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Portfolio Probe User`s Manual

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1. 08 4 9 Count Constraints 0 000 0 00000000 eee Teese eee Tere eee Alternative prices Multiple distance 45 CONTENTS 6 Wiis amp Seeger n Vea te i SS as ore A ee ee oaks penta oe me ce eh ro ee eee erario gh ch th ae omer E pia ance a Looe Rae Ro ee ee oA echt are aaa awd ea Beas hh Eon ee Boh eee eee ee 4 18 Constraint Penalties and Soft Constraint 5 Details of Random Portfolio Example 5 1 Performance Measurement Statiqd 5 2 Testing the Effect of Constraint Boundy Static Util Ge Ree rr a Ee ee G pane be eee ee eae ee eed Dynamic Utility o a sa a de eo a we ad 5 3 Test a Risk Mode Ex Ante Volatility Comparisonj 008 4 Ex Ante versus Realized Volatility 5 4 Performance Measurement Shadowing 5 0 Evaluating a Trading Strategy o 5 6 Going Farthed Optimizing Long Only Portfolio 6 1 Required Inputg ee eS eee ae eee rl i face Hse cee Sede Sp os hp Gp da te A de Se bo he amp ae 6 2 Examples for Passive Portfoliog 6 3 robe do io Examples for Active Portfolios a a The Information Ratio with a Tracking Error Constraint TON IAEA Tae eee yd Baas aia E ee E Gee dada 6 4 Utility free Optimi
2. gt varcon ml lt rbind c 1 2 1 4 c 1 3 1 5 gt varcon ml Ll 3 ay 1 2 1 4 tad LE 16 Suppose that there are three columns in vtable perhaps the variance argu ment is a three dimensional array with 3 slices in the third dimension Then the command var constraint varcon mi would say that the first variance benchmark combination is restricted to be between 1 2 and 1 4 the second variance benchmark is restricted to be between 1 3 and 1 5 and the third is unrestricted Now let s add some row names gt varcon m2 lt varcon mi dimnames varcon m2 lt list c 2 0 NULL varcon m2 E tl G2 2 1 2 1 4 o 1 3 1 5 NE NE The command var constraint varcon m2 says that the third variance benchmark is to be between 1 2 and 1 4 the first variance benchmark is restricted to be between 1 3 and 1 5 and the second variance benchmark is unconstrained Some examples with multiple variances can be found on page 153 14 4 Expected Return Constraints This section discusses the argument e alpha constraint 148 CHAPTER 14 ADVANCED FEATURES Simple use of this argument is discussed on page 69 Use of this argument is analogous to the advanced use of var constraint page 45 except that a one column matrix is a lower bound rather than an upper bound and indices refer to expected return benchmark combinations columns of atable rather than to variance benchmark combinations So for instance alpha
3. gt do call rbind randport eval randport4 FUN function x summary x number of assets S code note The command gt do call rbind some list is equivalent to the command gt rbind some list 1 some list 2 some list length some list Summary The summary method for random portfolios shows how many assets are in the portfolios and the number of times each asset appears in a portfolio gt summary randport5 port size 7 8 9 10 1 32 165 802 count assets stockC stockD stockA stockB stockE stockF stockI 1000 1000 987 975 973 972 972 stockG stockH stockJ 968 961 960 This shows us that out of the 1000 portfolios 802 contained all 10 assets 165 had 9 assets 32 had 8 assets and 1 had 7 assets We also see that stockC and stockD were both in all of the portfolios while stockJ was only in 960 of them 40 CHAPTER 3 GENERATING RANDOM PORTFOLIOS 3 4 Exporting Random Portfolios The deport function will write files containing the result of random portfolio The simplest use is gt deport randport2 1 randport2 csv This writes a comma separated file where the columns each correspond to one of the assets that appear in the object and the rows correspond to the portfolios or trades There are arguments that allow you to switch the meaning of rows and columns and to give a universe of assets which must include all of those appearing in the object See the help file for details Wri
4. 2 2 a Portfolio Problema a 101 CONTENTS 7 Creating Matriceg oaa a a a 129 SN N A at ak ee ee 129 131 131 11 1 Special Instruction 1 Long only when shorts exist 11 2 Special Instruction 2 Benchmark in long short optimization 131 133 133 ace et amp amp oa eer say ene sane on ah ee 134 A Sy Sec nat anda E cas 135 Ed Sh ee aa pa vo a adh ieee E 136 12 5 Testing Optimization Quality 136 13 Utility 139 A a ee 139 a E ee 140 EII ooo pe eee eee eee 141 Eb ao II dn ony aR ed 141 amp Diaries Sook hate odeeea arene Oi 141 a aaa aaa 142 ANA S a a A 142 143 oh ee BD E OR 143 dado ud a 144 cial ed a ad 145 A ahd 3 Aan 147 HO A a 148 DA PS a ee eee ey ee e y 149 ee We dd es on ee Gee dd dd 150 eee eh ee se ee y 150 152 II aa 153 O E 154 hit Dr a a dd dee 155 A RE 156 A E 161 LLG oR Ree ee eee ee 161 163 ee eee ee 163 A ae oe aiden Ge 163 15 3 Writing C or C Codage ca aa amp eke e a Bee ws i 163 A ee nee GO 164 CONTENTS List of Tables 2 1 Simple analyses of the MACD funds 14 2 Utilities codes for the utility table 10 LIST OF TABLES List of Figures ee ee ee 16 ataca E 17 o 20 oi 21 ina a ad 22 a da 23 r i i 23 25 25 26 26 27 2 11 PE measurement 200 turnover for 2007 bd 28 2 12 Performance measurement 200
5. If Portfolio Probe was installed in the default place then Portfolio Probe is loaded into an R session with gt library PortfolioProbe It is possible that the lib loc argument to library may be required In S PLUS it will depend on the particular installation but something similar is likely 1 5 ROAD MAP 15 1 5 Road Map We suggest that there is one of two frames of mind that you are likely to have e Conceptual primarily wanting to understand the task e Operational primarily wanting to do the task With the choice of two tasks that produces four possible routes through the document FigurdLT lis a graphical view of the suggestions Note that not all chapters appear in the maps A lot of the document can be reserved for reference as the need arises In particular if you are only generating random portfolios you can safely ignore several of the chapters If you are only interested in utility free optimization then Figure _2 shows a possible route through this document This supposes that you will impose a turnover constraint rather than providing trading costs for the assets 1 6 Typography Conventions Computer commands or pieces of commands are written in this font For example a variance argument is written as variance whenever it is the argu ment itself that is referred to Entire commands are written in the same font Commands are introduced by gt which is the prompt in the S language so that any ou
6. Linear constraints are used when there is aggregation across assets If a constraint only involves one asset at a time then there are other constraints that are more expedient for instance max weight or threshold Building Constraints To impose linear constraints you must specify at least two arguments you can accept default values of other arguments These two key arguments are e lin constraints e lin bounds The lin constraints argument gives the numbers or categories for each asset for each constraint lin bounds provides the lower and upper bound for each sub constraint Because the functions are picky about these arguments it is best to create them with the build constraints function The build constraints function takes a vector a matrix a factor or a data frame containing the information for the lin constraints argument The vector or matrix can be character numeric or logical A data frame is required if you have a mixture of types In this example we are giving a character matrix with two columns one for countries and one for sectors gt cons obj lt build constraints cbind country countryvec sector sectvec gt names cons obj 1 lin constraints bounds 4 8 LINEAR CONSTRAINTS 59 The result of build constraints is a list with two components The first component lin constraints is suitable to give as the lin constraints argu ment the bounds component is the template for an object to g
7. e upper trade max weight One typical use of the max weight argument is max weight 0 05 This limits the maximum weight of each asset to 5 of the gross value of the portfolio The weight is the absolute value of the monetary value of the position divided by the gross value So even in long short portfolios the absolute weights always sum to 1 The other typical use is to give max weight a named vector that states the maximum weight for each asset Assets that are not named are not restricted Example If you want to allow a few assets to have larger weights then you can create a vector of all the pertinent assets Suppose you want almost all assets to have weight no more than 5 and a few to have weight no more than 7 5 Then you could create the appropriate vector by gt maxw spec lt rep 0 05 length length prices gt names maxw spec lt names prices gt maxw spec spec assets lt 0 075 The maxw spec vector would then be given as the max weight argument A related argument is the control argument enforce max weight When this is TRUE the default and if there are positions in the existing portfolio that will break the maximum weight constraint then forced trades are created to make them conform to the maximum weight caution There are circumstances in which the max weight argument does not guar antee that the maximum weight in the final portfolio is obeyed if the weight is too large in the existing
8. ylim c 0 100 xlab Theoretical quantiles ylab Percentile gt abline 0 1 col blue You can see what such a plot would like with a purely random strategy with commands like gt plot ppoints 20 100 sort sample 0 100 20 replace TRUE gt abline 0 1 col blue 5 6 Going Farther e You can modify the scripts to use your own data e You can modify the functions that the scripts use Chapter 6 Optimizing Long Only Portfolios This chapter discusses commands for optimizing long only portfolios It applies to both active and passive portfolios The examples in this chapter are just sketches of what you are likely to want to do in practice Chapter 4 discusses all the constraints that may be imposed Some of the examples limit the number of assets that are held in the portfolio or traded These limits are convenient but threshold constraints page D2 may be better to use Chapter J covers trading costs it is usually a good idea to constrain trading in some way whether with trading costs or a turnover constraint 6 1 Required Inputs The trade optimizer function needs a price vector The prices argument needs to be a vector of prices with names that are the asset names The assets in prices control the assets that are used in the problem Other inputs such as the variance and linear constraints may contain additional assets which will be ignored Other than prices there is no single argum
9. Ltl 1 2 variance 0 1 benchmark i utility only 1 1 attr benchmarks 1 we wee Here are the tables from an optimization with the same variance and expected returns but with two benchmarks gt opt2 atable Cot 1 21 La 0 4 Lel 06 alpha 0 1 2 0 1 2 benchmark 8 8 8 9 9 9 attr benchmarks i Indexi Indexi Indexi spr spz spx gt opt2 vtable 1 2 G3 L 4 variance 0 1 0 al benchmark 8 8 9 9 utility only 1 1 1 1 attr benchmarks 1 Indexi Iudexi spx spz note The values in the alpha table variance table and utility table are zero based because these entities are sucked directly into C where indexing is zero based rather than one based as in the S language 143 Variance Constraints This section discusses the argument e var constraint Simple use of this argument is discussed on page 145 This argument also handles problems that are too complex for the bench constraint argument The var constraint argument can be 146 CHAPTER 14 ADVANCED FEATURES e a plain vector with one or more values with or without names e a one column matrix with or without row names e a two column matrix with or without row names Plain vectors are equivalent to one column matrices One column matrices are equivalent to two column matrices where the first column is all zeros The names on vectors or row names of matrices specify the zero based index of the column of the variance tab
10. S code note The names given inside the c function need to be put in quotes because they are not proper names of S objects S would try to interpret them as numbers if they were not in quotes If any element of the var constraint argument is named then they all need to be When there are no names the columns of the variance table are used in order A standard use of multiple variances is to optimize with one and check the tracking error or volatility with the other One good reason to evaluate the solution on a variance not used in the optimization is because of the bias created in the optimization process the tracking error or volatility that is achieved in the optimization is a downwardly biased estimate of what will be realized Using multiple variances in the optimization will reduce the amount of bias in any particular variance matrix though some bias will still remain Multiple Time Periods You may have forecasts for multiple time periods For example you may have GARCH forecasts of the variance matrix for several different time horizons You want the portfolio to do well at all of the time scales A key issue is how to weight the different time scales You may want different risk profiles at different time scales for instance expected returns may be given relatively more weight versus risk for short time horizons 14 7 MULTIPLICITY EXAMPLES 155 Credit Risk The variance matrices that we have talked about have all been
11. and it is as long as you pay attention to the details gt cbind A c 1 2 A 4 1 1 2 2 gt cbind A 1 2 A HEE Both of these forms can be useful but they give matrices that are oriented in different directions Debugging Burns 2009 contains a slightly expanded introduction to debugging and there are a number of additional sources Here is something to get you started Suppose we do something and get an error gt update soft op1 expected return new alpha Error in trade optimizer prices priceten variance varten Object new alpha not found 130 CHAPTER 10 PRACTICALITIES AND TROUBLESHOOTING Now in this instance it is clear that the problem is the new alpha that we tried to use Perhaps we mistyped the name or we haven t created it yet But the error could be very much more obscure it could be unable to find an object that we ve never heard of and possibly in a function that we didn t know we were using You can always do the following to get a sense of where the problem hap pened gt traceback 5 trade optimizer prices priceten variance varten eval expr envir enclos eval call parent frame update default soft op1 expected return new alpha update soft op1 expected return new alpha PNW Pp Sometimes just seeing the traceback can be enough to understand what went wrong Other times no For those other cases we can actually go into any of those function calls and
12. benchmarks lt c spx We will also need to create a utility table but in this case it is simple enough to create on the fly within the call to the optimizer Our optimizer call can look something like gt opt buav lt trade optimizer prices varian expected return alphas benchmark spx Ss var constraint c 1 100 vtable vtab buav utable rep c 0 1 c 3 3 We can check the tables that are created to see if it is doing what we want gt opt buav atable Lat alpha 0 benchmark 500 attr benchmarks i epr gt opt buav vtable Ltd 2 variance 0 0 benchmark 500 f utility only 1 0 attr benchmarks H apar e gt opt buav utable 1 alpha spot 0 variance spot 0 14 7 MULTIPLICITY EXAMPLES 153 destination 0 opt utility 1 risk aversion 1 wt in destination 1 The utility table we created says to maximize the information ratio We need to put a name on the object given as the variance constraint because we want to constrain the variance benchmark combination given in the second column of the variance table not the first column Rival Variance Forecasts If you have two or more variance matrices you can use these in a single op timization For example if you have a statistical factor model a fundamental factor model and a macroeconomic factor model you could use all three in the optimization Since these are created using different sources of information
13. long only TRUE gross value 1 4e6 The amount put into the portfolio is 1 4 million currency units The simplicity of this command means that it may not do what you want since it is dependent on default values for most arguments Minimize Tracking Error We can minimize the variance relative to a benchmark that is minimize track ing error To be concrete let s assume that the benchmark is the S amp P 500 with the asset name of spx The benchmark needs to be in the variance matrix If it isn t and you have the weights of the constituents refer to page 2 The benchmark need not be in the price vector unless trading the benchmark is allowed via the bench trade argument A simple version of minimizing the tracking error is gt op te1 lt trade optimizer prices varian af utility minimum variance long only TRUE gross value 1 4e6 benchmark spx This is almost surely too simple For example you may want to limit the number of assets in the portfolio 6 3 EXAMPLES FOR ACTIVE PORTFOLIOS 93 gt op te2 lt trade optimizer prices varian utility minimum variance long only TRUE gross value 1 4e6 benchmark spx port size 55 max weight 0 05 The op te2 object represents a portfolio containing up to 55 assets that ap proximately minimizes the tracking error among all portfolios of size 55 in which assets have a maximum weight of 5 If you are rebalancing an existing portfolio rather than crea
14. 2007 return percent Density Figure 2 12 Performance measurement 200 turnover for 2007 2008 o 2 o o gt 9 z oO Q y o e S 40 38 36 34 32 2007 2008 return percent Table 2 2 Performance measurement using initial positions LT 00 turnover 400 turnover 2007 return 2007 performance percentile 2007 2008 return 2007 2008 performance percentile 2 6 GOING FARTHER 29 doesn t provide any information on whether it is luck or skill Random portfolios can be used to help decide that The previous section discussed performance measurement using the shad owing method We can do the same thing to test a trading strategy One key difference is that we are doing it ex ante rather than ex post Plus we know the constraints and the trading exactly Another difference is that with per formance measurement we are interested in the particular portfolio that was actually held at the start of the period but with evaluating a strategy we want the performance to be good no matter where we start Operationally the real difference is that when evaluating a trading strategy we create a set of random portfolios at which to start and we see how the strategy does during the time period starting at each of those portfolios That is we repeat the shadowing process a number of times In the example we create 20 initial portfolios at the beginning of 2008 and see how the MACD strategy performs after 1
15. ADVANCED FEATURES gt aret crash lt c 20 6 9 3 gt aret prosper lt c 25 3 5 8 3 gt names aret medium lt anam4 gt names aret crash lt anam4 gt names aret prosper lt anam4 gt aret crash equities bonds commodities cash 20 6 9 3 gt aret prosper equities bonds commodities cash 25 0 3 5 8 0 3 0 Since we are doing asset allocation there are no prices for the assets and we want the answer to be the weights of the assets Hence we want to create a price vector that is all ones and set the gross value to a useful value gt aprice lt rep 1 4 gt names aprice lt anam4 gt aprice equities bonds commodities cash 1 i 1 El gt ten thou lt 10000 c 5 5 gt op medium lt trade optimizer aprice avar mediun aret medium gross value ten thou long only TRUE E utility mean variance risk aversion 02 gt op medium new portfolio 100 equities bonds commodities cash 27 86 20 83 28 69 22 62 Above we have performed a mean variance optimization with the medium data and then shown the weights of the resulting portfolio in percent The variance matrix is not positive definite but the optimizer doesn t care note though that no more than one zero variance should be in a single variance matrix In order to perform simultaneous optimization we need to put the three vari ance matrices into a three dimensional array and the expected return vectors into a
16. DETAILS OF RANDOM PORTFOLIO EXAMPLES This function merely calls the univariate MACD function from TTR multiple times and saves the signal part of the output The objects us price and us macd are of class xts In order to avoid assuming that xts is available we convert these to matrices gt us pricemat lt as matrix us price gt us macdmat lt as matrix us macd The only substantial change is that row names are created for the matrices R Commands to Do It All This describes one way of using the auxiliary files to reproduce the analyses We assume that there is a directory called topdir This has a subdirectory called test If the variance caching is done it will have another subdirectory called Varshr The auxiliary files from the website are placed in topdir R is started in test The scripts include an object called PP_LOC that you can change if you would like a different topology for the directories In the R session start off by making sure you are in the right place The result of gt getwd should be topdir test Then you want to install the regular functions and the graphics functions gt source pprobe functions01 R gt source pprobe graphFun01 R Now you need to build the basic data gt source pprobe R_data01 txt This retrieves data from the internet and builds the basic data objects This will fail if the first asset ID is not available but should work if others are miss
17. forced trade c ABC 8 DEF 15 says to sell at least 8 units lots of ABC and buy at least 15 units of DEF If you wanted to buy exactly 15 units of DEF then you would say forced trade c ABC 8 DEF 15 upper trade c DEF 15 Trades may also be automatically forced if the existing portfolio breaks maxi mum weight constraints see pageHM9 The positions argument can also force trades 4 7 Positions This section discusses the arguments e positions e tol positions The positions argument does not embody any constraints that can t be achieved with other arguments It exists because the constraints can sometimes be more conveniently expressed via positions The constraints that positions can do are e max weight e universe trade lower trade e upper trade e threshold e forced trade The key difference between positions and these other constraints is that positions is expressed in monetary value In examples we will assume the currency unit is dollars but it is really the currency unit that the prices vector is in The positions argument takes a matrix with rows representing assets and 2 4 or 8 columns Not all assets in the problem need to be present in positions but by default there will be a warning if not 4 7 POSITIONS 55 Portfolio constraints The first two columns of positions contain portfolio constraints Column 1 is the minimum amount of money that is allowed in the portfolio for each of the ass
18. in reality the name of the vector can be whatever you like The other two pieces of basic information are the variance matrix and the vector of expected returns neither of these are required You only need to give these if they are involved in a constraint To generate 100 random portfolios that have country and sector constraints no more than a 4 tracking error and no more than 55 assets the following command would do gt randport2 lt random portfolio 100 prices varian long only TRUE gross value 1e6 bench constraint c spx 0472 252 port size 55 lin constraints cntrysect constraint lin bounds cntrysect bounds S code note The first three arguments in the call that creates randport2 do not need to have the name of the argument specified because they are all in the order of the arguments in the definition of the function In contrast the call that creates randporti uses the argument name in all cases If there is any doubt then it is safest to give the argument by name The examples so far assume that there is no existing portfolio or that it doesn t matter The existing argument gives the current portfolio gt randport3 lt random portfolio 100 prices varian long only TRUE gross value 1e6 bench constraint c spx 0472 252 existing current portfolio port size 55 lin constraint cntrysect constraint lin bounds cntrysect bounds Sometimes it is more convenient to hav
19. long short O1 108 113 passive 221193 150 1152 position too small 28 positions argument BEZZ postions argument 57 ppoints function pprobe verify function 63 prices argument DJ 101 quadratic constraint 76178 20 quantile argument 48 153 160 quotes in S 74 154 R language instance of S language 14 R website 13 random portfolio 83 43 34 random portfolio summary B9 random weights 37 random portfolio function B3H43 45 random portfolio utility function 42 randport eval function B8H39 66 rbind function 29 read a file 109 read table function 09 real time optimization OGHTOS rep function 97 103 repeat S construct 106 return continuously compounded 11 log EI simple 11 rights issue 122 risk credit SHG market 155 risk aversion parameter 140 150 risk model testing 24 risk tolerance 140 risk aversion argument rival forecasts 153 1154 round lot 45 S code note 15 B3 BA BJ 0 61 63 A a 271 103 105 ere S function 171 abline as matrix 109 110 build constraints 53 c ra za pa cbind 61 9 Cfrag list 164 class A constraints realized 65 cov wt 10 debugger 130 deport 40 114 do call 89 drop 09 110 factor model stat 10 162 fitted 1412 head 33 intersect 61 library 4 ppoints pprobe verify 163 random portfolio 83H43 43 random port
20. names prices a gt trade optimizer long buy cost cost mat cost par c 1 1 5 The first command creates a two column matrix The first column relates to the items that are linear commission and spread the second column contains the cost involving the square root of the trade volume When this is used in an optimization the exponents for these columns are 1 and 1 5 respectively The second coefficient is 1 5 where naively you may expect it to be 0 5 because the square root rule says that the cost per share depends on the square root of the number of shares traded If you think in terms of calculus we are taking the integral of the cost function empirically found a power of 0 6 to fit a set of trades on large US stocks Since a square root is a power of 0 5 their findings suggest that trading costs grow slightly faster as the size of the trade increases If you were to use this power law then the cost par in the example above would be 1 and 1 6 instead of 1 and 1 5 9 4 On Scaling Costs Relative to Utility There are at least three things we want in the costs that we declare e The cost of a larger trade in an asset is appropriate relative to a smaller trade of that asset e The cost of trading one asset is well scaled relative to the cost of trading the other assets e The costs are appropriately scaled in light of the utility that is used The proper scaling of the costs relative to the utility whether
21. optimization only The risk aversion parameter is large relative to the values in penalty constraint In this case the optimizer may have problems getting the constraints to be obeyed e positions names The asset names in the positions argument are not the same as the universe of assets in the problem e random start random portfolios only The start sol argument is given which is ignored in random portfolio start sol is sometimes confused with existing the current portfolio e randport failure random portfolios only Fewer than the requested number of random portfolios were generated e start noexist The start sol argument is given but existing is not This is sometimes what is wanted but it might indicate that the meaning of start sol is confused with existing e turnover max The maximum turnover is too large to have an effect If you want to specify a minimum turnover but no maximum turnover then this warning is avoided by setting the maximum to Inf e utility switch optimization only The utility either given or default is switched to another utility e value range One or more of the ranges for monetary value gross value etc is narrow relative to the prices e variance list compact variance is given which means that there is no way to check that the assets are the same and in the same order as the assets in prices e zero iterations The iterations max argument can be set to zero but this is not the
22. rep 0 length long names names long zero lt long names short zero lt rep 0 length short names names short zero lt short names YN Y Y S code note These commands create two new vectors containing all zeros and with names that are the long assets for one of them and names that are the short assets for the other The rep function replicates its first argument These new vectors are then used as the lower trade and upper trade arguments 104 CHAPTER 7 OPTIMIZING LONG SHORT PORTFOLIOS gt opt listl lt trade optimizer prices varian gross value 1 4e6 net value c 1e4 2e4 lower trade long zero upper trade short zero universe trade c long names short names ntrade 50 The lower trade and upper trade arguments to trade optimizer are limits on the number of units e g lots or shares that can be traded Each asset can be given a different limit If what is passed in is a vector with names then any number of assets may be represented in any order So using long zero as the lower trade argument says that the assets named by long zero can not have negative trades that is can only be bought The same logic applies to the upper trade argument with short zero Another important piece of the command is setting universe trade so that assets that are in neither long names nor short names won t trade at all This example is a simple case the lower and upper bounds can be more complicated when
