 2002 and 2004 with 24 947 observations Table E presents a description of the data showing the average number of daily observations and the maximum maturity available for each year during the 1992 2004 period The time series of the spot oil price from 1992 to 2004 is displayed in Figure 1 where it is possible to appreciate the high volatility exhibited by oil Spot prices reach a maximum of 55 per barrel and the minimum is as low as 10 per barrel The spot price defined as the value of an expiring futures contract is usually unobservable and must be estimated The simplest way of doing it is to use the closest to maturity futures con tract as a proxy for the oil spot price A more rigorous
55. DATA The Kalman filter is a widely used methodology which recursively calculates optimal estimates of unobservable state variables given all the information available up to some moment in time Using maximum likelihood methods we can also obtain consistent estimates of model parameters In finance the Kalman filter has been used to estimate and implement stochastic models of interest rates commodities and other relevant economic variables In spite of its extensive use the literature has not stressed on the Kalman filters ability to use historical information when there are missing observations Most previous works have used complete panel data even at the cost of throwing away data on contracts not traded frequently or of aggregating data with close to but not identical maturities with evident loss of information This problem is particularly acute in markets with infrequent trading where contracts with specific maturities do not trade every day Below we show that a natural extension of the standard Kalman filter may be applied to jointly estimate the current term structure and its dynamics in markets with infrequent trading 4 1 Standard Kalman filter In this section we present a very brief description of the Kalman filter For a detailed explanation see for example Harvey 1989 Chapter 3 or Hamilton 1994 Chapter 13 Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe
56. DEN OBTENER DE LAS CAPACIDADES SERVICIOS Nombre del producto o servicio Nombre del usuario Capacidad de desarrollar metodolog as e Nuevos Servicios y Capacitaci n en gesti n del riesgo Instituciones financieras y regulatorias implementar soluciones en el mbito de la Ingenier a Financiera y Gesti n del Riesgo 2 9 CAPACIDAD INSTITUCIONAL PARA GESTION CIENTIFICO TECNOLOGICA describa las principales capacidades creadas o fortalecidas por el proyecto se alando expresamente cual es el caso Fortalecimiento del FinlabUC Laboratorio de Investigaci n Avanzada en Finanzas lo que permite abordar nuevos proyectos de investigaci n aplicada con nuevos desarrollos tecnol gicos asociados 2 10 FORMACI N DE PERSONAL FORMACION CIENTIFICO TECNOLOGICA DE PARTICIPANTES EN EL PROYECTO Tipo T tulo de Curso o taller o del Nombre del Disciplina Instituci n o empresa Mes y A o Ciudad Pa s 1 Proyectos de t tulos proyecto de t tulo o de la tesis participante disciplina en que realiz la en que el donde se realiz 2 Tesis de magister Fondecyt actividad de formaci n participante logr la formaci n 3 Tesis doctorales predominante ind quelos todos la formaci n ind quelos todos 4 Posdoctorados Modelo estoc stico FELIPE 14146082 K Finanzas Pontificia Universidad Marzo 2007 Santiago Chile multicommodity para la SEVERINO do Che din mica de precios de DIAZ contratos futuros Selecci n y estimaci n del mo
57. Dai and Singleton 2000 for interest rates which can be traced back to Vasicek 1977 and Langetieg 1980 However in contrast to the interest rates literature which usually assumes a stationary process for the underlying spot rate the Gaussian model for the spot price presented in this article is nonstationary as it is usually assumed in the commodities literature Journal of Futures Markets DOI 10 1002 tut Cortazar and Naranjo Even though one factor models may be able to explain a sizable fraction of the total price variance these models tend to fit observed futures prices and the term structure of the volatility of futures returns rather poorly The optimal number of factors that should be specified in a model depends on the stochastic behavior of the term structure of the specific commodity that is being modeled Cortazar amp Schwartz 1994 and on the complexity that the modeler is willing to accept l The N factor model presented in the following extends existing models of commodity prices to an arbitrary number of factors while providing simple analytic valuation formulas for futures prices This renders the model tractable and easy to implement and calibrate Moreover the model is Gaussian which allows the use of the Kalman filter to estimate unobserved state variables and the use of maximum likelihood techniques to calibrate model parameters However previ ous commodity literature has only focused in using one t
58. G Naranjo L An N factor Gaussian model of oil futures prices Journal of Futures Markets 2006 26 3 243 68 Andersen L Broadie M A primal dual simulation algorithm for pricing multi dimensional American options Management Science 2004 50 9 1222 34 G Cortazar et al Computers amp Operations Research 35 2008 113 129 129 51 Longstaff F Optimal recursive refinancing and the valuation of mortgage backed securities NBER Working Papers 10422 2004 52 Glasserman P Yu B Number of paths versus number of basis functions in American option pricing Annals of Applied Probability 2004 14 4 2090 119 53 Stentoft L Convergence of the least squares Monte Carlo approach to American option valuation Management Science 2004 50 9 1193 203
59. However there are serious problems when these methods are used in markets with sparse bond price data For example traditional curve fitting methods render unreliable estimates of the current term structure for days without a sufficient number of observations or without short or long term bond prices In addition a typical Kalman filter implementation assumes a complete panel of bond prices or yields which becomes problematic if there is a substantial number of missing observations as is the case in many emerging markets In this article we develop and implement a method for jointly estimating the current term structure and its dynamics in markets with infrequent trading We propose solving both issues by using a dynamic term structure model estimated from incomplete panel data To achieve this we modify the standard Kalman filter approach to deal with the missing observation problem We can then use historical price data and a dynamic model to estimate the current term structure With this approach we are able to obtain an estimate of the current term structure even for days with an arbitrary low number of price observations The proposed methodology can be applied to a broad class of continuous time term structure models with any number of stochastic factors To show the implementation of the approach for an emerging market with infrequent trading we estimate a three factor generalized Vasicek model using Chilean government bond price data The ap
60. INFORME FINAL GOBIERNO DE CHILE CONK YI C digo Proyecto D0311039 Nombre del Proyecto DESARROLLO DE HERRAMIENTAS COMPUTACIONALES PARA OPTIMIZAR LA GESTION DE CARTERAS DE INVERSION EN MERCADOS EMERGENTES APLICACION A LOS FONDOS DE PENSIONES EN CHILE Instituciones Participantes Pontificia Universidad Cat lica de Chile Otros Participantes AFP Habitat S A Dictuc S A RiskAmerica Director del Proyecto Gonzalo Cortazar Sanz Firma Fecha de emisi n 30 07 2007 GOBIERNO DE CHILE CONICYT FONDEF COMISION NACIONAL DE INVESTIGACION CIENTIFICA Y TECNOLOGICA FOMENTO AL DESARROLLO BERNARDA MORIN 495 CASILLA 297 V CORREO 21 FONO 3654400 FAX 6551394 CHILE CIENTIFICO Y TECNOLOGICO INDICE I PARTE ACTA DE TERMINO DEL PROYECTO I PARTE INFORME EJECUTIVO 1 RESUMEN EJECUTIVO CASTELLANO E INGLES 2 SINTESIS DE RESULTADOS 3 CAPACIDADES CIENTIFICO TECNOLOGICAS PRODUCTOS Y SERVICIOS DESARROLLADOS POR EL PROYECTO 4 RESULTADO EVALUACI N EX POST III PARTE INFORME DE GESTION OBJETIVOS DEL PROYECTO RESULTADOS IMPACTOS ACTUALES Y ESPERADOS EN EL MEDIANO PLAZO PLAN DE NEGOCIOS GESTION DEL PROYECTO Gre AN IV PARTE INFORME CIENTIFICO TECNOLOGICO INDICE INVESTIGACION Y DESARROLLO OTROS INFORMES TECNICOS EVALUACION CIENTIFICO TECNOLOGICA EVALUACION ECONOMICO SOCIAL oe D V PARTE ANEXOS Y APENDICES ANEXO 1 PLAN DE NEGOCIOS ANEXO 2 PLANES DE TRABAJO INICIAL Y EFECTIVAMENTE EJEC
61. MATION IN MARKETS WITH INFREQUENT TRADING GONZALO CORTAZAR EDUARDO S SCHWARTZ and LORENZO F NARANJO Pontificia Universidad Cat lica de Chile USA University of California at Los Angeles USA 3 New York University USA ABSTRACT There are two issues that are of central importance in term structure analysis One is the modelling and estimation of the current term structure of spot rates The second is the modelling and estimation of the dynamics of the term structure These two issues have been addressed independently in the literature The methods that have been proposed assume a sufficiently complete price data set and are generally implemented separately However there are serious problems when these methods are applied to markets with sparse bond prices We develop a method for jointly estimating the current term structure and its dynamics for markets with infrequent trading We propose solving both issues by using a dynamic term structure model estimated from incomplete panel data To achieve this we modify the standard Kalman filter approach to deal with the missing observation problem In this way we can use historic price data in a dynamic model to estimate the current term structure With this approach we are able to obtain an estimate of the current term structure even for days with an arbitrary low number of price observations The proposed methodology can be applied to a broad class of continuous time term structure models
62. MINISTRACION SUPERIOR TOTAL 506 417 122 873 229 912 153 632 ANEXO 4 INFRAESTRUCTURA Y BIENES ADQUIRIDOS POR EL PROYECTO Para cada obra de infraestructura o equipo cuyo valor facturado sea mayor a US 5 000 identifique los usos responsables y porcentaje de tiempo en que ser utilizado de modo que su uso sea coherente con la l nea de trabajo del proyecto El plan de mantenci n se debe realizar seg n est ndares 1 Listado de obras de infraestructura listado definitivo identificando el nombre de la obra caracter sticas superficie construida la unidad institucional que la utiliza y la direcci n del lugar en que se encuentra Nombre de la infraestructura Caracter sticas de la Superficie Unidad Institucional Direcci n calle N ciudad construcci n responsable 2 Listados de bienes equipos y otros listado definitivo de equipos identificando el nombre caracter sticas y c digo del equipo y de inventario precio facturado la unidad institucional a que est asignado el responsable de la unidad y la direcci n del lugar en que se encuentra MM Proyector Multimedia Nec vr670 73008 IV 4 Computadores Armados Intel Pentium 73736 73737 1 900 IV 73738 73739 continuaci n tabla anterior N Responsable nombre completo Unidad Direcci n calle N ciudad Usos Estimado de uso Institucional Gonzalo Cortazar Sanz Avda Vicu a Ma
63. MONEDA NACIONAL MM 276 RESUMEN DE INVERSIONES ITEM EN MONEDA Terreno Maquinaria y equipos po Otros definir TOTAL COSTO PROYECTO I D NMONEDA NMONEDA FINANCIAMIENTO NACIONAL EXTRANJERA PARTE V ANEXOS ANEXO 1 PLAN DE NEGOCIOS A continuaci n se describen los principales aspectos del Plan de Negocios en ejecuci n 4 1 Productos Se considera la comercializaci n de 3 servicios independiente Cada uno de ellos se distribuye via Internet como m dulos independientes de la plataforma RiskAmerica la que al incluir estos nuevos m dulos se distribuye bajo el nombre de RiskAmericaPlus Los 3 M dulos de Servicios ofrecidos son 4 1 1 M dulo SVC El SVC o Sistema de Valorizaci n de Carteras consiste en un m dulo del servicio RiskAmericaPlus cuyo objetivo es asignarle una TIR a cada nemot cnico solicitado El usuario env a v a Web un archivo indicando los nemot cnicos asociados a su cartera devolviendo el sistema la TIR que el modelo le asigna a cada uno La TIR del modelo depende de si el activo fue transado ese d a de cu l es la curva de referencia para el d a la que es actualizada diariamente y de la historia de spreads que este nemot cnico ha exhibido respecto de la curva en el pasado RiskAmerica RiskAmerica Portfolio mmodity Valorizaci n d ras Valorizaci n de Carteras Valorizaci n de Carteras Excepciones Usuario Gonzalo Cortazar
64. Monthly values of the extended Brennan and Schwartz 23 open mine according to historical copper pricing conditions from January 1999 December 2003 MM US 15 7 r 0 4 0 5 0 6 0 7 0 8 Spot Copper Price US e NPV Open Mine ROV 4 Closed Mine ROV Fig 8 Value of the open mine using ROV and NPV as a function of spot price for y 0 01 and v 0 1 time horizon it could be thought that current spot prices would not have a great effect on mine values Fig 7 shows this is not the case Doing comparative static analysis on how mine value changes with variations in the spot price or in any individual state variable or parameter value is rather straightforward For example Fig 8 shows how mine value increases with copper spot prices It is also interesting to note how mine values are convex because as mine value approaches zero the probability of abandoning the mine increases Finally the same figure compares mine value computed with the real option model to a simple net present value calculation which does not recognize operating flexibilities to abandon or close operations It can be seen that when spot prices are lower option values are greater and these two valuation methodologies diverge the most By the same token when prices are high flexibilities are not too valuable and both valuations converge Comparative static analysis for the value or for the optimal policy can e
65. ORIN 495 CASILLA 297 V CORREO 21 FONO 3654400 FAX 6551394 CHILE 1 OBJETIVOS DEL PROYECTO 1 1 OBJETIVOS GENERALES Programados El objetivo principal del proyecto es desarrollar herramientas aplicaciones y servicios computacionales que aprovechen en forma efectiva las tecnologias asociadas a internet para modernizar el sistema financiero nacional apoyando una mejor gesti n de carteras de inversi n en activos transados en el mercado nacional e internacional Los desarrollos se focalizar n preferentemente en la problem tica de los fondos de pensiones pero sus resultados impactaran la gesti n de otras carteras de inversi n como las administradas por compa as de seguros y fondos mutuos entre otros Esta modernizaci n se apoyar tanto en el estado del arte metodol gico mundial como en investigaci n cient fica que aborde la problem tica de mercados financieros poco profundos como el nacional con activos que se transan con una baja frecuencia thin markets lo que dificulta el uso de numerosos procedimientos y metodolog as utilizadas en los mercados desarrollados De este modo se pretende 1 apoyar una gesti n m s eficiente de las carteras al incluirse mayor informaci n relativa a retornos y riesgos involucrados 2 hacer un an lisis de estrategias de inversi n que apoye la asignaci n de activos asset allocation 3 apoyar funciones de medici n y gesti n del riesgo y 4 establecer un conjunto de benchm
66. Optimista 0 50 100 Descripci n Variable 2 Porcentaje de Cartera Administrada que es impactada por nuevas tecnolog as de Gesti n Optima de Cateras Valor M s Probable Descripci n Variable 3 Porcentaje de Activos en Cartera que es impactada por nuevas tecnolog as de Gesti n Optima de Cateras A O 1 2 3 4 5 6 7 8 9 10 INGRESOS Ingresos por ventas 0 0 297 535 772 891 1010 1010 1069 1129 Externalidad Positiva 1 0 O 10457 10457 10457 10457 10457 10457 10457 10457 Total ingresos 0 O 10 754 10 991 11 229 11 348 11 466 11 466 11 526 11 585 COSTOS Mano de obra calificada 0 0 226 357 487 552 616 646 676 676 Mano de obra no calificada 0 0 0 0 0 0 0 0 0 0 Insumos M N 0 0 24 26 29 32 35 39 43 43 Bienes de capital M N 0 0 14 15 17 19 20 23 25 25 Otros M N 0 0 12 13 15 16 18 19 21 21 Total costos 0 0 276 412 548 618 689 727 765 765 INVERSIONES En moneda nacional 0 0 0 0 0 0 0 0 0 0 En moneda extranjera 0 0 0 0 0 0 0 0 0 0 Total inversiones 0 0 0 0 0 0 0 0 0 0 COSTO I amp D parciales por a o En M N 290 219 0 0 En M E 0 0 0 0 Total I amp D todos los aportes 290 219 0 0 S lo lo solicitado a FONDEF 97 97 0 0 BENEFICIOS 290 219 10 478 10 579 10 681 10 729 10 778 10 740 10 761 10 821 Ingresos 0 0 10 754 238 238 119 119 0 0 0 Costos 0 0 276 136 136 70 71 38 38 0 Inversion O 509 509 6 6 6 6 6 6 6 Costo total I D 290 219 0 0 Fondef 97 97 0 0 Beneficio neto 290 290 10 987 108 108 55 54 32 32 6 568 06 9 244
67. PRODUCTO SERVICIO O VALOR ACTUAL CRITERIOS PARA LA FIJACI N DE PRECIO MODALIDAD DE LA TRANSFERENCIA PROCESO MM SVC Indices Portfolio MM 781 No existen consideraciones especiales Excedentes de Dictuc SA que es de propiedad de la PUC 2 6 TRANSFERENCIA TECNOLOGICA PRODUCTO SERVICIO PROCESO O INSTITUCI N EMPRESA PRODUCTORA PARTICIPO EN EL DESARROLLO DEL PROYECTO CONTRATO DE TRANSFERENCIA VENTA O LICENCIA SVC ndices Portfolio RiskAmerica Si MMS 300 Continuacion PRODUCTO ACCIONES FUTURAS COSTO estimado en MM de 1999 de las FUENTES DE FINANCIAMIENTO SERVICIO O diga cu les son las acciones recursos y plazos no desarrolladas ni prove dos por el acciones necesarias faltantes para la PROCESO nombre proyecto que permitir n la explotaci n de los resultados del proyecto transferencia SVC Contactos individuales y eventos de promoci n ndices Portfolio 2 7 VENTAS INSTITUCIONALES de productos y servicios del proyecto no consideradas en el pto 2 6 CONCEPTO qu se vendi A O que se efect o la venta MONTO en MM del a o 1999 2 8 CAPACIDADES TECNOLOGICAS Detalle las capacidades tecnol gicas creadas o mejoradas con este proyecto Ejemplos en APENDICE 1 CAPACIDADES TECNOL GICAS PRODUCTOS O SERVICIOS QUE SE USUARIOS DE LOS PRODUCTOS O PUE
68. Plan De Experimentos 15 08 2005 15 08 2005 15 08 2005 Definido Para J Assetallocation Experimentos Criticos Do Efectuados periodos 25 08 2005 25 08 2005 25 08 2005 Efectuados Assetallocation Prototipo Probado a nivel 7 Laboratorio Prototipo Probado A Nivel 30 09 2005 30 09 2005 30 09 2005 3 Laboratorio Assetallocation Prototipo probado a Nivel 27 Piloto Planta 30 09 2005 30 09 2005 30 09 2005 Assetallocationv Plan de Experimentos ele 15 10 2005 15 10 2005 15 10 2005 Plan De Experimentos Definido Riskmatrix Plan de Experimentos 1 definido Plan De Experimentos Definido Portfoliovalue 15 10 2005 15 10 2005 15 10 2005 Experimentos Cr ticos Dor Efectuados Experimentos Criticos Efectuados Portfoliovalue 25 10 2005 25 10 2005 25 10 2005 Pll Experimentos Criticos Br ee 25 10 2005 25 10 2005 25 10 2005 Experimentos Criticos Efectuados Riskmatrix Dise o de Prototipo 9 emitido 30 10 2005 30 10 2005 30 10 2005 Diseno De Riskmatrixv 1 Dise o de Prototipo 1 emitido 30 10 2005 30 10 2005 30 10 2005 Diseno De Portfoliovaluev1 Prototipo Probado a nivel 1 Laboratorio Prototipo Probado A Nivel Laboratorio Portfoliovalue 30 12 2005 15 03 2006 23 01 2006 Prototipo probado a Nivel 21 pijoto Planta 30 12 2005 15 03 2006 23 01 2006 Portfoliovaluev 1 Prototipo probado a Nivel Piloto Planta 30 12 2005 30 12 2006 30 12 200
69. Sample RMSE for the Year 2001 and Out of Sample RMSE for Years 2002 to 2004 for Copper Futures With the Use of Different Numbers of Factors 2001 In sample 2002 2003 2004 1F 2 46 2 36 1 37 5 88 2F 0 25 0 12 0 17 1 29 3F 0 16 0 07 0 08 0 44 reversion may make the model fit the oil futures well but not other commodity futures like metals To address this issue Table V presents results for the same model estimated with the use of copper futures for 1 2 and 3 factors In par ticular the data used in this estimation consist of daily copper futures traded at NYMEX from January 1992 to December 2001 The table dis plays out of sample RMSE for the years 2002 2003 and 2004 and also in sample RMSE for the vear 2001 It can be seen that results similar to those obtained for oil are found for copper suggesting that the model behaves well for some metals This is also consistent with the results reported by Schwartz 1997 As a last measure of robustness the model volatility term structure of futures returns is calculated from Equation 18 and compared to the empirical volatilities 67 7 obtained directly from observed futures prices a GAT ree log F t 7 F t At 7 a 29 i l Figure 7 shows for each model and panel the theoretical and empirical volatility term structures It can be seen that one and two factor models do not fit the empirical volatility term structure well Although
70. UCION TOTAL DEL PROYECTO incorpore planillas presupuestarias inicial y final detalladas por Item y fuente de financiamiento PLANILLA PRESUPUESTARIA INICIAL FINANCIAMIENTO ITEM COSTO INSTITUCIONES EMPRESAS U FONDEF TOTAL OTRAS ENTIDADES HONORARIOS 278 738 67 789 87 525 123 424 INCENTIVOS REMUNERACIONES SUBCONTRATOS 0 000 0 000 0 000 0 000 CAPACITACION 0 000 0 000 0 000 0 000 PASAJES Y VIATICOS 13 080 0 000 9 319 3 761 EQUIPOS 35 161 25 361 5 000 4 800 INFRAESTRUCTURA 29 700 29 700 0 000 0 000 SOFTWARE 0 000 0 000 0 000 0 000 FUNGIBLES 2 700 0 000 1 350 1 350 PUBLICACIONES Y 0 000 0 000 0 000 0 000 SEMINARIOS PROPIEDAD INTELECTUAL 126 500 0 000 125 000 1 500 GASTOS COMUNES 2 697 0 000 0 000 2 697 GASTOS GENERALES 9 070 0 000 0 306 8 764 GASTOS DE 11 704 0 000 0 000 11 704 ADMINISTRACION SUPERIOR TOTAL 509 350 122 850 228 500 158 000 PLANILLA PRESUPUESTARIA FINAL FINANCIAMIENTO ITEM COSTO INSTITUCIONES EMPRESAS U FONDEF TOTAL OTRAS ENTIDADES HONORARIOS 280 122 67 759 78 521 133 842 INCENTIVOS REMUNERACIONES SUBCONTRATOS CAPACITACION PASAJES Y VIATICOS 7 697 3 936 3 761 EQUIPOS 35 781 25 414 6 204 4 163 INFRAESTRUCTURA 41 519 29 700 11 819 SOFTWARE 0 169 0 169 FUNGIBLES 1 050 0 953 0 097 PUBLICACIONES Y SEMINARIOS PROPIEDAD INTELECTUAL 125 001 125 001 GASTOS COMUNES GASTOS GENERALES 3 374 3 309 0 065 GASTOS DE 11 704 11 704 AD
71. UTADO ANEXO 3 PLANILLAS PRESUPUESTARIAS INICIAL Y EJECUTADO ANEXO 4 INFRAESTRUCTURA Y BIENES DEL PROYETO ANEXO 5 PUBLICACIONES I PARTE ACTA DE TERMINO DEL PROYECTO A IDENTIFICA CI N DEL PROYECTO Nombre del Proyecto DESARROLLO DE HERRAMIENTAS COMPUTACIONALES PARA OPTIMIZAR LA GESTION DE CARTERAS DE INVERSION EN MERCADOS EMERGENTES APLICACION A LOS FONDOS DE PENSIONES EN CHILE C digo FONDEF del Proyecto D0311039 Director del Proyecto Gonzalo Cortazar Sanz Instituciones Beneficiarias Pontificia Universidad Cat lica de Chile Empresas participantes AFP Habitat S A Dictuc S A RiskAmerica Otras Instituciones participantes Montos comprometidos en contrato Fondef 158 00 millones Instituciones 122 85 millones Empresas 228 50 millones Otros millones B EJECUCI N 1 Fecha toma de raz n 06 12 2004 2 Plazo contractual en meses 28 meses 3 Fecha efectiva de inicio 06 12 2004 4 Fecha de t rmino efectiva 30 07 2007 5 Duraci n efectiva 31 meses 6 El proyecto tuvo una duraci n total de 31 meses El financiamiento por FONDFF se efectu durante 28 meses Montos efectivamente aportados Fondef 155 303 millones Instituciones 122 873 millones Empresas 229 913 millones Otros 0 millones e Costo Total del Proyecto El costo total del proyecto fue de 509 35 millones de pesos e Aportes de Fondef El monto total rendido y aprobado por FONDEF es de 153 632 730 ciento cincuenta
72. a os 13 24 TIR 3 53 M 29 20 Convexidad a os 123 67 Serie de Tiempo Graficar valor W al Grafico a cry Graficar Tipo v Ol Asimismo se pueden comparar y realizar an lisis entre los distintos ndices Ri skAmerica RiskAmerica SVC ndices Portfolio mmodity Mis Indices Indices Renta Fija Opciones a z Gr fico Yalor ll da Indices Renta Variable S Agregar Indices Constructor de Indices Comparador de Indices Seleccionar 66 Fecha Inicio Comparador de Indices Walor rv Fecha T rmino 11 0 Graticar ad o amp de de amp PERLE SEE PET oap eg 3 Indices Seleccionados RiskAmerica Plus m Gob Global Corp Global m IF Global Gob Global LH Global BR Global M FF Global M LH Global V Br Global Se entrega adem s la posibilidad de construir y ndices personalizados ario Gon RiskAmerica Indices Portfolio mmodity Constructor Descripcion de Indices Descripci n del Indice indice indice a El Indices Renta Variable Seleccionar Indice Fecha de Consulta 09 01 2007 E imposici n 09 01 2007 Caracter sticas 09 01 2007 Estadisticas 040142007 Hombre indice Valor 1 116 30 a ir ia Y Graficar Moneda ve o E Ticker indice 1 Dia 0 0267 Fecha Inicio 02 01 2006 MTD 0 61 W Papeles 2 YTD 0 61 Monto MMM 0 00 1 A o 9 98 Duraci n a os 976 DesvEst 1 a o 4 15 MM