23. that we hold these portfolios throughout the time period and compare the return of the actual portfolio to the distribution of returns from the random portfolios The information that we need in order to apply this method is e the constraints under which the fund operates While it would be best if we knew the constraints exactly approximations of the constraints can be used For the examples of this section two funds were created by starting with a specific portfolio at the start of 2007 and using the constraints e long only e the number of assets in the portfolio can be no more than 20 e the maximum weight for any asset is 10 Every day throughout 2007 and 2008 an optimal trade was made according to an MACD signal the default from the MACD function in the TTR R package 19 20 CHAPTER 2 RANDOM PORTFOLIO EXAMPLES Figure 2 1 MACD funds compared to random portfolios 2007 st o S SP 3 D o E a 8 S 5 2 a a 0 20 40 60 2007 return percent Table 2 1 Simple analyses of the MACD funds AT 00 nover 400 turnover 2007 return 2007 performance percentile 2007 2008 return 2007 2008 performance percentile Ulrich 2009 Note that this is meant neither to praise nor to pick on MACD it is merely an easy way of producing expected returns that you can replicate yourself if you wish The difference between the two funds is that one has a target of 200 annual turnover buys plus sells w
24. 0 4762577 121 stockB stockl stockJ 20091012 0 2862711 0 5049999 0 2087290 20091013 0 2868016 0 5121764 0 2010220 20091014 0 2667007 0 5273961 0 2059032 This works whether prices is just a vector or a matrix 38 CHAPTER 3 GENERATING RANDOM PORTFOLIOS 3 3 Working with Random Portfolios Small Selections You can use head to get the first few random portfolios and tail to get the last few These are generic functions in R Their random portfolio methods return an object that retains the class and other attributes of the original object These functions can be useful to inspect the portfolios to see if they look reasonable without printing hundreds or thousands of portfolios to the screen They can also be used to test commands such as the example immediately below Evaluating Portfolios The sister function to random portfolio is trade optimizer It can be of interest to see some of the values that the optimizer would return for each of the random portfolios The randport eval function does that for each of the random portfolios or trades in the object it finds what the optimizer says about it You can select which components of the output of trade optimizer to keep using the keep argument The result is a list as long as the random portfolio object and each component of that list is a list containing the kept components Here is a small example of keeping the portfolio variances gt randport eval head randport4 3 k
25. 12 5 Testing Optimization Quality The optimization algorithm is random pseudo random to be technical You get a consistent answer for any particular problem because the same random seed is used by default You can get different paths and hence probably different answers by setting the seed argument to NULL One way of getting a new optimization is new optobj lt update orig optobj seed NULL You can create a list of a number of solutions for comparison with commands similar to 12 5 TESTING OPTIMIZATION QUALITY 137 opt list lt vector list 10 for i in 1 10 opt list i lt update orig opt seed NULL summary unlist lapply opt list function x x results objective digits 7 summary unlist lapply opt list function x trade distance x orig opt digits 7 Y SNE y y Here we ve created an empty list and filled it up with solutions to the prob lem Finally we looked at the distribution of the achieved objective and the distribution of trade distances from the original solution S code note Some care needs to be used when doing seed NULL in a call to update If seed was used in the original call to trade optimizer then the result will be to use the default value of seed which is fixed rather than creating a new random seed The update function creates its own call to the function and then evaluates that call Saying arg NULL removes arg from the call if it was in the original call 138 CHA
26. 60 2 us pricemat us macdmat max weight 1 port size 30 gross value 1e6 long only TRUE vardir paste PPLOC Vardir sep Even though the variances aren t really used we still need to provide a valid location for the directory holding the variances because the function still pulls the variances in The next command finds the minimum variance that can be achieved with the given constraints We don t really care if we have the exact minimum or not so in the interests of speed the number of iterations that the optimizer is allowed is reduced to 3 gt rp30w10 minvar dat lt pp date minvar rownames us pricemat 251 695 us pricemat max weight 1 port size 30 gross value 1e6 long only TRUE iteration 3 vardir paste PPLOC Varshr sep The command to create the dynamic utility with the volatility constraint is gt rp30w10vv20 mvu2 dat lt pp date var meanvarutil rownames us pricemat 251 695 4 1 44 rp30w10 minvar dat 60 2 us pricemat us macdmat max weight 1 port size 30 gross value 1e6 long only TRUE throw error FALSE vardir paste PPLOC Varshr sep We want a constraint that is 120 of the minimum volatility but the constraint is actually on the variance the square of 120 is 1 44 5 3 Test a Risk Model The full set of commands for this task is in file pprobe_R risk txt A preliminary step is to create a vector stating where in the price d
27. Australia stockE 1 0312180 1 0847337 Japan stockF 1 2067453 1 0977745 Australia gt cons objmix bounds lower upper Risk1 Inf Inf Risk2 Inf Inf Country Australia Inf Inf Country Japan Int Inf Country Singapore Inf Inf Yet again the next step is to specify the bounds and then use the arguments Portfolio Constraints versus Trade Constraints The default behavior is to constrain the portfolio If you want the trade to be constrained rather than the portfolio then set the lin trade argument to TRUE You can have a mixture of constraints on the trade and on the portfolio The vector given as lin trade is replicated to have length equal to the number of columns in the lin constraint object If lin constraints has three columns then the argument lin trade c TRUE FALSE TRUE means that the constraint in the first column is on the trade the second column constrains the portfolio and the third column constrains the trade Net Constraints versus Gross Constraints By default constraints use the sum of weights or value or counts If you want the constraints on the gross that is the absolute value rather than the net then set the lin abs argument to be TRUE For example in a long short portfolio you may want to limit the net exposure to a sector but you may also want to limit the gross amount in the sector 64 CHAPTER 4 CONSTRAINTS The lin abs argument is replicated to have length equal to the number o
28. a long only portfolio is generally unambiguous That is not true of long short portfolios For long short portfolios there is not only the issue of the value within the portfolio but also how much cash is supporting the portfolio This matters when valuing the portfolio in order to compute returns and realized volatility The type nav and cash arguments allow control over this The cash argument which is only consulted when the type is nav can either be a single number or a vector with length equal to the number of rows in prices When type is nav and cash is not given or is NULL then it is computed to be the gross value at the first time point this is valid for some long short portfolios but certainly not all Here are some plausible calls gt valuation randport dollar neutral pricemat collapse TRUE type nav no leverage gt valuation randport dollar neutral pricemat collapse TRUE type nav cash some value gt valuation randport 120 20 pricemat collapse TRUE type nav cash 0 Random Weights If you want to have the random portfolios expressed in terms of weights rather than asset units then you can use the valuation function with the weight argument set to TRUE For example gt head valuation ranport pricemat weight TRUE 2 011 stockB stockC stockJ 20091012 0 0003892000 0 5046108 0 4950000 20091013 0 0003841843 0 5299070 0 4697089 20091014 0 0003536518 0 5233887
29. assets in the portfolio e 5 maximum weight for each asset e the sum of the 4 largest weights is no more than 15 Throughout this section the last three constraints are maintained but the first is sometimes changed Figure 2 6 compares the ex ante estimates of volatility from the two models with data up to the start of the fourth quarter of 2007 The estimates agree very well Figure 2 7 is a similar plot but for the fourth quarter of 2008 We would like to see high correlations between the estimated and realized volatilities for the portfolios Figure 2 8 shows the correlation for each quarter between the shrinkage estimate of volatility and the realized volatility The red dashed lines represent 95 confidence intervals for the correlation as found by a statistical bootstrap The pictures are very similar for factor models As we need to expect the predicted level of volatility in the last quarter of 2008 is severely lower than the realized volatility the actual volatilities tended to be more than twice the predicted Two more sets of random portfolios were generated One set was long short dollar neutral zero net value The other was 120 20 long short In both cases the same constraints of 30 assets 5 maximum weight and 15 sum of the 4 largest weights were imposed Figure 2 9 shows the quarterly correlations between the ex ante shrinkage estimate and the realized volatility for the dollar neutral portfolios Figure 2 10 shows th
30. bargain blindly we don t know the up side we are forgoing and we don t know the extent of the protection we are getting Random portfolios allow us to examine the trade off over historical periods For the examples of this section the constraints that are imposed are e long only e no more than 30 assets e no asset with a weight greater than 10 and some times e volatility no more than some value Figure 2 3 shows the distribution of mean variance utility with risk aversion 2 for portfolios that are held throughout 2007 The blue line indicates the distribution for portfolios that have the additional constraint that the volatility is no more than 8 as measured ex ante at the beginning of the year We see that we lose a lot of up side with the volatility constraint But this is merely a snapshot it tells us what happened to have occurred but it doesn t indicate what we can expect to occur 22 CHAPTER 2 RANDOM PORTFOLIO EXAMPLES Figure 2 3 Mean variance utility for calendar year 2007 o m o N o gt N fand i 9 8 o 9 o o o 0 5 0 0 0 5 1 0 1 5 2 0 Mean variance utility with risk aversion 2 Suppose that we are interested in utility over a quarter or for convenience 60 trading days We can move that 60 day window one day at a time and get different answers We have two years of data 2007 and 2008 in our example so we have roughly 440 windows There are at least two approaches One is
31. be maximum return 142 CHAPTER 13 UTILITY e The variance argument is not given or is NULL and distance is not being minimized e The utility is mean variance or mean volatility and risk aversion is 0 In the last of these items the stated utility in the output remains mean variance or mean volatility but a look at the definitions shows that it is really maximum expected return 13 6 Minimum Distance This is specified with utility distance This minimizes the distance from the portfolio to the specified target portfolio plus the trading costs More details are found in Section 6 4 13 7 Going Farther e Chapter 4 discusses utilities where multiple variances and or expected return vectors are given Chapter 14 Advanced Features This chapter will be of use if you have or want e Multiple variances e Multiple expected return vectors e Multiple benchmarks e Compact variances seldom recommended If you merely want constraints on multiple benchmarks that is easy and is discussed on page 72 14 1 Multiplicity The computation that is performed is really controlled by three entities the alpha table the variance table and the utility table In the standard case of no multiplicity these entities can safely be ignored When you do have multiplicity then understanding what these objects are saying is a good idea If you understand them then you can check to make sure that it is doing what you want done If i
32. be closed but no more than 3 then say close number c 2 3 This sort of constraint is unlikely to be useful in optimization but can be used to generate random portfolios that match what an actual trade has done 76 CHAPTER 4 CONSTRAINTS 4 17 Quadratic Constraints Quadratic constraints on the portfolio are not officially supported but can be imposed Quadratic constraints on the trade are not possible The acceptance of more than one variance means that trade optimizer and random portfolio can be tricked into allowing quadratic constraints on the portfolio Here are the steps for creating quadratic constraints e Add the constraint matrices to the variance e Specify the constraint bounds e Make sure the constraints are not benchmark relative e If doing optimization tell the optimizer not to include the constraints in the utility Add Constraints to the Variance Suppose you have a single variance matrix as per usual called varian and you have two quadratic constraint matrices called qmat1 and qmat2 You need to create a three dimensional array with three slices This is an easy operation in the S language but care needs to be taken that the assets match gt assetnam lt dimnames varian 1 gt varian3 lt array c varian qmatllassetnam assetnam qmat2 assetnam assetnam c dim varian 3 c dimnames varian list NULL S code note The c function features prominently in this comman
33. be tempted to impose that with a command like gt posmati 2 lt min 3e5 posmati 2 WRONG Note that there is no warning from this S has no way of knowing that you are doing something you don t really want to do What you do want to do is gt posmati 2 lt pmin 3e5 posmat1 2 right The min function returns just one number which is the minimum of all the numbers it sees The pmin function does a minimum for each element which is what we want in this instance 56 CHAPTER 4 CONSTRAINTS Trade constraints Trade constraints are imposed with the third and fourth columns of positions If you want to impose trade constraints without imposing portfolio constraints then you can just make the first two columns infinite For example if we want selling of each asset to be limited to 5000 and buying of each asset to be limited to 8000 then we could create a matrix like gt posmat3 lt cbind rep Inf length prices Inf 5000 8000 gt head posmat3 LA 2 L3 bad 1 Inf Inf 5000 8000 2 Inf Inf 5000 8000 3 Inf Inf 5000 8000 4 Inf Inf 5000 8000 5 Inf Inf 5000 8000 6 Inf Inf 5000 8000 gt dimnames posmat3 lt list names prices NULL These constraints are essentially identical to lower trade and upper trade except they are expressed in money rather than asset units and it is possible to impose forced trades with the positions constraints Forced constraints A t
34. check can fail to properly detect the problem How the current loop has avoided the problem is to explicitly state the prices vector and variance matrix that we want to use by putting them in the additional args argument We can compare the volatility estimates in a plot like gt plot 100 sqrt 252 rp30s15 facexante Q 4 1 100 sqrt 252 rp30s15 shrexante Q 4 Ex Ante versus Realized Volatility We compute the realized volatility of the random portfolios over each of the quarters gt rp30s15 realized Q lt empty8 gt for i in 1 8 rp30s15 realized Q il lt pp realvol valuation rp30s15 Q i11 us pricemat quarter ends i quarter ends i 1 collapse TRUE t To make a plot comparing a prediction and the realized do gt plot 100 sqrt 252 rp30s1i5 facexante Q 4 rp30s15 realized Q 4 gt abline 0 1 col blue We now compute correlation between predicted and realized volatility gt corshrreal lt numeric 8 gt names corshrreal lt names rp30s15 realized Q gt corfacreal lt corshrreal gt for i in 1 8 5 4 PERFORMANCE MEASUREMENT SHADOWING 89 corshrreal i lt cor rp30s15 shrexante Q i rp30s15 realized Q il method spearman corfacreal i lt cor rp30s15 facexante Q i rp30s15 realized Q i method spearman t ttt Actually this is computing the correlation between the predicted variance and t
35. constraint c 1 5 6 0 2 3 is equivalent to alpha constraint ac 2mat nam or alpha constraint ac 2mat where gt ac 2mat nam Let Est 15 6 Int 205 Let ac 2mat LI 2 i 23 Inf 2 8 6 Inf Y 5 14 5 Multiple Utilities This section does not pertain to generating random portfolios unless a utility constraint is imposed When there are multiple columns in the alpha or variance tables there are usually multiple utilities for any given trade In order to do an optimization we need to combine that set of numbers into just one number A common approach is to get a min max solution This process consists of always picking out the worst answer for each trade we then try to get the best of these worst values Another approach would be to take the median of the values for a trade This represents an optimistic attitude while min max is very pessimistic min max assumes that nature will do its worst against us The quantile argument allows us to choose either of these or something in between The useful range of values for quantile is 0 5 the median to 1 the maximum yielding the min max solution Each of the individual utilities for a trade is called a destination What goes into a destination is under user control For example if some destinations are deemed to be more important than others they can be given more weight with the dest wt argument 14 6 UTILITY TABLES 149 There are two sorts of
36. create a price vector at a specific time gt price 20061229 lt us pricemat 251 We create a return matrix from the price matrix gt us retmat lt diff log us pricemat We also need a specific variance matrix gt varshr Q1 lt var shrink eqcor us retmat 1 250 The next step is to generate random portfolios with all the constraints except the volatility constraint gt rp30w10 lt random portfolio number rand 10000 prices price 20061229 gross value 1e 06 long only TRUE port size 30 max weight 0 1 The utility with risk aversion 2 of these portfolios can now be computed for the year 2007 gt rp30w10 mvu2 2007 lt pp meanvarutil valuation rp30w10 ab us price 252 502 collapse TRUE 2 The same two steps generate portfolios calculate utility are now done with the volatility constraint 5 2 TESTING THE EFFECT OF CONSTRAINT BOUNDS 85 gt rp30w10v08 lt random portfolio number rand 10000 prices price 20061229 variance varshr Q1 gross value 1e 06 long only TRUE port size 30 max weight 0 1 var constraint 08 2 252 gt rp30w10v08 mvu2 2007 lt pp meanvarutil valuation rp30w10v08 us price 252 502 collapse TRUE 2 Caching Variances Many operations depend on a series of variance matrices Variance matrices are sometimes reasonably large so the memory used by a few hundred of them can become a problem One solution is to create a directory
37. expected returns should match for exam ple they can both be annualized or both be daily You also need such things as benchmark constraints to be scaled to match the variance Costs need to account for the time scale of the expected returns and whether or not the expected returns are in percent The variance matrix is singular While the Portfolio Probe optimization algorithm will happily handle singular matrices the results need not be useful If there are more assets than time points in the returns data then a factor model or shrinkage estimator rather than a sample variance should be used This problem is certainly more serious for long short portfolios but is problematic for long only portfolios as well If the true variance really is singular for example cash is one of the assets see page 156 then you should get good results The problem is if the variance matrix says that there are portfolios that are essentially riskless when they are not The variance matrix is not symmetric The optimizers inherently assume that the variance is symmetric They will get confused if this is not the case If what you are using as the variance is not symmetric then you can get a version that is with gt varian sym lt varian t varian 2 10 1 EASY WAYS TO BE WRONG 123 There is asynchrony in the data Global data should not be used at a higher frequency than weekly unless asyn chrony adjustments are made In addition to di
38. for allowance is 0 9999 that is one basis point away you may want to alter this depending on how important the tightness of these constraints is to you and on the size of the portfolio relative to the prices of the assets The allowance computation is unlikely to be what is desired for net value so it is recommended that net value always have length two Section 7 4 on page 106 gives more details about constraining the value of the portfolio There are two cases that are of particular interest dollar neutral portfolios and portfolios in the genre of 120 20 Dollar Neutral Portfolios A dollar neutral portfolio implies that the net value is zero This is a case where the rule that net value should only be given as an interval might be relaxed The argument net value 0 translates into an interval that is symmetric around zero and the radius of the interval is the gross value times one minus allowance Even so it is probably better to explicitly set your interval for the net value 120 20 and 130 30 Portfolios The simplest way to get these is to give constraints just like the stated aim long value 1 2 NAV short value 0 2 NAV or possibly a range can be given long value c 1 2 1 25 NAV short value c 0 2 0 25 NAV net value c 999 1 0 NAV 4 3 LIMITS ON ASSETS 49 4 3 Limits on Assets This section discusses the arguments e max weight e enforce max weight e universe trade e lower trade
39. for the second asset etc Then the specific variances 162 CHAPTER 14 ADVANCED FEATURES vech The vech representation e vartype Always required This is a vector of integers with length equal to the number of variance matrices represented The integers indicate the type of representation for each matrix there is no restriction on the combinations of types that may be used The codes for the formats are full matrix 0 simple factor model 1 full factor model 2 vech 3 e nvarfactors Required if any representation is a factor model either simple or full When given this must have length equal to the length of vartype The elements of this vector that correspond to a factor model must contain the number of factors for that model e varoffset Never required but may add some safety This is an integer vector containing the offset for the start of each variance matrix within the vardata component The first element is zero the second is the length of the data for the first variance matrix the third is the combined lengths of the first and second variance matrices etc The length of this vector is the number of matrices given so the element that would be the total length of vardata is not given Here is an example of putting an object produced by factor model stat from the BurStFin package into this format varfac lt factor model stat retmat out fact vdata lt c t varfac sdev varfac loadin
40. impact 115 market risk 155 max weight argument 49 102 123 maximize information ratio 03195 102 39 maximum expected return utility 41 maximum weight 49 mean variance optimization 96 104 mean variance utility 20 mean volatility utility 1441 median optimization 48 memory of the machine 161 INDEX min max optimization 48 050 L59 160 minimize tracking error 92 minimize variance 92 Minimum variance utility 141 miniter argument 42 missing values variance estimation 12 monetary constraints 5148 108 multiple expected returns 148 L50 160 multiple time periods 154 multiple variances 147 50HT61 net asset value B7 net value definition 48 nonlinear costs L16017 not optimizing starting solution I33 ntrade argument 51 93 97 04 objective function 42 163 operating system 164 optimization algorithm 135 distance 9198 mean variance 96 104 median 148 min max 148 1509 S960 not doing 13311134 objective 142 163 Pareto see min max optimiza tion real time OGHI0S utility free wrong BEHRA out trade argument B3 BA Pareto optimization see min max optimization passive portfolio 902193 50152 penalty for broken constraint 13 163 performance measurement 9 21 27 83181 polynomial costs 17H18 port size argument 52 93 95 102 portfolio long only B00 INDEX
41. is similar to putting it in a suitable command might be curgross lt valuation cur port prices total gross gt gt gt opt outl lt trade optimizer prices varian expected return alphas net value c 1e4 2e4 gross value curgross cash required ntrade 15 turnover cash required max weight 05 var constraint 04 2 252 existing cur port universe trade names cur port The comments about the universe trade and turnover arguments made about injecting money also apply here The formulations given for both injecting and extracting money are im plicitly assuming that the optimizer will want to trade at least the amount requested That doesn t have to be true A safer approach would be to give turnover an interval perhaps something like 106 CHAPTER 7 OPTIMIZING LONG SHORT PORTFOLIOS turnover cash required c 1 1 01 7 4 Money Constraints If you are looking for a control in the optimizer for leverage there isn t one The amount of leverage applied is determined by the fund manager leverage is external to the optimization The decision on leverage comes first which determines the gross value and the net value quantities that the optimizer does need to know The constraints on the long and short values are not just redundant to the gross and net value constraints Figure 71 shows that if we plot the gross and net value constraints then these form a rectangle that is diagonal to the l
42. it is reasonable to suppose that results may be better by using more than one of them The first task is to create a suitable variance object which will almost always be a three dimensional array gt varmult lt array c vstat vnam vnam vfund vnam vnam vmacro vnam vnam c length vnam length vnam 3 list vnam vnam NULL S code note A three dimensional array is a generalization of a matrix instead of a dim attribute that has length 2 it has a length 3 dim Likewise the dimnames is a list that has three components We need each of the variance matrices to be in a slice of the third dimension In the example above we are ensuring that the matrices we are pasting together all correspond to the same assets in the same order by subscripting with vnam You need to decide which quantile to optimize The best quantile to use probably depends on how the variance matrices relate to each other and on the type of optimization that you are performing Once this decision is made the optimization is performed as with a single variance gt opts m1 lt trade optimizer prices varmult long only TRUE gross value 1e6 expected return alphas bench constrain c spx 0572 252 quantile 7 This command is maximizing the information ratio with a tracking error con straint as is done on page 94 In this case the tracking error is constrained relative to each of the variance matrices The information ratio that is bei
43. list of assets that can be sold and a list of assets that can be bought Suppose that buy vec and sell vec are vectors giving the assets that are allowed to be bought and sold The first thing to do is to set up the constraints that make sure nothing on the buy list can be sold and nothing on the sell list can be bought 6 4 UTILITY FREE OPTIMIZATION 97 gt lower tr lt rep 0 length buy vec gt names lower tr lt buy vec gt upper tr lt rep 0 length sell vec gt names upper tr lt sell vec S code note These commands create two new vectors containing all zeros and with names that are the buy assets for one of them and names that are the sell assets for the other The rep function replicates its first argument Then these are used in the call to the optimizer In this case we assume that we want to minimize tracking error relative to spx gt op bhs lt trade optimizer prices varian long only TRUE gross value 1 4e6 benchmark spx existing cur port lower trade lower tr upper trade upper tr universe trade c buy vec sell vec The universe trade argument is restricting the assets that are traded to be only in the buy or sell list Assets not named are constrained to not trade You may wish to use the turnover argument to constrain turnover and other arguments 6 4 Utility free Optimization The idea of utility free optimization is that you have an ideal target portfolio and you would l
44. matrix gt avar all lt array c avar crash avar medium avar prosper c 4 4 3 gt dimnames avar all lt list anam4 anam4 NULL S code note A three dimensional array is a generalization of a matrix instead of a dim attribute that has length 2 it has a length 3 dim Likewise the dimnames is a list that has three components We need each of the variance matrices to be in a slice of the third dimension We also need the order of assets within each matrix to be the same but that is already true 14 7 MULTIPLICITY EXAMPLES 159 gt aret all lt cbind crash aret crash medium aret mediun prosper aret prosper gt aret all crash medium prosper equities 20 8 25 0 bonds 6 4 3 5 commodities 9 5 8 0 cash 3 3 3 0 Min Max Solution The most common approach to simultaneous optimization is to maximize the minimum utility also known as Pareto optimality To clarify each allocation will produce a utility for each scenario the optimizer only pays attention to the worst utility for an allocation with no regard to which scenario produces that utility it then finds the allocation that does best with respect to the worst utility The aim then is to never do really badly Before we do the actual optimization we need to do some ground work Vv op 0 lt trade optimizer aprice avar all aret all gross value ten thou long only TRUE utility mean variance iterations 0 warning message omit
45. of model gt rp30s15 facexante Q lt empty8 gt rp30s15 shrexante Q lt empty8 gt for j in 1 8 rp30s15 facexante Q j lt unlist randport eval rp30s15 Q j 1 additional args list prices us pricemat quarter ends j variance varfac Q j keep var values use names FALSE rp30s15 shrexante Q j lt unlist randport eval rp30s15 Q 31 additional args list prices us pricemat quarter ends j variance varshr Q j1 keep var values use names FALSE HERR We generally think in terms of volatility but it is actually daily variance that is saved into the objects 88 CHAPTER 5 DETAILS OF RANDOM PORTFOLIO EXAMPLES advanced The commands above ensure that a subtle but important problem is avoided The randport eval function looks at the image of how the random portfolio object was created the call attribute and uses that to create the actual data objects that are used The prices vector for the random portfolios was us pricemat quarter ends i In the current loop there is no i so either an arbitrary i will be used or there will be an error if no i is found We could have had our loop variable be i which would have been fine in this case but there is no guarantee that the loop variables correspond to the same thing There is a check that will warn you if it detects that the current data are different from the original If the data are largely static then the
46. on long only portfolios In long short portfolios the variance is a smaller quantity so it takes a larger risk aversion to have the same effect In general risk aversion is likely to be larger in long short situations than for long only 13 3 MEAN VOLATILITY UTILITY 141 13 3 Mean Volatility Utility Mean volatility utility is invoked with utility mean volatility With mean volatility utility we maximize al w CVwTVw C w where is the risk aversion parameter and w satisfies all of the constraints The mean volatility risk aversion does depend on annualization For example the risk aversion for annual data is V252 times the risk aversion for daily data But volatility aversion is invariant to whether or not the data are in percent 13 4 Minimum Variance Minimum variance utility is specified with utility minimum variance The minimum variance utility minimizes wi Vw C w over the w that satisfy the constraints This utility is used when e The utility argument is set to be minimum variance e The expected return argument is not given or is NULL and distance is not being minimized e The utility is either mean variance or mean volatility and risk aversion is Inf 13 5 Maximum Expected Return The maximum expected return utility is specified via utility maximum return With this utility we maximize aw C w where w must satisfy the constraints This utility is used when e The utility argument is set to
47. program perceives as an error In general these are longer versions of error or warning messages that you might receive 11 1 Special Instruction 1 Long only when shorts exist It is an error to declare the portfolio to be long only when there are short positions in the existing portfolio If you have short positions currently and you want to create a long only portfolio then you should do the following e Declare long only FALSE e Use positions to set the long only constraint for the portfolio For the last item you could alternatively use forced trade and lower trade but positions is the easier approach The positions argument could be built like gt posarg lt cbind rep 0 length prices Inf gt dimnames posarg lt list names prices NULL Then used like positions posarg 11 2 Special Instruction 2 Benchmark in long short optimization Naively using a benchmark with a long short portfolio is probably not what you want to do Creating a benchmark in Portfolio Probe and elsewhere is equivalent to being short the benchmark in the amount of the gross value If you have a portfolio in the genre of 120 20 then a benchmark is concep tually fine but operationally wrong If you use the benchmark argument you 131 132 CHAPTER 11 SPECIAL INSTRUCTIONS will be short 140 of the benchmark with a 120 20 portfolio while you want to be short 100 Whether you have a benchmark in the traditional sense or in the
48. retail Inf Inf sector telecom Inf Inf We could then use the arguments 60 CHAPTER 4 CONSTRAINTS lin constraints cons obj lin constraints lin bounds bounds csw gross value 1e6 If on the other hand we want bounds in terms of monetary value then we could do bounds csv lt cons obj bounds bounds csv 1 2 lt c 1e5 2e5 4e5 5e5 bounds csv 3 2 lt 2e5 bounds csv 5 lt c 3e5 6e5 bounds csv SPOS MP OM OMS lower upper country France 1e 05 4e 05 country Italy 2e 05 5e 05 country Spain Inf 2e 05 sector energy Inf Inf sector media 3e 05 6e 05 sector retail Inf Tar sector telecom Inf Inf Now the arguments would be lin constraints cons obj lin constraints lin bounds bounds csv gross value 1e6 lin style value In this case we need to give the lin style argument because we are using a non default value for it It is also possible to mix styles gt bounds csmix lt bounds csw gt bounds csmix 1 3 lt bounds csv 1 3 gt bounds csmix lower upper country France 1le 05 4e 05 country Italy 2e 05 5e 05 country Spain Inf 2e 05 sector energy Inf Inf sector media 0 3 0 6 sector retail Int Tar sector telecom Inf Inf So the arguments become lin constraints cons obj lin constraints lin bounds bounds csv gross value 1e6 lin style c value weight There is no need for any particular pattern of finite bounds In t
49. same as doing no optimization To get the start sol trade as the final trade set funeval max to zero or one 10 3 Cheatsheets Implied Ranges Table 10 1 lists the true meaning of a single number for those arguments that represent a range of values Note that close number is the outlier in that its single value means exactly that value 10 4 TROUBLESHOOTING 127 Table 10 1 Implied ranges of arguments set varas sesen Co x Inf must give the range 7 o 8 gt 8 Table 10 2 The meaning of threshold inputs threshold constraint trade sel trade buy portfolio short 0 0 ON portfolio long 0 0 a Threshold Inputs Table 10 2 shows the meaning of the threshold argument when given a matrix with each of the allowable number of columns a plain vector is the same as a one column matrix The subscript of the values in the table indicate the column of the value So x is from the first column and 22 is from the second column and so on A three column matrix can only be given in the case of a long only portfolio Positions Inputs Table 10 3 indicates the allowable number of columns for objects given as the positions argument and the meaning of those columns See Section 177 for more details 10 4 Troubleshooting This section lists a number of problems and possible solutions Utility Problems The utility stays the same when I change the risk aversion There are at leas
50. sense that you have a set of liabilities there are at least two choices e Put the portfolio of liabilities as a short asset in the portfolio and stop it from being traded e Use a variance matrix that is relative to the liabilities There are two functions in the BurStFin package that may be of use in this situation The var add benchmark function allows you to add a portfolio of assets to a variance matrix var relative benchmark transforms a variance matrix to be relative to one of its assets Chapter 12 Adjusting Optimization Speed and Quality The default control settings for the optimizer are a compromise between the time it takes to finish and the quality of the solution Optimization is usually done in one of two settings e Backtesting strategies e Production In backtesting the quality of the solution is much less of an issue than the speed of the optimization If the default setting seems too slow to you you can decrease the maximum number of iterations allowed Setting iterations max 5 is a possibility this is not unreasonably small for a lot of problems For production where time is not critical and quality is more of concern then an easy way to get better solutions is to add the argument stringency 1 This will take a few times longer to execute but will give at least as good of a solution The rest of the chapter goes into more detail about the options that you have and what they mean 12 1 Stay
51. should be one or more solutions that are used as starting values for the trade in optimization The current portfolio should be given in the existing argument You confuse the meaning of the lower trade or upper trade ar gument If you habitually abbreviate the lower trade argument to lower you may begin to think of the bound as being on the final portfolio rather than on the trade 124 CHAPTER 10 PRACTICALITIES AND TROUBLESHOOTING Bounds for linear constraints are not properly scaled Values in 1in bounds need to match the corresponding value in lin style If the bounds are in values but the style is in weight then there probably isn t a real constraint Alternatively if the bounds are in weights and the style is in value then the constraint is likely to be impossible You do not give the lin abs or lin trade argument when needed or reverse the meaning If the bounds are set for absolute constraints but lin abs is not given then the constraints may be inconsistent Perhaps even worse they may not be inconsistent and give an answer that on the surface seems okay You give just one trading cost argument but it is the wrong argu ment If you want costs to be the same whether the position is long or short and whether you are buying or selling you need to give the long buy cost argu ment If you give a different argument then the costs correspond to the named situation only and the costs in other situations are
52. single monetary argument that needs to be given it is manda tory that the monetary value of the portfolio be constrained somehow All of the monetary arguments are in the currency units that prices uses In all cases it is sufficient to only give turnover The argument turnover 12000 says that the buys plus sells of the trade can not exceed 12 000 currency units While this makes most sense when there is an existing portfolio that is not necessary The turnover can of course be constrained even when other monetary constraints are given The turnover can be expressed as an interval turnover c 11000 12000 When only one number is given the implied lower bound is zero How the other monetary arguments are used largely depends on whether or not the portfolio is long only Long only Portfolios If you want long only portfolios then you need to set the long only argument to TRUE the default is FALSE You can state the amount of money in the resulting portfolio by giving the gross value argument Ultimately this needs to be a range of allowable values You can give the range explicitly with a vector of two numbers gross value c 999900 1e6 Alternatively you can give a single number gross value 1e6 When a single number is given this is taken to be the upper bound the lower bound is computed via the allowance argument The default allowance is 0 9999 that is one basis point away So by default these two specifi
53. that is expected returns and variance or distance is the point where the science gets a bit fuzzy This is also a key issue controlling the merit of the optimization If the costs given the optimizer are too small then excessive trading eats up per formance If the given costs are too large then performance is hurt via lost opportunity Perfection is not necessary for the optimization to be of value but attempting perfection is probably not such a bad mission The added utility of a trade needs to compensate for the transaction cost Thus the costs should correspond to the expected returns over the expected holding period To be more specific suppose that we think it will cost 50 basis points to trade a certain asset If we are using daily data and our expected return for this asset is 10 basis points then we will have to have a holding period of 5 days in order to break even To amortize the cost it that is the 50 basis points should be divided by the number of days that we expect to hold this asset If we expect to hold it for 2 days the optimizer would see a cost of 25 basis points If we expect to hold it 10 days the optimizer would see a cost of 5 basis points The statement of this example might seem to imply that volatility is of no concern it is of concern Let s revisit the example from Section 9 3 120 CHAPTER 9 TRADING COSTS gt cost mat lt cbind commission names prices spread names prices 2
54. that this argument solves Suppose you have set a difficult problem for random portfolio and you want 1000 portfolios Suppose further that the first attempt was successful so clearly the problem is not impossible but the next one million attempts fail As far as you are concerned you are waiting forever gen fail times the number of portfolios requested is the maximum number of failed attempts allowed In our example you requested 1000 and the default value of gen fail is 4 so it would stop after 4000 failures and return the one random portfolio it successfully found and warn you that it didn t do so well with your request Note that it is seldom obvious whether a specific set of constraints is easy to satisfy difficult to satisfy or impossible 3 7 Adding a Utility Constraint The random portfolio function does not allow a constraint on the utility but random portfolio utility does If computation time is of concern then it can be better to just use random portfolio with constraints on the variance and expected returns However this may not be quite what you want There is not much difference between the functions in terms of how they are used The key difference is the objective limit argument The objective is the negative of the utility So if you want the utility to be at least 0 6 then you want the argument objective limit 0 6 The meaning of gen fail is the same but the other control arguments are those used with opti
55. the optimizer is to find the combination of inputs that achieves the best result for the objective function By convention optimizers find the minimum of objective functions The optimizer in trade optimizer follows the convention The objective in trade optimizer is the sum of two quantities the negative of the utility except minimum variance and minimum distance don t take the negative and the penalty for broken constraints The penalty for a broken constraint is the appropriate element of penalty constraint times a measure of how broken the constraint is The penalties for all of the constraints are summed to get the penalty The same process is done for generating random portfolios The difference is that the utility part is zero when random portfolio is used so there is only the penalty part that is minimized 15 3 Writing C or C Code The C code that underlies the optimization and random portfolio generation can be incorporated into your own C or C programs There are quite a few arguments to the C functions for optimization and some of the arguments involve a number of variables Thus it is far from trivial to write the program calling these functions 163 164 CHAPTER 15 DREGS Speed of execution is not a good reason to go completely to C the execution of the S code takes a fraction of a second and so is negligible for practical problems Perhaps the most likely reasonable reason to stay all in C is to d
56. this is quite a short time period from which to make generalizations 2 2 TESTING THE EFFECT OF CONSTRAINT BOUNDS Figure 2 4 Approximate minimum volatility 12 14 Minimum volatility 10 2007 2008 Figure 2 5 Mean variance utility from overlapping 60 day windows iQ o q o Density 0 1 0 2 03 0 0 10 5 0 Mean variance utility with risk aversion 2 23 24 CHAPTER 2 RANDOM PORTFOLIO EXAMPLES 2 3 Test a Risk Model We have a wide choice of risk models If we could compare results of risk models on a large number of realistic portfolios we could make informed decisions about the best risk models for particular tasks Random portfolios can provide those realistic portfolios in a convenient form This section merely provides a brief taster of what is possible The BurStFin R package contains two functions for estimating variance matrices see page 10 One estimates a statistical factor model the other shrinks the sample variance towards the equal correlation matrix Each function was given 250 days of returns and used default values of all arguments Since there are only 70 assets this is a rather atypical case for these estimators Often there are more assets than time points Even so the amount of shrinkage towards equal correlation ranged from about 17 to about 32 One thousand random portfolios were generated at the start of each of eight quarters The constraints were e long only e no more than 30
57. varian lt factor model stat retmat zero TRUE For assets with missing values this biases correlations toward zero rather than towards the average correlation as the previous command does 112 CHAPTER 8 GENERAL USE advanced The treatment of missing values can have a material impact on the opti mization and it can be beneficial to specialize the estimation to your particular application An easy way to do this is to return the factor model representation gt varfac lt factor model stat retmat out factor When the output argument is set to factor the result is a list with com ponents named loadings uniquenesses and sdev You can modify these to correspond to an approach appropriate for you Then you can create the vari ance matrix from the object gt varian lt fitted varfac This works because the result of factor model stat when it returns the factor representation varfac in this case has a class and that class has a method for the fitted generic function The result of this function is the variance matrix that is represented by the factor model Adding a Benchmark to the Variance Perhaps you have a variance matrix of the individual assets but the benchmark is not included in the variance You can add the benchmark to the variance ma trix if you have the weights for the constituents of the benchmark and all of the constituents are in your variance matrix Merely use the var add benchmark function from
58. way of doing the same thing is gt be lt 04 7 2 gt names bc lt gt be spx 0 0016 spx and then use the argument bench constraint bc Scaling Don t forget that this constraint is on squared tracking error not tracking error While tracking errors are almost always an annual number your data prob ably aren t annual For example if your variance is on a daily scale then you need to transform the constraint to the daily scale bench constraint c spx 04 2 252 Lower and Upper Bounds Lower bounds as well as upper bounds can be specified by giving a two column matrix For example bench constraint rbind spx c 02 04 72 would restrict spx tracking error to between 2 and 4 This is most likely of interest when generating random portfolios 72 CHAPTER 4 CONSTRAINTS Multiple Benchmarks If you want upper bounds on more than one benchmark merely give a numeric vector with the names of the benchmarks bench constraint c spx 04 2 R1000 0572 specifies no more than a 4 tracking error from spx and also no more than a 5 tracking error from R1000 To get both upper and lower constraints for multiple benchmarks you give a two column matrix with multiple rows For example bench constraint mult benchcon lowup where gt mult benchcon lowup L4 2 spx 0 0004 0 0016 R1000 0 0009 0 0025 R2000 0 0016 0 0064 Advanced Use When there are multiple variance matrices
59. 0 0 0 2e4 gt head posmat4 L 11 L2 G31 LA LSI Gel 0 7 GSI 1 Int Inf Inf Inf 0 0 0 20000 2 Inf Inf Inf Inf 0 0 0 20000 3 Inf Inf Inf Inf 0 0 0 20000 4 Inf Inf Inf Inf 0 0 0 20000 6 inf Int lt Inf Inf 0 0 0 20000 6 Inf Inf Inf Inf 0 0 0 20000 gt dimnames posmat4 lt list names prices NULL Tolerance Imagine a situation where asset ABC is restricted to be no more than 50 000 in the portfolio At time 1 the optimizer likes ABC and buys enough that it is close to that upper limit From time 1 to time 2 ABC dutifully does well so now it is above the 50 000 limit Assuming the limit remains the same for time 2 then the optimizer will force a sale of some ABC So we are performing a trade probably of a trivial amount for a rather arbitrary reason If there is a trade threshold constraint on selling ABC then it is even worse we are effectively forcing a sale of at least the threshold amount The tol positions argument solves this problem If tol positions is zero the default then the constraints are taken as stated If tol positions is positive then assets that violate portfolio constraints in positions by a value less than tol positions are not forced to trade In our example if the value of the position of ABC at time 2 is 50 203 then at least 203 worth of ABC will be forced to be sold if tol positions is less than 203 But there will not be a forced trade if tol positions is greate
60. 03 2009 attire cless gt 1 randportBurSt seed attribute begins 1 1142929704 1716596987 285978235 Each component of the list is a portfolio or trade which is a vector giving the number of asset units shares lots contracts for each asset that appears 3 2 Valuation Random portfolios are seldom interesting in their own right generally it is necessary to manipulate them to get the information we want This section and the next discuss several common tasks Simple Use The valuation method for random portfolios can give a variety of results re lating to the monetary content of the random portfolios The simplest usage is to give a vector of prices in addition to the random portfolio object gt head valuation ranport pricevec 2 111 stockB stockC stockJ 38 92 50461 08 49500 00 2 stockB stockI stockJ 28625 66 50497 42 20871 84 36 CHAPTER 3 GENERATING RANDOM PORTFOLIOS What we get is the amount of money in each position in each portfolio Any vector of prices that contains the assets in the random portfolio object can be used It can be the price vector used to create the random portfolios or some other price vector A matrix of prices can be given as well The columns should correspond to assets gt head LELI 20091012 20091013 20091014 2 20091012 20091013 20091014 valuation ranport pricemat stockB stockC stockJ 38 92 50461 08 49500 38 34 52882 52 46875 36 20 53
61. 1 lt trade optimizer prices varian a expected return alphas gross value 1 4e6 net value c 1e4 2e4 This puts 1 4 million currency units into the gross value of the portfolio and the net value is between negative 10 000 and positive 20 000 There are a few additional arguments that will be useful in practice If you are building a new portfolio then a more likely command is something like gt opt ir2 lt trade optimizer prices varian W expected return alphas gross value 1 4e6 net value c 1e4 2e4 port size 55 max weight 05 This new command adds port size to give a portfolio containing up to 55 assets It also ensures that no position will be more than 5 of the gross value To update an existing portfolio the command might be gt opt ir3 lt trade optimizer prices varian expected return alphas gross value 1 4e6 net value c 1e4 2e4 ntrade 15 max weight 05 F existing cur port turnover 5e4 Here both the number of assets in the trade and the turnover are limited and of course the existing portfolio is specified 7 2 EXAMPLES 103 Maximize Return with a Bound on the Variance If you have a target for the maximum variance that you want then you can add a constraint on the variance Suppose that volatility should be no more than 4 We need to transform the 4 into a variance and possibly go from annual to the frequency of the variance If the variance is on a daily frequency and cr
62. 11 spr The first row gives the zero based index of the expected return If there is only one set of expected returns then this is always zero If there are no expected returns this is negative one The second row gives the zero based index of the corresponding benchmark A negative number in this row means there is no benchmark There is a benchmarks attribute which is a character representation of the benchmarks When you are passing an object in as the atable argument the second row of the matrix itself is overwritten with the information in the benchmarks attribute which must be present Variance tables are essentially the same as alpha tables The first row of course corresponds to variances rather than alphas The other change is that there is a third row which states if this combination is only used in the utility value 1 or if it is used in constraints as well value 0 This information is used to more efficiently generate random portfolios As with alpha tables when you pass an object in as the vtable argument the second row is overwritten with the information given in the benchmarks attribute of the object 14 3 VARIANCE CONSTRAINTS 145 Here is an example of the alpha table and variance table from an opti mization that has two variance matrices three expected return vectors and no benchmarks gt optifatable 13 1 21 L 3 alpha 0 1 2 benchmark 1 l attr benchmarks a 44 qe ess gt optifvtable
63. 5 txt file 109 114 typography conventions 13 unchanging utility 27 universe trade argument 50 56 97 CoA 105 update function 136 upper trade argument Bd 97 04 utable argument 149 160 utility benchmark relative 152 constraint 42 definitions 139 destination 149 INDEX table 49 1150 59160 unchanging 127 zero 128 utility free optimization valuation function 65437 CA too small 123 value constraints 06 var add benchmark function 12 var constraint argument Z0 variance OHA adding benchmark 1412 argument IGIHI62 compact representation 1611162 constraint 70 103 45147 154 factor model see factor model minimize 92 missing value treatment 12 multiple LL 1501161 table 44 1145 149 vech representation 61 vech representation 161 version component 164 violation of constraints 65H66 volume average daily 50 61 113 vtable argument 144 warnings suppressing 112411126 weight constraint 9H destination 48 143 sum of largest constraint 74 73 write a file 40 114 wrong optimization 21H124 zero utility 28
64. 574 36 48750 stockB stockI stockJ 28625 66 50497 42 20871 84 28199 07 50358 50 19765 00 26625 10 52650 68 20555 60 The price matrix we used looks like gt pricemat 1 4 20091012 20091013 20091014 stockA stockB stockC stockD 27 63 19 46 11 67 5 79 29 47 19 17 12 23 5 85 27 86 18 10 12 39 5 43 Collapsing Values 2 Often we don t care about the individual positions but we do care about the gross value of the portfolios Use the collapse argument to get this gt valuation ranport pricevec collapse TRUE 1 100000 00 99994 92 99992 25 99993 29 This has length 4 because that is the number of portfolios in ranport Perhaps the most useful is to use collapse and a matrix of prices gt valuation ranport pricemat collapse TRUE 1 2 3 4 20091012 100000 00 99994 92 99992 25 99993 29 20091013 99795 86 98322 57 101966 21 106510 98 20091014 102360 56 99831 38 104858 49 107995 59 The result is a matrix with as many rows as the price matrix and as many columns as there are portfolios in the random portfolio object It is possible to get the net value rather than the gross gt valuation ranport pricemat collapse TRUE type net 1 2 3 L 4 3 2 VALUATION 37 20091012 1000 00 999 92 999 93 999 99 20091013 6045 86 2394 43 3872 21 2285 54 20091014 4860 56 5469 98 2364 57 1377 59 The type argument is only used when collapse is TRUE Net Asset Value The value of
65. BurStFin gt varian lt var add benchmark varian assets spx weights spx This function takes three arguments a variance matrix a vector of weights that should sum to 1 and a character string giving the name for the bench mark The weight vector needs to have names which are some of the asset names on the variance matrix The computation performed by this function is the preferred method of intro ducing a benchmark into a variance matrix for optimization See Burns 2003a for a study on approaches to incorporating benchmarks into the variance matrix for optimization 8 2 The Random Generation or Optimization The optimization process itself is covered in other chapters The basics are covered in chapters that depend on what you are doing e Random portfolio generation Chapter B e Long only portfolio optimization active passive or utility free Chapter e Long short portfolio optimization Chapter J 8 3 POST OPTIMIZATION 113 Trading costs are discussed in Chapter D while Chapter J covers constraints Chapters 0 and 14 have additional information Section QJ on page EI is particularly recommended caution If in optimization a penalty is imposed because one or more constraints are broken then the optimizer will produce a warning Do not ignore the warning It could well mean that there is a mistake in the specification You can re run the optimization perhaps with an increased number of i