73. ach without having to compromise rigorous modeling in order to obtain a solution Acknowledgments We thank professor Eduardo Schwartz UCLA the researchers of the FINlabUC Laboratorio de Investigaci n Avanzada en Finanzas Pontificia Universidad Cat lica de Chile and the participants of the 9th Annual International Conference of Real Options Paris June 2005 for helpful discussions We also thank the financial support of FINlabUC FONDECYT Grant 1040608 FONDEF Grant D0311039 and FUNDACION COPEC UNIVERSIDAD CATOLICA Grant PC00021 References 1 Black F Scholes M The pricing of options and corporate liabilities Journal of Political Economy 1973 81 637 54 2 Gibson R Schwartz ES Stochastic convenience yield and the pricing of oil contingent claims The Journal of Finance 1990 45 3 959 76 3 Schwartz ES The stochastic behavior of commodity prices implications for valuation and hedging The Journal of Finance 1997 52 3 923 73 4 Schwartz ES Smith JE Short term variations and long term dynamics in commodity prices Management Science 2000 46 893 911 5 Casassus J Collin Dufresne P Stochastic convenience yield implied from commodity futures and interest rates Journal of Finance 2005 60 5 2283 331 6 S rensen C Modeling seasonality in agricultural commodity futures Journal of Futures Markets 2002 22 393 426 7 Cortazar G Schwartz ES Implementing a stochastic model for oil futures prices Energy Economics 2
74. aciones Cortazar G Gravet M Urzua J 2008 The Valuation of Multidimensional American Real Options using the LSM Simulation Method Computers amp Operations Research Vol 35 2008 113 129 Cortazar G Schwartz E S Naranjo L 2007 Term Structure Estimation in Markets with Infrequent Trading International Journal of Finance and Economics por aparecer Cortazar G Naranjo L 2006 An N Factor Gaussian Model of Oil Futures Prices The Journal of Futures Markets Vol 26 No 3 March 2006 243 268 1 2 2 Documentos de Trabajo a n no publicados Cortazar G Milla C Severino F 2007 A Multicommodity Model of Futures Prices Using Futures Prices of One Commodity to Estimate the Stochastic Process of Another Cortazar G Bernales A Beuermann D 2007 Methodology and Implementation of Value at Risk Measures in Emerging Fixed Income Markets with Infrequent Trading 1 2 3 Tesis de Mag ster Finalizadas Modelo estoc stico multicommodity para la din mica de precios de contratos futuros Selecci n y estimaci n del modelo utilizando componentes principales comunes y filtro de Kalman FELIPE SEVERINO DIAZ Tesis de Mag ster en Ciencias de la Ingenier a Pontificia Universidad Cat lica de Chile 26 03 2007 Estimaci n de Spreads por Liquidez en un Mercado con Pocas Transacciones El Caso del Mercado de Bonos del Banco Central de Chile PEDRO MAT AS MORAL MESA Tesis de Mag ster en Cien
75. ain the value of a pure discount bond P x P x 1 exp u t x v t 5 Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe IJFE 317 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 IJFE 317 6 G CORTAZAR ET AL where ne a exp k T 6 kj N pi i l K T a 7 I Y FAY 1 exp k t 1 exp k t y 1 exp k 23 ay hk ki kj ki kj Sometimes it is convenient to work with the equivalent annualized spot rate From equation 5 we obtain R x 1 J log P x t e v t 8 which is a linear function of the state variables Therefore under the generalized Vasicek model spot rates also have the Gaussian distribution The value of a coupon bond B x t with maturity t ty and N coupons C paying at times t can therefore be computed as N B x 1 Y CiP x 1 9 i l The implied yield to maturity of a coupon bond maturing at 7 y x 7 is obtained solving the following equation N B x 1 Y Ci exp yti 10 i 1 Note that if C gt 0 Vie 1 N the relationship between B x 7 and y x 7 is one to one and continuous in the state variables However unlike spot rates y x T is not a linear function of the state variables and will not be normally distributed 4 KALMAN FILTER ESTIMATION WITH INCOMPLETE PANEL
76. all price information available as opposed to traditional approaches of aggregating data for a set of maturities A Kalman filter estimation procedure that allows for a time dependent number of daily observations is used to calibrate the model When applied to all daily oil futures price transactions from 1992 to 2001 the model performs very well requiring at least three factors to explain the term structure of futures prices but four factors to fit the volatility term structure The organization of the article is as follows The next section explains the N factor Gaussian model for the spot price of oil The Kalman filter methodology is then presented in an incomplete panel data setting Estimation results for oil futures prices are presented and a conclusion is provided OIL PRICE MODEL AND FUTURES VALUATION In this section an N factor Gaussian model for the spot price of a com modity is presented as well as its relation with other models commonly found in the commodities literature In addition valuation formulas for futures contracts and the theoretical volatility term structure of futures returns are obtained The Model The N factor model presented in this article generalizes existing two and three factor models commonly found in the literature Cortazar amp Schwartz 2003 Gibson amp Schwartz 1990 Schwartz 1997 Schwartz amp Smith 2000 to an N factor setting The model is based on the A N canonical representation of
77. ally the case because financial markets have innovations and new contracts with longer maturities are frequently introduced Traditional applications of the Kalman filter typi cally address this missing data problem by aggregating or discarding data with the consequent loss of information The missing data problem may be so relevant that some authors have chosen not to use the Kalman filter but to propose an alternative procedure to handle cases where the panel data are incomplete Cortazar and Schwartz 2003 propose a very simple estimation procedure and apply it to an incomplete panel of oil futures prices The methodology however does not make an optimal use of prices in the estimation of state variables as opposed to the Kalman filter and is unable to obtain parameter estimation errors An alternative procedure used in this article is to modify the tradi tional application of the Kalman filter to address incomplete panel data Journal of Futures Markets DOI 10 1002 fut conditions Sgrensen 2002 uses this procedure for a seasonal price model and Cortazar Schwartz and Naranjo 2003 use it for estimating the term structure of interest rates in an emerging market with low frequency transactions However this approach has not received much attention in the literature This article studies the ability of an N factor Gaussian model to explain the stochastic behavior of oil futures prices when estimated with the use of
78. ameter is small although very significant Figures 2 and 3 display the standard deviation of the estimation error calculated as the squared root of the diagonal elements of matrix P for two different state variables in the four factor model This is important when incomplete data sets are used As expected a more complete data set induces a lower estimation error but the estimation error of a particular In this article this parameter corresponds to the root mean squared error RMSE for the whole sample Journal of Futures Markets DOK 10 1002 fut ei OU Futures Prices state variable should depend on the availability of futures contracts for the specific maturity range represented by that particular state variable Figure 2 plots from January 1999 to December 2001 the estimation error of x which heavily depends on the availability of long term futures contracts It can be seen that the estimation error sharply increases for some dates corresponding to days when long term contracts are not traded In addition it shows that the standard deviation increases between two consecutive issues of long term contracts because a shorter remaining maturity provides less information of long term behavior Similarly Figure 3 displays the time evolution from January 1999 to December 2001 of the estimation error for state variable x which has the highest mean reversion among all state variables Because short term futures contra
79. ample illustrates that our approach provides much more stable curves than those obtained by curve fitting methods Table 3 presents in sample and out of sample error measures by maturity Out of sample error measures were calculated by re estimating the model using data from 1997 to 2000 and then comparing yield curves obtained from the model to observed yields for the year 2001 which was not used in the parameter estimation It can be seen that all errors are reasonably low while errors for short term bonds are larger than for long term bonds Out of sample errors are similar to in sample errors showing the stability of the model and its ability to be used in real world applications Finally we analyse the volatility term structure of spot interest rates and compare it to volatilities obtained directly from bond yields The theoretical volatility structure of interest rates which is Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe IJFE 317 IJFE 317 TERM STRUCTURE ESTIMATION 13 1 ae Bond Yields on 10 06 1999 3 E 6 5 gt o 4 q E Previous Day Observed Bond Yields 2 e Observed Bond Yields Model Term Structure 9 5 0 5 0 5 10 15 20 9j 11 Maturity Years 13 Figure 6 Estimated and observed coupon bond yields on 10 06 1999 15 Table 3 In sample and out of sample RMSE for the year 2001 17 Maturity range Years
80. ared errors but also by comparing empirical and model implied volatility term structures which is critical for valuing option like contingent claims and also for risk management applications In addition to defining the commodity price model a methodology to obtain parameter estimates must be chosen For Gaussian models such as the one presented in this article a closed form formula of the probability distribution of futures prices is known and parameters may be obtained by maximizing their likelihood function Therefore consis tent estimates of model parameters are obtained with their respective estimation errors Some difficulties must be addressed however to successfully apply these models to commodity markets For example most multifactor mod els are based on nonobservable state variables that must be estimated from observed prices Thus in addition to calibrating model parameters A different approach for modeling the spot price of a commodity is based on the Heath Jarrow and Morton 1992 no arbitrage model Cortazar amp Schwartz 1994 Miltersen amp Schwartz 1998 journal of Futures Markets DOL 10 1000Aut Cortazar and Naranjo which are assumed constant for each data set it is necessary to estimate state variables for each date When state variables and futures prices are related by a closed form formula it is possible to estimate the unobserved state variables by inverting the pricing formula Chen a
81. arks para diversas carteras de inversi n Todo lo anterior busca favorecer la gesti n e informaci n para directivos y usuarios y en ultimo t rmino la competitividad y desempe o de la industria No obtenidos Obtenidos no programados A n cuando desde un principio se esperaba no restringir el impacto del proyecto s lo a la industria de las AFP sino tambi n a la de otros actores de la industria financiera el impacto sobre la industria de los Fondos Mutuos fue m s fuerte y m s anticipado de lo programado Es as como desde marzo de 2006 los precios para todos los instrumentos de renta fija son entregados v a Internet a todos los fondos mutuos nacionales como precio oficial para ser utilizados en el c lculo diario de la cuota de todos los fondos mutuos seg n circular de la Superintendencia de Valores y Seguros y en acuerdo con la Asociaci n de Administradoras de Fondos Mutuos de Chile Asimismo ya se han iniciado contactos internacionales que pueden llevar a exportar algunas de las tecnolog as desarrolladas 1 2 OBJETIVOS ESPECIFICOS Programados El proyecto en sus diversos mbitos de acci n tiene los siguientes objetivos espec ficos 1 Generar conocimiento cient fico acerca del comportamiento de los mercados financieros nacionales y su interrelaci n con los mercados internacionales Esta investigaci n deber generar informaci n de inter s para Chile y para entender el comportamiento de los mercados en los diver
82. as instituciones y empresas patrocinantes 4 Sobre el FONDEF 5 7 OBSERVACIONES CONCLUSIONES Y RECOMENDACIONES INSTITUCIONALES IV PARTE INFORME CIENTIFICO TECNOLOGICO Y ECONOMICO SOCIAL GOBIERNO DE CHILE CONK YI C digo Proyecto D0311039 Nombre del Proyecto DESARROLLO DE HERRAMIENTAS COMPUTACIONALES PARA OPTIMIZAR LA GESTION DE CARTERAS DE INVERSION EN MERCADOS EMERGENTES APLICACION A LOS FONDOS DE PENSIONES EN CHILE Instituciones Participantes Pontificia Universidad Cat lica de Chile Otros Participantes AFP Habitat S A RiskAmerica Dictuc S A Fecha de emisi n 23 07 2007 FONDEF FOMENTO AL DESARROLLO ER COMISION NACIONAL DE INVESTIGACION CIENTIFICA Y TECNOLOGICA BERNARDA MORIN 495 CASILLA 297 V CORREO 21 FONO 3654400 FAX 6551394 CHILE IV PARTE INFORME CIENTIFICO TECNOLOGICO Y ECONOMICO SOCIAL 1 INDICE 1 1 Indice por Tema de Investigaci n 1 Modelaci n y Calibraci n de Procesos Estoc sticos para Precios de Instrumentos en Mercados con Paneles de Datos Incompletos 2 Metodolog as de Valorizaci n de Derivados escritos sobre Subyacentes con procesos Completos 3 Metodolog as de Medici n de Spreads 4 Metodolog as de Medici n de Riesgos y de Asignaci n de Activos Asset Allocation para Carteras de Inversi n 5 Modelaci n y Calibraci n Conjunta de Procesos Estoc sticos de M ltiples Activos 1 2 ndice por Documento de Resultados 1 2 1 Public
83. asily be performed for any of the state variables strengthening the ability of the LSM method to study the behavior of an investment project for different scenarios 4 3 An alternative implementation for multi dimensional settings In the previous sections we have shown a simple implementation of the LSM approach for solving a real options model with a three factor price process As stated previously one of the main contributions of this approach is the computation of the expected continuation value by regressing discounted future option values on a linear combination of functional forms of current state variables The way these functional forms are chosen is not straightforward and as is discussed in this section it may become an important issue in high dimensional settings 126 G Cortazar et al Computers amp Operations Research 35 2008 113 129 9 8 7 Ane e Chebyshev cross products e Reduced Form wu 5 10 4 3 2 1 0 r r 0 10 20 30 40 50 60 Number of Regressors Increasing order Fig 9 RMSE as a function of the number of regressors for Chebyshev Polynomials and for the reduced base form using only futures Longstaff and Schwartz 19 propose for multidimensional implementations of their method the use of basic functions from Laguerre Chebyshev Gegenbauer Jacobi polynomials or the simple powers and cross products of the state variables used in th
84. carteras de inversi n 2 3 COMPETITIVIDAD DE LOS PRODUCTOS SERVICIOS Y O PROCESOS MEJORADOS agregue m s columnas si es necesario PRODUCTO SERVICIO VENTAJA naturaleza del MONTO valor actual del beneficio en PRODUCTO SERVICIO O PROCESO CON QUE DESVENTAJAS O PROCESO nombre beneficio MM COMPITE nombre breve descripci n exprese cuantitativamente M dulo SVC Calidad de Informaci n Valor mejor informaci n usuario Sistemas Internos otros proveedores Dependencia externa M dulo Indices Calidad de Informaci n Valor mejor informaci n usuario Sistemas Internos otros proveedores Dependencia externa M dulo Portfolio Calidad de Herramientas Valor mejor informaci n usuario Sistemas Internos Dependencia externa 2 4 MERCADO PRODUCTO SERVICIO OFERTA ACTUAL DEMANDA ESTIMADA DE y A OS CANALES DE i O PROCESO nombre Volumen MM 2007 Volumen MM 2007 PARTICIPACI N COMERCIALIZACI N i A ESPERADA DE Indique unidades Indique unidades MERCADO M dulo SVC 240 Usuario Mes 180 720 Usuario Mes 240 60 4 RiskAmerica M dulo Indices 36 Usuario Mes 10 720 Usuario Mes 120 60 4 RiskAmerica M dulo Portfolio 18 Usuario Mes 8 720 Usuario Mes 210 60 4 RiskAmerica 2 5 VALORIZACI N ECON MICA DE LOS RESULTADOS DEL PROYECTO Indique en forma sint tica Los detalles y justificaci n incorp relos en el plan de negocios
85. ch were made in the natural resource sector mainly because of its high irreversible investments and the well developed commodity futures markets Even though real option models like the one we just described have been successful in capturing many managerial flexibilities in general they have considered very simple specifications of the price risk process hindering the use of this approach in real world applications This simple risk specification represented the state of the art in commodity price modeling when this approach was developed more than two decades ago Since then much research has been done to capture in a better way the commodity price stochastic process but real option models have not kept pace with this research probably in part due to the added complexity to obtain numerical solutions in a multi factor setting In this section we extend the Brennan and Schwartz 23 model to include a multifactor specification for uncertainty model which in later sections will be solved numerically Commodity price processes differ on how convenience yield is modeled and on the number of factors used to describe uncertainty Early models i e Brennan and Schwartz 23 assumed a constant convenience yield and a one factor Brownian motion Later on mean reversion in spot prices began to be included as a response to evidence that futures 1 Later in the paper we add to this notation the subscript to indicate a simulated path G Cortazar et al
86. ci n de Si objetividad rigurosidad y compromiso de permanente innovaci n que puedan comunicar los proveedores de los mismos Es por esto que m s que proteger desarrollos la estrategia de protecci n consiste en comunicar en forma cre ble estos atributos y en ofrecer innovaciones permanentes dif ciles de replicar por otros proveedores 2 13 A IDENTIFICACION DE EVENTOS EVENTOS RELACIONADOS CON EL PROYECTO EN QUE PARTICIPO PERSONAL DEL PROYECTO N Titulo o nombre del evento Tipo de Evento Pais Ciudad Fecha de Fecha t rmino 1 Congreso pais ciudad d nde inicio del del evento 2 Seminario donde se se realiz el evento 3 Taller realiz el evento 4 Curso evento 5 Simposio 6 Mesa Redonda a especificar Clasificaci n 1 C amp T 2 de Negocios 3 de Difusi n 4 de Capacitaci n Otros especificar Evento organizado por 1 el proyecto Otros especificar Seminario Internacional de Innovaci n Chile Santiago 29 03 2007 29 03 2007 2 3 UC Fondef Financiera Lanzamiento RiskAmercaPLUS 4th Annual Conference of Asia Pacific India Gurgaon 20 06 2007 22 06 2007 1 Association of Derivatives APAD MDI Latin American Meeting of the Econometric M xico Ciudad de 02 11 2006 04 11 2006 Society M xico 4 2006 FMA Annual Meeting EEUU Salt Lake City Beijing 11 10 2006 14 10 2006 5 2006 Far Eastern Meeting of the Econometric 1 China 9 07 2006 12 07 2006 Society Rh
87. cias de la Ingenier a Pontificia Universidad Cat lica de Chile 17 01 2006 Metodolog a e Implementaci n de M todos de VALUE AT RISK en Mercados de Renta Fija con baja Frecuencia de Transacciones ALEJANDRO ADRIAN BERNALES SILVA Tesis de Mag ster en Ciencias de la Ingenier a Pontificia Universidad Cat lica de Chile 23 12 2005 Modelos Estoc sticos de Precios de Commodities y Estimaci n Conjunta de la Din mica de dos Commodities Mediante el Filtro de Kalman CARLOS IGNACIO MILLA GONZALEZ Tesis de Mag ster en Ciencias de la Ingenier a Pontificia Universidad Cat lica de Chile 23 12 2005 1 2 4 Memorias de T tulo Finalizadas Bonos Corporativos una Revisi n del Mercado y una Aproximaci n a un M todo Pr ctico de Valorizaci n Memoria Escuela de Ingenier a Pontificia Universidad Cat lica de Chile CLAUDIO EDUARDO HELFMANN SOTO 31 12 2005 Valorizaci n de Instrumentos Financieros en Mercados con Pocas Transacciones An lisis de una Metodolog a Basada en un Modelo Din mico para la Tasa Cero Real en Chile JOSE LUIS MANIEU ESPINOSA Memoria Escuela de Ingenier a Pontificia Universidad Cat lica de Chile 12 08 2005 Decisiones de Asset Allocation en Carteras de Inversi n de las AFP Aplicaci n del Modelo de Black amp Litterman RODRIGO ALFONSO IBANEZ VILLARROEL Memoria Escuela de Ingenier a Pontificia Universidad Cat lica de Chile 26 07 2005 1 2 5 Presentaciones en Congresos Acad mico
88. ckenna N 4860 USOS 1 Docencia 2 Investigaci n 3 Servicios 4 Capacitaci n 5 Asesor as Otros describir 3 PLAN DE MANTENCI N El contrato de finiquito incluir el plan de mantenimiento operaci n y cuidado de equipos y mantenci n de obras as como los seguros de rigor Nombre del N inventario Actividades principales Periodo entre Responsable equipo de mantenci n mantenciones ANEXO 5 PUBLICACIONES Cortazar G Gravet M Urzua J 2008 The Valuation of Multidimensional American Real Options using the LSM Simulation Method Computers amp Operations Research Vol 35 2008 113 129 Cortazar G Schwartz E S Naranjo L 2007 Term Structure Estimation in Markets with Infrequent Trading International Journal of Finance and Economics por aparecer Cortazar G Naranjo L 2006 An N Factor Gaussian Model of Oil Futures Prices The Journal of Futures Markets Vol 26 No 3 March 2006 243 268 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 IJFE 317 PROD TYPE COM ED KAVYASHREE pp 1 17 col fig NIL 3B2 PAGN MVIMALA SCAN VIMALA INTERNATIONAL JOURNAL OF FINANCE AND ECONOMICS 2 WILEY _ Int J Fin Econ 12 1 17 2007 InterScience DISCOVER SOMETHING GREAT Published online in Wiley InterScience www interscience wiley com DOI 10 1002 ijfe 317 TERM STRUCTURE ESTI
89. co Central pero que exhiben distinta liquidez ej PRC 8 a os versus PRC a 10 a os Tesis de Mag ster Estimaci n de Spreads por Liquidez en un Mercado con Pocas Transacciones El Caso del Mercado de Bonos del Banco Central de Chile PEDRO MAT AS MORAL MESA Tesis de Mag ster en Ciencias de la Ingenier a Pontificia Universidad Cat lica de Chile 17 01 2006 Memorias de Titulo Bonos Corporativos una Revisi n del Mercado y una Aproximaci n a un M todo Pr ctico de Valorizaci n Memoria Escuela de Ingenier a Pontificia Universidad Cat lica de Chile CLAUDIO EDUARDO HELFMANN SOTO 31 12 2005 Valorizaci n de Instrumentos Financieros en Mercados con Pocas Transacciones An lisis de una Metodolog a Basada en un Modelo Din mico para la Tasa Cero Real en Chile JOSE LUIS MANIEU ESPINOSA Memoria Escuela de Ingenier a Pontificia Universidad Cat lica de Chile 12 08 2005 Tema 4 Metodolog as de Medici n de Riesgos y de Asignaci n de Activos Asset Allocation para Carteras de Inversi n Existe una extensa literatura de c mo medir los riesgos financieros en una cartera e inversi n Value at Riks Tracking error etc y de c mo tomar decisiones de Asignaci n de Activos Asset Allocation que permitan mejorar el proceso de inversiones Sin embargo para poder resolver estos problemas en mercados emergentes como el chileno con ausencia de transacciones hace falta modificar procedimientos y generar informaci
90. convenience yield which implies a constant cost of carry Brennan amp Schwartz 1985 Even though this widely used and simple model has the advantage of being very tractable it has some undesirable properties like exhibiting a constant volatility term structure of futures price returns Empirical evidence suggests however that the volatility term structure of futures prices is a decreasing function of maturity Bessembinder Coughenour Seguin Smoller 1996 which may be expluined by the existence of mean reversion in commodity prices Bessembinder et al 1995 l To address this issue several authors have proposed different one factor models that take into account mean reversion in commodity prices Laughton amp Jacoby 1993 1995 Ross 1997 Schwartz 1997 However an empirical implication of all models that consider a single source of uncertainty is that futures prices for different maturities should be perfectly correlated which defies existing evidence To account for a more realistic model of commodity prices two and three factor models have been proposed Cortazar amp Schwartz 2003 Gibson amp Schwartz 1990 Hilliard amp Reis 1998 Schwartz 1997 Schwartz amp Smith 2000 The advantage of using more factors in mod eling the spot price process and the cost of carry is that a better fit to observed futures prices may be obtained This goodness of fit can gener ally be observed not only in terms of mean squ