66. Chapter 3 Generating Random Portfolios The random portfolio function generates a list of portfolios or the trades for them that satisfy constraints but pay no attention to utility 3 1 The Command To generate random portfolios you give random portfolio the number of ran dom portfolios that you want to generate the basic information for the problem and the constraints that you would like There is also the out trade argument which controls whether it is the random portfolio the default or the trade which is output At a minimum you need to specify the vector of prices and the amount of money in the portfolio One possibility is gt randporti lt random portfolio prices prices long only TRUE gross value 1e6 Of course this is not very interesting You are likely to want more than one random portfolio and to have non trivial constraints imposed S code note The notation 1e6 means 1 000 000 that is a one followed by six zeros The prices always required needs to be a vector of positive numbers that has names which identify the assets in the problem Here is an example of the first few values of a suitable price vector gt head pricevec stockA stockB stockC stockD stockE stockF 27 63 19 46 11 67 5 79 5 15 20 99 33 34 CHAPTER 3 GENERATING RANDOM PORTFOLIOS The assets named in prices define the universe of assets for the problem In ex amples the prices argument is usually given a vector called prices
67. E volatility names prices sqrt ave daily volume names prices gt gt cost amort lt cost mat expected holding names prices gt gt trade optimizer long buy cost c0st amort cost par c 1 1 5 This adds a command which divides what might be called the raw costs by how long we expect to hold each asset The result is the amortized costs The holding period should be in the same time units as the expected returns If we have daily returns then an asset that we expect to hold 25 days should have 25 as its expected holding period However that same asset should have an expected holding period of about 0 1 when we are using annualized returns That the expected returns are noisy suggests that the costs should be in creased from this rational value to reduce trading There is another sense of scaling costs By default trade optimizer sums the costs from all of the trading and then divides by the gross value of the port folio This is the natural thing to do as the expected returns and the variance are also scaled by the gross value when the utility is computed However you do have the option to scale by the trade value or not to scale the costs at all this is controlled by the scale cost argument 9 5 Costs Due to Taxes Taxes present a cost structure that is entirely different from trading costs re sulting from the market In particular the cost will be different depending on whether you are bu
68. ICITY EXAMPLES 151 gt sqgrt 252 optsi var values 1 0 009029673 0 009029442 gt opts1 utility values 1 3 235515e 07 3 235350e 07 In this example the tracking errors are the same at 90 basis points and the utilities are equal as well A reasonable complaint about this optimization is that it treats both bench marks equally even though the new benchmark is speculative while the current one is known precisely It is probably more reasonable to insist on a smaller tracking error against the current benchmark A difference in the tracking errors is easily achieved with the addition of an other argument The utility table needs to be changed slightly from its default So we recover the utility table from the original optimization make the change then do a new optimization gt utabl lt optsi utable gt utabl alpha spot 0 variance spot 0 destination 0 opt utility 0 risk aversion 1 wt in destination 1 gt utab1 6 1 lt 1 4 gt utabl Li 2 alpha spot 0 0 variance spot 0 0 destination 0 0 opt utility 0 0 risk aversion La wt in destination 1 4 PRRORRO All that we have done is change the weight in destination value for the first current benchmark from 1 to 1 4 Now the optimization yields gt opts2 lt trade optimizer prices varian long only TRUE gross value 1e6 existing cur port utility minimum variance benchmark c spx newspx quantile 1 utable utab1 gt sqrt 252
69. NULL gt op credl lt trade optimizer prices var mult var constraint c 1 18 The first command creates the three dimensional array containing the two types of risk taking care to make sure the data line up properly The second com mand does the optimization The variance constraint is a named vector where the name is the zero based index to the third dimension of the variance array In this case the name is 1 because we are constraining the second variance But op cred1 is not the optimization that we are aiming for The credit risk is used in the utility as well as being constrained We need to tell the optimizer not to use credit risk in the utility by passing in a utable argument We can make a simple change to the utility table from the optimization we ve just done gt op credi utable 1 52 alpha spot 0 0 variance spot 0 1 destination 0 1 opt objective 1 1 156 CHAPTER 14 ADVANCED FEATURES risk aversion i 1 wt in destination 1 1 gt uticred lt op credi utable gt uticred 3 2 lt 1 gt uticred Cea 2 alpha spot 0 0 variance spot 0 destination 0 i opt objective 1 1 risk aversion al 1 wt in destination 1 1 gt gt op cred2 lt trade optimizer prices var mult var constraint c 1 18 utable uticred The change we made was to say that the column with credit risk in it shouldn t go into a destination One indication that we are doing it right is that t
70. PTER 12 ADJUSTING OPTIMIZATION SPEED AND QUALITY Chapter 13 Utility Utility is ignored in random portfolios but see Section B 7 however it is central to optimization The descriptions of the utilities use a few entities e The weight vector w e The vector of expected returns a e The variance matrix V e The transaction costs C w In long only portfolios all of the elements of w are non negative and they sum to 1 In long short portfolios the weights may be negative and it is the absolute values of the weights that sum to 1 that is the weight of an asset is its value in the portfolio divided by the gross value of the portfolio C w is actually an abuse of notation for the transaction cost Transaction cost obviously depends on the existing portfolio Cost is not strictly a function of weight unless the gross value for the portfolio is constant 13 1 Maximum Information Ratio This utility which is the default if there is both a variance and expected returns is specified with the argument utility information ratio or utility exocost information ratio The generic information ratio problem is to maximize al w Vwi Vw over all vectors w that satisfy the constraints 139 140 CHAPTER 13 UTILITY This inherently assumes zero costs There are two approaches at least to incorporating costs The first is to subtract the trading cost from the expected value aTw C w Vwi Vw This is what
71. Portfolio Probe User s Manual Burns Statistics 19th November 2010 Copyright 2003 2010 Burns Statistics Limited All rights reserved http www burns stat com Edition 3 2010 November 18 S PLUS is a registered trademark of TIBCO Contents Dad dd did 12 Necessary Toolg d aaa a pace OO SaaS Oe oe eae te KE E id om made weed eas ne OO OH ce aro E a Bea eE ada ph putes ea de e Cie cee eee oe eee ood esa heen vies OHO a eh a ed aes Saree Oe A mao ot Bese a ene Ae o a Fea ee ea eee a eh hs serie rodeos a de didas e ida a pa ab AR AI et Bt ed Scar OS Eine eo Ba elas dt Sid he Et E pa SS mame E 6 OOK reer eee tT eee ree ee er ere ee te eee oe Serre ee Sere Te A ee ae A edad E rs E EEE hs aaa Be A ee e EEE 13 13 13 14 14 15 15 19 19 21 24 27 27 29 4 CONTENTS 4 Constraint A II e a dat lower trade and upper trade 44 Number of Assets o Number of Assets to Trade Number of Assets in the Portfolio Portfolio Thresholds Summary of Threshold Input 4 6 Forced Traded 2 e eee owe ee eT eee Ee ETE eee ee Summary of positions inputg 0004 4 8 Linear Constrainth ooo a eee eee Se ee ESA ete ies SPE SER TEET ALi ade Seek Inspecting Linear Constraints
72. The optimizer makes this process quite painless even when you want a fair amount of control over the trade The examples given below assume that the cash management is via the gross value An alternative would be to control the longs and shorts directly You would use the long value and the short value arguments for this more specific control Injecting Money into a Portfolio If you want to perform optimization with a constraint on the variance see page 103 then a command that will do this while only increasing existing positions is curgross lt valuation cur port prices total gross gt gt gt opt ini lt trade optimizer prices varian expected return alphas gross value curgross newcash net value c 1e4 2e4 ntrade 15 max weight 05 var constraint 04 2 252 existing cur port turnover newcash universe trade names cur port The key points here are setting the gross value to the desired new amount and forcing the turnover to equal the amount by which the gross is increased this means that no positions can be reduced Setting the trade universe to be the assets that are in the current portfolio ensures that no new positions are opened Ifit is okay to open new positions then do not give the universe trade argument You can also increase turnover any amount that you want in order to do rebalancing as well as managing cash Extracting Money out of a Portfolio Taking money out of the portfolio
73. Y 12 4 Improving Quality The easiest and generally most effective approach to improving quality is to set stringency to 1 You can improve quality by increasing iterations max and fail iter If the converged component of the result is FALSE then it is probably the case that the best solution has not been found You can restart the optimization from where it left off gt opti lt trade optimizer prices varian gt opt2 lt trade optimizer prices varian start sol opt1 For more thorough optimization increase iterations max and fail iter gt opt3 lt trade optimizer prices varian start sol opt1 iterations 100 fail 10 Another way of doing this same thing would be gt opt3 lt update opti start sol opti iterations 100 fail 10 S code note If an object is a list and has a component named call that is an image of the command that created the object then the update function will recom pute the call and change any arguments that are given in the call to update The update function can be used on the results of trade optimizer and random portfolio A natural thing to do if you want a quality solution is to increase the maxi mum number of iterations to something like 100 or 1000 That is not the best approach For very large problems 100 iterations might be useful but increasing the number of runs and increasing the number of iterations slightly is usually a better approach
74. a constraint that says that the largest weight of any one asset is 5 This is conceptually exactly the same as max weight 05 4 15 COST CONSTRAINTS 75 However there are some operational differences The key difference is that using sum weight ensures that either the constraint is satisfied or there is a penalty imposed for breaking the penalty But it will be somewhat slower when sum weight is used See page 49 for the possibility of max weight failing to enforce the constraint 4 15 Cost Constraints This section discusses the argument e limit cost This is unlikely to be of use with optimization limit cost puts lower and upper constraints on the cost This can be useful for mimicking actual trading with random portfolios If this is given then it must be a length two vector giving the allowable range of costs limit cost c 2 3 2 35 or possibly something like limit cost opt obj results cost c 99 1 01 4 16 Number of Positions to Close This section discusses the argument e close number The minimum and maximum number of positions to close in the existing port folio can be specified with the close number argument If a single number is given then exactly that number are to be closed So close number 3 is really shorthand for close number c 3 3 Note that this is different than other arguments where a single number does not imply a lower bound To specify that at least 2 positions should
75. ably also want to limit the number of assets to trade as well One possible command is gt op ini lt trade optimizer prices varian long only TRUE expected return alphas turnover newcash bench constraint c spx 0 04 2 252 max weight 05 existing cur port ntrade 5 lower trade 0 The key features in this command are setting the turnover argument to the amount of money to be injected and setting the lower trade argument to zero You may also want to use the port size argument to restrict the number of assets in the new portfolio Alternatively you could set the universe trade argument to be the names in the current portfolio meaning that no additional assets will be added to the portfolio 6 6 ASSET ALLOCATION 99 Extracting Money out of a Portfolio To raise cash from the portfolio you want the optimizer to do no buying This is achieved with the following command gt op outi lt trade optimizer prices varian long only TRUE expected return alphas turnover cash required ntrade 5 bench constraint c spx 0 04 2 252 max weight 05 existing cur port upper trade 0 The pertinent parts of this command are setting the turnover argument to the amount of cash that is required and setting upper trade to zero The formulations given for both injecting and extracting money are im plicitly assuming that the optimizer will want to trade at least the amount requested That doesn t have to be tr
76. al optimization In addition to needing the utable argument the quantile argument needs to be 1 in order to get a min max solution op minmax lt trade optimizer aprice avar all aret all gross value ten thou long only TRUE utility mean variance utable utab3 quantile 1 op minmax new portfolio 100 equities bonds cash 0 91 34 82 64 27 gt 5 A look at the utility values shows typical behavior of a min max solution the solution has the same utility for two of the scenarios General Simultaneous Optimization Min max optimization protects against the very worst outcome However it may really be buying too much insurance We can change the quantile ar gument to a number that is less than 1 but only numbers that are at least one half are sensible gt op q7 lt trade optimizer aprice avar all aret all gross value ten thou long only TRUE utility mean variance utable utab3 quantile 7 gt op q7 new portfolio 100 equities bonds cash OG FO 32 70 66 54 In this case of only three utility values the quantity it is minimizing is a weighted average of the two worst utilities For this example there is very little change from the min max solution and it seems to go more conservative rather than less conservative We can adjust the risk aversion but when utable is given we need to use the force risk aver control argument By default the risk aversion in the utility table that is passed in is
77. andom Portfolio Examples This chapter outlines the code that underlies the examples given in Chapter 2 The files that contain the exact commands are available on the website There are at least two uses for this e You can reproduce the analyses in Chapter 2 e If you have a task similar to one of the examples you should be able to do what you need to do by mimicking what is done here The analyses are somewhat simplified they ignore such things as dividends stock splits and other corporate actions The analyses also depend on some custom functions that can be found in file pprobe_functions01 R A set of adjusted closing prices on a non random selection of large US stocks was retrieved from the internet The set of commands for creating the data is given in file pprobe_R_data01 txt The first step is to read in the set of identifiers gt us selection lt scan us selection txt what The function used next depends on the TTR package Ulrich 2009 If you don t already have it installed on your machine then you need to do gt install packages TTR Some of these instructions may need to be changed if you are using S PLUS rather than R The command above is one of those The next step is to get the price data gt us price lt pp TTR multsymbol us selection 20060101 20090101 The price data are then used to calculate the MACD matrix gt us macd lt pp TTR multmacd us price 81 82 CHAPTER 5
78. ata the quarters end gt quarter ends lt c 251 312 375 438 502 563 a 627 691 755 gt names quarter ends lt paste rep 2006 2008 c 1 4 4 Q c 4 1 3 sep We create the template for a number of objects that we ll fill in gt empty8 lt vector list 8 gt names empty8 lt tail names quarter ends 8 We make a return matrix using the price matrix gt us retmat lt diff log us pricemat 5 3 TEST A RISK MODEL 87 Ex Ante Volatility Comparisons We load the package that contains the variance estimators installation of this package is described on page 1T0 gt library BurStFin Now we initialize lists to hold the variances and then estimate the variances Next gt varshr Q lt empty8 gt varfac Q lt empty8 gt for i in 1 8 varshr Q i lt var shrink eqcor us retmat seq to quarter ends i 1 length 250 varfac Q i lt factor model stat us retmat seq to quarter ends i 1 length 250 t ttet we create a set of random portfolios for each quarter gt rp30s15 Q lt empty8 gt for i in 1 8 rp30s15 Q i11 lt random portfolio 1000 us pricemat quarter ends i gross value 1e6 long only TRUE port size 30 max weight 05 sum weight c 4 15 4 What we want from the random portfolios is the predictions of their volatility So we create an object to hold the predictions from each type
79. attributes rp all lt attributes rp1 3 6 Unsatisfiable and Difficult Constraints Not all sets of constraints can be achieved Obviously there are no portfolios that satisfy a variance that is smaller than zero or even smaller than the minimum variance given the other constraints If you set random portfolio such a task it is bound to fail There is a trade off between returning quickly when asked the impossible and being successful when asked the merely difficult There is of course a default value for this trade off but you can adjust it for specific circumstances There are a number of control arguments that say how hard to work In terms of impossible constraints the most important is init fail This says how many separate attempts to make before quitting when there have been no portfolios successfully found 42 CHAPTER 3 GENERATING RANDOM PORTFOLIOS For each attempt there is a trio of arguments that control how hard to work within the attempt iterations max gives the maximum number of iterations before stopping fail iter is the maximum number of consecutive iterations allowed that fail to make progress miniter gives the minimum number of iterations allowed even if fail iter says to quit Of course if a portfolio is found that satisfies all the constraints then the attempt is declared successful and stops no matter what the value of miniter The remaining argument of this ilk is gen fail Let s start with the prob lem
80. beta stocks or buy low beta stocks We ll create a constraint that does more of this than its opposite in terms of asset count We build the constraint object and the bounds lcc4 lt build constraints cbind HighBeta beta gt 1 lbc4 lt 1cc4 bounds 1bc4 1 1 lt 3 5 1bc4 VVV Y lower upper HighBeta FALSE 3 5 Inf HighBeta TRUE Inf Inf Our arguments are lin constraints lcc4 lin constraints lin bounds lbc4 lin style count lin trade TRUE 4 10 Alpha Expected Return Constraints This section describes the argument e alpha constraint 70 CHAPTER 4 CONSTRAINTS The alpha constraint argument bounds the expected return of the optimized portfolio In its simplest use it is merely a single number that gives the mini mum for the expected return For example alpha constraint 1 1 An upper bound as well as a lower bound can be specified by giving a two column matrix For instance alpha constraint cbind 1 1 2 3 will restrict the expected return to be between 1 1 and 2 3 Note that if a benchmark is given then this will constrain the expected return relative to the benchmark This is just the usual expected return of the portfolio minus the expected return of the benchmark Advanced use of alpha constraint chiefly when more than one vector of expected returns is given is discussed on page 147 4 11 Variance Constraints This section discusses the argument e var constraint The va
81. bounds as well as upper bounds on the distances then you need to give dist bounds a two column matrix with each row corresponding to a distance For example dist bounds cbind c 0 1 1e5 0 2 c 0 2 2e5 0 3 Each of the arguments dist style dist trade and dist utility can either have length equal to the number of distances or length one if the argument is the same for all arguments For example in the example above dist trade and dist utility both have their length one default value The dist prices argument can be a list of price vectors to match the dist center argument 4 14 Sums of Largest Weights This section discusses the argument e sum weight Consider the constraint that the largest n weights should sum to no more than some limit this is handled by the sum weight argument For example if you want the 5 largest weights to sum to no more than 40 and the 10 largest weights to sum to no more than 70 the argument would look like sum weight c 5 4 10 7 The values in the vector given to sum weight are the limits the names of the vector are the numbers of weights in the sums S code note The names within the c function must be inside quotes as they are not legal S object names Another way of getting the same thing would be gt sumw arg lt c 4 7 gt names sumw arg lt c 5 10 Then sumw arg would be given as the sum weight argument The argument sum weight c 1 05 creates
82. cations of gross value are equivalent In general there is no problem with a constraint this tight the key thing is how wide the range is relative to the prices of the assets There will be a warning if the interval is seen to be too narrow In the case of long only portfolios net value and long value are synonyms for gross value so you can give any one of these three Long short Portfolios The arguments gross value net value long value and short value con trol the value of the portfolio To be clear The long value is the amount of money in positive positions The short value is the amount of money in negative positions this is meant to be a positive number but the absolute value of negative numbers is taken for 48 CHAPTER 4 CONSTRAINTS the short value The gross value is the sum of the long and short values The net value is the long value minus the short value There are two minimal sets of these arguments e gross value and net value e long value and short value Here are some examples gross value 1e6 net value c 200 3000 OK long value 6e5 short value 5e5 OK gross value 1e6 net value c 0 2e5 long value 6e5 OK gross value 1e6 long value 6e5 not OK neither pair These four arguments are logically each of length two giving an allowable range If they only have length one then the second value is computed by multiplying the given value by the allowance argument The default value
83. ce of the portfolio Page 140 has more on the risk aversion parameter A command to do a simple mean variance optimization is gt op mvi lt trade optimizer prices varian long only TRUE expected return alphas gross value 1 4e6 utility mean var risk aversion 7 This command does not include a benchmark As noted above if the benchmark argument is given then the mean and variance will be relative to the benchmark If you do have a benchmark an alternative approach is to do the optimization in the absolute sense and add a constraint on the tracking error The tracking error constraint is placed on the optimization with a command like gt op mv2 lt trade optimizer prices varian long only TRUE expected return alphas gross value 1 4e6 utility mean var risk aversion 7 bench constraint c spx 0 04 2 252 For an explanation of the value given for bench constraint see page 71 In mean variance optimization the risk aversion parameter is quite impor tant Mean Volatility Optimization Mean volatility optimization is just like mean variance optimization except that the value of the utility argument is utility mean volatility and the risk aversion argument will usually have a substantially different value Buy Hold Sell List It is not absolutely necessary to create expected returns for an actively managed portfolio Instead you can minimize the variance or tracking error using a
84. cemat us pricemat 1 251 alphamat us macdmat frac turnover 2 252 max weight 1 port size 20 do warn c novar FALSE utility switch FALSE A plot can now be performed the den lt density 100 pp simpret randser 20 w10 t200 value c 1 252 plot the den main xlab 2007 return percent xlim range the den x 100 serret200 07 abline v 100 serret200 07 col blue lwd 2 ie ial i a 90 CHAPTER 5 DETAILS OF RANDOM PORTFOLIO EXAMPLES A percentile can also be calculated gt round mean pp simpret randser 20 w10 t200 value c 1 252 lt serret200 07 100 5 5 Evaluating a Trading Strategy The code for this section is in file pprobe R_tradeStrat txt The first step is to generate a set of random portfolios 20 of them in this case to serve as the starting points gt rp 20 w10 2008 lt random portfolio 20 Y us pricemat 502 gross value 1e7 long only TRUE max weight 1 port size 20 Doing the optimization and then the random mimicking for each of the starting portfolios is all done in the following function gt evalst 20 w10 t200 2008 lt pp evalstrat 100 id rp 20 w10 2008 us pricemat 1 502 us macdmat frac turnover 2 252 max weight 1 port size 20 verbose 2 do warn c utility switch FALSE novariance FALSE The percentiles for a point in time can be plotted gt plot ppoints 20 100 at sort evalst 20 w10 t200 2008 percentile 61
85. combination of expected return and variance is used and the information ratio is the utility in all cases 14 7 Multiplicity Examples This section provides a few examples of using multiple variances expected re turns and benchmarks Dual Benchmarks Dual benchmarks can be used for example to incorporate predictions about a future rebalancing of the benchmark Due to the relative movements that have occurred since the last rebalance we may have predictions of what assets will enter and exit the benchmark and with what weight The portfolio can be optimized against both benchmarks this allows the portfolio to be rebalanced to some extent against the new benchmark while still having protection relative to the current one This can mean a cheaper rebalance if it is done before others do their rebalancing If our benchmark is spx and our prediction of the post rebalance benchmark is newspx then minimizing variance relative to the two benchmarks can be done as gt optsi lt trade optimizer prices varian long only TRUE gross value 1e6 existing cur port utility minimum variance benchmark c spx newspx quantile 1 Often a min max solution is found for dual benchmarks To get such a solution set quantile to 1 The min max solution says that we want to minimize the worst tracking error It is typical though not always true that the utility in the optimal solution is the same for both benchmarks 14 7 MULTIPL
86. containing indi vidual files of each variance that can each be loaded as needed The solution shown here works for R it would probably need to be adjusted some to work in S PLUS The code that creates the files of variances may be found in file pprobe_R_varCache txt Here is code to create the files gt for i in 250 700 JJd lt rownames us retmat i JJn lt paste varshr JJd sep assign JJn var shrink eqcor us retmat seq to i length 250 save list JJn file paste Varshr JJn rda sep rm list JJn cat done with JJd n The following fragment from a function shows how the variance files can be used for i in seq along dates this date lt dates i vnam lt paste varprefix this date sep attach paste vardir vnam rda sep this rand lt random portfolio variance get vnam detach In each iteration the file containing the desired variance matrix is attached the variance is used and then the file is detached The function has a couple of arguments vardir and varprefix that allow some flexibility Dynamic Utility The first dynamic utility object does not involve the variances The following call sets the window size to 60 trading days and the risk aversion to 2 among other things 86 CHAPTER 5 DETAILS OF RANDOM PORTFOLIO EXAMPLES gt rp30w10 mvu2 dat lt pp date meanvarutil rownames us pricemat 251 695
87. cted returns Costs need not be symmetric for buying and selling You may for example want the cost of increasing positions to be more than the cost of decreasing them because of increased liquidity risk and or because of the implicit cost of needing to close the position again 9 2 Specifying Costs Costs are specified similarly in trade optimizer and random portfolio When generating random portfolios costs will be ignored unless limit cost is given or the utility is constrained The specification of costs is very flexible while simple costs can be given easily To allow for asymmetry in costs there are four cost arguments e long buy cost e long sell cost e short buy cost e short sell cost 115 116 CHAPTER 9 TRADING COSTS This allows differences in the costs for long positions versus short positions as well as differences for selling and buying If there is no difference between buying and selling and between long and short positions then give only the long buy cost argument caution If you give one of the cost arguments other than long buy cost but no others then the named situation will be the only case where there is cost That seems unlikely to be what you want Minimally costs must be given for all of the assets that are allowed to trade in the problem if costs are given at all Costs may be given for any number of assets as long as all the tradeable assets are included Linear Costs Linear costs ar
88. d When c is used on matrices as in its first occurrence in the command the dimensions are stripped and the result is a plain vector of all of the elements of the three matrices The c function can be used with lists as well as with atomic vectors For the dimnames of the resulting three dimensional array we need to have a length three list where the final component is ignored and hence can be NULL In this case we need all of the arguments to c to be lists so we need to give it list NULL Just giving an argument of NULL would not have had the effect that we wanted Impose Constraint Bounds The next step is to impose the constraint bounds This will be done with the var constraint argument as discussed in Section 43 The constraints in this case will be on the second and third variances that is on indices 1 and 2 Also note that the bounds are in terms of the weights there is not a choice to put the bounds in terms of monetary value 4 17 QUADRATIC CONSTRAINTS 77 If you are generating random portfolios without a utility constraint and you are not using a benchmark then you can proceed to the actual computation The intervening steps only apply if you are using a benchmark or a utility Dummy Run The next two steps are aided by performing a dummy run of optimization something like gt qc temp lt trade optimizer prices varian3 iterations 0 It is best to include the variance constraints in this run
89. d volatilities of asynchronous data The Journal of Derivatives 5 4 diBartolomeo 2003 diBartolomeo D 2003 Portfolio management under taxes In Satchell S and Scowcroft A editors Advances in Portfolio Con struction and Implementation Butterworth Heinemann Grinold and Kahn 2000 Grinold R C and Kahn R N 2000 Active Port folio Management McGraw Hill Gupton et al 1997 Gupton G M Finger C C and Bhatia M 1997 CreditMetrics TM Technical Document http www riskmetrics com Kallberg and Ziemba 1983 Kallberg J G and Ziemba W T 1983 Com parison of alternative utility functions in portfolio selection problems Man agement Science 29 1257 1276 Michaud 1998 Michaud R O 1998 Efficient Asset Management Harvard Business School Press Ulrich 2009 Ulrich J 2009 TTR Technical Trading Rules http CRAN R project org package TTR R package version 0 20 1 167 Index 120 20 portfolio 24 ES abbreviation of function arguments abline function allowance argument 47 48 alpha table 44 145 alpha constraint argument 70 alphas see expected returns amortize cost 19 annualization information ratio 94 risk aversion 40 41 argument abbreviation 54 array three dimensional 153 158 61 as matrix function 09 110 asset allocation 99 assets used in optimization 01 00 asynchronous data 11 23 atable 49 atabl
90. destination weight Each utility has a weight within its destination The within destination weights can be used to create a weighted average of utilities within a single destination they can also be used to modify the utility as demonstrated in the dual benchmark example which starts on page 50 The weights in the dest wt argument on the other hand are weights among the destinations which are used along with the quantile argument to control the selection of the objective from among the destinations For example in scenario analysis see page 56 dest wt could be the probability assigned to each scenario note The answer when dest wt is not given is in general slightly different than when dest wt is given and all of the weights are equal The actual problem that is done is mostly specified by the utility table 14 6 Utility Tables The utility table is a matrix that gives the combinations of variances expected returns and benchmarks that are to be used and what to do with them Here is the utility table from our example of two variances and three expected returns gt optifutable LI LA L 3 LAI LSI 0 8 alpha spot 0 El 2 0 1 variance spot 0 destination 0 opt utility 1 risk aversion 1 1 RRRRO RRRNO RRROR RRRpE wt in destination FPRPRrRPOoOrRPN The first row is the expected return this is the zero based index of the columns of atable That is it indicates a particular expected return benchmark combi nation B
91. e You may want to reduce the time that an optimization takes because you are doing a large number of optimizations The only viable way to reduce time is to reduce the iterations max argument For a lot of problems 5 iterations is likely to optimize enough that backtests will give the right indication Zero iterations will almost surely be too few but even one iteration may not be so terrible as to be useless 12 3 THE OPTIMIZATION PROCESS 135 12 3 The Optimization Process This section briefly describes the optimization algorithm so that the suggestions for improving quality in the next section will make sense The optimization process can be broken into three levels e sub algorithms e iterations e runs The algorithm for trade optimizer is generally described as a genetic algo rithm That is a simplification Each iteration performs several sub algorithms some of them genetic plus simulated annealing and others can be described as greedy algorithms A run is a series of iterations The run ends when too many consecutive iterations fail to find an improvement That number of failures is controlled by the fail iter argument which has a default of zero So by default any iteration that fails to improve will cause the run to end If fail iter is 1 then it takes two consecutive iteration failures to end the run The default is to perform a single run Multiple runs are done if the stringency argument is positive The stri
92. e argument 44 backtest 27 bench constraint argument 710172 pA 65 1153 benchmark add to variance 12 alpha table 44 145 constraint 94105 22 153 dual 65 501152 variance table 44 145 benchmark for long short I3TH132 benchmark relative utility 152 bug report 164 build constraints function 538 BurStFin package 10 buy hold sell list D697 103 c function KA 76 04 C or C code G63H164 168 call component 136 cash flow LOS5HI0A CAUTION BJKIA GICOM LS C3 131 162 cbind function 61 129 Cfrag list function 164 cheatsheet 264127 class function I close number argument Z5 comma separated file 40 109 MA compact variance representation 61 compare to another program 134 constraint absolute linear 63164 all 5 benchmark 04495 122 153 building linear B8 63 close number 75 cost 75 count 67169 country B4 distance 2174 expected return 69 47 148 linear E7H67 liquidity BOWS 55 long side linear 64 market capitalization 62 monetary 45 48 numerical linear 61 penalty for breaking 78 79 113 163 portfolio size 93 102 portfolio value 06 quadratic 6478 20 sector 34 short side linear 64 soft 78H79 sum of largest weights 74 75 threshold 5253 56 INDEX tracking error BA MUA PA 95 153 trade 63H64 trade number 3 turn
93. e correlations for the 120 20 portfolios Random portfolios give you the tools to select the best variance estimators for the types of portfolios that you hold and allow you to see how good that is 2 3 TEST A RISK MODEL 25 Figure 2 6 Shrinkage versus factor model estimates of volatility for 2007 Q4 10 O Shrink to equal correlation 12 5 13 0 13 5 14 0 14 5 15 0 1 12 5 13 0 13 5 14 0 14 5 15 0 15 5 Statistical factor model Figure 2 7 Shrinkage versus factor model estimates of volatility for 2008 Q4 Shrink to equal correlation 24 26 28 30 22 22 24 26 28 30 Statistical factor model 26 CHAPTER 2 RANDOM PORTFOLIO EXAMPLES Figure 2 8 Quarterly correlations shrinkage versus realized long only 0 6 0 7 0 8 Correlation shrink with realized 0 5 0701 07Q2 07Q3 0704 0801 08Q2 08Q3 08Q4 Figure 2 9 Quarterly correlations shrinkage versus realized dollar neutral 0 50 0 55 0 60 Correlation shrink with realized 0 45 07Q1 07Q2 07Q3 0704 0801 08Q2 08Q3 08Q4 2 4 PERFORMANCE MEASUREMENT SHADOWING 27 Figure 2 10 Quarterly correlations shrinkage versus realized 120 20 0 75 Correlation shrink with realized 0 65 0 55 07Q1 07Q2 07Q3 0704 0801 08Q2 08Q3 0804 2 4 Performance Measurement Shadowing Section P I presents the static method of performance measurement While that method is more powerful than peer groups and benchmarks we can often do even better If we know
94. e given with a vector or a one column matrix of costs For example to have costs of 30 basis points for all trades the command would be gt opti lt trade optimizer prices i long buy cost prices 003 To have costs of 50 basis points for short sales and costs of 30 basis points for all other trades the command is gt opt2 lt trade optimizer prices long buy cost prices 003 short sell cost prices 005 The in these commands and subsequent ones refers to additional arguments that you want in the optimization The advantage of linear costs is that they are easy to think about and easy to write down Linear costs are undoubtedly more popular than they otherwise would be because computational engines have generally been unavailable for more realistic models of cost Nonlinear Costs Costs are unlikely to really be linear It can be quite useful to have non integer exponents in the trading cost functions This is allowed via the cost par ar gument cost par is a vector giving the exponents for the cost function Consider gt cost poly 1 21 3 4 ABC 0 00 0 01 0 02 0 00 DEP 0 03 0 02 0 01 0 01 gt trade optimizer long buy cost cost poly gt trade optimizer long buy cost cost poly cost par 0 3 9 2 SPECIFYING COSTS 117 These two calls to trade optimizer are equivalent though there would be a slight computational efficiency advantage to the first
95. e mult argument should be given If for instance all of the lot sizes are 100 then you would do gt deport opti mult 100 1 opti cey When there are different lot sizes you should give a vector containing the lot size of each asset gt deport opti mult lotsizes i opti cey If you want the file to be in monetary units then the mult argument should be prices gt deport opti mult prices to txt i opts txt This command writes a tab separated file which in this case is called opti txt The what argument can be given to control what information is put into the file see the help file for details Chapter 9 Trading Costs This chapter covers trading costs It is possible to constrain costs so costs can be used when generating random portfolios This is only likely when imitating a series of trades with random trades Costs are a tricky part of optimization but are extremely important 9 1 Background If there were any justification for the attitude that optimization is hard it should be on account of trading costs However the problem of trading costs is fundamental to fund management you have to confront the issue whether or not you use an optimizer There are numerous sources of trading costs commissions bid ask spread and market impact are commonly considered Less obvious costs can be such things as capital gains taxes if a position is sold and the cost of imperfectly known expe
96. e the trades rather than the portfolios If you want the trades just set the out trade argument to TRUE gt randtrade3 lt random portfolio 100 prices varian long only TRUE gross value 1e6 bench constraint c spx 04 2 252 existing current portfolio F port size 55 lin constraint cntrysect constraint lin bounds cntrysect bounds out trade TRUE S code note The full name of the out trade argument must be given This is unlike almost all other arguments where only enough of the first portion of the name needs to be given to make it unique among the arguments to the function For 3 2 VALUATION 35 a full explanation of argument matching in S see Burns 1998 page 19 or Burns 2009 If the existing argument is not given or is NULL then it doesn t matter which value out trade has the output is the same in either case The result of a call to random portfolio is a list where each component of the list is a portfolio or trade The object has a number of attributes including a class attribute randportBurSt Here is a small example gt random portfolio 2 priceten gross value 1e5 long only TRUE port size 3 max weight 5 1 stockA stockB stockJ 1337 1481 6484 2 stockB stockF stockH 1274 2382 3100 arer cali random portfolio number rand 2 prices priceten gross value 1e 05 long only TRUE port size 3 max weight 0 5 attr timestamp 1 Thu Oct 29 11 36
97. eated from returns in decimal as opposed to percent then the command would look like gt opt vc1 lt trade optimizer prices varian expected return alphas gross value 1 4e6 net value c 1e4 2e4 ntrade 55 max weight 05 var constraint 04 2 252 This is not doing quite what is advertised in the headline but may actually be more appropriate This is really maximizing the information ratio with a constraint on the variance If the best information ratio has a variance that is smaller than the bound then it will be the best information ratio that is chosen If you truly want the maximum return given the variance bound then you need to say so explicitly gt opt vc2 lt trade optimizer prices varian expected return alphas gross value 1 4e6 net value c 1e4 2e4 ntrade 55 max weight 05 var constraint 04 2 252 utility maximum return Minimize Variance Given a Long List and a Short List Suppose you don t have specific predictions for the returns of the assets but you do have a set of assets that you predict will go up and a set you think will go down That is you have a buy hold sell list In this case you can force only buys from the long list sells from the short list and minimize the variance of the portfolio If long names and short names are vectors containing the asset names that are predicted to go up and down then the computation can proceed along the lines of long zero lt
98. ed on a portfolio Give constraints realized the portfolio object the result of a call to trade optimizer and the constraints that are of interest gt constraints realized opt ir2 cbind country countryvec lower upper realized nearest violation country France Inf Inf 0 020106 Inf NA country Italy Ini Inf 0 156728 Inf NA country Spain Inf Inf 0 157020 Inf NA You may also use the additional linear constraint arguments like lin abs and lin trade to specify the nature of the constraints you want gt constraints realized opt ir2 cbind country countryvec lin abs TRUE lin style value lower upper realized nearest violation country France Inf Inf 433494 Inf NA country Italy Inf Inf 296702 ut NA country Spain Inf Inf 669666 Inf NA Inspecting Linear Constraints The summary for portfolios includes the result of constraints realized on the portfolio as well as listing the effective values of the arguments that control the type of constraints If you are doing optimization you can get this with opt obj lt trade optimizer summary opt obj If you are generating random portfolios you can do rp obj lt random portfolio randport eval rp obj subset 1 FUN summary Setting the subset argument to 1 means that we are only going to evaluate the first random portfolio An example for random portfolios is gt tail randport eval rp obj6 FUN summary subset 1 1 valuati
99. eep var values 013 1 var values 11 382 3576 123 2 var values 1 147 6476 137 3 var values 1 134 6368 In this case where we are returning only one number per portfolio it makes more sense to coerce this to a numeric vector gt unlist randport eval head randport4 3 keep var values F use names FALSE 1 382 3576 147 6476 134 6368 Keep in mind that these values are ex ante predictions they may or may not have much relation to realized variance note In randport eval the optimizer is called using the same names of objects as was used when the random portfolio object was originally created Objects with these names must be visible at the time that randport eval is used If any of 3 3 WORKING WITH RANDOM PORTFOLIOS 39 these objects has changed then it is the current value rather than the original value that is used caution Additional arguments or changes to arguments may be given to randport eval so that what the optimizer does is not exactly what random portfolio did If you are making a change to an argument then you need to use the exact same abbreviation if any as in the original call to random portfolio There is a FUN argument to randport eval that if given applies that func tion to each of the portfolio objects that are created For example we could do gt randport eval head randport4 3 FUN summary Or perhaps a more useful command along the same lines
100. ent that is required However there are two concepts that are required the amount of money in the portfolio has to be constrained somehow and there has to be something that provides a utility Monetary Value The turnover or the amount of money in the portfolio needs to be constrained Details can be found on page 45 Utility At least one of the arguments e variance 91 92 CHAPTER 6 OPTIMIZING LONG ONLY PORTFOLIOS e expected return e dist center must be given Utility free optimization minimizing the distance to a target portfolio can be done using dist center and related arguments See Section 6 1 to do this Assuming distance is out of the picture If expected return is not given then the utility is the minimum variance Ifa variance matrix is not given then the utility is the maximum expected return If both are given then the default utility is to maximize the information ratio It can be safest to explicitly state the utility that you want Optimizing a long only portfolio without a variance or a distance is very unlikely to be a good idea 6 2 Examples for Passive Portfolios This section contains some examples that build in complexity Expected returns do not enter into optimization for passive portfolios and they need not be given Minimize the Variance of the Portfolio This is the simplest optimization to perform gt op mvi lt trade optimizer prices varian utility minimum variance
101. ere is a fixed cost for trading any of an asset then you need to give a two column matrix with the second column containing all zeros gt intercept only lt cbind intercepts 0 This is most sensible if the intercepts are not all equal otherwise it is almost the same as constraining the number of assets to be traded If the portfolio is long short then there can be more than one type of trad ing cost for a single asset For example an asset that is long in the existing portfolio can be sold heavily enough that it becomes short There is then the issue of whether or not both intercepts should be counted in the trade The doubleconst control parameter determines which is done If doubleconst is FALSE the default then only the first intercept is used If doubleconst is TRUE then both intercepts are counted 9 3 Power Laws present an argument that the trading cost per unit of an asset should be commission plus half the bid ask spread plus some constant times the square root of the quantity the amount to be traded divided by the average daily volume The last term involving the square root comes from an inventory model They suggest that the constant should be on the order of the daily volatility of the asset Sample code implementing this model might look like gt cost mat lt cbind commission spread names prices 2 volatility names prices sqrt ave daily volume 9 4 ON SCALING COSTS RELATIVE TO UTILITY 119
102. es y Chapter 8 General Use y Chapter 3 Generate Random Y Chapter 4 Constraints y Chapter 5 Example Code y Chapter 10 Practicalities Chapter 8 General Use a Long only Chapter 7 Long short Chapter 4 Constraints l Chapter 9 Trade Cost Chapter 10 Practicalities 16 TYPOGRAPHY CONVENTIONS 17 Figure 1 2 Possible route for utility free optimization Chapter 8 General Use Chapter 6 Long only Optimization especially Section 6 4 Chapter 4 Constraints Chapter 10 Practicalities 18 CHAPTER 1 ORIENTATION Chapter 2 Random Portfolio Examples This chapter shows some of the more common applications of random portfolios Chapter B contains the code that created each of these examples These examples use daily adjusted price data for an arbitrary but non random selection of 70 US large cap stocks for the years 2006 through 2008 2 1 Performance Measurement Static Suppose we have a particular fund that we would like to evaluate We can generate a number of random portfolios that obey the constraints of the fund The fund manager might have selected any one of the random portfolios that we generate We can compare the selection the fund manager did make with the selections of the proverbial monkeys The static method of performance measurement is to create a set of random portfolios that satisfy the fund s constraints at the start of the period assume
103. ets Column 2 is the maximum amount of money So this includes the max weight constraint If both positions and max weight are given then whichever is the stronger for each asset is the actual constraint n days of daily volume in portfolio On page 50 there is an example of imposing liquidity constraints on the trade Here we want to impose liquidity constraints on the portfolio Suppose that we want to restrict positions in the portfolio to be no more than 8 days of average volume Since positions is expecting monetary value rather than the number of asset units we first need to transform to money gt liquid value lt ave daily volume names prices prices gt any is na liquid value trouble if this is TRUE There are a couple things to note here We are assuming that the volume and the prices refer to the same asset units we don t want one to be in shares and the other in lots We ll see on page 56 why the occurrence of missing values would be upsetting Now we are ready to build our matrix to give to the positions argument If we have a long only portfolio then the matrix can be gt posmati lt cbind 0 8 liquid value If we have a long short portfolio then it would be gt posmat2 lt 8 cbind liquid value liquid value Let s do one more supposition Suppose that we have a long only portfolio and we want no position to be larger than 300 000 as well as having the liquidity constraint You might
104. f columns in the lin constraint object As with similar arguments lin abs can be a vector lin abs c TRUE TRUE FALSE Let s look at the four possible combinations of the values of lin trade and lin abs These are portfolio net not absolute the default portfolio gross absolute trade net not absolute e trade gross absolute For long only portfolios the first two are the same since all positions are positive An extreme example is to enforce all four types on the same constraint You want to name the constraints properly so that you can keep track of them gt con 4c lt build constraints cbind c tn countryvec ae c ta countryvec c pn countryvec c pa countryvec The result of the above command can then be used like lin constraints con 4c lin constraints lin trade c TRUE TRUE FALSE FALSE lin abs c FALSE TRUE The lin abs argument will automatically be replicated to be FALSE TRUE FALSE TRUE Of course we could have given all four values Long side Constraints and Short side Constraints This section applies to long only portfolios for trade constraints but not portfolio constraints For long short portfolios it applies to both trade constraints and portfolio constraints A long side constraint only involves assets that have long positions the short positions are effectively treated as zero for the purposes of the constraint Likewise a short side constraint only looks at asse
105. fferent opening hours asyn chrony can be caused by assets that do not trade often The illiquid assets will have stale prices A model that adjusts for asynchrony due to different opening hours is presented in Burns et al 1998 The expected return for a benchmark is not equal to the weighted average of the expected returns of its constituents This problem is pointed out in Michaud 1998 Minor violations are unlikely to have much affect but differences do distort the optimization The signs of covariances in the variance matrix are reversed for assets that are intended to be short While this could produce valid results if properly done it is confusing and totally unnecessary Portfolio Probe was developed with long short portfolios in mind bizarre manipulations can be eliminated Input Mangling While many mistakes are caught by the software it is quite possible to confuse the inputs One check is to see if the optimizer is using the objective that you intend You can do this by gt opt 1fobjective utility 1 mean variance risk aversion 0 02 The name of an argument is inadvertently dropped For example you add a starting solution to a command but forget to name it start sol so the optimizer thinks it is a different argument Most arguments are checked for validity but in some cases the checks may not be good enough You confuse the meaning of start sol with existing The start sol argument
106. folio utility 42 randport eval B8 39 66 rbind 29 read table 09 rep 87 103 summary 39 113 tail B83 traceback 130 164 trade optimizer 01H108 update 36 valuation I14 var add benchmark 112 S language debugging 2930 includes R 14 multi line command 15 S PLUS website 3 scale cost argument 20 scenario analysis 56160 sector constraint 34 sense making 121 short value definition 47 short side linear constraint 64 shrinkage see Bayesian simple return 11 small portfolio 28 soft constraint 78 79 split stock 22 172 stale prices 123 start not optimizing 33 134 start sol argument 123 34 statistical factor model 10 0012 5661 162 stock split 122 stringency argument 133 36 sum of largest weights constraint KAS summary function 59 13 tab separated file 109 14 tail function 88 taxes 18 20 three dimensional array 153 153 161 threshold argument 52153 127 threshold constraint positions argument 56 time decay 20 tol positions argument 57 traceback function 30 164 tracking error 151 constraint B4 112 64195 minimize 92 trade too trivial 23 trade optimizer function PTHTOS trading costs see costs trading strategy testing 27 29 transaction costs see costs troubleshooting 2729 turnover see trade value argument turnover argument 47 93 95 102 10
107. gainst the current benchmark and no more than 5 against our prediction of the benchmark after it is rebalanced Minimizing relative to two or more benchmarks is however somewhat more complicated that is discussed on page 50 Also a little more complicated is to get a benchmark relative utility but a constraint on the portfolio variance page 52 shows how to do that Maximize Benchmark relative Information Ratio If you want the the utility in this case the information ratio to be relative to a benchmark then you merely need to give the benchmark argument gt op ir5 lt trade optimizer prices varian long only TRUE expected return alphas gross value 1 4e6 benchmark spx You get substantially different behavior depending on whether you use benchmark or bench constraint e bench constraint does not affect which utility is used 96 CHAPTER 6 OPTIMIZING LONG ONLY PORTFOLIOS e benchmark changes the utility You can use both in which case benchmark changes the utility as usual but there is also a constraint on the tracking error Mean Variance Optimization The default objective is to maximize an information ratio whenever both vari ance and expected returns are given If you want to perform a mean variance optimization instead then you need to specify the utility argument The generic mean variance problem is to maximize the expected return of the port folio minus the risk aversion times the varian
108. gs varfac uniqueness varfac sdev 2 vlist lt list vardata vdata vartype 1 nvarfactors ncol varfac loadings a OM ae Oye yh The vdata object is a vector of the pertinent numbers in the correct order Note that the loadings are first multiplied by the standard deviations then this matrix is transposed Likewise the uniquenesses need to be scaled by the squared standard deviations Since this is producing a simple factor model the type of variance is 1 Finally the number of factors is given as the number of columns of the loadings matrix The vlist object is ready to be passed in as the variance argument caution Be careful when annualizing a factor model representation You want each element of the actual variance to be multiplied by some number such as 252 but not every number in the representation is multiplied by the same thing Chapter 15 Dregs This chapter contains topics that don t seem to belong anywhere else 15 1 Portfolio Probe Constituents The pprobe verify function is useful for seeing if the installation on a partic ular machine worked okay It is far from a full test suite it merely checks that all of the functions are present and that one particular problem optimizes okay This can also be used to see which version of Portfolio Probe is present 15 2 The Objectives The objective function of an optimization has a numeric value as its result and a set of inputs that can be changed The job of