91. ction to delay investments to expand capacity to reduce costs through learning among many others 8 11 The introduction of multifactor price models into these real option models with many interacting flexibilities increases the difficulty of solving them making traditional numerical approaches like the finite difference methods clearly inadequate There has been however new research on using some sort of computer based simulation procedures for solving American options which coupled with an increasing availability of computing power has been successfully applied to solving multifactor financial options 12 18 One of the most promising new approaches in this literature is the LSM method proposed by Longstaff and Schwartz 19 which has been tested for some financial options of limited complexity 20 22 In this paper we explore the applicability of the LSM method in a multidimensional real option setting We extend the Brennan and Schwartz 23 one factor model for valuing a copper mine initially solved using finite difference methods to include a more realistic three factor stochastic process for commodity prices more in line with current research We implement the LSM method and discuss how complexity may be reduced Numerical results show that the procedure may be successfully used for multidimensional models notably expanding the applicability of the real options approach The remainder of this paper is organized as follows Section 2
92. ction of reserves Results are very similar to those reported in 23 4 2 Results for the three factor extension of the Brennan and Schwartz 23 model We now report the solution to the Brennan and Schwartz 23 model extended to include the Cortazar and Schwartz 7 three factor commodity price model The parameter values used are those reported in Table 1 We now assume a 30 year concession horizon and three opportunities to switch operation states per year To value the mine for a particular date say April the 14th 1999 we must first determine the values of the state variables So yo Vo corresponding to that date which are 0 64 0 198 and 0 244 respectively Following the implementation procedure described in Section 4 1 we obtain a value for the open mine of MMUS 15 64 and for the closed mine of MMUS 15 52 To explore how mine value changes according to variations in price conditions we solve for the value of the mine for a 5 year time span Results are reported in Fig 7 It is interesting to note that mine value exhibits mean reversion Even though it is well known that copper prices do exhibit mean reversion which is captured in the three factor model given that a mine produces copper during a long G Cortazar et al Computers amp Operations Research 35 2008 113 129 125 32 30 28 12 r r r r r r r r r 01 99 08 99 02 00 09 00 03 01 10 01 05 02 11 02 06 03 12 03 Dates Fig 7
93. ctions data from January 1997 to December 2001 As noted in Section 4 the Kalman filter considers measurement errors in the observations For simplicity we assume that the error variance covariance matrix R is diagonal Also we aggregate bonds into five groups depending on their maturities the first group includes the discount bonds with maturities up to year and the next four groups include coupon bonds with maturities ranging from 1 to 5 years from 6 to 10 years from 11 to 15 years and from 16 to 20 years respectively Bonds within each group are assumed to have measurement errors with the same standard deviation e Ei Es Es and s respectively With these assumptions 18 different parameters must be estimated Table 2 presents parameter estimates and their respective estimation errors Note that all the parameters are statistically significant though the mean reversion coefficient of the first factor is very small suggesting that this factor follows a process which is close to a random walk Note that the correlation between the factors is very high which may lead us to believe that two factors could be sufficient to explain the dynamics of the yield curve However we find that with one and two factors the total in sample RMSE is 0 52 and 0 35 respectively compared with 0 12 obtained using Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe IJFE 317 11 13 15 17 19
94. cts usually trade every day there are no abnormal spikes like the ones observed in Figure 2 This standard deviation presents a decreasing seasonal pattern with a 1 month cycle following monthly emissions of short term futures contracts Model Robustness Besides the stability of parameter estimates the performance of this model and estimation procedure is measured by analyzing the fit to the observed futures prices term structure and the empirical volatility term structure of futures returns Figures 4 and 5 show the fit of the models for two dates These spe cific dates were chosen as examples of market conditions when oil 20 Futures Price S 0 2 4 6 8 t0 Maturity Years FIGURE 4 Estimated and observed oil futures prices on 06 30 1998 The figure displays the theoretical term structure of futures prices on 06 30 1998 with the use of one two three and four factor models and compares it to observed futures prices when the term structure is in contango Journal of Futures Markets DOI 10 1002 fut AO Futures Price Cortazar and Naranjo Maturity Years FIGURE 5 Estimated and observed oil futures prices on 04 11 2001 The figure displays the theoretical term structure of futures prices on 04 1 1 2001 with the use of one two three and four factor models and compares it to observed futures prices when the term structure is in backwardation TABLE iHi In Sample RMSE and Bias for Pane
95. del proyecto financiado parcialmente por proyecto Fondef DOOI1024 orientado a c mo determinar la mejor curva de precios de hoy y su din mica a trav s del tiempo en presencia de un n mero limitado de precios producto de la falta de liquidez del mercado La estrategia propuesta consiste en calibrar un modelo multifactorial para la din mica de los precios y utilizar filtros de Kalman calibrados con paneles incompletos Publicaciones Cortazar G Schwartz E S Naranjo L 2007 Term Structure Estimation in Markets with Infrequent Trading International Journal of Finance and Economics por aparecer Cortazar G Naranjo L 2006 An N Factor Gaussian Model of Oil Futures Prices The Journal of Futures Markets Vol 26 No 3 March 2006 243 268 Presentaciones en Congresos Acad micos Cortazar G Schwartz E S Naranjo L 2006 Term Structure Estimation in Markets with Infrequent Trading Latin American Meeting of the Econometric Society LAMES ITAM Ciudad de M xico Nov 2 4 2006 Cortazar G Schwartz E S Naranjo L 2006 Term Structure Estimation in Markets with Infrequent Trading 2006 FMA Annual Meeting Salt Lake City Oct 11 14 2006 Cortazar G Schwartz E S Naranjo L 2006 Term Structure Estimation in Markets with Infrequent Trading 2006 Far Eastern Meeting of the Econometric Society Beijing July 9 12 2006 Cortazar G Schwartz E S Naranjo L 2006 Term Structure Estimat
96. del proyecto presentados en su Formulaci n bajo estos supuestos fueron FLUJO NETO Ingresos 0 0 10 754 238 238 119 119 0 0 0 Costos 0 0 276 136 136 70 71 38 38 0 Inversion 0 509 509 6 6 6 6 6 6 6 Costo total I D 290 219 0 0 Fondef 97 97 0 0 Beneficio neto 290 290 10 987 108 108 55 54 32 32 6 568 06 9 244 18 90 50 14 Nota M N moneda nacional M E moneda extranjera Como se plante anteriormente la estimaci n de los par metros anteriores est sujeta a bastante incertidumbre Sin embargo se puede se alar que desde la formulaci n del proyecto uno de los par metros se ha incrementado significativamente al aumentar el par metro Fondos Potenciales subi al a o 2007 a m s de MM US 100 000 es decir 2 5 veces mayor que el valor estimado en la Formulaci n Esto permitir a dividir por 2 5 alguno de los indicadores anteriores por ejemplo suponer que el Incremento Marginal de la Rentabilidad fuera 0 1 en vez de los 0 25 asumidos originalmente y conservar el valor de los indicadores originales Debido a lo anterior una evaluaci n conservadora mantendr a los indicadores econ mico sociales presentados en la Formulaci n inicial DESCRIPCION DE LA SITUACION SIN PROYECTO Para analizar la situaci n sin proyecto se ha hecho el supuesto que en caso de no realizarse este proyecto otro con similares objetivos pero sin algunas de las sinergias presentes en este proyecto se desarrollar a con un retraso de s
97. delo utilizando componentes principales comunes y filtro de Kalman Estimaci n de Spreads por PEDRO 13551671 6 Finanzas Pontificia Universidad Enero 2006 Santiago Chile Liquidez en un Mercado MAT AS Gat licade Chile con Pocas Transacciones MORAL MESA El Caso del Mercado de Bonos del Banco Central de Chile Metodolog a e ALEJANDRO Finanzas Pontificia Universidad Diciembre 2005 Santiago Chile Implementaci n de ADRIAN Gatolicade Chile M todos de VALUE AT BERNALES RISK en Mercados de SILVA Renta Fija con baja Frecuencia de Titulo de Curso o taller o del proyecto de t tulo o de la tesis Tipo 1 Proyectos de t tulos 2 Tesis de magister 3 Tesis doctorales 4 Posdoctorados Nombre del participante Disciplina disciplina Fondecyt predominante en que realiz la ind quelos todos Instituci n o empresa actividad de formaci n Ciudad Pa s donde se realiz la formaci n ind quelos todos Mes y A o en que el participante logr la formaci n TE Transacciones A A A OA ON Modelos Estoc sticos de Precios de Commodities y Estimaci n Conjunta de la Din mica de dos Commodities Mediante el Filtro de Kalman Bonos Corporativos una Revisi n del Mercado y una Aproximaci n a un M todo Pr ctico de Valorizaci n Valorizaci n de Instrumentos Financieros en Mercados con Pocas Transacciones An lisis de una Metodolog a Basada en un Mode
98. dicals Inc i sia WILEY Published online in Wiley InterScience www interscience wiley com InterScience 244 Cortazar and Naranjo INTRODUCTION The valuation and hedging of commodity contingent claims has received a great amount of attention by both academics and practitioners and has become an important area of financial economics Inextricably inter woven with this issue is the modeling and estimation of the stochastic behavior of commodity prices The practical implication of having better models and estimation methodologies is that commodity producers and consumers and also financial intermediaries may implement sound investment and risk management strategies with vast economic implica tions On the contrary the application of naive models can lead to unre liable results which may include heavy financial losses to corporations Culp amp Miller 1994 Among commodities traded in financial markets oil is one of the most important and has been studied in the literature extensively Its relevance induces innovations in financial markets generating new oil contingent claims that should be used in the estimation of the stochastic behavior of oil prices One important source of information for the study of oil prices is the futures market Oil futures markets have included in recent years new futures contracts with longer maturities up to 7 years When new contracts are introduced there is no historical inf
99. e act e tanto como oferente y como contraparte de estas aplicaciones No obtenidos 2 RESULTADOS DEL PROYECTO 2 1 PRODUCTOS SERVICIOS Y O PROCESOS Detalle seg n tabla NOMBRE DESCRIPCI N TIPO DE TIPO DE ESTADO DEL MEDIDAS DE RSULTADO INNOVACI N DESARROLLO PROTECCI N P PROGRAMDO I INESPERADO M dulo Computacional V a WEB que Valoriza Comercializci n No es posible de ser Instrumentos de Renta Fija PortfolioValue patentado M dulo Indices M dulo Computacional V a WEB que Entrega Comercializci n No es posible de ser Informaci n de Referencia de Mercado patentado PortfolioBenchmarks M dulo Portfolio M dulo Computacional V a WEB que entrega Comercializci n No es posible de ser herramientas de gesti n para Carteras de Inversi n patentado PortfolioRisk AssetAllocation RiskMatrix 2 2 PAQUETE TECNOL GICO PRODUCTO PROCESO o PAQUETE TECNOLOGICO enumere el conjunto de elementos que compone el paquete tecnol gico asociado al producto o proceso desarrollado SERVICIO nombre M dulo SVC Set integrado de herramientas computacionales v a Internet orientadas a la valorizaci n de instrumentos financieros M dulo Indices Set integrado de herramientas computacionales v a Internet orientadas a generar y entregar informaci n relativa al comportamiento de los mercados financieros M dulo Portfolio Set integrado de herramientas computacionales v a Internet orientadas a apoyar la gesti n de
100. e internacionalizaci n de los servicios 4 5 Ventas Actuales y Proyecci n de Resultados Futuros Las ventas esperadas de los 3 m dulos de Servicios para el a o 2007 alcanzan MM 198 las que debieran incrementarse en los a os futuros hasta alcanzar un monto esperado de MM 570 el a o 2010 5 GESTION DEL PROYECTO 5 1 PLAN DE TRABAJO EFECTIVAMENTE REALIZADO vs PLAN DE TRABAJO INICIAL Comentario El proyecto se encuadr bastante bien en el Plan de Trabajo inicial propuesto Los principales cambios fueron la extensi n de la fecha final del proyecto en 1 mes y que algunos de los resultados programados sufrieron un reordenamiento en el tiempo La raz n principal de estos cambios fueron de acuerdo con cambios en la percepci n de los requerimientos de los potenciales clientes Ense anzas obtenidas Qu considerar a para mejorar la formulaci n y ejecuci n de un pr ximo plan 5 2 GASTO EJECUTADO vs PRESUPUESTO INICIAL Comentario El proyecto se ajust bastante bien al presupuesto inicial total presentado con peque os ajustes en algunos tems Ense anza obtenida Qu considerar a para la formulaci n y ejecuci n de un pr ximo presupuesto 5 3 INSTITUCIONES PARTICIPANTES A AL INICIO DEL PROYECTO APORTES comprometido en contrato CONCYT INSTITUCION pesos de 1996 A F P Habitat 103 500 Riskamerica Dictuc SA 125 000 PUC O REO B DURANTE EL PROYECTO INSTITUCION ROL APORTES reales OBSERVA
101. e estimation of long term interest rates becomes unreliable Also without a sufficient number of transactions an over parameterization of traditional models can occur We propose to solve the problems of term structure estimation in markets with infrequent trading by using also past price information to infer the current term structure This requires a dynamic model of the stochastic behaviour of interest rates to be able to mix current and past prices in a meaningful way Some dynamic models in particular multifactor ones use a limited number of unobservable factors to summarize the stochastic behaviour of the whole yield curve in a way that is sufficiently accurate but also tractable These unobservable state variables together with the model parameters must be estimated using observable bond price information In the following sections we present an estimation methodology based in the Kalman filter that may be successfully used to estimate the term structure in markets with infrequent trading To illustrate our estimation methodology we will consider a generalized Vacisek model for the instantaneous risk free interest rate Our methodology may be used however with other interest rate models such as a one factor CIR model Cox et al 1985 a multifactor CIR model Duffie and Kan 1996 or general exponential affine models Dai and Singleton 2000 among others A generalized Vasicek model is a multifactor mean reverting Gaussian model of the in
102. e proyecto debiera inducir que las administradoras de fondos de pensiones pueden mejorar la gesti n de sus carteras y de esta manera obtener mayores retornos de sus inversiones sin incrementar el riesgo asumido El principal impacto econ mico social del proyecto es el Incremento Marginal de la Rentabilidad aplicado a una Fracci n de los Fondos Potenciales que pudieran verse beneficiados con las herramientas de gesti n de carteras desarrolladas Por ltimo se debe estimar el adelantamiento de los flujos en n mero de a os que representa la realizaci n del proyecto comparado con la situaci n base sin proyecto i e se estima que si no hubiera habido proyecto otro similar se hubiera desarrollado teniendo el mismo impacto despu s Los cuatro par metros anteriores son dif ciles de estimar y de ellos depende el resultado de los indicadores econ micos sociales En la Evaluaci n ex ante presentada en la formulaci n del proyecto se asumieron los siguientes valores para estos par metros Incremento Marginal de la Rentabilidad 0 25 Fracci n 25 de las administradoras 60 de los activos 15 Fondos Potenciales MMUS 40 000 Adelantamiento 1 a o si no se hace el proyecto los flujos se realizan 1 a o despu s Con lo que el beneficio econ mico social por a o se estim en US 15 millones Se puede se alar que desde la formulaci n del proyecto uno de los par metros se ha incrementado significativamente al aumentar
103. ear to switch between operating states This is an approximation to the continuous time Brennan and Schwartz model which assumes an infinite concession time and infinite opportunities per year to switch operating states 124 G Cortazar et al Computers amp Operations Research 35 2008 113 129 Table 8 Open and closed mine value as a function of spot price Spot price US Ib Mine value finite Mine value difference method reported in 23 Simulation method Open Closed Open Closed 0 4 4 15 4 35 4 2 4 4 0 5 7 95 8 11 7 93 8 12 0 6 12 52 12 49 12 51 12 49 0 7 17 56 17 38 17 51 17 31 0 8 22 88 22 68 22 8 22 6 0 9 28 38 28 18 28 29 28 09 1 0 34 01 33 81 33 89 33 69 0 90 Critic Price US open close abandon 0 00 T T T T T T T T T T T T 150 140 130 120110100 90 80 70 60 50 40 30 20 10 Reserves Fig 6 Critical prices for opening closing or abandoning a mine as a function of reserve level obtained using the LSM method Table 8 compares the finite difference values reported in 23 with those obtained using the above simulation procedure The mine and market parameters used are those reported in 23 It can be seen that the simulation method converges to the known finite difference solution Our simulation procedure may also provide the optimal operating policy Fig 6 shows the critical prices for aban doning opening a closed mine and closing an open mine as a fun
104. ecision at each point is taken by maximizing market value among all available alternatives At time 7 given that the concession ends the value of both the open and the closed mine is zero Vr x a Q Wr x 0 0 0 VO Vo 22 Then at 1 T At there is no time left to change the operating policy so there is no need to estimate an expected continuation value So the market values are Vr ar x Q Max CF S7_as q 0 VQ 23 Wr_ar x Q Max CF Sr_ar q K2 0 VQ 24 Then at t T 2At we must estimate the expected continuation value We regress the discounted mine value on a linear combination of functional forms of the state variables L X for each inventory level Q Vra X Qe F494 Wy AX Oye 14044 Lra Ray 0 7240 l8w 0 7241 e 25 Once the optimal coefficients are found we can estimate the expected continuation values at t T 2At Gy o r 2 1Gw o r 2411 Lr 24r X l v o 7 2a1l4w 0 T 2A01 26 122 G Cortazar et al Computers amp Operations Research 35 2008 113 129 Table 2 Expected and realized value of an open mine as a function of the operating policy Expected value Optimal decision Realized value CF S 4 Gy 0 gar X 00 Continue open V x Q CF S q Vizar x 0 O gAt e O ADAr K MAt G y 0 x 00 Close V x Q K MAt W ya x 0 QO e 0 20 Ar 0 Abandon Vi x Q 0 Table 3 Expected and realized
105. ed maturities The vector of observations z is then at Zi E A2 where z and z are m x 1 and m x 1 vectors containing the observed yields of discount and coupon bonds respectively Of course either m or m can be zero but not both at the same time The parameters of the measurement equation are H d H d i 0 4 o ti d 3 wel eb we A4 u t d y 0 he sent mt A H 9 x 7 X vd WRye 1 Ti By Kerb T Egel di d o a y O A se x A YV 1 Tine Dem gt Tint Xdi 1 The gradient of the yield with respect to state variables can be computed by differentiating implicitly equation 10 with respect to the state variables Lon exp u c x v t 1 Lon exp y x os E exp x os a gt j l Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe A6 IJFE 317 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 41 49 51 IJFE 317 16 G CORTAZAR ET AL so that M T dy x t 1 Wexpul x v t M OX Xj exp WX 1 The remaining parameters to be specified belong to the covariance matrix of measurement errors In this paper we assume that this covariance matrix is diagonal and can only have five different parameters 7 s Es and Es The first of them corresponds to the variance of measurement errors of d
106. el par metro Fondos Potenciales subi al a o 2007 a m s de MM US 100 000 es decir 2 5 veces mayor que el valor estimado en la Formulaci n Esto permitir a dividir por 2 5 alguno de los indicadores anteriores por ejemplo suponer que el Incremento Marginal de la Rentabilidad fuera 0 1 en vez de los 0 25 asumidos originalmente y conservar el valor de los indicadores originales Debido a lo anterior una evaluaci n conservadora mantendr a los indicadores econ mico sociales presentados en la Formulaci n inicial siendo sus flujos netos FLUJO NETO Ingresos Costos Inversion Costo total I D Fondef Beneficio neto Nota M N moneda nacional M E moneda extranjera 568 06 9 244 18 90 50 14 0 0 509 219 97 290 10 754 276 509 0 0 10 987 238 136 238 136 119 70 6 55 119 71 6 54 38 32 COBIERNO DE CHILE CONK YI FONDEF FOMENTO AL DESARROLLO CIENTIFICO Y TECNOLOGICO III PARTE INFORME DE GESTI N C digo Proyecto D0311039 Nombre del Proyecto DESARROLLO DE HERRAMIENTAS COMPUTACIONALES PARA OPTIMIZAR LA GESTION DE CARTERAS DE INVERSION EN MERCADOS EMERGENTES APLICACION A LOS FONDOS DE PENSIONES EN CHILE Instituciones Participantes Pontificia Universidad Cat lica de Chile Otros Participantes AFP Habitat S A Risk America Dictuc S A Fecha de emisi n 23 07 2007 COMISION NACIONAL DE INVESTIGACION CIENTIFICA Y TECNOLOGICA BERNARDA M
107. els and in any market with infrequent trading a very common situation in many emerging markets Our approach is currently being used by a consortium of financial and academic institutions in Chile to estimate the Chilean term structure of interest rates The results are updated daily at the website RiskAmerica com Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe IJFE 317 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 IJFE 317 TERM STRUCTURE ESTIMATION 15 APPENDIX In this appendix we describe in detail how to apply the methodology developed in Section 4 to the generalized Vasicek model introduced in Section 3 with an incomplete panel data set of discount and coupon bond yields The transition equation of the state variables under a generalized Vasicek model is independent of the observations and the associated terms appearing in equation 12 are At o O1OnP in diag 1 k Ad c Q En At Al 2 At OnO1Pny 00 o n where diag x stands for a diagonal n x n matrix whose i i element is x At is the time interval at which yields observed and other parameters are the ones appearing in equation 4 Let m and m be the number at time of observed discount and coupon bonds respectively and ta y 1 and t ae 1 the sets containing their respective associat
108. erson ucla edu Copyright 2007 John Wiley amp Sons Ltd IJFE 317 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 IJFE 317 2 G CORTAZAR ET AL representations of the yield curve limiting the number of parameters and giving more stability to the term structure For the modelling of the term structure dynamics the main concern is the movement of the term structure across time To address this issue one alternative is to model the stochastic movement of the spot rate and then to use no arbitrage arguments to infer the dynamics of the term structure Examples of this approach include one factor mean reverting models Vasicek 1977 two factor models Brennan and Schwartz 1979 multifactor extensions of the Vasicek model Langetieg 1980 single factor general equilibrium models Cox et al 1985 and multi factor extensions of the CIR model Duffie and Kan 1996 among many others Another approach is to use the whole current term structure as the input to the model and no arbitrage arguments to infer 1ts stochastic movement Ho and Lee 1986 Heath et al 1992 Even though these type of models use all the information contained in the current term structure they are more difficult to calibrate Once a dynamic model of interest rates is proposed the estimation method that will be used must be chosen One possibility is to estimate the model using a