109. hares then round lotting is automatic 4 2 Monetary Value of the Portfolio This section discusses the arguments e gross value e net value e long value e short value e turnover e long only e allowance 45 46 CHAPTER 4 CONSTRAINTS Table 4 1 Summary of constraints Arguments Constraint Section gross value net value monetary value of the portfolio long value short value allowance turnover tumnover buys plas se f ong omty mo short values if TRUE EA max weight maximum weight in the portfolio per asset AI universe trads restrii assets to he traded EJ lower trade lower and upper bounds on the upper erase cemunterohaner ants to wade _ mirado __number of assets to trade port size imimber of assets in the portfolio al threshold threshold constraints on trade and portfolio trades that must be done positions otherwise available constraints tol positions expressed in monetary terms lin constraints lin bounds linear and or count constraints lin trade on the portfolio lin abs and or the trade lin style lin direction alpha constraint bound on expected 4 scot the porto bound on squared tracking error dist center dist style dist bounds distance from one or more portfolios dist trade dist utility dist coef sum weight maximum of the sum of a specified A Mm ot the argot wets IT ER 4 2 MONETARY VALUE OF THE PORTFOLIO 47 While there is no
110. hat lin style count When lin style is count then count constraints user defined integer constraints are produced This can only be used with a categorical column of lin constraints a count constraint on a numerical column will trigger an error Let s consider some examples Suppose that we have a long short portfolio and we want an equal number of long positions and short positions from the UK and we want one more French long position than French short position We have no preference regarding Germany Then we would create a bounds object like gt lbci lower upper Country France 0 5 1 5 Country Germany Inf Inf Country UK 0 2 0 5 Our call would then include the arguments lin bounds 1bc1 lin style count We can accept the default values of lin abs and lin trade because we want a net constraint on the portfolio We can infer from the bounds object that the lin constraints argument contains just one constraint that is called Country Notice that in this context lin abs FALSE lin direction 0 long positions count as one but short positions count as negative one Also notice that we are giving bounds that are between integers you don t want to give bounds for count constraints that are integer Now suppose that we want exactly one German position in the portfolio and we want one or two French positions Our bounds object should now look like gt 1bc2 lower upper Country France 0 5 2 6 Cou
111. have no lots of ABC if less than 7 lots are held Portfolio thresholds are specified similarly to trade thresholds except they are in the third and fourth columns of the matrix instead of the first and second Here is an example gt thresh mi lt cbind 0 0 rbind ABC c 7 7 DEF c 5 8 gt thresh mi 1 21 3 4 ABC 0 0 7 7 DEF 0 Oo 5 8 If thresh ml is given as threshold then there should be at least 7 units of ABC either long or short if it is in the portfolio at all Also DEF should be at least 5 short or at least 8 long or not in the portfolio In this case there are no trading threshold constraints since the first two columns are all zero Summary of Threshold Inputs e A vector or a one column matrix symmetric trading constraints e A two column matrix symmetric or asymmetric trading constraints e A three column matrix first two columns are trading constraints third column is portfolio constraint for long only portfolios It is an error to give a three column matrix with long short portfolios e A four column matrix first two columns are trading constraints third and fourth columns are portfolio constraints 54 CHAPTER 4 CONSTRAINTS 4 6 Forced Trades This section discusses the argument e forced trade The forced trade argument is a named vector giving one or more trades that must be performed The value gives the minimal amount to trade and the names give the assets to be traded For example
112. he utility values component of op cred2 has length one while it is length two for op cred1 Since we only want one variance in the utility the utility that we want will have length one Multiple Scenarios Scenario analysis creates some hypotheses about what might happen in the future and then tries to pick the best course of action The assumption is that the future will look like one of the scenarios or perhaps some combination of scenarios Our first step is to create some data We will create a medium scenario a crash scenario and a prosperous scenario The probability that these numbers are reasonable is close to zero gt anam4 lt c equities bonds commodities cash gt acor medium lt matrix c 1 2 1 2 1 1 1 1 1 3 3 gt acor medium equities bonds commodities equities 1 0 0 2 Qd bonds 0 2 19 0 1 commodities 0 1 0 1 1 0 gt acor crash lt matrix c 1 3 3 3 1 2 3 42 1 35 39 gt acor crash LI 1 21 3 ft 1 0 0 3 03 2 0 3 1 0 0 2 3 1 03 0 2 1 0 gt acor prosper lt matrix c 1 4 3 4 1 2 3 2 1 3 3 14 7 MULTIPLICITY EXAMPLES 157 gt acor prosper Ct 1 2 R3 LJ 1 0 02 03 EX tt 1 0 Ge Jl Ss 0 18 gt avol medium lt c 20 8 12 avol crash lt c 35 16 20 avol prosper lt c 15 5 7 avar medium lt rbind cbind t acor medium avol medium avol medium 0 0 avar crash lt rbind cb
113. he expected returns and the positions in your 7 5 REAL TIME MONITORING 107 Figure 7 1 Constraints on gross net long and short values N oO Gross value 12 oO 2 LO O pes 179 Net value 2 wt m 4 5 6 7 8 Long value 108 CHAPTER 7 OPTIMIZING LONG SHORT PORTFOLIOS portfolio If you are willing to assume that the suggested trades always happen then you could merely have cur port lt cur opt new portfolio The technique here might be characterized as looking for the next small thing the size of the trade and the maximum number of assets to trade should probably be small 7 6 Going Farther Tasks that you might want to undertake To add trading costs to your optimization command see Chapter J on page I5 To add constraints see Chapter Jon page 45 To review common mistakes see Section 0 1 on page 21 To improve your solution or examine the quality of solutions you are get ting go to Chapter 2 on page 133 To export your solution to a file see Section B 3 Jon page 13 To perform optimizations with multiple variances and or multiple ex pected return vectors see Chapter 4 If you want a benchmark with a long short portfolio see Section 11 2 Chapter 8 General Use The core functionality of Portfolio Probe is generating random portfolios and optimizing trades More detailed description of these activities can be found in other chapters This chapter focuses on periphera
114. he input et cetera note The retmat object holds log returns also known as continuously compounded returns You may hear strong opinions on whether simple returns or log returns are preferred In mean variance optimization neither is entirely correct however the results are likely to be quite similar especially if daily or higher frequency returns are used caution Do not use daily returns when your data include assets from markets that close at different times global data in particular The asynchrony of the re turns means that the true correlations are higher than those that will be es timated from the data Thus optimizations will be distorted Weekly is the highest frequency that should be used with data that are substantially asyn chronous Note though that techniques do exist to adjust for asynchrony see Burns et al 1998 caution There is the possibility of confusion with the word factor In S an object that is of type factor is something that is categorical For example a vector of the sector or country of assets will at some point at least be a factor when used as a linear constraint A factor model does not contain any factors in this sense A large amount of the effort in factor model stat is dealing with missing values The default settings are reasonable for long only optimizations Possi bly a more appropriate matrix for benchmark relative analyses and long short optimization would be gt
115. he realized volatility Since we are using Spearman rank correlations it doesn t matter whether we use variance or volatility the ordering is unchanged when we convert from variance to volatility Next we bootstrap the correlations and compute 95 confidence intervals gt corboot Q lt rep list numeric 5000 8 gt names corboot Q lt names empty8 gt for i in 1 5000 this bootsamp lt sample 1000 1000 replace TRUE for j in 1 8 corboot Q j1 1 i lt cor rp30s15 shrexante Q j this bootsamp rp30s15 realized QL j1 this bootsamp method spearman corshrreal 95ci lt sapply corboot Q quantile probs c 025 975 t vtt 4 4 4 The confidence intervals aren t all that important but they provide a nice ex ample of showing how easy it is to perform statistical bootstrapping The dollar neutral and 120 20 portfolios are done analogously The key difference is in the call to valuation to get the realized volatility The type and cash arguments become important the assumption of how much cash there is makes a difference to the returns and hence to the volatility 5 4 Performance Measurement Shadowing The commands for this section are in file pprobe R_perfMeas_shadow txt The first thing is to create the optimized run as shown in Section B I Then the random runs that imitate the optimal run are done gt randser 20 w10 t200 lt pp serial rand 200 init 20 w10 pri
116. hile the other has a 400 turnover target Figure 2 1 shows the results for 2007 Both funds did reasonably well The situation changes if we extend the analysis through 2008 Figure 2 2 displays those results Table 2 T1 shows the numerical values for the analyses The percentile shows where the return of the actual fund falls among the returns of the random portfolios By convention performance percentiles are near zero for good results and near 100 for bad results One failing of this analysis is that the random portfolios are not guaranteed to obey the constraints at the end of the period Even so the results are likely to be useful and the effort to create the analysis is minimal The more elaborate analysis in Section 2 4 overcomes the constraint violation problem as well as being substantially more sensitive The number of random portfolios that were generated was 1000 This is not 2 2 TESTING THE EFFECT OF CONSTRAINT BOUNDS 21 Figure 2 2 MACD funds compared to random portfolios 2007 and 2008 o a o st a EN Sa a YN D Q o Q e 2 o 2007 2008 return percent enough to make the density plots completely smooth but is more than enough for the substantive part of the task at hand 2 2 Testing the Effect of Constraint Bounds Constraints are put on portfolios as a form of insurance We are willing to give up some up side in order to be protected from down side In general we are striking that
117. his example the only sector we are bounding is media Once a bounds matrix has been set up it can be used when building new constraint objects Suppose that we want to change from sectors to industries we can build new constraints like 4 8 LINEAR CONSTRAINTS 61 gt cons obj2 lt build constraints cbind country countryvec industry industvec bound bounds csw S code note In the cbind command we are assuming that countryvec and industvec have the same assets in the same order A safer approach would be gt ci inam lt intersect names countryvec names industvec gt cbind country countryvec ci inam industry industvec ci inam The bounds component of cons obj2 will have the same bounds for the coun tries as bounds csw and will have infinite bounds for the industries A bounds object that is actually used may contain extraneous bounds for example bounds for sectors when only countries are being constrained How ever it is an error not to give bounds for all of the constraints represented in lin constraints Only categorical constraints have been discussed so far We now look at numerical constraints Numerical Constraints Risk Factors Constraining risk factors is perhaps the most common numerical linear con straint In a fixed income portfolio duration would be an example Whether constraints are numerical or categorical the first step is to use build constraints gt cons obj3 lt build cons
118. hreshold constraints with one of these arguments though both arguments can be used in one problem Trade Thresholds Trade threshold constraints force trades to be at least a certain amount of an asset if it is traded at all The argument threshold c ABC 4 DEF 23 requests that at least 4 units lots perhaps of asset ABC and at least 23 units of DEF be bought or sold if they are traded at all This is not the same as forcing a trade of a certain size that is discussed in Section 6 When the threshold argument is a vector or a one column matrix then the constraint is taken to be symmetric Another way of stating the constraint given above would be threshold rbind ABC c 4 4 DEF c 23 23 where what we are giving to threshold looks like 4 5 THRESHOLD CONSTRAINTS 53 gt rbind ABC c 4 4 DEF c 23 23 pA 2 ABC 4 4 DEF 23 23 Give a two column matrix when you want different constraints for buying and selling for at least one of the assets threshold rbind ABC c 5 4 DEF c 0 23 The command above states that if ABC is sold then at least 5 units should be sold If ABC is bought then at least 4 units should be bought If DEF is bought then at least 23 units should be bought There is no threshold constraint if DEF is sold Portfolio Thresholds A portfolio threshold constraint states the minimum number of units of an asset that should be held in the portfolio For example you may prefer to
119. ike to get as close to that target as possible while still obeying all of your portfolio constraints Expected returns are not used for this and you don t have to have a variance matrix One of the simplest practical possibilities is a call like gt op di lt trade optimizer prices dist center target wts utility distance existing current portfolio turnover 30000 gross value 1e6 long only TRUE The key parts of this call are the dist center argument that takes a rep resentation of the target portfolio the utility argument and the turnover argument The call is taking advantage of the default value of dist style which is weight The target wts object should be a named numeric vector that sums to 1 where the names are asset ids Some sort of constraint on turnover is almost surely desirable That could be accomplished using costs but the turnover argument is a much easier option Another constraint that is quite likely in this setting is a tracking error constraint For this we do need a variance and the bench constraint argument must be specified 98 CHAPTER 6 OPTIMIZING LONG ONLY PORTFOLIOS gt op d2 lt trade optimizer prices dist center target wts utility distance existing current portfolio turnover 30000 gross value 1e6 long only TRUE variance varian bench constraint c spx 0 0472 252 In this case we are specifying a maximum tracking error of 4 assuming the variance was produced with da
120. ily data see Section 4 12 for more details The default style of distance as controlled by the dist style argument is weight In this style distance is considered to be the sum of absolute differences in the weights of the new portfolio and the target portfolio This is often how we conceive of distance but it has the disadvantage that the answer need not be unique An option is to add the argument dist style sumsqwi This changes the distance to be the weighted sum of squared differences in weights where the weights for the sum of squares are the inverse of the weights for the target portfolio Assets not in the target portfolio are ignored in this formulation of distance This form of distance may or may not be more pleasing Section 1 13 has more information on the arguments that control distance 6 5 Managing Cash Flow When you either are required to raise cash or have cash to put into the portfolio you can use optimization to find a good trade If there is significant cash flow then this process has the benefit of keeping the portfolio closer to optimal than it otherwise would be thus reducing the need for large periodic rebalances The examples given below assume an active portfolio merely eliminate the expected return argument if yours is passive Injecting Money into a Portfolio If you want to put more money into an existing portfolio then the main thing to say is that you don t want any selling You prob
121. ind t acor crash avol crash avol crash 0 0 tE Y EA SES S code note The last command creating avar crash does several things at once First in the center of the command it multiplies the correlation matrix times the volatility vector which multiplies each row of the correlation matrix by the corresponding volatility Then it transposes the matrix and multiplies again by the volatility vector this is now multiplying what originally were the columns of the correla tion matrix by their corresponding volatilities Then it binds on a column of all zeros for cash and finally binds on a row of all zeros Below you can see the result of all this manipulation once the asset names are put onto the variance matrix gt avar prosper lt rbind cbind t acor prosper avol prosper avol prosper 0 0 gt dimnames avar medium lt list anam4 anam4 gt dimnames avar crash lt list anam4 anam4 gt dimnames avar prosper lt list anam4 anam4 gt avar medium equities bonds commodities cash equities 400 32 0 24 0 0 bonds 32 64 0 9 6 0 commodities 24 9 6 144 0 0 cash 0 0 0 0 0 0 gt avar crash equities bonds commodities cash equities 1225 168 210 0 bonds 168 256 64 0 commodities 210 64 400 0 cash 0 0 0 0 gt avar prosper equities bonds commodities cash equities 225 0 30 31 5 0 bonds 30 0 25 FO 0 commodities 31 5 T 49 0 0 cash 0 0 0 0 0 0 gt aret medium lt c 8 4 5 3 158 CHAPTER 14
122. ing This currently produces data for 69 stocks rather than the original 70 BNI has been dropped If you are going to use the cached variance matrices then execute the file that contains those commands gt source pprobe R_varCache txt The static method of performance measurement is done with gt source pprobe R_perfMeas_static txt To test constraint bounds do gt source pprobe R_bounds txt Risk is explored with 5 1 PERFORMANCE MEASUREMENT STATIC 83 gt source pprobe R_risk txt The shadowing method of performance measurement is done with gt source pprobe R_perfMeas_shadow txt Backtests are evaluated with gt source pprobe R_varCache txt 5 1 Performance Measurement Static The file pprobe R_perfMeas_static txt gives the full set of commands for these examples Step number one is to create the initial portfolio This was done by opti mizing a portfolio gt init 20 w10 lt trade optimizer us pricemat 251 expected return us macdmat 251 port size 20 max weight 1 gross value 1e6 long only TRUE utility maximum return do warn c novar FALSE The 200 turnover path of optimal portfolios was created with gt ser 20 w10 t200 lt pp serial opt init 20 w10 pricemat us pricemat 1 251 alphamat us macdmat frac turnover 2 252 max weight 1 port size 20 do warn c novar FALSE utility switch FALSE keep port TRUE An objec
123. ing at a Given Solution A special type of optimization is to just return an object where the solution is the same as the starting value At first glance this seems like a useless operation but in fact there are many reasons to do this here are a few e See the cost of a solution 133 134 CHAPTER 12 ADJUSTING OPTIMIZATION SPEED AND QUALITY e See constraint violations of a solution e Get expected returns variances and so on of random portfolios this is what randport eval does e Ensure that another program is doing the same problem e Examine the utility of your current portfolio or of a proposed trade trade optimizer makes this easy Give the trade that you want to test as the start sol argument and set funeval max to zero or one An asset allocation example is gt aa gsi lt trade optimizer aprice avar mediun aret medium gross value ten thou long only TRUE utility mean variance risk aversion 05 start sol e60b40 funeval 0 gt valuation aa gs1 weight equities bonds 0 6 0 4 gt aa gsi utility value 1 2 08 gt aa gslfalpha value i 6 4 gt aa gsi var value 1 169 6 Note that setting iterations max 0 is not the same This still allows the pre iteration and post iteration operations and the result could be substantially different from the starting solution If the existing portfolio is where you want to stay then you can use the argument ntrade 0 12 2 Reducing Time Us
124. ive as lin bounds The lin constraints object must have column names so build constraints will add column names if they are not already there The column names are used to keep track of which bounds go with which constraint gt cons obj lin constraints 1 3 country sector ABC Spain media DEF France retail GHI Italy energy gt cons obj bounds lower upper country France Inf Inf country Italy Inf Inf country Spain Inf Inf sector energy Inf Inf sector media Inf Inf sector retail f Inf sector telecom Inf Inf The typical use of the bounds component is to create a separate object out of it and then change any of the infinite values desired Bounds and lin style What the bounds look like depends on the style that you pick for them Bounds on linear constraints may be in terms of either weight or monetary value The lin style argument controls this a value of weight versus a value of value lin style also accepts a third value count which is discussed on page 57 If we want bounds in terms of weights the default then we could create the bounds matrix something like bounds csw lt cons obj bounds bounds csw 1 2 lt c 1 2 4 5 bounds csw 3 2 lt 2 bounds csw 5 lt c 3 6 bounds csw Vo NE NE ON lower upper country France 0 1 0 4 country Italy U 2 0 5 country Spain Inf 0 2 sector energy Inf Inf sector media 0 3 0 6 sector
125. ively you could look at an individual component of the object such as the trade gt opti trade The prices that summary uses to value the portfolio are by default the prices used in the optimization You can give summary different prices gt summary opti price new prices You can also get the valuation of the individual assets as well as overall valuation using the valuation function gt valuation opti gt valuation opti trade TRUE The default behavior of valuation when given a portfolio object is to give information on the portfolio The trade can be examined instead by setting the trade argument to TRUE When valuation is given a vector the prices must be given gt valuation opti trade price new prices Prices are optional when portfolio objects are given the prices used during the optimization are used if none are specified Export the Trade The other task to be performed once you are satisfied with the optimization is to place the information elsewhere The deport function does this export is a logical name for this functionality but that has a different meaning in R The simplest use of this function is just to give the name of the portfolio object gt deport opti 1 opti csv The return value of the function is a character string giving the name of the file that was created If the optimization was performed in terms of lots but you want the file to reflect number of shares th
126. ization of active portfolios This is not absolutely true see page 96 for an example Additionally there needs to be a decision on how to balance expected return with risk Maximize the Information Ratio The easiest optimization that involves both expected returns and variance is to maximize the information ratio The information ratio of a portfolio is the expected return of the portfolio divided by the standard deviation of the port folio The standard deviation is the square root of the variance This is a 94 CHAPTER 6 OPTIMIZING LONG ONLY PORTFOLIOS natural quantity in mean variance land to maximize we re seeking the best risk adjusted return A command to maximize the information ratio is gt op iri lt trade optimizer prices varian long only TRUE expected return alphas gross value 1 4e6 The alphas object is a vector of expected returns of the assets When both variance and expected returns are given the default objective is to maximize the information ratio In this example 1 4 million currency units are put into the portfolio The information ratio is traditionally put into annualized terms If the variance and expected returns in your command are not annualized then you can annualize the information ratio by multiplying the utility by the square root of the number that you would use to annualize the variance and expected returns For example for a daily frequency the annualized information ratio wo
127. l tasks In particular there are hints about using the S language 8 1 Setting Up Data Before computations can be performed there needs to be data in place on which to do the computations Data that we may need include prices existing positions in the portfolio expected returns variances membership of assets in various groups such as countries and sectors and so on Prices and Other Imports You will probably need to import prices into S You may need to get other data into S as well A typical command to create a price vector is gt prices lt drop as matrix read table prices txt sep t or gt prices lt drop as matrix read table prices csv sep depending on whether your file is tab separated or comma separated Other file formats can be handled as well S code note These commands are assuming that there are two items on each line of the file an asset name and a number which is the price of a given quantity of the asset The read table function returns a type of S object called a data frame In S data frames and matrices are both rectangles of entries A matrix must have all of its entries the same type for example all numbers or all character 109 110 CHAPTER 8 GENERAL USE strings A data frame has the freedom to have different types of entries in different columns one column could be numeric and another categorical In this example read table creates a data frame that has one col
128. le When there are no names then the first column s are implied in order So all six of the following are specifying the same constraints var constraint vc vec var constraint vc imat var constraint vc 2mat var constraint vc vec nam var constraint vc imat nam var constraint vc 2mat nam where gt vce vec 11 24 1 5 gt vc 1mat Lyi 1 1 2 4 2 2 5 gt vc 2mat E 1 1 2 is 0 2 4 2 O 1 5 gt vc vec nam 0 1 2 4 1 5 gt vc imat nam Esti 2 4 1 5 vc 2mat nam Esid Dd 0 Oo 2 4 1 0 4 5 Veo All of these are saying the variance benchmark combination in the first column of the variance table has an upper bound of 2 4 and the variance benchmark combination in the second vtable column is to be no more than 1 5 If you were to give the equivalent of vc vec nam directly you would do var constraint c 0 2 4 1 1 5 A very similar command means something quite different 14 4 EXPECTED RETURN CONSTRAINTS 147 var constraint cbind 2 4 1 5 means that the first variance is constrained to be at most 1 5 but no less than 2 4 In this case you will get an error but you can not depend on getting an error when you confuse the direction of matrices Two column matrices give lower bounds as well as upper bounds on the variance Lower bounds are most likely to be useful in generating random portfolios but could potentially be of interest in optimization as well Consider the matrix
129. lue might look like dist center target values dist style value dist bounds 25000 while using shares might look like dist center target shares dist style shares dist bounds 25000 The dist trade argument takes a logical vector and allows you to constrain the distance of the trade rather than the portfolio The dist utility argument also takes a logical vector TRUE values of this imply that you want to use it as the objective in optimization For more on this see Section 6 4 Alternative prices By default distance is defined by the prices vector You don t have to define it that way You can include a value for dist prices that gives different prices to be used in the computations of distances The dist style argument takes a value with custom before the style when you want to use the dist prices values For example you might have dist center target shares dist style customshares dist prices some price vector dist bounds 35000 Multiple distances It is possible to have more than one distance constraint In this case the dist center argument needs to be a list For example you might have ar guments that look like dist center list target1 wt target2 share target3 wt dist style c weight shares weight dist bounds c 0 2 2e5 0 3 74 CHAPTER 4 CONSTRAINTS Here we have three distances declared the first and third in terms of weight and the second in terms of shares If you want lower
130. ly slightly more than 1 neg dest wt optimization only There is at least one negative destina tion weight in the utility table This is almost surely wrong unless the problem is outside typical portfolio optimization neg risk aversion optimization only There is at least one negative risk aversion value Risk aversion is traditionally thought to be non negative but you could have a special situation no asset names One or more objects do not have asset names and hence require that they be in the same order as the assets in prices Some objects are allowed to not have asset names in order to facilitate quick experiments asset names are always recommended for production situations noninteger forced There is a forced trade that is constrained such that the trade must be non integer This could be something you want to do but can mean that something is wrong with the specification of constraints 126 CHAPTER 10 PRACTICALITIES AND TROUBLESHOOTING e nonzero penalty optimization only The penalty is non zero meaning that not all constraints are satisfied e notrade The formulation does not allow any trading If you really want no trading then the recommended approach is to set ntrade 0 e novariance optim optimization only An optimization other than with a distance is requested but no variance is given This can be a reasonable operation for some long short situations but is generally suspicious e penalty size
131. mall If there are only a few assets allowed in the portfolio the sum of allowable maximum weights can be less than 1 The error of max weight being too small will be caught in simple cases However it is possible for the test to be fooled by a combination of restrictions on trading Another possibility is that the ranges of the money values gross value etc are too narrow 10 5 S LANGUAGE PROBLEMS AND SOLUTIONS 129 The optimizer is not performing a trade that I am insisting upon via the lower trade or upper trade argument to trade optimizer Use the forced trade argument see Section 1 6 What if I don t have prices You can do an asset allocation and effectively get weights back See page D9 I m trying to generate random portfolios with constraints I know work but it fails You can set the S language seed to some number and try the call to random portfolio again For example gt set seed 123 10 5 S Language Problems and Solutions This section provides a little help with some of the aspects of the S language that you might be most likely to bang your head against Some hints for the R beginner on the Tutorials page of the Burns Statistics website might also be of help Creating Matrices There are a number of places in this document where we create a matrix This process generally involves either cbind or rbind cbind as in column bind and rbind as in row bind That sounds simple enough
132. measures of market risk There are other sources of risk credit risk for example that could be included in an optimization For more on credit risk see for example While we ll speak of credit risk here keep in mind that other forms of risk could be used instead or in addition Suppose that we have var market and var credit that represent the two types of risk While it may be possible to scale the credit risk so that it is com parable to the market risk it may be easier to accept that they are different If the two forms of risk are not comparable then you can perform the optimization with the market risk as you normally would and put a constraint on the credit risk When deciding how to constrain the credit risk the first thing to do is explore its limits One end of the range is when credit risk is ignored entirely in the optimization the other end is when the credit risk is minimized with no concern for the market risk gt op cred0 lt trade optimizer prices var market gt trade optimizer prices var credit funeval 0 start sol op cred0 var value 1 26 02844 gt trade optimizer prices var credit utility minimum variance var value 1 11 92467 So the range of interest is about 12 to 26 If we decided to constrain the credit risk to 18 then we would do gt var mult lt array c var market anam anam var credit anam anam c length anam length anam 2 list anam anam
133. mething that provides a utility Monetary Value The turnover or the amount of money both long and short in the portfolio needs to be constrained Details can be found on page 43 Utility At least one of the arguments e variance 101 102 CHAPTER 7 OPTIMIZING LONG SHORT PORTFOLIOS e expected return e dist center must be given The dist center argument and its mates are used to do utility free optimization that is to minimize the distance to an ideal target portfolio This is likely to be more common for long only portfolios but there is no reason not to do it for long short portfolios as well If you do want to do it see Section 6 4 which assumes long only Assuming distance is out of the picture If expected return is not given then the utility is the minimum variance If a variance is not given then the utility is the maximum expected return If both are given then the default utility is to maximize the information ratio 7 2 Examples This section presents some examples Feel free to skip over those that don t apply to you These examples do not include arguments for the cost of trading see Chapter D Nor does it cover several of the constraints see Chapter Maximize the Information Ratio If you have a set of expected returns and a variance then maximizing the information ratio is a natural thing to do and it is the default This can be done simply with a command similar to gt opt ir
134. mization There are two forms of failure e the objective does not achieve objective limit e the objective is okay but there are other broken constraints The objfail max argument controls how many of the first type are allowed If objfail max 1 the default and the first try does not achieve the objective limit then an error is triggered The calls look like gt rpi lt random portfolio 100 the prices gt rui lt random portfolio utility 100 0 6 the prices 3 8 GOING FARTHER 43 3 8 Going Farther Tasks that you might want to undertake e Chapter discusses how to specify the constraints e To review common mistakes see Section QJJon page 121 e Chapter D shows the commands used for examples of random portfolio tasks 44 CHAPTER 3 GENERATING RANDOM PORTFOLIOS Chapter 4 Constraints This chapter covers the constraints that can be imposed for generating random portfolios and optimization random portfolio and trade optimizer use the exact same set of constraint arguments 4 1 Summary of All Constraints Table Z_J lists the possible constraints along with the arguments used to achieve them In addition to these explicit constraints there is the implicit constraint of trading integer numbers of asset units Round Lots Trading is only done in integer numbers of asset units except when there is an existing position that is non integral Thus if the prices are given for lots as opposed to s
135. n allows a name to be put on the value The 0 04 is squared to transform the tracking error into a variance This is then divided by 252 in order to go from the annual value to daily which is the frequency of the variance we are assuming in this example If the variance were 6 3 EXAMPLES FOR ACTIVE PORTFOLIOS 95 annualized and created from returns in percent then the bench constraint argument would have been c spx 472 note The benchmark argument is not given in the previous command If it had been then the information ratio that was optimized would be relative to the benchmark A more realistic command might be gt op ir3 lt trade optimizer prices varian long only TRUE expected return alphas gross value 1 4e6 turnover 1e5 ntrade 15 bench constraint c spx 0 04 2 252 max weight 05 existing cur port port size 60 This command includes the existing portfolio has a limit on the turnover and specifies that the final portfolio should contain no more than 60 assets You can easily add constraints to other benchmarks as well just make the bench constraint argument longer gt op ir4 lt trade optimizer prices varian long only TRUE expected return alphas gross value 1 4e6 turnover 1e5 ntrade 15 bench constraint c spx 0 04 2 252 newspx 0 0572 252 max weight 05 existing cur port port size 60 In this example we are constraining the tracking error to be no more than 4 a
136. ng optimized is the 70 quantile of the set of information ratios using each of the variances 154 CHAPTER 14 ADVANCED FEATURES If you are doing a variance constraint for instance maximizing the informa tion ratio in a long short portfolio as on page 03 then the approach is slightly different You need to make the variance constraints argument the proper length yourself gt opts m2 lt trade optimizer prices varian expected return alphas gross value 1e6 net val c 1e4 2e4 ntrade 55 max weight 05 var constraint rep 0472 252 3 If the value given as the variance constraint weren t replicated then only the first variance would be constrained The var constraint argument is the more general Benchmark constraints actually create variance constraints the bench constraint argument only ex ists for convenience When more than one variance is given bench constraint puts the constraint on each of the variances More control can be gained over variance constraints by putting names on the vector given The names need to be the zero based index of the columns of the variance table see page 45 For example to constrain the first variance to a volatility of 4 and the third to a volatility of 5 the command would be gt opts m3 lt trade optimizer prices varian gt expected ret alphas gross val 1e6 net val c 1e4 2e4 ntrade 55 max weight 05 var constraint c 0 0472 252 2 0572 252
137. ngency criterion is satisfied if the best trade has been found again in different runs stringency times For example suppose stringency is 1 After each run we have the best found so far If the result of the new run is the same the same assets traded the same number of units traded as the best found previously then the stringency criterion is met and the process stops If stringency were 2 then it would take three runs with the same result to end the process When stringency is positive there are three types of runs e initial runs e final runs e non convergence run Initial runs perform the optimization as stated The number of initial runs is determined by the runs init control argument If the stringency hasn t been satisfied in the initial phase then the final runs are started The key difference is that only assets that were involved in trades in the initial phase are eligible to trade in this phase The number of final runs is controlled by runs final If the stringency is still not satisfied at the end of the final runs then one more run is done This run starts with the best solution found in any of the previous runs The restriction to trading as in the final runs remains in place The maximum number of iterations allowed in this run is a multiple of the maximum number of iterations for the other runs that multiple is given by the nonconverge mult control argument 136 CHAPTER 12 ADJUSTING OPTIMIZATION SPEED AND QUALIT
138. ntry Germany 0 5 1 5 Country UK Inf Int We would have arguments lin bounds 1bc2 lin style count lin abs TRUE We are adding lin abs to the call because we want an absolute constraint If we want exactly two long UK positions and we want no short French positions then we need to do a bit more work The extra work is because we now have two constraints a long side constraint and a short side constraint so we need two columns in lin constraints not the one column that we have been using We start by building the appropriate object 4 10 ALPHA EXPECTED RETURN CONSTRAINTS 69 gt lcc3 lt build constraints cbind Country longc countryvec Country shortc countryvec Now we get the bounds matrix that we need gt lbc3 lt 1cc3 bounds gt 1bc3 gt 1bc3 3 lt c 1 5 2 5 gt 1bc3 4 2 lt 5 gt 1bc3 lower upper Country longc France Inf Inf Country longc Germany Inf Inf Country longce UK 1 5 2 5 Country shortc France Inf 0 5 Country shortc Germany Inf Inf Country shortc UK Int Inf Now we ll use arguments lin constraints lcc3 lin constraints lin bounds 1bc3 lin style count lin direction c 1 1 Logical values are allowed to define constraints Logicals are probably most likely to be used in count constraints Here we have a somewhat contrived example as if the others aren t where we want to lower the beta of the portfolio We lower beta if we sell high
139. o real time optimization within a C or C environment However doing real time optimization in S is quite easy as shown on page 108 If you are determined there is a function in the package that can help you write C code The Cfrag list function will write a file that has the C initialization of variables given in an S language list The list that you would want to do this for is an intermediate result of an optimization For example gt op clist lt trade optimizer prices varian intermediate Clist gt Cfrag list op clist 1 3 file test c The test c file now holds a fragment of C code initializing most of the variables that are passed into the C function that does the optimization This can be of use in two ways either to initialize some variables that will always be the same or at least mostly the same in your application or to test your implementation as you are writing it To create a C program you essentially need to recreate the trade optimizer function The S language code for trade optimizer is the documentation for how to write the C code 15 4 Bug Reporting Send email to patrick burns stat com to report any bugs that you find A report should include e The operating system including the version that you are using e The version of R or S PLUS that you are using e Give the version component optimization or attribute random portfo lios For example gt op version C code S code p
140. olBurSt which is for objects created by trade optimizer If deport is given an object without a class then an error occurs because there is not a default method However the methods can be used directly 3 5 COMBINING RANDOM PORTFOLIO OBJECTS 41 3 5 Combining Random Portfolio Objects You may want to combine some random portfolio objects Suppose you have objects named rp1 rp2 and rp3 resulting from calls to random portfolio You would like these to be in one object as they all have the same constraints or perhaps they have slightly different constraints but you want them all in the same analysis The c function will put them all together gt rp all lt c rp1 rp2 rp3 But not all is well gt deport rp all Error in deport rp all no applicable method for deport gt summary rp all Length Class Mode 1 45 none numeric 2 45 none numeric 3 1 45 none numeric 4 45 none numeric Even though rp all is basically correct it doesn t have the class that the other objects have Without the class generic functions like summary deport and valuation don t work as expected gt class rp all lt class rpi gt deport rp all 1 rp all csu Once the class is put on the object we can operate as usual Almost If you want to use randport eval then you need the call attribute as well In that case you could give the big object all of the attributes of one of the original objects gt
141. on trade fraction of gross gross net long short 2 00683544 0 01830419 1 01256982 0 99426563 constraints realized constraints realized linear lower upper realized nearest violation Country France 300 0 26 09 28 09 NA 4 9 COUNT CONSTRAINTS 67 Country Germany o 100 Zi ie 20 OF NA Country UK 100 2000 175 39 75 39 NA lin trade 1 FALSE lin abs 1 FALSE lin style 1 value lin direction HAJ o An example for optimization is gt tail summary op obj6 valuation trade fraction of gross gross net long short 0 05179854 0 02020499 0 03600176 0 01579677 constraints realized constraints realized linear lower upper realized nearest violation Cntry France 300 0 0 00 297 6500 2 350e 00 NA Cntry Germany 0 0 100 00 11 4600 1 146e 01 NA Cntry UK 100 0 2000 00 782 1500 6 822e 02 NA Beta 0 9 0 95 0 9001 1 060e 04 NA lin trade 1 FALSE lin abs 1 FALSE lin style Cntry Beta value weight lin direction 1 O Notice that if a linear argument is the same for all constraints then just one value is printed On the other hand if an argument has more than one value lin style in the example above then all values are printed including names of the constraints 4 9 Count Constraints This section continues the description of arguments e lin constraints e lin bounds e lin trade e lin abs e lin style 68 CHAPTER 4 CONSTRAINTS e lin direction but with the restriction t
142. one Polynomial costs are discussed below A more telling use of cost par is gt trade optimizer long buy cost cost poly cost par c 0 1 1 5 2 34 In this case if ABC trades 30 lots and DEF trades 25 lots then the cost for ABC will be 0 01 30 02 301 5 3 586 The cost for DEF will be 03 02 25 01 251 01 2523 17 952 When cost par is used there is a restriction that all of the cost matrices must have the same number of columns as cost par In actuality there is no loss of generality as columns of zeros can be added to matrices as required and cost par can be the union of all of the desired exponents We ll see in the next section that in a long short portfolio when a trade makes an asset go from long to short or vice versa that there is a choice of having the intercept counted in both instances or just the first There is no such choice when using cost par it doesn t know about zero being the intercept and the full cost is always used Polynomial Costs This subsection describes creating polynomial costs Using the more general cost par argument is usually preferred so feel free to skip this subsection Polynomial costs are given by a matrix The order of the polynomial is one less than the number of columns in the matrix The first column contains the intercept the second column has the linear coefficients the third column has the quadratic coefficients etc caution Do no
143. ong value and short value axes The constraints displayed in the figure are the gross value between 10 and 12 and the net value between 1 and 2 Constraints on the long value and the short value will be vertical and horizontal sections so adding these change the shape of the allowable region 7 5 Real Time Monitoring You may want to perform real time optimization The prices will change from minute to minute and if your expected returns change continuously as well then constantly updating the optimization could be valuable Such a process is very easy to set up Here is an example gt repeat prices lt update prices user defined function alphas lt update alphas user defined function cur port lt update portfolio user defined function cur opt lt trade optimizer prices varian expected return alphas turnover 5e4 ntrade 5 existing cur port if length cur optftrade deport cur opt what trade Ed J S code note In S repeat performs an infinite loop Generally the body of a repeat loop will contain a test of when the loop should be exited In this case there is no such test the optimizations will continue to be performed until you explicitly interrupt the loop It would be easy enough to use the date function to check the time of day and exit the loop after the close of trading This example has three functions that you would need to write These functions update the prices t
144. opts2 var val 1 0 008227913 0 009735447 gt opts2 utility val 1 3 761031e 07 3 761069e 07 The utility values are still equal but now the utility represents something dif ferent for the current benchmark than for the new benchmark so the tracking 152 CHAPTER 14 ADVANCED FEATURES errors are different now they are 82 basis points for the current benchmark and 97 for the new benchmark There is not a limit on the number of benchmarks that you can use just add the names to the vector given as the benchmark argument If you have an active portfolio you may want to put on constraints relative to each benchmark This is done precisely the same as with a single benchmark constraint Adding a second benchmark constraint is just like adding any other constraint it does not create multiple utilities An example is given on page 05 Benchmark relative Utility and Absolute Variance Constraint We saw on page DZ that it is easy to have a utility in absolute terms on the portfolio with a benchmark constraint The reverse problem benchmark relative utility with a constraint on portfolio volatility is just as easy to state but not as easy to do We need to create a variance table to pass in as the vtable argument It needs a column with the benchmark to use in the utility and a second column without the benchmark for the constraint We can do this with gt vtab buav lt rbind c 0 0 c 0 1 c 1 0 gt attr vtab buav
145. ormed before this condition occurs then there is no convergence for the run When stringency is positive the convergence is a little more complicated cost intercept nonzero One or more of the cost arguments have non zero values for the intercept This only applies when polynomial costs are given This may well be correct but it is easy to forget the intercept column in the cost matrix dist prices One or more of the assets that are in prices are not in a component of dist prices and hence are assumed to be zero dist style Since it is easy to make a mistake between giving shares or values to the dist center argument a check is made to see if what is stated in dist style fits into the gross value range that is given or inferred dist zero One or more of the assets in a component of dist center have a non zero value but a price of zero in dist prices extraneous assets One or more objects contain data on assets that are not in prices and hence not of use in the computation ignore max weight One or more assets in the existing portfolio break their maximum weight constraint and either maximum weights are not forced to be obeyed or the trade can not be forced to be large enough See page 49 infinite bounds All bounds in lin bounds are infinite meaning there are no bounds at all max weight restrictive The vector of maximum weights is very re strictive That is the sum of a subset of the maximum weights is on
146. ortfolio Probe has been written to work with either version of S This document assumes you are using S as opposed to using C code Portfolio Probe uses only very general features of S so it should run the same in any version of R or S PLUS When this document says S it means either R or S PLUS the term S should not be construed to mean only S PLUS Some of the examples explicitly assume R is being used the same effect would be done slightly differently in S PLUS Programming experience is not mandatory whatever your objective there is likely to be an example provided in this manual that is close to your case The present document assumes knowledge of S to the level of Some Hints for the R Beginner a brief structured introduction which can be found in the Tutorials section of http www burns stat com Commands beyond that level are included and explained While it is reasonably easy to start using Portfolio Probe there is a lot of room to grow Portfolio Probe s flexibility and the power of S can carry you a long way 1 3 Installing the Software If you are using R and your machine has access to the internet then you can install Portfolio Probe with gt install packages PortfolioProbe repos http www portfolioprobe com R See the Frequently Asked Questions in the User area of www portfolioprobe com if a firewall blocks this command or if you are using S PLUS 1 4 Loading the Software
147. ortgen _BurSt 1 17 trade optimizer 023 gt attr randport version C code S code randport_BurSt 1 17 random portfolio 010 e A full description of the problem what happens and what you wanted to happen e IfS creates an error then give the results of the call gt traceback This displays the state that S was in when the error occurred 15 4 BUG REPORTING 165 e If possible send the data for a small problem that reproduces the bug e If the bug is sporadic and uses random portfolio then also give the random seed from an object where the bug is observed gt attr ranport seed A reward is given to the first to report a bug in the code or documentation Ideas for improving the ease of use or functionality are always welcome 166 CHAPTER 15 DREGS Bibliography Almgren et al 2005 Almgren R Thum C Hauptmann E and Li H 2005 Equity market impact Risk Burns 1998 Burns P 1998 S Poetry http www burns stat com Burns 2003a Burns P 2003a On using statistical factor models in optimiz ing long only portfolios Working paper Burns Statistics http www burns stat com Burns 2003b Burns P 2003b Portfolio sharpening Working paper Burns Statistics http www burns stat com Burns 2009 Burns P 2009 The R Inferno Tutorial Burns Statistics http www burns stat com Burns et al 1998 Burns P Engle R and Mezrich J 1998 Correlations an
148. ory type source repos http www burns stat com R Once the package is installed for your version of R you can do gt library BurStFin in each session in which you want to use its functionality this command might be slightly more complicated if you are using S PLUS The factor model stat function in BurStFin creates a variance matrix using a factor model The command to get a variance matrix can be as simple as gt retmat lt diff log price history gt varian lt factor model stat retmat Alternatively a variance that shrinks towards the equal correlation matrix can be computed with gt varian lt var shrink eqcor retmat 8 1 SETTING UP DATA 111 S code note Here price history is a matrix of asset prices with the columns corresponding to the assets and the rows corresponding to times the most recent time last The calculation produces a matrix of returns which we call retmat in this case The vector that is used as the prices argument could be the final row of the price history matrix The log function returns an object that looks like its input but each element of the output is the natural logarithm of the corresponding element of the input The diff function when given a matrix performs differences down each col umn The first row of the result is the second row of the input minus the first row of the input the second row of the result is the third row of the input minus the second row of t
149. over 93 95 102 unsatisfiable 41 42 utility 2 variance 0 103 145 1470 54 violation 65H66 weight 19H75 constraint bounds testing BI 22 SA 186 constraints realized function 65 continuously compounded return 11 conventions typography 15 Convergence 125 costs 115 amortization 19 asymmetric 115 constraint 75 linear 16 nonlinear 6H piecewise linear 20 polynomial 17H18 power law I8 119 relative to expected returns 1197 20 relative to noise 136 square root LI8 119 taxes 118 20 count constraint 67H69 country constraint BA cov wt function 10 credit risk 55H50 csv file 40 109 MA currency 121 data frame 09 debugger function 30 debugging S 29130 decay time 20 deport function 40 deport functionexport function 114 dest wt argument 143 49 destination of utility 149 weight 118 149 distance constraint 72H74 optimization 169 do call function B9 do warn argument 117 1244128 dollar neutral portfolio BA 43 doubleconst argument I8 drop function 109 110 dual benchmarks duration 61 enforce max weight argument 49 error produced by S 164 examples 921109 10211100 56 760 existing argument 123 expected returns 93 multiple 148 150 1150 relative to costs LIS 120 factor model full OU fundamental 153 macroeconomic 153 simple 161 102 stati
150. pn Italy 0e 00 1e 05 92422 7578 NA c pn Spain 1e 05 0e 00 76340 23660 NA c pa France 3e 05 4e 05 394621 5379 NA c pa Italy 3e 05 4e 05 304686 4686 NA c pa Spain 3e 05 4e 05 400398 NA 398 This contains a matrix where each row has information on a sub constraint The final column indicates the size of violations if all of the elements in this column are NA then there are no violations A negative violation means that the realized value is below the lower bound and a positive violation means that the realized value is above the upper bound The next to last column gives proximity of the realized value to the nearest bound if there is not a violation The same convention in terms of sign is used a negative number means the realized is closer to the lower bound The last two columns have exactly one missing value for each row In this example the sizes of the violations are relatively trivial However they would not be trivial in terms of their affect on the objective if you were optimizing any violation means that the optimizer has been concentrating on getting a feasible solution a solution that merely satisfies the constraints and may not be near the best feasible solution At times there is a non zero penalty that is of trivial size relative to the utility you need not worry in such cases 66 CHAPTER 4 CONSTRAINTS Evaluating Un imposed Constraints It is possible to look at the value of constraints that were not impos
151. portfolio A warning may not be given since max weight merely limits the extent of trading of the assets One case is if the control argument enforce max weight is FALSE When this argument is TRUE the default then forced trades are automatically built 50 CHAPTER 4 CONSTRAINTS to make the positions conform to their maximum weights This is done for the maximum of the range of the gross value If the range for the gross value is large and the resulting portfolio has a gross value substantially smaller than the maximum allowed gross then some maximum weights could be broken It is also possible that a maximum weight is broken by enough that the trade can not be forced to be large enough In this case the trade is forced to be as large as possible if enforce max weight is TRUE If the maximum weight is the same for all assets then you can ensure that it is obeyed by using sum weight well either it s obeyed or you see a warning about it To just constrain the largest weight the sum weight argument see Section 14 would look like sum weight c 1 1 This restricts the largest weight to 10 universe trade The universe trade argument should be a vector of character strings contain ing some of the asset names Those will be the only assets allowed to trade If this argument is NULL the default then there is no constraint on the assets to trade except possibly for benchmarks lower trade and upper trade The lowe