109. es de gran impacto En primer lugar dio origen a 3 nuevos servicios de gesti n del riesgo que se distribuyen por Internet a trav s de RiskAmerica Servicio SVC que valoriza diariamente instrumentos de renta fija con pocas transacciones Servicio Indices que entrega benchmarks con el comportamiento del mercado para diversas clases de activos y Servicio Portfolio que entrega informaci n de retorno riesgo y performance para carteras del mercado y propias Estos servicios ya est n siendo utilizados por muchas instituciones financieras del pa s Adem s el proyecto gener publicaciones en revistas y presentaciones en congresos internacionales y apoy la realizaci n de tesis de mag ster y memorias de t tulo fortaleciendo las actividades del FINlabUC Laboratorio de Investigaci n Avanzada en Finanzas de la Pontificia Universidad Cat lica de Chile ABSTRACT The main objective of this project is to develop tools software applications and services that use Internet technologies to provide better portfolio management for domestic and world market assets and modernize the Chilean financial markets Although the new developments are primarily focused on pension funds results can also be used for other portfolios like mutual funds or those managed by insurance companies This market modernization is supported by standard state of the art methodologies as well as new specific research on thin markets a characteristic of Chilean financial
110. etorni Fecha Inicio Fecha Termino Calculadora Frontera Eficiente B a i 01 01 2006 Retorno Portafolio 10 312 96 al Desviaci n Portafolio Restricciones a El d WA V Restringir Venta Corta ps Wo Seleccione be wa En An lisis de Desempe o se entregan herramientas para calcular carteras el performance de carteras definidas por el usuario Usu rtazar RiskAmerica l z RiskAmerica S Portfolio mmodity Informaci n de Mercado CEE ioco ae Proyectos o was En nn Riesgo Absoluto Proyectos FondoC_H x ee Riesgo Relativo Indicadores de Retornos Anualizados da A Asset Allocation Opciones ar FondoC_H An lisis de Desempe o Fecha Inicio 02012008 ES Medidas Performance Fecha T rmino 07 02 2007 E Libre de Riesgo IF Global Be Mercado IPSA ee E ssiaXJap_yEN MM Cash_USD E oor O E Em_EUR Ol euro MM Europa EUR E Japon YEN E Ltam_too GB Pacwap2 YEN 1 Ret 1 a o 2 Ret 2 a os 3 Ret 3 a os Matriz de Indicadores a 2 Ret 1 a o Ret 2 a os Ret 3 a os PsiaXJap_YEN 21 26 21 51 16 27 Cash_USD 5 3 4 42 3 47 DOLAR 442 2 46 2 73 Eu_Em_EUR 15 28 33 05 29 44 EURO MET MET 394 cma EMO zone l Dan nar Dar l 4 2 Clientes Los clientes potenciales para los resultados de este proyecto son principalmente instituciones financieras a las que se
111. f bond prices for a wide range of different maturities makes it easy to extract a term structure of spot rates that explains observed prices Moreover in some countries such as the United States zero coupon bonds Strips of different maturities are individually traded In many emerging markets however bonds trade infrequently so that for every particular day there are bond prices for only a few maturities This missing observation problem makes it difficult and sometimes impossible to estimate the term structure using only current data In this article we develop a methodology for using an incomplete panel data of bond price observations to estimate the current term structure We use an extended Kalman filter approach to estimate a dynamic multi factor model of interest rates using the panel data with missing observations The Kalman filter estimation provides not only the parameters of the model but also the time series of the factors The approach jointly estimates the current term structure and its dynamics The model can be used to value and hedge all types of interest rate derivatives including bonds with embedded options This methodology also allows us to estimate the term structure for days with an arbitrary small number of traded bonds We implement the approach using a three factor generalized Vasicek 1977 model and Chilean government bond data The methodology however can be implemented with a broad class of dynamic interest rate mod
112. folio 9mmodity Ri skAme rica Herramientas Portfolio Informaci n de Mercado Conjunto de herramientas que permite estudiar en detalle distintas carteras publicas del mercado financiero nacional entregando informaci n de rentabilidades composici n y performance Medici n de Riesgo Absoluto Herramienta que permite calcular y realizar distintos an lisis de Value at Risk a las carteras construidas por el usuario www riskamerica com Fono 56 2 354 4086 354 4823 Fax 56 2 552 1608 E mail info riskamerica com Medici n de Riesgo Relativo Herramientas que permiten calcular y realizar distintos an lisis de Tracking Error y Value at Risk Relativo a carteras y benchmarks construidos por el usuario Asset Allocation Herramienta que permite realizar an lisis de Asset Allocation para conjuntos de activos definidos por el usuario Medici n de Performance Conjunto de herramientas que permiten analizar y comparar el desempe o de carteras entregando indicadores de retorno riesgo y performance para un periodo de tiempo espec fico 4 RESULTADO DE EVALUACION EX POST DESARROLLADA POR LA INSTITUCION Tal como se plantea en la Formulaci n del proyecto el beneficio social principal de este proyecto se genera al contribuir a la modernizaci n del mercado financiero nacional Un resultado exitoso en cuanto a nuevas herramientas de gesti n de carteras como es el caso de lo ocurrido con est
113. hat because the current estimation procedure uses a very large number of data points it will reject any hypothesis of no structural change On the other hand it may be inter esting to use as much information as possible from the whole futures term structure in testing for structural change The structural break is tested by comparing the time series covari ance matrix of state variables for cach sample and then computing a Wald test statistic In particular the Wald statistic is computed with the use of the GMM procedure applied to the time series dynamics of each state variable and carried out for each model This generates Wald statistic values of 456 05 171 70 115 19 and 97 51 for the four three two and one factor models respectively Given that critical values at 95 for this test are 18 31 12 59 7 81 and 3 84 respectively the null hypothesis of no structural change is clearly rejected In addition the presence of volatility parameters in the valuation formula of futures prices might also cause empirical and theoretical volatility term structures to diverge Because the estimation is performed by improving the likelihood function of price innovations there is always a trade off between improving the pricing of futures contracts and the time series volatility estimation A larger number of factors gives more flexibility to adjust first and second moments simultaneously hence explaining why the four factor model outperforms
114. ical results show that one and two factor models fail to accu rately fit observed prices and the volatility term structure The three factor model while explaining market prices very well overestimates the short term volatility in some panels which may be attributed to a structural change in oil prices occurring after 1997 Finally the four factor model performs well explaining the stochastic behavior of oil prices In general the model works well in estimating the term structure of oil futures prices and the volatility term structure of oil futures returns As such the model could be useful for oil producers and consumers and also financial intermediaries like futures traders in valuing and hedging oil contingent claims For example the model could be used to value oil linked financial contracts with option like characteristics or to implement long term hedging strategies with existing futures contracts APPENDIXA This Appendix deduces Equation 17 svith the use of Equation 16 Because the conditional distribution for the spot price Sp is lognormal it follows that l E9S exp E20 4 teu 30 where Y log S EY Cl E2 x uT and Var Yp 1 x Cove x 1 Journal of Futures Markets DOI 10 1002 fut 266 Cortazar and Naranjo From Equation 3 the conditional moments of x are T t E x e Tx edr A 0 T Cov xy f e FOX e7 dr 0 where dt dw dw Thus
115. iendo investigaci n reconocida internacionalmente Es en este contexto que independientemente de la evoluci n comercial de RiskAmerica sta se debe mantener ligada estrechamente a la investigaci n acad mica que se realiza en la Pontificia Universidad Cat lica de Chile a trav s del FINlabUC Laboratorio de Investigaci n Avanzada en Finanzas 4 4 Comercializaci n La distribuci n de los resultados se realizar a trav s de la plataforma WEB RiskAmerica la que comunic al mercado la incorporaci n de estos servicios expandidos como RiskAmericaPlus Asimismo se deben intensificar el plan de contactos comunicaci n y capacitaci n de usuarios potenciales a trav s de encuentros y presentaciones y distribuci n de material gr fico Adicionalmente se debe seguir explorando oportunidades de internacionalizaci n de los servicios A continuaci n se entrega copia parcial de material desarrollado 4 5 Ventas Actuales y Proyecci n de Resultados Futuros Las ventas esperadas de los 3 m dulos de Servicios para el a o 2007 alcanzan MM 198 las que debieran incrementarse en los a os futuros hasta alcanzar un monto esperado de MM 570 el a o 2010 A continuaci n se presentan los ingresos y egresos proyectados por los pr ximos 7 a os as como el VAN de los excedentes descontados al 12 anual En forma muy conservadora no se supone valor residual alguno Se estima un VAN al 12 de MM 781 6 para la explotaci n comercial de los resultad
116. implicity that the risk premium is constant However this could be extended to any linear function of the state variables to reflect a possible correlation between spot prices convenience yields and interest rates Casassus amp Collin Dufresne 2005 Joumal of Futures Markets DOI 10 1002 ut Cortazar and Naranjo where h h L p t a t h e t K LKL B Kg t LB dw dw Qdt LAY LE Q EL and Y LY From the analysis of Dai and Singleton 2000 it follows that th re exists an affine transformation T that rewrites any model to another one with the maximum number of parameters that can be econometri cally identified In particular if the reversion matrix K is restricted to have termwise different eigenvalues then an arbitrary N factor model can always be transformed to a model like the one presented in Equations 1 and 2 For example consider the Gibson and Schwartz 1990 model logS h 8 KZ B dt Xdw 9 where h 1 0 K i o B a 7 sia Ea 2 and 0 ka 0 7 lp di d Qdt E Pa p 1 To obtain the relationship between the two models the following affine transformation is applied over the original state variable vector L o 10 Pa a 1 20 a t where L and 0O 1 x afk The new vector x corresponds to the state variables in the new model This transformation is invertible and therefore it establishes a one to one cor
117. informaci n de riesgo de retornos y de performance de carteras p blicas de AFP y de Fondos Mutuos como se muestra en las siguientes p ginas web Ri eA erica Indices Portfolio mmodity a 2 Retornos Riesgo Performance Fecha Inicio 03 01 2003 ES Indicadores de Performance da Fecha T rmino 23 03 2007 E Fondo A Riesgo AbSoTuIO Fondos Fondo A lv Riesgo Relativo Libre de Riesgo IF Global Iw Asset Allocation An lisis de Desempe o Mercado Ponderado Globa Walor Normalizado 1 2 3 4 5 Tla E BANSANDER MM currum E hssimrar O PLANMITAL O PRODA E SANTAMARIA E Sis Ponderado E Sis Promedio Ame l 1 Shape 2 Modigliani 3 Jensen 4 Treynon 5 Inf Ratio Matriz de Indicadores ar Sharpe Modigliani Jensen Treynon Inf Ratio BANSANDER 182 7 94 116 829 16 O CUPRUM 18 7844 ose s26 183 __ HABITAT 1 78 zna oos 812 151 PLANMITAL 178 778 027 800 _ 152 PROMDA 177 7718 o42 sox jf 15 JC 174 758 158 701 14 1 if SANTAMARIA r ani nar nes VI ara En Riesgo Absoluto se entregan herramientas para calcular el Value at Risk de carteras propias tanto por m todos param tricos como por simulaci n hist rica RiskAmerica Ri skAmerica Value at Risk Ingreso de Proyecto gt mmodity Histograma Yariaci
118. ing Average Yield standard Years observations frequency yield deviation Pure discount bonds 0 1 1115 89 70 5 81 2 04 Coupon bonds 1 1 5 377 30 33 6 46 1 83 1 5 2 5 426 34 27 6 29 1 45 2 5 3 5 443 35 64 6 20 1 17 3 5 4 5 642 51 65 6 15 1 17 4 5 5 5 519 41 75 6 36 1 12 5 5 6 5 550 44 25 6 36 0 87 6 5 7 5 766 61 63 6 33 0 91 7 5 8 5 921 74 09 6 22 0 81 8 5 9 5 451 36 28 6 31 0 80 9 5 10 5 584 46 98 6 31 0 65 10 5 11 5 268 21 56 6 30 0 72 11 5 12 5 458 36 85 6 21 0 67 12 5 13 5 262 21 08 6 20 0 64 13 5 14 5 507 40 79 6 14 0 60 14 5 15 5 269 21 64 6 10 0 71 15 5 16 5 311 25 02 6 13 0 61 16 5 17 5 269 21 64 6 18 0 60 17 5 18 5 309 24 86 6 32 0 53 18 5 19 5 404 32 50 6 32 0 53 19 5 20 533 42 88 6 26 0 60 Total 10384 Trading frequency is defined as the number of days for which there is a transaction of a given bond over all available trading days P Continuous compounding 45 Standard deviation of observed yields generally decreases as bond maturity increases which is consistent with mean reversion in interest rates Figure 4 illustrates the sparseness or infrequent trading of daily bond transactions in Chile by showing for each day during the second semester of 2001 when a bond was traded or not The panel data shown are clearly incomplete a condition that is critical in the choice of the estimation methodology 5 2 Estimation results We estimate the three factor Vasicek model parameters using bond price transa
119. ing coupon bonds and we are interested in the volatility of spot rates Second the panel data contain many missing observations To address these problems we aggregate the data in groups according to their maturity The first group contains bonds with one to two years of maturity and so on Then for each date we take the average yield of all the bonds in a given group and we compute the volatility of daily changes of these yields In addition we compute the average duration of the bonds in each group To compare this empirical volatility to model spot volatilities we assume that the volatility of each group represents the volatility of a discount bond with maturity equal to the average duration in the group Figure 7 shows the term structure of spot volatilities from the model and from the empirical estimates Comparing this figure with Figure 3 we observe that our model volatilities are much closer to the empirical volatilities than those obtained using the curve fitting methods 6 CONCLUSION The estimation of the term structure of interest rates is a critical issue not only from a theoretical point of view but also for all market participants including banks regulators and financial institutions It is an essential ingredient in the valuation and hedging of all fixed income securities It is also necessary for financial planning and for implementing monetary policy In economies with well developed and liquid financial markets the existence o
120. inuation value Given that the expected continuation value depends on future outcomes the procedure must work its way backwards starting from the end of the time horizon T Starting with the last price in each path given that at expiration the expected continuation value is zero the option value in T for the price path w can be computed as C Sr w Max Q S7 A 0 16 One time step backward at t T At the process is repeated for each price path but now expected continuation value must be computed It is important to notice that at this last time step the expected continuation value may be computed using the analytic expression for a European option The main contribution of the LSM method is to compute the expected continuation value for all previous time steps by regressing the discounted future option values on a linear combination of functional forms of current state variables Given that the way these functional forms are chosen is not straightforward in most of the paper we use simple powers of all state variables monomials and their cross products which is the most common implementation of the method found in the literature In the last section of the paper we revisit this decision and provide alternative functional forms which in our tests have shown to be computationally efficient in multidimensional settings In particular let L with j 1 2 M be the basis of functional forms of the state variable Sp_
121. ion in Markets with Infrequent Trading 15th annual meeting of the European Financial Management Association Madrid June 28 July 1 2006 Cortazar G Naranjo L 2005 An N Factor Gaussian Model of Oil Futures Prices 2005 FMA Annual Meeting Chicago October 12 15 http www fma org Chicago ChicagoProgram htm Cortazar G Schwartz E S Naranjo L 2004 Term Structure Estimation in Low Frequency Transaction Markets A Kalman Filter Approach with Incomplete Panel Data March 2004 EFA 2004 Maastricht Meetings Paper No 3102 Maastricht August 18 21 http ssrn com abstract 567090 Tema 2 Metodolog as de Valorizaci n de Derivados escritos sobre Subyacentes con procesos Completos Discusi n A n cuando existen m ltiples procedimientos para valorizar opciones de tipo Americano cuya estrategia ptima de ejercicio no es evidente la complejidad para su implementaci n crece exponencialmente con la dimensi n del problema a resolver Dado que los modelos de precios actuales requeridos para representar adecuadamente la din mica de tasas de inter s y de precios son multifactoriales estas metodolog as tradicionales son en la pr ctica inutilizables para valorizar los activos derivados escritos sobre estas tasas de inter s o precios La estrategia de resoluci n propuesta consiste en adaptar metodolog as recientes que usan un m todo que combina simulaci n de Montecarlo forward con resoluci n backward de rboles const
122. is paper For example 1f the state variables were only two X and Y a simple order two expected continuation value function would have six regressors namely G X Y Go GX Y 43XY 44X sY 30 Although this procedure for specifying the regression basis has the benefit of being simple and theoretically con vergent 22 52 53 in high dimensional settings it may induce numerical problems due to the least squares regression instability 21 and performance problems due to the high number of regressors An alternative to the described procedure for specifying the base that we have tested is to take advantage of the structure of the problem to be solved Thus given that optimal exercise of options depends on expected spot prices and volatilities instead of using as regressors powers of all state variables it could be better to use functions on futures European options or bond prices which have economic meaning Recent independent work has shown the potential of this approach for implementing multidimensional financial derivatives For example Andersen and Broadie 50 include as regressors European call options and their powers for valuing a multi stock option and Longstaff 51 value the prepayment option on a term structure string model with 120 state variables using closed form par price bonds and their powers We are not aware however of any use of a similar approach in the real options literature Thus our alternative implementa
123. iscount bonds The remaining four parameters correspond to the variance of coupon bonds for maturities ranging between 1 to 5 years 6 to 10 years 11 to 15 years and 16 to 20 years respectively Therefore the covariance matrix of measurement errors is R 0 0 R A7 R A8 where R diag amp and R diag are diagonal matrices ACKNOWLEDGEMENTS We thank Kenneth Singleton Stephen Schaefer Alfredo Iba ez researchers of the FINlabUC Laboratorio de Investigacion Avanzada en Finanzas Pontificia Universidad Catolica de Chile and seminar participants at Verona and the 2004 European Finance Association meetings in Maastricht for helpful comments and suggestions Gonzalo Cortazar acknowledges the financial support of FONDECYT Grant No 1040608 and FONDEF Grants No D0311039 and D0011024 This is a revised version of a previous working paper entitled Term Structure Estimation in Low Frequency Transaction Markets A Kalman Filter Approach with Incomplete Panel Data NOTES See the working paper version of this article for details on these methods The coupon bonds considered here are amortizing bonds paying semi annually equal coupons These instruments are described in more detail in Section 5 3 In a mean reverting model every perturbation is on average reduced by half in log 2 k units of time 4 The canonical form proposed by Dai and Singleton 2000 for Gaussian interest rates allows for the pos
124. ith Infrequent Trading 6066335 1 Cortazar Sanz Gonzalo 1 Using futures prices of one commodity to estimate the stochastic process of another 6066335 1 Cortazar Sanz Gonzalo 1 Term Structure Estimation in Markets with Infrequent Trading JJ SE of Oil Futures Prices 9 6066335 1 Cortazar Sanz Gonzalo 1 The Valuation of Multidimensional American Real Options using the LSM Simulation Method 6066335 1 Cortazar 10 6066335 1 Cortazar Sanz Gonzalo 1 Term Structure Estimation in Low Frequency Transaction Markets A Kalman Filter Approach with Incomplete Panel Data 1 Copie el N correspondiente de la tabla anterior Duraci n si fue asistente N Horas 2 N RUT Apellido Apellido Nombres Instituci n Pa sde Lugar de Fecha Fecha Tema Disciplina C digo de Paterno Materno de Origen Origen Trabajo en de de Fondecyt actividades del Chile Inicio t rmino predominan proyecto te en las que particip Schwartz Greenwald Eduardo UCLA EEUU En Chile y 2004 2007 Supervisi Finanzas Research Saul Via Web n Research 14 COOPERACION INTERNACIONAL Y NACIONAL COLABORACION DE EXPERTOS 1 3 IMPACTOS ACTUALES Y ESPERADOS DE MEDIANO Y LARGO PLAZO 3 1 IMPACTOS ECONOMICO SOCIALES PRODUCIDOS Mejor valorizaci n de carteras de inversi n para todos los Fondos Mutuos del Pa s y para algunas otras instituciones financieras del pa s lo
125. kJap_YEN Latam_Loc Japon_YEN Cash_USD FsiaXJap_YEN 0 20 0 0 2 0 4 0 6 0 8 1012 141618202224 TE Tracking Error 0 12179 Descomposici n Analisis ATE Posici n 0 Ps 0 62432 ATE TE Incremental Posicion Bench ne Indice 1 AsiaXJap_YEN 0 01091 0 01033 2 Cash USD 0 00003 0 00002 0 00001 En Asset Allocation se entregan herramientas para calcular carteras con combinaciones riesgo retorno ptimas RiskAmerica RiskAmerica SVC Indices Portfolio Informaci n de Mercado Frontera Eficiente Riesgo Absoluto Ingreso de Proyectos a Frontera Eficiente a Riesgo Relativo A Proyecto FondoC_H v e Markowitz Asset Allocation Frontera Eficiente Matriz Yarianza Covarianza a gt Seleccione HIST Mat_1s w An lisis de Desempe o Fecha Inicio Fecha Termino 12 er 20 01 01 2006 31 12 2008 paisaje Retorno Esperado a z Risk merica Plus Seleccione HIST Retorn M Fecha Inicio Fecha Termino Calculadora Frontera Eficiente B a _ 0101 2006 31 12 2006 Retorno Portafolio 10 312 3 al Desviaci n Portafolio 1 35 Jos Restricciones a a as wi 1 87 Y Restringir Venta Corta we Ole Seleccione bel W3 3 28 196 En An lisis de Desempe o se entregan herramientas para calcular carteras el performance de carteras definidas por el usuario RiskAmerica RiskAmerica SVC Indices Portfolio
126. l requiring at least three factors to explain the term structure of futures prices but four factors to fit the volatility term structure The model also performs very well for daily copper futures transactions from 1992 to 2001 and for out of sample daily oil futures transactions from 2002 to 2004 2006 Wiley Periodicals Inc Jel Fut Mark 26 243 268 2006 The authors thank the researchers of the FINIabUC Laboratorio de Investigaci n Avanzada en Finanzas Pontificia Universidad Cat lica de Chile for helpful discussions and excellent research assistance Most of the work was completed while Lorenzo Naranjo was at the FINlabUC at the Pontificia Universidad Cat lica de Chile They also acknowledge the financial support of FONDECYT Grant No 1040608 FONDEF Grant No D0311039 and Fundacion COPEC Universidad Cat lica Grant No PC00021 Correspondence author Departamento de Ingenier a Industrial y de Sistemas Pontificia Universidad Cat lica de Chile Vicu a Mackenna4860 Santiago Chile e mail gcortazafing puc cl Received September 2004 Accepted July 2005 m Gonzalo Cortazar is a Professor at the Departamento de Ingenier a Industrial y de Sistemas Escuela de Ingenier a Pontificia Universidad Cat lica de Chile in Santiago Chile Lorenzo Naranjo is a Ph D candidate at the Stern School of Business at New York University in New York The Journal of Futures Markets Vol 26 No 3 243 268 2006 O 2006 Wiley Perio
127. lation It can be seen that as time evolves from 0 to T the state variables that describe the three factor dynamics for copper price 120 G Cortazar et al Computers amp Operations Research 35 2008 113 129 Continuatiion Value Simulations a Regression Fig 2 Implementation of the LSM in the simple copper mine discounted continuation values for all N simulated paths and expected continuation function computed from a regression on powers of the spot copper price Spot Price Z m A o State Variables i i i i i i i i i i i i i i i i i i H i i i i i H i i 1 i H i 1 i 1 i 1 i H 7 OPEN CLOSED Fig 3 State space representation of the Brennan and Schwartz 23 model x w S y v w evolve following different paths At any point in time and for any value of the three state variables the mine may have any amount of copper reserves between zero and the initial reserves Qmax In addition the mine at that point may be open or closed with market values V x Q or W x a Q respectively For each state of the system and for each operating policy there is an associated cash flow for the mine For example when the mine is open and the operating policy is to remain open during At years producing q the cash flow CF is CF S q qAt S A t 20 Recall that for any price model the spot price depends on the state variable
128. lo Din mico para la Tasa Cero Real en Chile Decisiones de Asset Allocation en Carteras de Inversi n de las AFP Aplicaci n del Modelo de Black amp Litterman 13922911 8 13548272 2 Finanzas Pontificia Universidad Cat lica de Chile Finanzas Pontificia Universidad Cat lica de Chile Pontificia Universidad Cat lica de Chile Finanzas CARLOS IGNACIO MILLA GONZALEZ Pontificia Universidad Cat lica de Chile CLAUDIO EDUARDO HELFMANN SOTO JOSE LUIS MANIEU ESPINOSA RODRIGO ALFONSO IBANEZ VILLARROEL Diciembre 2005 Santiago Chile Diciembre 2005 Santiago Chile Agosto 2005 Santiago Chile Julio 2005 Santiago Chile 2 11 DIFUSION Y PUBLICACIONES DE RESULTADOS PUBLICACIONES RELACIONADAS CON CONTENIDOS DEL PROYECTO Tipo Nombre Publicaci n Nombre Nombre autor es RUT 1 libro 2 cap de libro del cap libro del art revista del libro o revista cuando en la columna anterior 1 3 art revista 4 manuales del manual t cnico de otros sea un cap o un art t cnicos Otros especificar Term Structure Estimation in Computers amp Operations Research G ravet M 9908534 Markets with Infrequent Trading esa J aes The Valuation of International Journal of Finance and Cortazar G 60663351 1 Multidimensional American Real Economics Naranjo L 12931431 1 Options using the LSM Simulation Method 3 3 An N Factor Gaussian Model of The Jour
129. lo un a o Este supuesto de evaluaci n busca reflejar la din mica actual de modernizaci n financiera que est teniendo el mercado que si bien est convergiendo a los est ndares de mercados desarrollados en algunas reas como las que aborda este proyecto no lo hace con suficiente velocidad No se est condicionando la modernizaci n del sistema chileno a este proyecto sino que se est suponiendo que al hacerlo se adelantan los resultados con el consecuente beneficio para los usuarios La evaluaci n considera que tanto los ingresos como los costos obtenidos por este desarrollo privado son equivalentes a la situaci n con proyecto pero que se obtienen un a o despu s Adem s la inversi n inicial es superior ya que para obtener los sistemas adecuados estos deben ser adquiridos y desarrollados por consultores externos en el extranjero quienes deben estudiar el comportamiento del mercado nacional para luego desarrollar desde cero los productos DESCRIPCION DE LA SITUACION CON PROYECTO La situaci n con proyecto considera que el proyecto adelanta los beneficios econ micos sociales y que el costo de desarrollo es menor en este proyecto que en su eventual competencia MEMORIA DE CALCULO DE LA EVALUACION ECONOMICO SOCIAL SIN PROYECTO La informacion aqui contenida se extrae de la hoja SITUACION SIN PYTO En millones de pesos Variables Criticas Var 1 Var 2 Var 3 Unidad m3 kg I ton etc Descripci n Variable 1
130. ls A 1992 2001 B 1992 1996 Panel A 1F 2F 3F 4F Panel B 1F 2F 3F 4F Panel C 1F 2F 3F 4F and C 1997 2001 RMSE 92 1 46 0 51 0 29 3 70 0 87 0 31 0 16 7 12 1 46 0 53 0 35 Bias 0 0004 0 0004 0 0002 0 0001 0 0009 0 0000 0 0000 0 0000 0 0010 0 0003 0 0003 0 0002 futures term structures exhibited strong contango or backwardation It can be seen that one and two factor models cannot fit observed futures prices accurately on these dates whereas three and four factor models fit them very well Table IHH shows the root mean squared error RMSE and bias of model futures estimates for in sample data It can be seen that futures Journal of Futures Markets DOI 10 1002 fut RMSE e IF a 2F 3F 4F Maturity Years FIGURE 6 Root mean squared errors for each model The figure is obtained by calculating the in sample root mean squared errors RMSE by maturity for the 35 existing oil futures contracts and for cach model with the use of Panel A 1992 2001 price estimations are unbiased for all models and panels and exhibit a RMSE of less than 1 for three and four factor models For example in ancl A considering an average spot oil price of 21 the RMSE corre sponds to an error of only 0 07 for the four factor model On the other hand the one factor model exhibits a high RMSE ranging from 3
131. m trico Matriz Yarianza Covarianza Simulaci n Hist Seleccione Fecha Inicio HIST Mat_1a v Fecha T rmino 01 0 Analisis de Tracking Error a gt ATE Posici n 0 ATE Posici n 1 Todos TE Incremental ATE Posici n Bench A TE Posici n 0 TE Incremental IPSA IF Global LH Global Gob Global Corp Global YEN USA USD UK_EUR PackJap_YEN Latam_Loc Japon_YEN Europa_EUR EURO Cash_USD FsiaJap_YEN 0 20 0 0 2 0 4 0 6 0 8 A TE Posici n 1 A TE Posici n Bench 1 01 2 1 41 6 18202224 TE Tracking Error 0 12179 Descomposici n Analisis A ES A TE A TE Posici n Bench Posici n 0 Px 1 AsiaXJap_YEN 0 01091 0 01033 0 62432 2 Cash_USD 0 00003 0 00002 0 00001 En Asset Allocation se entregan herramientas para calcular carteras con combinaciones riesgo retorno ptimas RiskAmeri 2 RiskAmerica SVC Indices Portfolio Commodity Informaci n de Mercado Frontera Eficiente Riesgo Absoluto i Ingreso de Proyectos ar Frontera Eficiente cry Riesgo Relativo J Proyecto FondoC_H Y Markowitz AssetAllocation Frontera Eficiente Matriz Yarianza Covarianza 2 x A I Seleccione HIST Mat_1 lt ae E Fecha Inicio Fecha Termino 12 4 E 13 20 26 33 3112 2006 Desviacion Retorno Esperado a 2 RiskAmerica Piu Seleccione HIST R
132. markets which imposes difficulties in the use of many methodologies common in developed markets The goals of the project are 1 to induce a more efficient portfolio management which uses better risk and return information 2 to analyze investment strategies and support the asset allocation process 3 to improve risk management and measurement 4 to define a set of portfolio benchmarks thus the goal is to help portfolio owners and managers by providing better information and management tools and improving industry competition and performance The economic and social impacts of this project are extremely high considering that the AFP system manages over US 100 billions Thus marginal increases on returns due to better tools and information for managing funds and controlling risks create a huge wealth most of it transferred to fund members increasing their welfare This return increase would in turn stimulate the economy increasing savings and improving resource allocation The social project evaluation recognizes that there is an unsatisfied demand for technologies for risk management and that there are no available products in international markets that satisfy national market requirements The project had multiple results with great impact in economic social scientific technological and institutional scope First it originated three new risk management services distributed by RiskAmerica using Internet SVC Service which prices
133. mp Scott 1993 Duffie amp Singleton 1997 Pearson amp Sun 1994 In this case it is assumed that futures prices are observed without measurement error However for any given date the number of available prices is generally higher than the number of state variables that need to be estimated Therefore it must be assumed that observed prices have some degree of measurement error which has to be assigned across the different contracts Moreover it may be desirable to include as many observed futures prices as possible inthe estimation process of the state variables Onc of the most successful econometric procedures that takes into account the above issues is the Kalman filter a widely used estimation methodology that can handle multifactor models with nonobservable state variables and measurement errors In addition it is capable of using a large price pancl in the estimation process avoiding the necessity of making an arbitrary selection of contracts to include in the estimation The Kalman filter has been used in finance to estimate state variables of commodity price models by Schwartz 1997 Schwartz and Smith 2000 Manoliu and Tompaidis 2002 and Sgrensen 2002 among others Traditional implementations of the Kalman filter normally assume a complete panel data set This implies that for all given dates in the esti mation sample prices for the same set of contracts with the same matu rities must be observed This is not norm
134. n mero de alumnos que se especializan en finanzas a nivel de postgrado Fortalecimiento de Relaciones con Sector Productivo Fortalecimiento de las relaciones de Cooperaci n Internacional ESPERADOS Incremento en los fortalecimientos institucionales anteriores 3 4 IMPACTOS AMBIENTALES PRODUCIDOS No existen ESPERADOS No existen 3 5 IMPACTOS REGIONALES PRODUCIDOS No existen ESPERADOS No existen 4 PLAN DE NEGOCIOS A continuaci n se describen los principales aspectos del Plan de Negocios en ejecuci n 4 1 Productos Se considera la comercializaci n de 3 m dulos de Servicios M dulo SVC M dulo Indices M dulo Portfolio 4 2 Clientes Los clientes de los servicios ofrecidos son instituciones financieras AFP Bancos Fodnos Mutuos Corredoras Cias de Seguros etc y organismos reguladores Banco Central Superintendencias etc 4 3 Factores de xito El principal Factor de xito es la capacidad de generar y comunicar una reputaci n de objetividad rigurosidad y compromiso de permanente innovaci n para los Servicios ofrecidos Para ello se hace necesario mantener activo un equipo de investigadores haciendo investigaci n reconocida internacionalmente 4 4 Comercializaci n La comercializaci n nacional se realizara a trav s de la plataforma RiskAmerica la que comunic al mercado la incorporaci n de estos servicios expandidos como RiskAmericaPlus Se est n explorando oportunidades d
135. n confiable relativa al comportamiento de las distintas clases de activos entre otros aspectos Documentos de Trabajo a n no publicados Cortazar G Bernales A Beuermann D 2007 Methodology and Implementation of Value at Risk Measures in Emerging Fixed Income Markets with Infrequent Trading Tesis de Mag ster Estimaci n de Spreads por Liquidez en un Mercado con Pocas Transacciones El Caso del Mercado de Bonos del Banco Central de Chile PEDRO MAT AS MORAL MESA Tesis de Mag ster en Ciencias de la Ingenier a Pontificia Universidad Cat lica de Chile 17 01 2006 Metodolog a e Implementaci n de M todos de VALUE AT RISK en Mercados de Renta Fija con baja Frecuencia de Transacciones ALEJANDRO ADRIAN BERNALES SILVA Tesis de Mag ster en Ciencias de la Ingenier a Pontificia Universidad Cat lica de Chile 23 12 2005 Memorias de T tulo Decisiones de Asset Allocation en Carteras de Inversi n de las AFP Aplicaci n del Modelo de Black amp Litterman RODRIGO ALFONSO IBANEZ VILLARROEL Memoria Escuela de Ingenier a Pontificia Universidad Cat lica de Chile 26 07 2005 Tema 5 Modelaci n y Calibraci n Conjunta de Procesos Estoc sticos de M ltiples Activos Durante el desarrollo del proyecto se hizo evidente que en algunas situaciones se hace conveniente utilizar informaci n de precios de ciertos instrumentos financieros para estimar de mejor manera el precio de otro instrumento que no fue transado pe
136. n filter then works recursively using the previous estima tions First the one step ahead prediction at time t of the state variables a a and its error variance covariance matrix P given all information up to time t are computed Xi lj A X FC 21 P 1 I APA Q 22 This allows for the calculation of one step ahead prediction of observed variables Lua H Xy d 23 These calculations only consider the dynamic properties of state variables and are not affected by the dimension of the vector of observ able variables The prediction error or innovation v and its associated ariance covariance matrix F are v i 24 ao H P H R 25 Joumal of Futures Markets DOI 10 1002 tut Cortazar and Naranjo When futures contracts are used as observations the measurement equation gives the futures price as a function of state variables and its maturity allowing all observed prices to be used in the estimation process With the use of Equation 17 the m row vectors of the matrix H and the elements of the vector d can be computed In addition the covariance matrix of measurement error R must be parametrized For this purpose the Babbs and Nowman 1999 approach is followed where it is assumed that all measurement errors are independent and have the same variance inducing a diagonal covariance matrix R Although this assumption could be relaxed there is a tradeoff betwee
137. n relativa a retornos y riesgos involucrados 2 hacer un an lisis de estrategias de inversi n que apoye la asignaci n de activos asset allocation 3 apoyar funciones de medici n y gesti n del riesgo y 4 establecer un conjunto de benchmarks para diversas carteras de inversi n Todo lo anterior busca favorecer la gesti n e informaci n para directivos y usuarios y en ultimo t rmino la competitividad y desempe o de la industria El impacto econ mico y social de este proyecto es extremadamente alto considerando que s lo el sistema de AFPs administra m s de 100 billones de d lares por lo que incrementos marginales en las rentabilidades que se derivar an de la disponibilidad de mejores herramientas e informaci n para gestionar los fondos y controlar sus riesgos crear an una gran riqueza que ser a capturada en su gran mayor a por los afiliados aumentando el bienestar de los pensionados Asimismo este incremento en el valor de los fondos impulsar a el desarrollo de la econom a nacional aumentando el ahorro nacional y mejorando la asignaci n de recursos La evaluaci n social reconoce por una parte que hay una demanda insatistecha por tecnolog as y servicios de gesti n del riesgo y por otra la inexistencia de productos en el mercado internacional que aborden la problem tica espec fica del mercado nacional El proyecto gener m ltiples resultados en mbitos econ mico sociales cient fico tecnol gicos e institucional
138. n the accuracy and the complexity of the model in terms of the number of parameters that need to be estimated The optimal estimates are then computed in what is called the update step a x X P HF v 26 P P g P H F H P 27 Note that the above calculations can be performed even if the num ber of observations varies with time and that the accuracy of the estima tion measured by the variance of the estimation error increases with the number of observations that are available to update the filter The estimation of model parameters W is obtained by maximizing the log likelihood function of innovations l l log LOW 5 log F 3 5 vi F lv 28 1 t where Y represents a vector containing unknown parameters EMPIRICAL RESULTS Data E The data used in this study consist of all daily light sweet crude oil futures prices traded at NYMEX from January 1992 to December 2004 There are currently 35 contracts traded for different maturities ranging from to 30 months and 3 4 5 6 and 7 years However from 1992 to 1996 the maximum maturity traded at NYMEX was only 4 years In 1997 new contracts were introduced to include maturities up to 7 years Journal of Futures Markets DOI 10 1002 fut ON Futures Prices TABLE Average Number of Daily Observations and Maximum Maturity Available of Light Sweet Crude Oil Futures Contracts for Panel A 1992 2001 B 1992 1996 C 1997 2001 and
139. nal mean Xy fx f Xq 1 i H x E X 1 1 24 where H 0 0x f x The prediction step equations are the same as before The update step equation under the extended Kalman filter is then XK Xqi 1 Pi HLF v 25 P Pai Py HF H Pai 26 where F HP H R 27 Vy Zp fR 28 An explanation on how to apply the extended Kalman filter to coupon bond yields can be found in the Appendix 5 EMPIRICAL RESULTS To illustrate our methodology we estimate a three factor generalized Vasicek model using Chilean government bond data The data used consist of inflation protected bonds the most liquid fixed income instrument traded in Chile Thus we are modelling the behaviour of real as opposed to nominal interest rates The choice of the Vasicek model seems appropriate for modelling real rates which might become negative whenever the rate of inflation exceeds the nominal interest rate Given that most of the outstanding bonds trade only sporadically the Chilean government bond market can be characterized as a market with infrequent trading and is used to test our term structure estimation methodology In the following sections we describe the data and analyse the estimation results based on in sample and out of sample yield errors and on the ability of the model to fit the observed term structure of volatilities 5 1 Data description The data consist of all transactions at the Santiago Stock Excha
140. nal of Finance 51 5 1011 1029 Langetieg TC 1980 A multivariate model of the term structure Journal of Finance 35 1 71 97 Lund J 1994 Econometric analysis of continuous time arbitrage free models of the term structure of interest rates Working Paper The Aarhus School of Business Lund J 1997 Non linear Kalman filtering techniques for term structure models Working Paper The Aarhus School of Business Mcculoch JH 1971 Measuring the term structure of interest rates Journal of Business 44 1 19 31 Mcculoch JH 1975 The tax adjusted yield curve Journal of Finance 30 3 811 830 Nelson CR Siegel AF 1987 Parsimonious modeling of yield curves Journal of Business 60 4 473 489 Nowman KB 1997 Gaussian estimation of single factor continuous time models of the term structure of interest rates Journal of Finance 52 1695 1706 Nowman KB 1998 Continuous time short rate interest rate models Applied Financial Economics 8 401 407 ksendal B 1998 Stochastic Differential Equations An Introduction With Applications 5th edn Springer Berlin New York Pearson ND Sun T S 1994 Exploiting the conditional density in estimating the term structure an application to the Cox Ingersoll and Ross model Journal of Finance 49 4 1279 1304 Pennacchi GG 1991 Identifying the dynamics of real interest rates and inflation evidence using survey data Review of Financial Studies 4 1 53 86 Schwartz ES 1997 The stochastic behavi
141. nal of Futures Markets Cortazar G 60663351 1 Oil Futures Prices Naranjo L 12931431 1 Ciudad y Pa s P ginas Editorial C digo Disciplina Clasificaci n de Edici n o donde se edit o para cap de libro o ISBN ISSN ISI disciplina Fondecyt 1 Cient fica Publicaci n public art de revista Q predominante 2 Tecnol gica 3 Difusi n Otras especificar 01 2008 OXFORD 113 129 PERGAMON ISSN 0305 0548 Computaci n ENGLAND ELSEVIER SCIENCE LTD 2007 por aparecer CHICHESTER JOHN WILEY SSN 1076 9307 Finanzas 1 ENGLAND W amp SONS INC SUSSEX PO19 8SQ HOBOKEN USA NJ 07030 03 2006 243 268 ISSN 0270 7314 Finanzas JOHN WILEY amp SONS INC 2 12 PROTECCION DE RESULTADOS a PATENTES Titulo Disciplina Pais Paginas Observaciones Estado Fecha Tipo de patente disciplina pais donde cantidad de 1 Solicitada Otorgamiento Nacional Internacional Fondecyt se solicit la p ginas de la 2 Otorgada Indique pa s es donde rige predominante patente patente AUTORES CONTINUACION PATENTES N patente Rut para nacionales y Apellido Paterno Apellido Materno Nombres Due o de la patente cuadro extranjeros residentes en anterior Chile b OTRAS FORMAS DE PROTECCION Tipo de protecci n Establecida si o no Resultados del proyecto El valor comercial de los resultados del proyecto se basan fuertemente en la reputa
142. nge from January 1997 to December 2001 1243 days of pure discount bonds and semi annual amortizing coupon bonds issued by the Chilean government Pure discount bonds are usually denominated Pagare Reajustable Banco Central PRBC bonds and semi annual amortizing coupon bonds are called Pagare Reajustable con Cupones PRC bonds Both type of bonds are inflation protected with payments brought to real terms using monthly inflation Table 1 summarizes the data It can be noted that pure discount bonds have maturities of less than year while coupon bonds have maturities ranging from 1 to 20 years Trading frequency is defined as the number of days for which we have at least one transaction of a bond of a specific maturity over all available trading days A trading frequency of 20 means that at least one bond with that maturity was traded an average of 50 days per year From Table 1 we see that for most maturities the trading frequency ranges from 30 to Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe IJFE 317 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 IJFE 317 10 G CORTAZAR ET AL Table 1 Description of the data daily transactions of Chilean government inflation protected pure discount and coupon bonds from January 1997 to December 2001 Maturity range Number of Average trad
143. o FONDEF COMISION NACIONAL DE INVESTIGACION CIENTIFICA Y TECNOLOGICA FOMENTO AL BERNARDA MORIN 495 CASILLA 297 V CORREO 21 FONO 3654400 FAX 6551394 CHILE DESARROLLO CIENTIFICO Y TECNOLOGICO 1 RESUMEN EJECUTIVO Describa en no m s de una p gina la problem tica u oportunidades que llev a formular este proyecto origen su desarrollo los resultados logrados y la proyecci n del mismo El objetivo principal del proyecto es desarrollar herramientas aplicaciones y servicios computacionales que aprovechen en forma efectiva las tecnolog as asociadas a Internet para modernizar el sistema financiero nacional apoyando una mejor gesti n de carteras de inversi n en activos transados en el mercado nacional e internacional Los desarrollos se focalizan preferentemente en la problem tica de los fondos de pensiones pero sus resultados impactan la gesti n de otras carteras de inversi n como las administradas por compa as de seguros y fondos mutuos Esta modernizaci n se apoya tanto en el estado del arte metodol gico mundial como en investigaci n cient fica que aborde la problem tica de mercados financieros poco profundos como el nacional con activos que se transan con una baja frecuencia thin markets lo que dificulta el uso de numerosos procedimientos y metodolog as utilizadas en los mercados desarrollados De este modo se pretende 1 apoyar una gesti n m s eficiente de las carteras al incluirse mayor informaci
144. o servicio Portfolio provee una completa gama de herramientas que permiten hacer m s eficiente la gesti n integral de carteras de inversi n en Chile Asignaci n de Activos Gesti n del Riesgo y Medici n de Performance En el competitivo mercado financiero actual el manejo adecuado de una cartera de inversi n requiere un monitoreo constante de la composici n comportamiento y exposici n de sta as como la comparaci n de su desempe o con el de otras carteras de mercado Con el fin de facilitar est s tareas y apoyar integralmente las decisiones de inversi n y gesti n de carteras hemos desarrollado una completa familia de herramientas relacionadas con las reas de Asset Allocation An lisis de Riesgo y Medici n de Performance Est s herramientas le permiten a nuestros usuarios cuantificar y monitorear de una manera r pida y f cil el desempe o y exposici n de sus carteras y a la vez dise ar y evaluar eficientemente las distintas estrategias de inversi n Adicionalmente estas herramientas pueden ser usadas en conjunto con nuestra familia de ndices permitiendo a nuestros usuarios representar adecuadamente el mercado financiero local y gestionar as de manera eficiente y confiable sus carteras de inversi n e wwwzriskamerica com Fono 56 2 354 4086 354 4823 Fax 56 2 552 1608 E mail info riskamerica com RiskAmerica SVC Port
145. oadie M Glasserman P Pricing American style securities using simulation Journal of Economics Dynamics and Control 1997 21 8 1323 52 Andersen L A simple approach to the pricing of Bermudian swaptions in the multi factor libor market model Journal of Computational Finance 2000 3 5 32 Haugh M Kogan L Approximating pricing and exercising of high dimensional American options a duality approach Cambridge MA MIT 2001 Longstaff FA Schwartz ES Valuing American options by simulation a simple least squares approach The Review of Financial Studies 2001 14 1 113 47 Stentoft L Assessing the least squares Monte Carlo approach to American option valuation Review of Derivatives Research 2004 7 2 129 68 Moreno M Navas J Review of Derivatives Research 2003 6 2 107 28 Clement E Lamberton D Protter P An analysis of a least squares regression method for American option pricing Finance and Stochastics 2002 6 4 449 71 Brennan MJ Schwartz ES Evaluating natural resources investments Journal of Business 1985 58 2 135 57 Majd S Pindyck R The learning curve and optimal production under uncertainty Rand Journal of Economics 1989 29 1110 48 Trigeorgis L The nature of option interactions and the valuation of investments with multiple real options Journal of Financial and Quantitative Analysis 1993 1 20 McDonald R Siegel D The value of waiting to invest Quarterly Journal of Economics 1986 101 707 27 Majd S Pindyck R Time to
146. obustez te rica de los mercados m s exigentes TATAMI M AT COPEC UN e 5 magona o nevesmiaci n HABITAT OPEC UNIVERSIDAD CAT LICA Seguridad y Confianza www riskamerica com Fono 56 2 354 4086 354 4823 Fax 56 2 552 1608 E mail info riskamerica com Ri skAmerica RiskAmerica SVC Indices Portfolio mmodity Servicios RiskAmerica RiskAmerica Estimaciones de estructuras de tasas de inter s y herramientas de valorizaci n y an lisis para los distintos instrumentos y transacciones del mercado de renta fija nacional SVC Servicio diario de valorizaci n y asignaci n de tasas para los distintos instrumentos de renta fija del mercado nacional ndices Familia de ndices del mercado nacional que permite tener referentes adecuados para la medici n y comparaci n de rendimientos de las distintas carteras de inversi n Portfolio Completa gama de herramientas que permiten hacer m s eficiente la gesti n integral de carteras de inversi n en Chile Asignaci n de Activos Gesti n del Riesgo y medici n de Performance Commodity Ingenier a Financiera para la evaluaci n y gesti n eficiente de proyectos y activos expuestos a las variaciones de los precios de commodities SS SS www riskamerica com Fono 56 2 354 4086 354 4823 Fax 56 2 552 1608 E mail info riskamerica com Ri skAr Y SVC Nuestro
147. ocate cor The valuation of multidimensional American real options using the LSM simulation method Gonzalo Cortazar Miguel Gravet Jorge Urzua Departamento de Ingenier a Industrial y de Sistemas Escuela de Ingenier a Pontificia Universidad Cat lica de Chile Vicu a Mackenna 4860 Santiago Chile Available online 22 March 2006 Abstract In this paper we show how a multidimensional American real option may be solved using the LSM simulation method originally proposed by Longstaff and Schwartz 2001 The Review of the Financial Studies 14 1 113 147 for valuing a financial option and how this method can be used in a complex setting We extend a well known natural resource real option model initially solved using finite difference methods to include a more realistic three factor stochastic process for commodity prices more in line with current research Numerical results show that the procedure may be successfully used for multidimensional models expanding the applicability of the real options approach Even though there has been an increasing literature on the benefits of using the contingent claim approach to value real assets limitations on solving procedures and computing power have often forced academics and practitioners to simplify these real option models to a level in which they loose relevance for real world decision making Real option models present a higher challenge than their financial option counterparts because of two
148. odity In this paper we extend and solve the well known Brennan and Schwartz 23 model for valuing natural resource investments Other papers on natural resource investments include 40 45 among many others 2 2 The Brennan and Schwartz 23 Model The valuation of a copper mine in 23 laid the foundations for applying option pricing arbitrage arguments to the valuation of natural resource investments In the model the value maximizing policy under stochastic output prices considers the optimal timing of path dependent American style options to initiate temporarily cease or completely abandon production We now describe the optimization problem in a general framework for valuing a switching option Consider the Brennan and Schwartz 23 model as a switching option with value V x j and cash flows CF x j at time ft which depend on a vector of N state variables x x x and the state of production j The model considers that there are K states of production and the switching option can move from one state j to another i paying the corresponding switching cost C x This state switches can be made at any of T 1 stages witht o f tr We assume for simplicity that the process for the state variables can be risk adjusted and that markets are complete Thus we can use the standard option pricing technique which means that the switching option can be valued as the discounted expectation under the risk neutral probability
149. olio ommodity ndices Completa familia de ndices que provee una cobertura total del mercado nacional a trav s de ndices agregados y detallados para los distintos instrumentos plazos clasificaciones de riesgo y monedas del mercado local Bonos de Gobierno Bonos Corporativos Letras Hipotecaras Intermediaci n Financiera Acciones Comparador de ndices Herramienta que permite comparar y descargar caracter sticas y estad sticas de los distintos ndices a lo largo del tiempo Constructor de ndices Herramienta que permite crear ndices propios considerando conjuntamente instrumentos espec ficos e ndices de mercado Permitiendo de esta manera a nuestros usuarios construir y monitorear diariamente ndices que reflejen sus carteras de inversi n o cualquier cartera que deseen representar Carga de ndices Adicional a la construcci n de ndices propios los usuarios pueden cargar ndices para as poder compararlos y utilizarlos como inputs de las distintas herramientas de RiskAmericaPlus Mis ndices Herramienta que le permite a los usuarios manejar una selecci n propia de ndices construidos cargados o provistos por RiskAmerica y trabajar diariamente con estos a trav s de distintas herramientas de an lisis www riskamerica com Fono 56 2 354 4086 354 4823 Fax 56 2 552 1608 E mail info riskamerica com Risk merica Indices Portfolio mmodity Portfolio Nuestr
150. om Fono 56 2 354 4086 354 4823 Fax 56 2 552 1608 E mail info riskamerica com Indices Portf dio 6mmodity RiskAmeric a ndices Nuestro servicio Indices entrega una completa familia de ndices del mercado nacional proveyendo as referentes adecuados para la medici n y comparaci n de rendimientos de las distintas carteras de inversi n Los mercados desarrollados poseen cientos de indicadores que permiten monitorear sus comportamientos realizar an lisis hist ricos y comparar los rendimientos relativos de las distintas carteras de inversi n mejorando as la capacidad de an lisis y fomentando el crecimiento y profundidad de estos Con el objetivo de fomentar el crecimiento y transparencia del mercado chileno y proveer indicadores confiables y precisos del comportamiento de ste hemos construido una completa familia de ndices la cual representa el mercado nacional en su totalidad Estos ndices son construidos en base a los precios de nuestro Servicio de Valorizaci n de Carteras los cuales siguen diariamente los movimientos de mercado y poseen volatilidades estables y consistentes para los distintos papeles lo que garantiza la estabilidad y correcto comportamiento de los distintos ndices www riskamerica com Fono 56 2 354 4086 354 4823 Fax 56 2 552 1608 E mail info riskamerica com Ri UN neric Ri skAmer ica Herramientas ndices Portf
151. om Fono 56 2 354 4086 354 4823 Fax 56 2 552 1608 E mail info riskamerica com RiskAmerica RiskAme rica ndices Portfolio ommodity El Servicio de Valorizaci n de Carteras consiste en una aplicaci n web a trav s de la cual nuestros usuarios pueden cargar diariamente sus carteras y descargarlas valorizadas en un plazo no mayor a 5 minutos Las tasas de valorizaci n son actualizadas diariamente a partir de la informaci n de transacciones de la Bolsa de Comercio de Santiago quedando accesibles para nuestros usuarios a partir de las 14 45 horas Es posible valorizar carteras en horario continuado y tambi n para fechas pasadas Servicio de Valorizaci n de Carteras El servicio consiste en una valorizaci n integral del instrumento entregando toda la informaci n generada referente a ste Tasa de Valorizaci n Precio en Porcentaje del Valor Par Precio en pesos Plazo al Vencimiento Duraci n Convexidad Clasificaci n de Riesgo TIR Base Spread Clase de Valorizaci n Se incluye el Precio en pesos del instrumento que puede ser evaluado para una posici n est ndar permiti ndole as a nuestros usuarios ponderarlo por las posiciones reales de su cartera Adicionalmente se entrega la Clase de Valorizaci n lo que otorga un mejor control sobre la informaci n y la capacidad de poder monitorear eficientemente las tasas si as se desea www riskamerica c
152. only prices for that particular day and the dynamics of the interest rate process We have not included the previous day curve in Figure 6 because it is almost identical to the curve shown The model s long term yields for the current day for which there is no data are very close to the observed previous day long term yields Comparing Figure 6 with Figure 2 which Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe IJFE 317 11 13 15 17 19 21 23 25 27 29 31 33 35 Color in Web B W in Print 37 39 41 43 45 47 49 51 IJFE 317 12 G CORTAZAR ET AL Table 2 Parameter estimates and standard errors from daily transactions of Chilean government inflation protected pure discount and coupon bonds from January 1997 to December 2001 Ki 0 00050 0 00012 Ko 1 11455 0 01681 K3 2 16431 0 05362 0 0 01747 0 00019 0 0 29298 0 00466 03 0 32780 0 00647 Pa 0 91042 0 01258 Paj 0 84189 0 02376 paz 0 97121 0 00246 A 0 00056 0 00002 do 0 01599 0 00418 da 0 05213 0 01836 0 05614 0 02654 Ed 0 00225 0 00014 A 0 00225 0 00004 a 0 00079 0 00001 E 0 00027 0 00001 a 0 00038 0 00001 Bond Yields on 01 09 1997 Observed Bond Yields Yield 9 Model Term Structure Maturity Years Figure 5 Estimated and observed coupon bond yields on 01 09 1997 corresponds to the same date this ex
153. or of commodity prices implications for valuation and hedging Journal of Finance 52 3 923 973 Schwartz ES Smith JE 2000 Short term variations and long term dynamics in commodity prices Management Science 46 893 911 Sorensen C 2002 Modeling seasonality in agricultural commodity futures Journal of Futures Markets 22 393 426 Svensson LEO 1994 Estimating and interpreting forward interest rates Sweden 1992 1994 Working Paper National Bureau of Economic Research Vasicek OA 1977 An equilibrium characterization of the term structure Journal of Financial Economics 5 2 177 188 Vasicek OA Fong HG 1982 Term structure modeling using exponential splines Journal of Finance 37 2 339 356 Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe IJFE 317 wry SEHR A MA a at ae ers nae AN N FACTOR GAUSSIAN MODEL OF OIL FUTURES PRICES GONZALO CORTAZAR LORENZO NARANJO This article studies the ability of an N factor Gaussian model to explain the stochastic behavior of oil futures prices when estimated with the use of all available price information as opposed to traditional approaches of aggre gating data for a set of maturities A Kalman filter estimation procedure that allows for a time dependent number of daily observations is used to calibrate the model When applied to all daily oil futures price transactions from 1992 to 2001 the model performs very wel
154. ormation In addition not all futures contracts trade every day If a complete data set of prices is to be used some prices need to be discarded or aggregated with great information loss Thus it would be desirable that the estimation proce dure uses all price information available Oil prices are very volatile have a high degree of mean reversion Bessembinder Coughenour Seguin amp Smoller 1995 and exhibit complex dynamics Thus it is important to analyze the number of risk factors required to model this stochastic behavior Cortazar amp Schwartz 1994 Several models of the stochastic process followed by commodity prices have been proposed in the literature They differ in how they speci fy spot price innovations and how they model the cost of carry The cost of carry represents the storage cost plus the interest paid to finance the asset minus the net benefit that accrues to the asset holder if any In the com modities literature the benefit received by the commodity owner but not by the futures contract owner is called the convenience yield Brennan 1958 1991 Deaton amp Laroque 1992 Gibson amp Schwartz 1990 Routledge Seppi amp Spatt 2000 Working 1949 which is commonly represented as a dividend vield Journal of Futures Markets DOI 10 1002 fut vic OH Futuros Prices Early models of commodity prices assume a one factor geometric Brownian motion for the spot price with a constant interest rate and
155. os del proyecto Montos en MM 2007 2008 2009 2010 2011 2012 2013 Mod SVC 180 200 0 220 0 240 240 240 240 Mod Indices 10 47 0 85 0 120 120 120 120 Mod Portfolio 8 75 0 142 0 210 210 210 210 Total Ing 198 322 447 570 570 570 570 Costos Fijos 120 120 120 120 120 120 120 Costos Variables 65 34 106 26 147 51 188 1 188 1 188 1 188 1 Total Costos 185 34 226 26 267 51 308 1 308 1 308 1 308 1 Excedentes 12 66 95 74 179 49 261 9 261 9 261 9 261 9 VAN 12 781 6 ANEXO 2 PLANES DE TRABAJO INICIAL Y EFECTIVAMENTE EJECUTADO Tipo Nombre Resultado Fechas Programada Reprogramada Lograda Convenios Contraparte Convenios Con 06 03 2005 06 04 2005 06 04 2005 Contrapartes Descripci n Plan de Experimentos Dio definido Plan De Experimentos 30 06 2005 30 06 2005 30 06 2005 Definido A o Www riskportfolio A Dise o de Prototipo 19 emitido 30 06 2005 30 06 2005 30 06 2005 Diseno De Pagina Web 7 Experimentos Criticos Efectuados Experimentos Criticos 30 06 2005 15 09 2005 15 09 2005 Efectuados 7 Www riskportfolio Prototipo Probado a nivel Dio Laboratorio Prototipo 30 06 2005 30 09 2005 30 09 2005 Www riskportfolio Probado 7 A Nivel Laboratorio Prototipo probado a Nivel 19 Piloto Planta 30 06 2005 30 09 2005 30 09 2005 Www riskportfolio comv 1 Dise o de Prototipo 7 emitido Dise o De 01 08 2005 01 08 2005 01 08 2005 Assetallocationv 1 Plan de Experimentos definido 7
156. os S A Santander Santiago S A Administradora General de Fondos Scotia Sudamericano Administradora de Fondos Mutuos S A Zurich Administradora General de Fondos S A Tipo Nombre ABN AMRO Bank Banco BICE Banco de Chile Banco de Cr dito e Inversiones Banco de la Naci n Argentina Banco del Desarrollo Banco del Estado de Chile Banco do Brasil Banco Falabella Banco Internacional Banco Paris Banco Penta Banco Ripley Banco Santander Santiago Banco Security BankBoston BBVA Citibank N A CorpBanca Deutsche Bank HNS Banco HSBC Bank Chile JPMorgan Chase Bank Scotiabank Sud Americano The Bank of Tokyo Mitsubishi Ltd Nombre Seguros Aseguradora Magallanes BICE Vida Cardif Chilena Consolidada COFACE Consorcio Nacional Continental Corredora Security S A Cruz del Sur Eurovida ING Interamericana ISE Liberty MAPFRE Mutual de Carabineros Mutual de Seguros de Chile Renta Nacional Royal amp Sunalliance Seguros de Vida la Construcci n SA 4 3 Factores de Exito El principal Factor de xito es la capacidad de generar y comunicar una reputaci n de objetividad rigurosidad y compromiso de permanente innovaci n para los Servicios ofrecidos Para ello se hace necesario mantener activo un equipo de investigadores hac
157. our 66 71 Plazo a os 13 24 TIR 3 53 E 9 Convexidad a os 123 67 Serie de Tiempo Graficar Tipo E El a Grafi Al cy ratico Tambi n se pueden cargar ndices generados externamente erica Mis Indices Mis Indices Indices Renta Fija Indices Renta Variable Constructor de Indices Comparador de Indices Familia de Indices Familia de ndices Descripci n de ndices Portfolio Q mmodity Comparador de indices ario Gon J Fecha de Consulta Hombre Dent Fija 2 El cob scr El cob cero Drenta Variable 0 Ohuices Construidos 4 nuevo indice RF_1 RV Dindices Cargados 25 asmarzo07 ES Valor 1 263 76 1 928 86 1 001 39 NA NA NA 1 D a 0 000 0 014 1 0695 NA NA NA Agregar a Ea Mis Indices Estad sticas MTD YTD 1A o 034 188 7 98 078 144 7 59 0 24 D14 014 NA NA NA NA NA NA NA NA NA DesvEst 1 a o 1 07 2 23 5 23 NA NA NA 5 33 2 64 0 00 NA NA NA He Instr 10 173 1 00 NA NA NA Caracter sticas Monto Durac Plazo MMM a os a os 2 010 41 629 03 NA NA NA NA 2 54 4 47 0 00 NA NA NA 3 06 4 47 0 00 NA NA NA Servicio 3 M dulo Portfolio Este m dulo entrega informaci n de de riesgo retorno y performance tanto de carteras existentes del mercado como de carteras propias En Informaci n de Mercado entrega informaci n
158. oyecto por un plazo no inferiora 3 a os b El uso de la infraestructura y equipamiento asociado al proyecto en el apoyo a proyectos de I amp D o servicios C amp T con alto impacto econ mico social C La valorizaci n comercializaci n y transferencia de los resultados del proyecto que se requiera para maximizar los impactos d La protecci n de los resultados as como el beneficio en t rminos razonablemente onerosos para la instituci n a partir de las rentas que de ellos se obtengan INSTITUCION _ BENEFICIARIA NOMBRE INSTITUCION BENEFICIARIA NOMBRE INSTITUCION BENEFICIARIA NOMBRE Pontificia Universidad Cat lica de Chile NOMBRE REPRESENTE LEGAL NOMBRE REPRESENTE LEGAL NOMBRE REPRESENTE LEGAL Carlos Vio Lagos VicerrectoryAdjunto de Investfaci n y Doctorado FIRMA FIRMA J e LAS INSTITUCIONES BENEFICIARIAS DECLARAN ESTAR EN CONOCIMIENTO Y DE ACUERDO CON EL CONTENIDO TOTAL DE ESTE INFORME Y QUE LOS DATOS REGISTRADOS EN ESTA DECLARACI N CORRESPONDEN A UN RESUMEN DE LOS DETALLADOS EN L GOBIERNO DE CHILI II PARTE INFORME EJECUTIVO CONK C digo Proyecto D0311039 Nombre del Proyecto DESARROLLO DE HERRAMIENTAS COMPUTACIONALES PARA OPTIMIZAR LA GESTION DE CARTERAS DE INVERSION EN MERCADOS EMERGENTES APLICACION A LOS FONDOS DE PENSIONES EN CHILE La informaci n entregada en esta parte del documento debe ser s lo la que puede ser de dominio p blic
159. pacto del proyecto Las instituciones declaran que de acuerdo a su evaluaci n de impacto el proyecto ha generado y est en proceso de generar los siguientes impactos e Cient fico Tecnol gico e Obtenido Nuevas metodolog as de valorizaci n y gesti n del riesgo principalmente para mercados con pocas transacciones como son los mercados de econom as emergentes como la chilena Esto se ha traducido en publicaciones tesis de mag ster y presentaciones en conferencias acad micas internacionales En proceso de obtenci n Nuevas publicaciones en preparaci n orientadas a formas de gestionar carteras de inversi n y a modelos multi activos 10 e Econ mico Social e Obtenido Mejor valorizaci n de carteras de inversi n para todos los Fondos Mutuos del Pa s y para algunas otras instituciones financieras del pa s lo que transparenta los mercados y permite una mejor competencia y asignaci n de recursos financieros En proceso de obtenci n Mejores decisiones de inversi n y de gesti n del riesgo para carteras de inversi n de las instituciones financieras del pa s a medida que vayan adoptando las herramientas que actualmente est n en fase de prueba e Institucional Obtenido Fortalecimiento del FINlabUC Laboratorio de Investigaci n Avanzada en Finanzas tanto en actividad reconocimiento y equipamiento Fortalecimiento del programa de Mag ster en Ciencias de la Ingenier a con incremento en el n mero de alumnos q
160. presents the problem to be solved It describes the classic Brennan and Schwartz 23 real option model of a natural resource investment and how we extend it to include a multifactor model of commodity prices A brief explanation on the real options approach for valuing investments is also included Section 3 presents the proposed computer based simulation procedure Section 4 discusses the results of the numerical solution to the original and to the extended Brennan and Schwartz model and some implementation issues for high dimensional models Finally Section 5 concludes 2 The problem 2 1 The Real options approach to valuation Real option valuation ROV can be understood as an adaptation of the theory of financial options to the valuation of investment projects ROV recognizes that the business environment is dynamic and uncertain and that value can be created by identifying and exercising managerial flexibility Options are contingent claims on the realization of a stochastic event with ROV taking a multi path view of the economy Given the level of uncertainty the optimal decision path cannot be chosen at the outset Instead decisions must be made sequentially hopefully with initial steps taken in the right direction actively seeking learning opportunities and being prepared to appropriately switch paths as events evolve ROV presents several improvements over traditional discount cash flow DCF techniques First it includes a bette
161. proach however may be used in any market with infrequent trading as is the case in many emerging markets The next section explains the shortcomings of static term structure estimation methods when there is sparse data In Section 3 we present the generalized Vasicek model that will be used for illustrating our methodology Section 4 presents the standard Kalman filter method and shows how it can be used in an incomplete panel data setting Section 5 presents empirical results of applying the methodology to the Chilean government bond market and Section 6 concludes Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe IJFE 317 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 E a ae S 2 5 e wW 49 51 IJFE 317 TERM STRUCTURE ESTIMATION 3 2 SHORTCOMINGS OF STATIC TERM STRUCTURE ESTIMATION MARKETS WITH INFREQUENT TRADING Term structure estimation has been traditionally implemented with static models that only use current bond prices or yields without regard to past information Some methods like Nelson and Siegel 1987 and Svensson 1994 assume a parametric functional form for the forward rates Other methods for example McCuloch 1971 1975 and Fisher et al 1994 use non parametric spline based interpolation methods to calculate the term structure Empirical evidence shows that in
162. r assessment of the value of strategic investments and a better way of communicating the rationale behind that value In most traditional DCF valuations a base value is calculated Then this base value is adjusted heuristically to capture a variety of critical phenomena Ultimately the total estimated value may be dominated by the adjustment rather than the base value With ROV the entire value of the investment is rigorously captured Conceptually this includes the base value and the option premium obtained from actively managing the investment and appropriately exercising options G Cortazar et al Computers amp Operations Research 35 2008 113 129 115 Second ROV provides an explicit roadmap or optimal policy for achieving the maximum value from a strategic investment Most traditional investment valuations boil down to a number and perhaps a set of assumptions underlying that number However the management actions required over time to realize that value are not clearly identified With ROV the value estimate is obtained specifically by considering these management actions As a result ROV indicates precisely which events are important and the necessary actions required to achieve maximum value There is a broad literature on ROV and how to maximize contingent claim value over all available decision strategies Among them Majd and Pindyck 24 include the effect of the learning curve by considering tha
163. reflecting more uncertainty on the true value of the state variables In any case the estimation of the state variables takes into account the whole variance covariance structure among observations 4 3 Kalman filter with a non linear measurement equation When applying the Kalman filter to coupon bond yields or prices we usually obtain a non linear measurement equation In this case the extended Kalman filter which applies to non linear measurement and or transition equations must be used We will briefly describe the mathematics of the extended Kalman filter Since under the generalized Vasicek model which has been used to illustrate the methodology the transition equation is a linear function of the state variables we restrict the analysis to the case where only the measurement equation is a non linear function of the state variables 4 Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe IJFE 317 1 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 41 49 51 IJFE 317 TERM STRUCTURE ESTIMATION 9 Let the measurement equation be a non linear function of the state variables Z fX vs v N O R 23 with f R gt R a continuous and differentiable function The extended Kalman filter when only the measurement equation is non linear is obtained by linearizing f x around the conditio
164. respondence between the state variables of the two models obtaining logS Ix u Lo a 11 dx Kx dt Edw 12 0 0 0 l TAN where K X pa and dw dw Q di Pas 0 Kk 0 oa Pa ave n p The reversion matris is assumed to have termw ise different eigenvalues to obtain simpler valuation formulas for futures contracts Journal of Futures Markets DOI 10 1002 fut The existing relationship between the variance covariance parame ters of the two models is given by LOX LE Q Y L Assuming a con stant risk premium A for each state variable the risk adjusted process of the Gibson and Schwartz 1990 model is dx KZ B X dt E dw 13 Under the equivalent martingale measure the drift of the spot price of a commodity must be equal to the difference between the instanta neous interest rate r which is assumed constant by Gibson and Schwartz 1990 and its instantaneous convenience yield so the following rela tion must hold poid r 14 The risk adjusted process of the new model is dx Kx A dt Ldw 15 Then it can be shown that the relationship between the risk premi ums of the two models is A L Under this new representation it follows from Equations 11 and 14 that it is not possible to estimate u a and A independently One of the three parameters or a linear combination of them must be exoge nously specified In the article Gibson and Schwartz 1
165. rix x is a n x 1 vector d is a m x 1 vector and v is a m X 1 vector of serially uncorrelated Gaussian disturbances with mean 0 and covariance matrix R with dimension is m x m Under this assumptions ie will be considered an incomplete panel data set To see why the Kalman filter still may be used with incomplete panel data sets note that given a vector of state variables X and a covariance matrix P _ of the estimation errors the filter first calculates a prediction of the state variables X _ and of the covariance matrix P _ of the errors using equations 14 and 15 For this calculation only the dynamic properties of the state variables are used which do not depend on the number of observable variables The filter then incorporates the new information given by the vector of observable variables z The same equations 16 and 17 can then be used to calculate optimal estimates of the state vector X and of the covariance matrix P As mentioned before since the Kalman filter computes at every date the conditional expectation X E _ x z the estimates can still be computed even if the number of observations vary with time Of course the greater the number of observations available to update the filter the better the accuracy of the estimation This is reflected in a lower variance of the estimation error When a reduced number of observations is available at some date the estimation error and its variance will be greater
166. ro que hist ricamente ha exhibido retornos correlacionados parcialmente entr s El inicio de esta l nea de investigaci n est siendo muy prometedora permitiendo mejorar sustancialmente los modelos para la din mica de precios que inicialmente consideraban s lo una familia de instrumentos Documentos de Trabajo atin no publicados Cortazar G Milla C Severino F 2007 A Multicommodity Model of Futures Prices Using Futures Prices of One Commodity to Estimate the Stochastic Process of Another Tesis de Magister Modelo estoc stico multicommodity para la din mica de precios de contratos futuros Selecci n y estimaci n del modelo utilizando componentes principales comunes y filtro de Kalman FELIPE SEVERINO DIAZ Tesis de Mag ster en Ciencias de la Ingenier a Pontificia Universidad Cat lica de Chile 26 03 2007 Estimaci n de Spreads por Liquidez en un Mercado con Pocas Transacciones El Caso del Mercado de Bonos del Banco Central de Chile PEDRO MAT AS MORAL MESA Tesis de Mag ster en Ciencias de la Ingenier a Pontificia Universidad Cat lica de Chile 17 01 2006 Modelos Estoc sticos de Precios de Commodities y Estimaci n Conjunta de la Din mica de dos Commodities Mediante el Filtro de Kalman CARLOS IGNACIO MILLA GONZALEZ Tesis de Mag ster en Ciencias de la Ingenier a Pontificia Universidad Cat lica de Chile 23 12 2005 Presentaciones en Congresos Acad micos Cortazar G Milla C Severino F
167. ruidos a partir de estas simulaciones Publicaciones Cortazar G Gravet M Urzua J 2008 The Valuation of Multidimensional American Real Options using the LSM Simulation Method Computers amp Operations Research Vol 35 2008 113 129 Presentaciones en Congresos Acad micos Cortazar G Gravet M Urzua J 2005 The Valuation of Multidimensional American Real Options using the LSM Simulation Method 9th Annual International Conference Real Options Theory Meets Practice Real Options Group and EDC Paris Paris June 23 25 http www realoptions org AcademicProgram academicprogram2005 html Tema 3 Metodolog as de Medici n de Spreads Discusi n Existen diversos tipos de spreads o diferenciales de precios o tasas entre los activos m s deseados por el mercado los que se transan a mayores precios o equivalentemente descontados a las menores tasas y el resto Este diferencial se puede deber a la existencia de riesgos de cr dito o de liquidez que explican que inversionistas racionales s lo los adquieren en la medida que se transen a un descuento relativo los mejores activos del mercado Durante la realizaci n del proyecto se desarrollaron metodolog as de estimaci n de spreads para diversos instrumentos de deuda bonos de empresas letras hipotecarias dep sitos a plazo Bonos de Reconocimiento etc as como estimaciones de los spreads de liquidez presentes entre activos de la misma familia Bonos Ban
168. s Cortazar G Milla C Severino F 2007 A Multicommodity Model of Futures Prices Using Futures Prices of One Commodity to Estimate the Stochastic Process of Another 4th Annual Conference of Asia Pacific Association of Derivatives APAD Gurgaon India June 20 22 2007 Cortazar G Schwartz E S Naranjo L 2006 Term Structure Estimation in Markets with Infrequent Trading Latin American Meeting of the Econometric Society LAMES ITAM Ciudad de M xico Nov 2 4 2006 Cortazar G Schwartz E S Naranjo L 2006 Term Structure Estimation in Markets with Infrequent Trading 2006 FMA Annual Meeting Salt Lake City Oct 11 14 2006 Cortazar G Schwartz E S Naranjo L 2006 Term Structure Estimation in Markets with Infrequent Trading 2006 Far Eastern Meeting of the Econometric Society Beijing July 9 12 2006 Cortazar G Milla C Severino F 2006 Using futures prices of one commodity to estimate the stochastic process of another INFORMS Hong Kong International 2006 Hong Kong June 25 28 2006 Cortazar G Schwartz E S Naranjo L 2006 Term Structure Estimation in Markets with Infrequent Trading 15th annual meeting of the European Financial Management Association Madrid June 28 July 1 2006 Cortazar G Naranjo L 2005 An N Factor Gaussian Model of Oil Futures Prices 2005 FMA Annual Meeting Chicago October 12 15 http www fma org Chicago ChicagoProgram htm Cortazar G Gra
169. s Wiener process Commodity holders are assumed to receive in addition to the price return a convenience yield which does not accrue to the holder of a financial instrument contingent on copper i e a futures contract This convenience yield C is assumed to be proportional to the spot price thus the risk adjusted process for commodity prices may be written as dS r c dt o dz 5 S with r being the risk free interest rate The initial amount of copper reserves is Qmax and the mine produces at a constant rate of q so there are R feasible states of reserves where R O max qAt Also the mine may be open closed or abandoned so there are 3 R states of production The cost of switching between states depends on K K2 and M with K being the cost of closing an open mine K2 being the cost of opening a closed mine and M the annual cost of maintaining a closed mine The mine is abandoned at no cost when market value reaches zero The unit cost of production is A thus the cash flow when the mine is open is CF S q S A 1 where t includes annual income and royalty tax payments In addition there is an annual property tax amounting to a fraction 21 or Ap of market value depending on whether the mine is open or closed When closed the mine has no earnings but incurs in a maintenance annual cost of M 2 3 Extending the Brennan and Schwartz 23 Model Initial applications of the real options approa
170. s section we have that the switching option now depends on three state variables 2 Cortazar and Schwartz 7 is an extension of the Schwartz 3 model for commodity prices and shares some of its good properties like mean reversion while ensuring positive prices Other commodity price models could have been used including square root processes stationary models or general affine models 49 118 G Cortazar et al Computers amp Operations Research 35 2008 113 129 Table 1 Parameter values of the Cortazar and Schwartz 7 three factor commodity price model calibrated using all copper futures traded between 1991 and 1998 at NYMEX Parameters Value 21 0 032 42 0 392 43 0 193 a 1 379 K 2 850 V 0 007 01 0 257 02 0 906 03 0 498 P12 0 215 P23 0 841 P13 0 229 30 25 a E 20 z 5 15 O gt 10 Model Volatility 5 E A Observed Volatility 0 0 5 1 15 2 2 5 3 3 5 4 Maturity years Fig 1 Empirical and theoretical volatility term structure using the Cortazar and Schwartz 7 three factor commodity price model calibrated using all copper futures traded between 1991 and 1998 at NYMEX Even though this model may be solved with traditional finite difference methods is solved much more efficiently using the simulation method shown in the following sections 3 Implementation 3 1 An introduction to the LSM method We propose solving multidimensional problems like
171. s x i e S f x In particular for the three factor Cortazar and Schwartz 7 model used in this paper we have S f x h x with h 1 0 0 21 Also as noted previously the mine may be open closed or abandoned and may switch from one operating state to another incurring in fixed costs Fig 4 summarizes the cash flows of an open mine which will either remain open be closed or abandoned during time t Fig 5 shows the same information but for a closed mine 3 In Section 2 2 the status of the mine open or closed was indicated using the variable j G Cortazar et al Computers amp Operations Research 35 2008 113 129 121 Open Mine Operating Policy Cash Flow at t Value at t At Continue Open CF S 9 V ar x 0 0 qAt V x 0 0 gt Close K MAt Wi a X Q Abandon 0 Veras Wisar 0 Fig 4 Cash flows and value of an open mine as a function of the operating policy Closed Mine Operating Policy Cash Flow at t Value at t At Open CF S 9 Ko Visa Xx 0 0 gAt W x Q _ Continue Closed M At War X Q Abandon 0 Visar Wisar 0 Fig 5 Cash flows and value of a closed mine as a function of the operating policy As described earlier after simulating all price paths from time zero to time T the method requires making optimal decisions starting at time T and then working backwards until time zero is reached The optimal d
172. se focalizar n preferentemente en la problem tica de los fondos de pensiones pero sus resultados impactar n la gesti n de otras carteras de inversi n como las administradas por compa as de seguros y fondos mutuos entre otros Esta modernizaci n se apoyar tanto en el estado del arte metodol gico mundial como en investigaci n cient fica que aborde la problem tica de mercados financieros poco profundos como el nacional con activos que se transan con una baja frecuencia thin markets lo que dificulta el uso de numerosos procedimientos y metodolog as utilizadas en los mercados desarrollados De este modo se pretende 1 apoyar una gesti n m s eficiente de las carteras al incluirse mayor informaci n relativa a retornos y riesgos involucrados 2 hacer un an lisis de estrategias de inversi n que apoye la asignaci n de activos asset allocation 3 apoyar funciones de medici n y gesti n del riesgo y 4 establecer un conjunto de benchmarks para diversas carteras de inversi n Todo lo anterior busca favorecer la gesti n e informaci n para directivos y usuarios y en ultimo t rmino la competitividad y desempe o de la industria Objetivos Espec ficos 1 Generar conocimiento cient fico 2 Realizar desarrollos tecnol gicos 3 Generar informaci n 4 Desarrollar mecanismos de transferencia 5 Formar investigadores y profesionales especializados 8 Objetivos y Resultados No obtenidos No hay 9 Apreciaci n de im
173. sibility of common eigenvalues in matrix K To obtain simpler analytical formulas for the prices of pure discount bonds we impose the condition that all eigenvalues are different but this restriction may easily be relaxed 5 We assume for simplicity that risk premiums are constant but this could be extended to any linear function of the state variables 6 For example see Lund 1994 1997 Duan and Simonato 1999 Geyer and Pichler 1999 Babbs and Nowman 1999 de Jong and Santa Clara 1999 and de Jong 2000 7 For example see Schwartz 1997 Schwartz and Smith 2000 and Sorensen 2002 8 See for example Pennacchi 1991 and Dewachter and Maes 2001 9 An exception is Sorensen 2002 who has applied Kalman filter for incomplete panel data in the commodity markets 0 Cortazar and Schwartz 2003 discuss this issue and propose an alternative approach that does not use the Kalman filter to deal with this problem of missing observations and apply it to commodity futures 11 The state space representation of the generalized Vasicek model is described in the Appendix 12 See for example Oksendal 1998 13 Additional information can be found in Harvey 1989 14 For example under a CIR model the resulting transition equation is also nonlinear See Lund 1994 1997 Duan and Simonato 1999 Geyer and Pichler 1999 and Chen and Scott 2003 15 In this analysis we assume the general case of an incomplete panel data setting
174. sos mercados emergentes Realizar desarrollos tecnol gicos que se expresen en nuevas metodolog as y en herramientas aplicaciones y servicios computacionales que tengan un impacto significativo en el manejo de los recursos de los fondos de pensiones nacionales Este impacto se producir a trav s de a la optimizaci n de las carteras de inversi n de las administradoras sujetas a las regulaciones existentes y b de mejoramientos sist micos a nivel de la industria producto del mayor conocimiento de los riesgos asociados a los diversos activos lo que debiera permitir ajustar regulaciones y restricciones de inversi n de modo de poder limitar las exposiciones al riesgo deseadas incurriendo en un menor costo en t rminos de rentabilidad Generar informaci n en la forma de indicadores de gesti n comparativa que sea objetiva y confiable y que apoye la modernizaci n transparencia y desarrollo del mercado financiero nacional Desarrollar mecanismos de transferencia efectiva de resultados de modo de maximizar el impacto sobre el sistema productivo del pa s y las oportunidades comerciales del proyecto asegurando su sustentabilidad Formar investigadores y profesionales especializados en aplicaciones financieras de alto nivel Una condici n necesaria para sustentar el desarrollo de la industria de las aplicaciones financieras en Chile y de colaborar de ese modo a modernizar el mercado financiero es contar con el capital humano capacitado qu
175. stantaneous spot interest rate which extends Vasicek 1977 This generalized formulation goes back to Langetieg 1980 and is also analysed in Babbs and Nowman 1999 It considers n stochastic mean reverting factors represented by the vector x of dimension n x 1 that define the instantaneous interest rate r r l x 1 The vector of state variables x is governed by the following stochastic differential equation dx Kx dt XEdw 2 where K diag k and X diag o are n x n diagonal matrices with entries that are strictly positive constants and different Also dw is a nx 1 vector of correlated Brownian motion increments such that dw dw Q dt 3 where the i j element of Q is p 1 1 the instantaneous correlation of state variables 7 and j Under this specification the state variables have the multivariate normal distribution and each of them reverts to 0 at a mean reversion rate given by k Thus according to equation 1 the instantaneous interest rate reverts to a long term value given by the constant 6 Note that this is a canonical model in the sense that it contains the minimum number of parameters that can be econometrically identified see Dai and Singleton 2000 By assuming constant risk premiums A the risk adjusted process for the vector of the state variables is dx A Kx dt Edw 4 where is a nx 1 vector of constants Applying standard no arbitrage arguments we obt
176. ste en un m dulo del servicio RiskAmericaPlus cuyo objetivo es asignarle una TIR a cada nemot cnico solicitado El usuario env a v a Web un archivo indicando los nemot cnicos asociados a su cartera devolviendo el sistema la TIR que el modelo le asigna a cada uno La TIR del modelo depende de si el activo fue transado ese d a de cu l es la curva de referencia para el d a la que es actualizada diariamente y de la historia de spreads que este nemot cnico ha exhibido respecto de la curva en el pasado RiskAmerica RiskAmerica Indices Portfolio mmodity Valorizaci n de Carteras Valorizaci n de Carteras Valorizaci n de Carteras B Excepciones Usuario Gonzalo Cortazar Instituci n RISKAMERICA Seleccione el archivo que contiene la informaci n a valorizar Archivo Erowse Fecha 27 03 2007 m Seleccione el tipo de servicio O Servicio TasasMercado O Servicio TasasMercado y Valorizaci n O servicio Valorizaci n Descargar Estructura de Tasas Descargar Archivos Hist ricos Servicio 2 M dulo ndices Este m dulo entrega informaci n referida al comportamiento del mercado financiero Esta descripci n se realiza en t rminos de distintas familias y clases de activos incluy ndose tanto renta fija como variable R skAmeric a RiskAmerica Indices Portfolio mmodity Informaci n Publicaciones Contacto Acerca Responsabilidades DEBE Valores al 23 03 2007 a gt Gr
177. structure Journal of Finance 35 71 97 Laughton D G amp Jacoby H D 1993 Reversion timing options and long term decision making Financial Management 22 225 240 Laughton D G amp Jacoby H D 1995 The effects of reversion on commodity projects of different length In L Trigeorgis Ed Real options in capital investments Models strategies and applications pp 185 205 Westport CT Praeger Journal of Futures Markets DOI 10 1002 tut 268 Cortazar and Naranjo Lund J 1994 Econometric analysis of continuous time arbitrage free models of the term structure of interest rates working paper Aarhus Denmark Aarhus School of Business Lund J 1997 Non linear Kalman filtering techniques for term structure models working paper Aarhus Denmark Aarhus School of Business Manoliu M amp Tompaidis S 2002 Energy futures prices Term structure models with Kalman filter estimation Applied Mathematical Finance 9 21 43 Miltersen K R amp Schwartz E S 1998 Pricing of options on commodity futures with stochastic term structures of convenience yields and interest rates Journal of Financial and Quantitative Analysis 33 33 59 Pearson N D 8 Sun T S 1994 Exploiting the conditional density in esti mating the term structure An application to the Cox Ingersoll and Ross model Journal of Finance 49 1279 1304 Pennacchi G G 1991 Identifying the dynamics of real in
178. t accumulated production reduces unit costs Trigeorgis 25 combines real options and their interactions with financial flexibility McDonald and Siegel 26 and Majd and Pindyck 27 optimize the investment rate and He and Pindyck 28 and Cortazar and Schwartz 29 consider two optimal control variables The ROV approach has been used to analyze uncertainty on many underlying assets including exchange rates 30 costs 31 and commodities 32 Real asset models have included natural resource investments environmental new technology adoption and strategic options among others 32 35 Recently real options analysis is gradually advancing into the domain of strategic management and economic organization Bernardo and Chowdry 11 analyze the way in which the organization learns from its investment projects A related model is presented in 36 They study the choice between a small and a large project where choosing the small project allows one to re invest later in the large project Lambrecht and Perraudin 37 introduce incomplete information and preemption into an equilibrium model of firms facing real investment decisions Miltersen and Schwartz 38 develop a model to analyze patent protected R amp D investment projects when there is imperfect competition in the development and marketing of the resulting product Finally Murto et al 39 present a modeling framework for the analysis of investments in an oligopolic market for a homogenous comm
179. te curve a formulation with state variables Journal of Financial and Quantitative Analysis 34 1 131 157 Dewachter H Maes K 2001 An admissible affine model for joint term structure dynamics of interest rates Working Paper Katholieke Universiteit Leuven Duan J C Simonato J G 1999 Estimating and testing exponential affine term structure models by Kalman filter Review of Quantitative Finance and Accounting 13 2 111 135 Duffie D Kan R 1996 A yield factor model of interest rates Mathematical Finance 6 379 406 Duffie D Singleton KJ 1997 An econometric model of the term structure of interest rate swap yields Journal of Finance 52 4 1287 1321 Fisher M Nychka D Zervos D 1994 Fitting the term structure of interest rates with smoothing splines Working Paper Federal Reserve Board of Governors Geyer ALJ Pichler S 1999 A state space approach to estimate and test multifactor Cox Ingersoll Ross models of the term structure Journal of Financial Research 22 1 Hamilton JD 1994 Time Series Analysis Princeton University Press Princeton NJ Harvey AC 1989 Forecasting Structural Time Series Models and the Kalman Filter Cambridge University Press Cambridge Heath D Jarrow R Morton A 1992 Bond pricing and the term structure of interest rates a new methodology for contingent claims valuation Econometrica 60 77 105 Ho TSY Lee S 1986 Term structure movements and the pricing of interest rate contingent claims Jour
180. terest rates and inflation Evidence using survey data Review of Financial Studies 4 53 80 Ross S A 1997 Hedging long run commitments Exercises in incomplete market pricing Economic Notes by Banca Monte 26 99 132 Routledge B R Seppi D J amp Spat C S 2000 Equilibrium forward curves for commodities Journal of Finance 55 1297 1339 Schwartz B S 1997 The stochastic behavior of conmadliis prices Implica tions for valuation and ee The Journal of Finance 52 923 973 Schwartz E S amp Smith J E 2000 Short term variations and long term dynamics in commodity prices Management Science 46 893 911 Schwarz EV amp Szakmary A 1994 Price discovery in petroleum markets Arbitrage cointegration and the time interval of analysis Journal of Futuros Markets 14 147 167 Serletis A 1992 Unit root behavior in energy futures prices Energy Journal 13 119 128 Sorensen C 2002 Modeling seasonality in agricultural commodity futures Journal of Futures Markets 22 393 4206 Vasicek O 1977 An equilibrium characterization of the term structure Journal of Financial Economies 5 177 188 Working H 1949 The theory of price of storage American Economic Review 39 1254 1262 Available online at www sciencedirect com ome e COMPUGELrS ScienceDirect operations les research ELSEVIER Computers amp Operations Research 35 2008 113 129 www elsevier com l
181. the extended Brennan and Schwartz model using the LSM method To illustrate the LSM method proposed in Longstaff and Schwartz 19 we consider throughout this section a very simple copper mine that may extract all available resources instantaneously at any moment during the concession period Also copper prices are considered in this section to follow a one factor model In the next section we will show how to implement the extended Brennan and Schwartz three factor model Consider a simplified copper mine in which all reserves Q may be instantaneously extracted at any point in time incurring in a unit production cost of A The copper spot price S is assumed to follow a one factor geometric Brownian motion dS r c dt adz 14 Sr with r the risk free interest rate and c the convenience yield G Cortazar et al Computers amp Operations Research 35 2008 113 129 119 The method starts by simulating a discretization of Eq 14 Si 1 r c At S 1 S 104 Ate 15 with Ar the time interval in years and e a random variable with a standard normal distribution Then Eq 15 is simulated through time obtaining a price path The process is repeated N times and a price matrix S with N price paths over a time horizon T is obtained Like in any American option valuation procedure the optimal exercise decision at any point in time is obtained as the maximum between the immediate exercise value and the expected cont
182. the three factor one in fitting the volatility term structure CONCLUSIONS This article studies the ability of an N factor Gaussian model to explain the stochastic behavior of oil futures prices when estimated with the use of all available price information In recent years oil futures markets The authors thank the referee for pointing this out Journal of Futures Markets DOt 10 1002 fut HR iai i Oil Futures Prices 265 have included new futures contracts with longer maturities that do not have historical prices In addition not all futures contracts trade every day To include all data without discarding or aggregating prices a Kalman filter estimation procedure that allows for a time dependent number of observations is used The model is calibrated with the use of all daily light sweet crude oil futures prices traded at NYMEX during the 10 year period from January 1992 to December 2001 The model is estimated with the use of one two three and four factors for the three different panels from 1992 to 2001 from 1992 to 1996 and from 1997 to 2001 It is found that most parameter estimates are significant and stable across different panels as opposed to the long term growth rate and most risk premium parameters which are not Moreover out of sample errors for the 2002 2004 period are similar to in sample errors supporting the stabil ity of the model In addition the model also performs well for copper futures Empir
183. the volatility term structure in Panel B 1992 1996 fits the The authors thank the referee for this comment r ae 7 Journal of Futures Markets DOI 10 1002 fut 1007 266 1 O pued pue 9661 7661 4 Pued 1007 7661 Y pued u HENUOS yara 10 sumpa saan Ny Jo AUpHeOA joudu ay pue Ppoul 4983 10 SUINIAS saunyny Jo sunjonsys Wd JNELOA EIMA 24 syuasaad saanBy 94 SUINA SIAMINY LO Jo 2INIHNAIS Wa Anyreyon 23uNDid s3024 ume s1634 Anane py L 9 5 t 7 1 0 20 0 oz 01 r 2 E or 07 09 POOL 08 oe 3 usd g puvd tero Y jeutg NDEO Cortazar and Naranjo three factor model very well it does not perform well in Panel C 4 1997 2001 as it overestimates the short term volatility This translates into an overestimated short term volatility for Panel A as it is the union of Panels B and C On the contrary the four factor model closely fits the 3 empirical volatility term structure across all panels 3 A possible explanation for the failure of the three factor model to fit the volatility term structure might be a structural break in the volatility of oil futures returns during the 1992 2001 period One way to test this i hypothesis is to compare model estimates between the two different peri ods 1992 1996 and 1997 2001 This could easily be done by comparing the likelihood functions and seeing whether this difference is significant One may be concerned however t
184. thod Mo Volatility Structure of Interest Rates 1997 2001 AS A 12 Svensson Method Volatility Structure o A Empirical Volatility from Bond Yields 10 8 6 Volatility 4 2 0 1 5 3 5 5 5 7 5 9 5 11 5 13 5 Maturity Years Figure 3 Empirical volatilities of interest rates in Chile and volatilities obtained from daily estimations of the term structure between 1997 and 2001 using the Svensson 1994 method Copyright 2007 John Wiley amp Sons Ltd Int J Fin Econ 11 000 000 2007 DOI 10 1002 ijfe IJFE 317 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 JFE 317 TERM STRUCTURE ESTIMATION 5 volatilities is not consistent with mean reversion in interest rates it implies very high volatilities for long rates Moreover the Svensson volatility estimates are much higher than the empirical estimates obtained directly from bond prices suggesting that missing observations induce unreliable rate estimates Similar results are obtained when using other curve fitting methods like Nelson and Siegel 1987 3 THE GENERALIZED VASICEK DYNAMIC TERM STRUCTURE MODEL As was shown in the previous section traditional static term structure estimation only incorporates current bond price or yield observations without regard to past information When long term bond prices are not available th
185. time series of bond prices Chan et al 1992 Broze et al 1995 Brenner et al 1996 Nowman 1997 1998 Andersen and Lund 1997 Alternatively state variables and parameters may be estimated from a panel of bond prices with different maturities Chen and Scott 1993 Pearson and Sun 1994 Duffie and Singleton 1997 Even though there are obvious benefits of calibrating a model using a panel with a large number of price observations the richer the data set the larger the estimated measurement errors These errors arise from the inability of a model with a limited number of factors to perfectly explain a large number of contemporaneous prices A powerful and widely used methodology to optimally estimate unobser vable state variables from a noisy panel data is the Kalman filter Recent applications of this methodology to dynamic models of interest rates include Lund 1994 1997 Ball and Torous 1996 Duan and Simonato 1999 Geyer and Pichler 1999 Babbs and Nowman 1999 and Chen and Scott 2003 The advantage of using the Kalman filter on a panel data is that it jointly uses all present and past price information Maximum likelihood methods can then be used to estimate the parameters of the model Both type of methods proposed in the literature curve fitting for estimating the current term structure and Kalman filtering for dynamic models have been successfully applied to markets for which there is a sufficiently complete price data set
186. tion in its simplest specification boils down to computing the expected continuation value function N Gy x 40 X ECS 31 i l where E S is the expected spot price under the risk adjusted measure i e the future price Our tests show that using this reduced base specification we can obtain similar valuation accuracy in a simpler way than using polynomials of state variables For example we solved a three factor European option with known analytic solution with two alternative implementations of the LSM approach Chebyshev functions and futures prices Fig 9 computes the RMSE as a function of the number of regressors showing that using futures requires less regressors for any giver error level Using less regressors for estimating the continuation function has many computational benefits including reducing CPU processing time which could be critical for high dimensional implementations For example we performed another test solving the extended three factor price model Brennan and Schwartz mine obtaining valuations within 1 for both LSM implementations while calculation time increased with the number of regressors as shown in Fig 10 These results suggest that if calculation time is an issue it is worth exploring alternative implementations of the LSM approach G Cortazar et al Computers amp Operations Research 35 2008 113 129 127 450 400 350 300 250 200 150 100 Time Index
187. ue se especializan en finanzas a nivel de postgrado Fortalecimiento de Relaciones con Sector Productivo Fortalecimiento de las relaciones de Cooperaci n Internacional En proceso de obtenci n Incremento en los fortalecimientos institucionales anteriores e Ambiental Obtenido No existen En proceso de obtenci n No existen Plan de trabajo El Plan de trabajo se estructur en torno al desarrollo de 5 subtemas que se denominaron PortfolioValue PortfolioBenchmarks PortfolioRisk RiskMatrix AssetAllocation todos los cuales en conjunto reciben la denominaci n RiskPortfolio Estos subtemas se estructuraron como subproyectos de I amp D dando origen a Tesis de Mag ster y Memorias de T tulo Presentaciones a Congresos Publicaciones M dulos computacionales y finalmente 3 servicios SVC Indices y Portfolio Las instituciones declaran que el plan de trabajo que representa las actividades del proyecto se encuentra en el ANEXO 2 de este informe Infraestructura y bienes adquiridos por el proyecto Las instituciones beneficiarias declaran tener inventariados todos los bienes adquiridos por el proyecto y declarados en ANEXO 4 de este Informe los que est n a cargo de personal de la instituci n y se encuentran asignados a las unidades institucionales que se indican en ese Anexo 11 Plan de Continuidad Las instituciones se comprometen a a La mantenci n y consolidaci n de las l neas de investigaci n asociadas al pr
188. uropean Journal of Operations Research 2004 157 2 486 500 Paddock J Siegel D Smith J Option valuation of claims on physical assets the case of offshore petroleum leases Quarterly Journal of Economics 1988 103 3 479 508 Cortazar G Casassus J Optimal timing of a mine expansion implementing a real options model The Quarterly Review of Economics and Finance 1998 38 755 69 2 Smith J McCardle K Valuing oil properties integrating option pricing and decision analysis approaches Operations Research 1998 46 198 217 Smith J McCardle K Options in the real world lessons learned in evaluating oil and gas investments Operations Research 1999 47 1 15 Lehman J Valuing oilfield investments using option pricing theory SPE Hydrocarbon Economics and Evaluation Symposium Proceedings 1989 p 125 36 Trigeorgis L A real options application in natural resource investment Advances in Futures and Options Research 1990 4 153 64 Laughton DG Jacoby HD Reversion timing options and long term decision making Financial Management 1993 22 3 225 40 Laughton DG Jacoby HD The effects of reversion on commodity projects of different length In Trigeorgis L editor Real options in capital investments models strategies and applications Westport Praeger Publisher 1995 Cortazar G Schwartz ES Implementing a real option model for valuing an undeveloped oil field International Transactions in Operational Research 1997 4 2 125 37 Cortazar
189. v 0 qa xo 0 Closed to Abandon MAt Gw o 1 X 0 Table 7 Restrictions on initial state variables and parameters of the Cortazar and Schwartz 7 model to induce a one factor price process similar to the Brennan and Schwartz 23 model Cortazar Schwartz model Brennan Schwartz model Yo 2 K vo yo A3 j a 1 vo yo r c la N 23 0 a 1 K 1 y 0 01 oO 02 0 03 0 Pr 0 P23 x0 p13 0 Table 6 shows how to find the critical state variables to close an open mine to open a closed mine or to abandon from an open or from a closed mine 4 Results 4 1 Results for the one factor Brennan and Schwartz 23 model In this section we validate our proposed approach by applying it to the one factor Brennan and Schwartz 23 real options model and comparing the results to those originally reported using traditional finite difference methods A simple way of validating our approach is to see the one factor price process as a particular case of the more general three factor process In this way by restricting some parameter values we can perform a better test on the algorithm by using the same computer program to solve both models Table 7 shows how the Cortazar and Schwartz 7 three factor model may be restricted to behave as the one factor model used in Brennan and Schwartz 23 The simulation program computed 50000 price paths assuming a maximum extraction time of 50 years with three opportunities per y
190. value be the maximum among the alternatives and the final value at time using actual realizations of the price simulation instead of expected values to avoid biases due to the Jensen s inequality at time 1 Table 3 shows the same information but when the mine is initially closed This procedure is repeated from t T 2At until t At At t At mine values are averaged over all price paths to provide an initial estimate of the expected continuation value for the mine 1 Ss A Gy 9 qAr 1 0 X F PD Va x 0 Q gAtye A Ar es w 1 i Gw 0 0 8 gt War alo Qe HM en o 1 Tables 4 and 5 show the initial mine values depending on the initial status and operating policy of the mine Finally to determine the optimal operating policy the method must find the critical state variables x which equate expected present values for different operating decisions G Cortazar et al Computers amp Operations Research 35 2008 113 129 123 Table 5 Closed mine values as a function of the initial operation decision Open Wo x O K CF So q Gy 9 gar 1 0 Continue closed Wo x Q MAt Gw o r 0 x Abandon Wo x 0 0 Table 6 Conditions to determine critical state variables x for switching mine operation Open to Closed CF x q Gy o qAra X K MAt Gw o t x Closed to Open MAt Gw o 1 X K2 CF x q Gy o qart X Open to Abandon CF x q G
191. value of a closed mine as a function of the operating policy Expected value Optimal decision Realized value K2 CF S q Gr g gar e X 0 Open W x 0 Q K2 CF S 0 q Viar x O Q Arye MAt Gw o r x Continue closed W x Q MAt W 4a1 x Q e 070044 0 Abandon W x Q 0 Table 4 Open mine values as a function of the initial operation decision Continue open Vo x Q CF So q Gy o gart 0 X Close Vo x 0 K MAt Gw o r 0 x Abandon V x Q 0 Thus the expected continuation value at time t T 2Ar as a function of the price state vector x may be computed For example the value of an open mine with Q units of resources conditional on the state vector x would be M Gv or20100 dy 9 7 2a toa 27 j l Given that we can compute the expected continuation value we are now able to obtain the optimal operating decisions by maximizing current cash flows plus the present value of expected continuation values For example when the mine is open there are three available operating alternatives to continue open to close down operations or to abandon the mine Adding current cash flows to discounted expected continuation values for each of the three alternatives the decision maker may choose the best course of action Table 2 shows for each of the three alternatives the expected present value at time the optimal decision should this expected present
192. vet M Urzua J 2005 The Valuation of Multidimensional American Real Options using the LSM Simulation Method 9th Annual International Conference Real Options Theory Meets Practice Real Options Group and EDC Paris Paris June 23 25 http www realoptions org AcademicProgram academicprogram2005 html Cortazar G Schwartz E S Naranjo L 2004 Term Structure Estimation in Low Frequency Transaction Markets A Kalman Filter Approach with Incomplete Panel Data March 2004 EFA 2004 Maastricht Meetings Paper No 3102 Maastricht August 18 21 http ssrn com abstract 567090 2 INVESTIGACI N Y DESARROLLO REALIZADA POR EL PROYECTO Durante el desarrollo del proyecto se realizaron m ltiples investigaciones cient ficas las que fueron luego incorporadas a diversas aplicaciones computacionales En lo que sigue se enumera una serie de Temas o Problem ticas que dieron origen a resultados cient ficos originales que quedaron documentados Para cada Tema se incluye una breve descripci n del problema y resultado y c mo stos quedaron documentados en t rminos de Publicaciones Documentos de trabajo Tesis de Mag ster Memorias de T tulo y o Presentaciones en Congresos Acad micos Internacionales Tema 1 Modelaci n y Calibraci n de Procesos Estoc sticos para Precios de Instrumentos en Mercados con Paneles de Datos Incompletos Discusi n Se perfecciona metodolog a cuya investigaci n se inici en fecha anterior al inicio
193. vo 8 Otros definir pa 0 9 pee PRE E MANO DE OBRA NO CALIFICADA Moneda Extranjera 7 Tipo de cambio E Para todos los productos y todas las Unidades de Negocio osto total anual osto total anual Costo mensual Costo total mensual Moneda Nacional Moneda Extranjera N Puestos Cargos unitario MM MM MM Obreros y Jornaleros Otros definir A A OE ee INSUMOS PARA LA PRODUCCION Moneda Extranjera L 3 Tipo de cambio EH Para todos los productos y todas las Unidades de Negocio osto total anua osto total anua Insumos para la Tipo de unidad a Costo Pa Cantidad mensual Costo total mensual Moneda Nacional Moneda Extranjera producci n considerar Unidades MM MM MM Insumos de Oficina 500000 Promedio e a BIENES DE CAPITAL Moneda Extranjera AA Tipo de cambio OA Para todos los Productos y todas las Unidades de Negocio osto total anual osto total anual Bienes de Tipo de unidad a Moneda Nacional Moneda extranjera En considerar Cantidad Costo Unitario MM MM N computadores 500000 4 Mantenci n Oficinas 1000000 10 0 0 AA AAA AAA A O OTROS COSTOS Moneda Extranjera IO Tipo de cambio OA Para todos los Productos y todas las Unidades de Negocio osto total anual osto total anual Tipo de unidad a Moneda Nacional Moneda Extranjera Otros costos considerar Cantidad Costo Unitario MM MM NN a ca A a TOTAL COSTOS ANUALES
194. way to estimate the spot price is to set T t in Equation 17 The price of the closest to maturity contract and the theoretical spot price derived from the four factor model are similar but not identical as shown in Figure 1 The data could also be divided with alternative criteria like structural breaks in oil prices Serletis 1992 Journal of Futures Markets DOI 10 1002 tut Cortazar and Naranjo 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 FIGURE 1 Time series of the oil spot price The figure displays the historic evolution of the theoretical oil spot price calculated from the four factor model with the use of parameter estimates from Panel A and compares it with the closest to maturity futures contract from January 1992 to December 2004 The spot price although very similar is not identical to the closest to maturity futures contract Parameter Estimation The model is estimated in Table H with the use of one two three and four factors for three different panels Panel A from 1992 to 2001 Panel B from 1992 to 1996 and Panel C from 1997 to 2001 The remaining data are used later for robustness tests of the model The Kalman filter described earlier is used to estimate unobserved state variables and parameters are obtained by maximizing the likelihood function of futures price innovations Parameter estimates are shown in Table 11 with standard errors in parentheses From this table
195. wo and three factor models without extending its use to a general N factor setting In this model the spot price process of the commodity can be described as log S Fx pt 1 where x isan X vector of state variables and p the long term growth rate is a constant The vector of state variables x follows the process dx Kx dt 2 dw 2 where 0 0 0 a 0 0 k 20 o md yal 0 0 0 K 0 0 T are n X n diagonal matrices with entries that are positive constants Also dw isan X 1 vector of correlated Brownian motion increments such that dw dw Q dt where the i j element of Q is p 1 1 the instantaneous correlation between state variables i and j This model specification implies that the state variables have a mul tivariate Normal distribution The frst state variable follows a random walk inducing a unit root in the spot price process Each of the other state variables reverts to zero at a mean reversion rate given by k In Journal of Futures Markets DOI 10 1002 fut Ltd S Affine i 011 Futures Prices order to compare the present model to traditional models found in the commodities literature x has been exogenously set to zero By assuming a constant risk premium A the risk adjusted process for the vector of state variables is j dx A Kx dt Xdw 3 where Aisan X 1 vector of real constants Instead of modeling the risk free interest rate and the
196. x t A T t l il E2 x T i e7 i T 1 33 e TOs t gt 1 2 N oT t i j 1l de Cov2 x T x T ETT i J 34 OP H 1j l TT i j The valuation formula 17 is obtained by inserting Equations 33 and 34 into Equation 30 BIBLIOGRAPHY Babbs S TL amp Nowman K B 1999 Kalman filtering of generalized Vasicck term structure models Journal of Financial and Quantitative Analysis 34 115 130 Bessembinder IL Coughenour J F Seguin P J amp Smoller M M 1995 Mean reversion in equilibrium asset prices Evidence from the futures term structure Journal of Finance 50 361 375 Bessembinder H Coughenour J F Seguin P J amp Smoller M M 1996 Is there a term structure of futures volatilities Reevaluating the Samuelson hypothesis Journal of Derivatives 4 45 58 Brennan M J 1958 The supply of storage American Economic Review 48 50 72 Brennan M J 1991 The price of convenience and the valuation of commod ity contingent claims In D Lund amp B ksendal Eds Stochastic models and option values pp 33 71 Amsterdam Elsevier Science Brennan M J amp Schwartz E S 1985 Evaluating natural resources invest ments Journal of Business 58 135 157 Casassus J amp Collin Dufresne P 2005 Stochastic convenience yield implied from commodity futures and interest rates Journal of Finance 60 2283 2331 Chen R R amp
197. y Even though observed prices indicate that markets seem to have behaved similarly on both dates the model estimates that the yield of a 19 year coupon bond changed by almost 1 in a day The extrapolated 19 year yield is clearly inaccurate Curve fitting methods provide unstable estimates of long rates when no long term bonds are traded Instability of term structure estimates can be measured by comparing the volatility term structure implied by the model with the empirical volatilities obtained from the time series of yields It is well known that the term structure of volatilities is downward sloping due to mean reversion in interest rates This means that the volatility of long rates obtained from the model should be lower than the volatility of short rates Figure 3 plots the volatility of interest rates calculated from daily estimations of the term structure in Chile between 1997 and 2001 using the Svensson 1994 method It can be seen that this term structure of Bond Yields 10 06 1999 8 6 4 A Observed Bond Yields Yield S xe Svensson Method Bond Yields e Previous Day Observed Bond Yields 2 Previous Day Svensson Method Bond Yields 0 T T T 0 5 10 15 20 Maturity Years Figure 2 Coupon bond yields for two consecutive dates 10 05 1999 and 10 06 1999 estimated from Chilean government inflation protected discount and coupon bond data using the Svensson 1994 me

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