152. ptimizer that the final two variances are not to be used in the utility We do this by modifying the utility table from the dummy run Utility tables are discussed on page 49 gt qc utiltab lt qc temp utable gt qc utiltab 1 21 63 alpha spot 0 0 0 variance spot 0 1 2 destination 0 1 2 opt objective 1 1 1 risk aversion 1 1 I wt in destination 1 1 1 gt qc utiltab destination lt c 0 1 1 gt qc utiltab Lay G3 E alpha spot 0 0 0 variance spot 0 1 2 destination 0 f 1 opt objective 1 1 risk aversion 1 1 1 wt in destination 1 1 1 Actual Computation Now we are ready for the actual computation If we are generating random portfolios we ll have a command like gt qc rp lt random portfolio 100 prices varian3 vtable qcvtab var con Alternatively if we are optimizing we d do something like gt qc opt lt trade optimizer prices varian3 vtable qcvtab utable qc utiltab var con 4 18 Constraint Penalties and Soft Constraints This section applies only to optimization it can not apply to generating random portfolios Virtually all of the constraints are enforced via penalties That is instead of the optimization algorithm minimizing the negative utility it minimizes the negative utility plus penalties for whatever constraints are violated Solutions that obey all of the constraints experience no penalties The algorithm actively attempts
153. quarter 2 quarters and the whole year We have the constraints e long only e no more than 20 names in the portfolio e no asset weight greater than 10 e 200 turnover target Figure 2 13 shows how good MACD was for these constraints during the first quarter of 2008 It has close to zero value on average that is it is neither good nor bad However for a specific starting portfolio it tends to either significantly outperform or significantly underperform If the strategy truly had no effect then the percentiles would fall roughly along the diagonal line That the per centiles tend to be zero or one hundred is an indication of the high power of the technique Each strategy is compared to only 100 random runs but adding more random runs would at most move the percentiles slightly FigureP 14 shows how the strategy did for the first half of the year It did exceptionally well the strategy outperformed all random paths for all but one of the starting portfolios in the exception it beat almost all the random paths The picture for the whole year Figure 15 is almost at the other extreme For the whole year the strategy significantly underperformed with almost all of the starting points From those plots we don t know if the amount of outperformance or under performance is large or trivial To learn that we can look at the return of the strategy relative to the mean return of the random paths That is we compare the strategy return
154. r than 203 Note that this tolerance only applies to assets that are already in the existing portfolio Summary of positions inputs e 2 column matrix lower and upper constraints on the value of portfolio positions e 4 column matrix 1 2 constraints on portfolio positions 3 4 constraints on the value of sells and buys in the trade e 8 column matrix 1 2 constraints on the portfolio positions 3 4 con straints on the trade 5 6 threshold constraints on the values of trades 7 8 threshold constraints on the values in the portfolio Note portfolio trade trade portfolio 4 8 Linear Constraints This section describes the arguments 58 CHAPTER 4 CONSTRAINTS e lin constraints e lin bounds e lin trade e lin abs e lin style e lin direction These arguments create linear constraints on the portfolio and or the trade These arguments can also impose count constraints a topic that is discussed in the next section Portfolio Probe makes a distinction between numeric constraints and con straints based on categories An example of the first is a bound on twice the value of asset ABC minus 1 5 times the value of asset DEF minus the value of asset GHI An example of the second is bounds on the value from each country in the asset universe Most other optimizers will expand such constraints into a matrix of zeros and ones you could do this also but the method used here is more compact as well as easier for the user
155. r constraint argument constrains the value of the variance In its most common usage it is merely a number giving the upper bound for the variance Using the argument var constraint 1 4 imposes an upper bound of 1 4 on the variance The next most common usage is a two column matrix that gives a range of values that the variance is allowed to be in For example var constraint cbind 1 2 1 4 says that the variance is to be at least 1 2 but no more than 1 4 The value given looks like gt cbind 1 2 1 4 11 1 2 WIL 14 More advanced use of this argument predominantly when multiple variances are given is discussed on page 45 4 12 TRACKING ERROR BENCHMARK CONSTRAINTS 71 4 12 Tracking Error Benchmark Constraints This section discusses the argument e bench constraint The bench constraint argument is used to constrain squared tracking errors There has to be information on not only the numerical bound that you want but the benchmark as well The benchmark needs to be an asset in the variance but it doesn t need to be in the prices unless benchmark trading is allowed Ultimately bench constraint creates a variance constraint it is just a more convenient way of specifying some commonly desired constraints Single Upper Bound If you want just an upper bound then give a named numeric vector of length one For example bench constraint c spx 0472 specifies a 4 tracking error from spx Another
156. r trade and upper trade arguments are given in asset units shares lots or whatever As their names suggest they limit the number of units that can be sold or bought Like max weight they can be given a single unnamed number or a vector with names corresponding to assets Assets not named are not restricted Values in lower trade can not be positive and values in upper trade can not be negative This would be forcing the asset to trade You can force trades but not with these arguments see forced trade page B or possibly positions page B One use of these arguments is to create a selection of assets to buy and a selection of assets to sell as seen on page 03 for long short portfolios and on page 96 for long only portfolios More traditional uses include imposing liquidity constraints and constraining the amount of particular assets in the final portfolio Liquidity Constraints Suppose that you want to constrain trades to a certain fraction of average daily volume The commands you would use to do this would be similar to gt liquidity lt 0 25 ave daily volume Then you could use this like upper trade liquidity lower trade liquidity 4 4 NUMBER OF ASSETS 51 The process of constraining the liquidity merely involves computing the max imum amount of each asset that should be traded setting the upper trade argument to this amount and setting the lower trade argument to the nega tive of that amount In thi
157. rade The argument ntrade 25 says that no more than 25 assets will be traded If you do not want a limit on the number of assets traded then set ntrade to the size of the universe or larger The default is to have no limit If you desire a minimum number of assets traded then give ntrade a length two vector For example ntrade c 4 25 52 CHAPTER 4 CONSTRAINTS states that the number of assets traded needs to be between 4 and 25 inclu sive A minimum number to trade is most likely to be useful when threshold constraints Section 4 5 are used as well Otherwise trading just one unit of an asset counts The minimum number to trade is more useful for generating random port folios than in optimizing Number of Assets in the Portfolio The port size argument constrains the number of assets in the constructed portfolio Like ntrade this can be given as either a single number meaning the upper bound or as a length 2 vector giving the range The argument port size 50 means no more than 50 assets may be in the portfolio while port size c 50 50 means that exactly 50 assets need to be in the portfolio 4 5 Threshold Constraints This section discusses the argument e threshold The threshold argument controls two types of threshold constraints trade thresholds and portfolio thresholds Threshold constraints may also be specified with positions page B2 Note that in any one problem you can only declare t
158. rade is forced either if the range for the portfolio does not include the current position or if the trading range does not include zero This substitutes for using forced trade Forced trades from positions cause the output of trade optimizer to have a component named positions forced which will show the forced trades created Universe constraints If there is a missing value NA in any of the first four columns of positions then the corresponding asset is not allowed to trade It is an error to have a missing value in the fifth through eighth columns These constraints could be achieved by using the universe trade argument Be careful not to let missing values stray in by accident The summary of optimized portfolios tells how many assets positions forces not to trade Threshold constraints The 5th through the gth columns of positions are for threshold constraints All four of these columns need to be given if any are given Note also that if these columns are given then the threshold argument can not be given You can put null constraints in columns that you don t want to have influ ence The most extreme possibility in this regard is if you want to use positions to only impose a portfolio threshold on a long only portfolio Suppose you want only positions that have at least 20 000 then you could build the positions matrix like 4 8 LINEAR CONSTRAINTS 57 gt posmat4 lt cbind rep Inf length prices Inf Inf Inf
159. ressing warnings page 124 3 Cheatsheets page 26 4 Troubleshooting page 127 5 S Language Problems and Solutions page 129 10 1 Easy Ways to Be Wrong The phrase garbage in garbage out has gone slightly out of fashion but it still has currency What follows are a few possibilities for garbage Data Mangling The prices are not all in the same currency It doesn t matter which currency is used but there needs to be only one 121 122 CHAPTER 10 PRACTICALITIES AND TROUBLESHOOTING There are prices for the wrong quantity For example the optimization is thought of in terms of lots but the prices for some or all of the assets are for shares There are stock splits rights issues etc that have not been accounted for Stock splits and rights issues that are not caught can mean that your existing holdings are wrong in the optimization A missed split in the price history degrades variance estimates that are based on returns created from the price history The variance is of prices not returns The variance matrix of the returns of the assets is needed If the variance is of the prices the results will be seriously wrong The functions in BurStFin that create variance matrices from data give a warning if prices are used The variance and or expected returns are not properly scaled The scaling that is chosen does not matter but it needs to be consistent The time scales of the variance and the
160. s a similar average market capitalization as an index then you can impose this as a linear numerical constraint We assume here that the index is weighted by market capitalization and we have the weights for the index The first step is to use build constraints as before gt cons mcap lt build constraints cbind Mcap index wts The possibly surprising part is what the bound looks like The value for this constraint assuming a weight style as opposed to value that the index has is the sum the squared weights gt mcap target lt sum index wts 2 The bounds you use might for example be 95 and 105 of the target 4 8 LINEAR CONSTRAINTS 63 Mixing Numerical and Categorical Constraints There is one tricky part when you have both numerical and categorical linear constraints Up until now we have used a matrix or a vector as the first argument to build constraints A matrix has to have all of its elements of the same type but now we want a mix of types numeric for the numerical constraints and character logical or factor for the categorical constraints Now we need to use a data frame not a matrix to represent our constraint data gt cons objmix lt build constraints data frame Risk1 betal Risk2 beta2 Country country gt head cons objmix lin constraints Risk1 Risk2 Country stockA 1 3119461 0 8972907 Australia stockB 0 9886379 0 9784834 Singapore stockC 1 1688637 0 7135961 Japan stockD 0 8160228 1 1195767
161. s example it is assumed that ave daily volume is a vector of numbers that has names which are the names of the assets similar to the prices vector These two vectors need not have the same assets If there are assets in the daily volume vector and hence in liquidity that are not in prices then these will be ignored If prices has assets that are not in liquidity then the missing assets will not be limited by the upper trade and lower trade arguments caution Make sure that the units for the volume are the same as those for the prices An easy mistake would be to try to limit trading to 10 of daily volume but instead limit it to 10 times daily volume because the volume is in shares while the prices are in lots Another sort of liquidity constraint is to limit the amount of assets in the portfolio to n days of average volume This is more complicated since we want the bounds on the portfolio but here we have constraints on the trade This is a case where using positions is an easier and safer approach See page 55 for an example of getting this constraint using positions 4 4 Number of Assets This section discusses the arguments e ntrade e port size Often these two constraints should be thought of as convenient shortcuts to threshold constraints page 52 Number of Assets to Trade The ntrade argument controls the number of assets that may be traded Gen erally it is given as a single number meaning the maximum number to t
162. see the state at the time of the error But we can only do that if the ground was prepared before the error happened Generally it is easy enough to reproduce the error We want to set the error option to save the contents of the function calls and not just what the calls were gt options error dump frames gt update soft op1 expected return new alpha Error in trade optimizer prices priceten variance varten Object new alpha not found gt debugger Message Error in trade optimizer prices priceten Object new alpha not found Available environments had calls 1 update soft op1 expected return new alpha 2 update default soft op1 expected return new alpha 3 eval call parent frame 4 eval expr envir enclos 5 trade optimizer prices priceten variance varten Enter an environment number or O to exit Selection 5 Browsing in the environment with call trade optimizer prices priceten variance varten Called from debugger look ind Browse 1 gt Here we have selected frame 5 the call to trade optimizer and at this point the objects that that function has created are available to us to explore You can do Browse 1 gt 1s to see what objects are there When you want to exit just type c as in continue Browse 1 gt c Chapter 11 Special Instructions This chapter contains instructions for how to overcome specific problems For example how to do something that the
163. some or all of the positions of the portfolio at the beginning of the test period random portfolios can give us a much more sensitive test of performance The information that we need for this method is e the constraints under which the fund operates e some or all of the positions at the start of the time period e the known or approximate turnover of the fund throughout the period With this method we imitate the trading of the fund throughout the period We get one random path by means of a series of random trades starting with what we know about the initial portfolio We then repeat that a number of times Let s revisit the example we saw in Section B I We know the full portfolio at the start of the period and we know the typical daily trading We use this knowledge to get the distribution of returns with zero skill Figure 2 1 shows the results for the year 2007 with 200 turnover while Figure 2 12 shows the 2007 2008 results for 200 turnover The strategy is very clearly outperforming chance The results are even more extreme for 400 turnover Table 22 shows the numerical results 2 5 Evaluating a Trading Strategy It is quite common to perform a backtest of a strategy However the backtest itself only shows how much the strategy makes or loses in the given situation It 28 CHAPTER 2 RANDOM PORTFOLIO EXAMPLES Figure 2 11 Performance measurement 200 turnover for 2007 Y S oO 3 N S S o z B 0 5 10 15 20 25
164. stical LOHIA 161 factor vs factor 11 factor model stat function 10 162 fail iter argument 42 136 feasible solution 65 file read 109 write 40 114 fitted function 112 force risk aver argument 150 forced trade automatic 49 positions argument 56 forced trade argument 54 104 funeval max argument 134 garbage 121 GARCH gen fail argument 42 generating random portfolios B3 43 genetic algorithm 135 global data OJ 123 gross value definition 48 gross value argument 47 head function 38 illiquid assets 123 import data into S T0910 170 information ratio annualizing 94 definition 93 maximize 931195 102 39 init fail argument HI intermediate argument 164 intersect function 61 inventory model 18 iterations max argument 42 134 leverage 106 library function 4 limit cost argument 75 lin abs argument 63 lin bounds argument 8 lin constraint argument 58 lin direction argument 64 lin style argument lin trade argument 63 linear constraint E 7H67 building 53H63 liquidity 5061 55 15 log return IJ long and short lists 03 long value definition 7 long only portfolio 91 100 long only with shorts 31 long short portfolio LO1WHTO0S 18 long side linear constraint 64 lot round 45 size 14 lower trade argument 50 97 104 machine memory 161 market
165. t forget the intercept gt WRONG lt cbind linear coefs quad coefs gt okay costs lt cbind 0 linear coefs quad coefs Because it is easy to forget the intercept a warning is issued if there are non zero values in any intercept If you really do have non zero elements in cost intercepts and you don t want to see the warning about it then you can set the do warn argument to c cost intercept nonzero FALSE Only the first part of the name needs to be given enough needs to be given so that it is unambiguous relative to the menu of names that can be given do warn Consider this example matrix given as the costs 118 CHAPTER 9 TRADING COSTS gt cost poly 1 13 pA D3 1 4 ABC 0 00 0 01 0 02 0 00 DEF 0 03 0 02 0 01 0 01 If ABC trades 30 lots and DEF trades 25 lots then the cost for ABC will be 0 01 30 02 30 18 3 The cost for DEF will be 03 02 25 01 257 01 25 150 53 Note that the matrix needs to have enough columns to encompass the highest order polynomial lower order polynomials will have zeros for their higher order coefficients caution You should make sure that each polynomial that you give makes sense through out the allowable trading range of the asset Negative costs for instance are often not reasonable Though there are special circumstances such as crys tallizing a loss for tax purposes where negative costs are appropriate If your costs are just intercepts th
166. t isn t doing what you want automatically then you can tell it to do what you do want If you are generating random portfolios but not optimizing then you can skip parts of this chapter The recommended route in this case is e Section T4 2 Alpha and Variance Tables e Section 4 3 Variance Constraints e Section 1 4 Expected Return Constraints Table 11 1 describes arguments to trade optimizer and random portfolio that are useful when dealing with multiplicity All of these arguments have defaults you only need to give values for these if the default behavior is not what you want 143 144 CHAPTER 14 ADVANCED FEATURES Table 14 1 Arguments for multiple variances and expected returns veable atable utable quantile states which quantile of the multiple objectives to use dest wt gives weights to the multiple objectives if desired 14 2 Alpha and Variance Tables The purpose of the alpha table is to state what combinations of expected return vectors and benchmarks will be used in the problem Likewise the variance table identifies variance benchmark combinations that will be used An alpha table is a matrix with two rows and an arbitrary number of columns In simple cases the alpha table will look like gt optobj atable 1 alpha 0 benchmark attr benchmarks 1 we Or if a benchmark is given it will look something like gt optobj atable Feil alpha 0 benchmark 500 attr benchmarks
167. t two possibilities 128 CHAPTER 10 PRACTICALITIES AND TROUBLESHOOTING Table 10 3 The column order for the positions argument trade buy threshold value portfolio short threshold value portfolio long threshold value trade sell threshold value 1 The objective is neither mean variance nor mean volatility it could be maximizing the information ratio minimizing variance or maximizing re turn Check the objective utility component of the output 2 You are passing in a value for the utable argument In this case you need to set force risk aver to TRUE in order for the risk aversion argument to override the value that is already in the utility table Pm minimizing the variance and the utility is exactly zero The problem is almost certainly that the objective is to maximize the informa tion ratio You can explicitly set the objective This does not occur if neither the expected return nor utable arguments are given It can happen if you give a vector of expected returns that are all Zero Portfolio Problems The optimizer is suggesting trades in which some of the assets trade a trivial amount Use the threshold argument see Section 5 The optimal portfolio has positions that are very small Use the threshold argument see Section 5 I m building a portfolio and the optimizer is having problems getting the monetary value of the portfolio right One possibility is that the max weight argument is too s
168. t with 400 turnover was also created Now we generate random portfolios that obey the constraints at the start of the time period gt rp251 20 w10 lt random portfolio 1000 us pricemat 251 port size 20 max weight 1 gross value 1e6 long only TRUE We then calculate the random portfolio returns over the period gt rp251 20 w10 ret07 lt pp simpret valuation rp251 20 w10 us pricemat c 251 nrow us pricemat 252 collapse TRUE We also need to calculate the return of the optimal fund throughout 2007 gt serret200 07 lt ser 20 w10 t200 value 251 ser 20 w10 t200 value 1 1 84 CHAPTER 5 DETAILS OF RANDOM PORTFOLIO EXAMPLES For this one time period and amount of turnover the only thing left to do is the comparison The percentile can be computed as gt round 100 mean rp251 20 w20 ret07 lt serret200 07 A visual representation can be done like gt plot density 100 rp251 20 w20 ret07 gt abline v 100 serret200 07 col blue If you want to save plots then you can open an appropriate graphics device before you perform the graphics commands and close the graphics device after wards For example gt pdf file myplots pdf gt plot density 100 rp251 20 w20 ret07 gt abline v 100 serret200 07 col blue gt dev off 5 2 Testing the Effect of Constraint Bounds The complete set of commands is given in pprobe_R_bounds txt Static Utility First we
169. taken as what is wanted gt op q7ri lt trade optimizer aprice avar all aret all gross value ten thou long only TRUE utility mean variance utable utab3 quantile 7 risk aversion 01 force risk aver TRUE gt op q7ri new portfolio 100 equities bonds cash 1 52 65 41 33 07 14 8 COMPACT VARIANCE OBJECTS 161 14 8 Compact Variance Objects Usually the full variance matrix is given In the case of multiple variances a three dimensional array is given where the third dimension represents the different variance matrices Computationally full matrices are the fastest so the preferred format is full matrices as long as the memory of the machine on which the optimization is being done is large enough This will almost certainly be the case unless the universe of assets is very large or there is a large number of variance matrices caution Unlike when a full matrix is given there is no mechanism when giving com pact variances for the optimizer to ensure that the assets are in the correct order within the variance The user needs to take full responsibility that the asset order in the variances is the same as in the prices argument This in cludes benchmarks which enjoy some automatic treatment when full matrices are given In addition to full matrices there are three formats supported e simple factor models These have formula AAT Y where A is a matrix of loadings with dimensions that are the number of assets b
170. ted gt utab3 lt op Ofutable gt utab3 Li 52 3 LA LS LEI 0 7 0 2 1 9 alpha spot 0 1 0 al 2 0 1 2 variance spot 0 0 0 1 1 1 2 2 2 destination 0 1 2 3 4 5 6 T 8 opt utility 0 0 0 0 0 0 0 0 0 risk aversion 1 1 1 1 1 1 1 1 1 wt in destination 1 1 l 1 al 1 1 1 1 The default behavior is to get all of the combinations of expected returns and variances in this case we have three of each so there are 9 combinations For our scenario analysis though we want only the combinations where the ex pected returns and the variances match three combinations In order to get the behavior that we want we need to provide a suitable matrix as the utable argument We could build a matrix from scratch but it is slightly easier to start with the one that is wrong There are three changes that we need to make to this matrix We need to select the three columns that we want we need to reset the destinations so that they are numbered from zero through one less than the number of combinations and we should specify the risk aversion parameter that we want utab3 lt utab3 c 1 5 9 utab3 destination lt 0 2 utab3 risk aversion lt 0 02 utab3 VM Y y 160 CHAPTER 14 ADVANCED FEATURES LI L 21 1 3 alpha spot 0 00 1 00 2 00 variance spot 0 00 1 00 2 00 destination 0 00 1 00 2 00 opt utility 0 00 0 00 0 00 risk aversion 0 02 0 02 0 02 wt in destination 1 00 1 00 1 00 At this point we are ready to do the actu
171. terations to see if the optimizer finds a solution that does not break any constraints If that doesn t work you should probably investigate if you have imposed constraints that don t allow a solution Section O I on page 21 lists some common mistakes On the other hand this method of treating constraints allows backtests to proceed even when the constraints that are imposed are infeasible for some of the time periods The answers that break the constraints while not optimal may be at least sometimes reasonable 8 3 Post Optimization This section is specific to optimization If you are generating random portfolios then the analogous discussion is Section B3 Once you have performed an optimization there are two tasks remaining see what the trade is and export the data so that the trade can be implemented Explore the Trade Information on the optimization can be viewed using the summary function gt opti lt trade optimizer prices varian long only TRUE gross val 1e6 gt summary opti The summary function provides information on e the utility costs and penalties for broken constraints e opening and closing positions etc e valuations of the portfolio and trade e realization of linear and count and distance constraints You can get some more specific information by printing the optimal portfolio object which can be done by just typing its name gt opti 114 CHAPTER 8 GENERAL USE Alternat
172. then specifying benchmark con straints is more complicated This is discussed on page 145 4 13 Distance This section discusses the arguments e dist center e dist style e dist bounds e dist trade e dist utility e dist prices These arguments impose distance constraints Let s start with a simple example Suppose we have a target portfolio that has the following weights gt targwt lt c Asset1 4 Asset2 3 Asset3 2 Asset4 1 4 13 DISTANCE 73 Suppose we want the resulting portfolio to have weight distance no more than 0 5 so the buys plus sells of the trade to get from the result to the target would be no more than 50 of the gross value of the portfolio To impose that constraint we would use arguments dist center targwt dist style weight dist bounds 5 The example above computes distance as the difference in weights The distance can alternatively be computed as the difference in monetary values These two are equivalent if the gross value is constrained to a narrow range although the weight formulation takes slightly more time to compute However if the gross value is allowed a wide range then there is a difference the value distance will favor a gross value close to the gross value of the target portfolio while the weight distance will be indifferent to the gross value For a value distance you can give either the values of the assets in the target portfolio or the shares in the target portfolio Using va
173. there is an existing portfolio It is possible that using the positions argument page 4 would be easier in such a case If for example the existing portfolio has a long position in an asset that is on the short list you may wish to force a sell in the asset that is at least as large as the existing position The commands above will not do that you need to resort to either the forced trade argument or the positions argument gt opt list2 lt trade optimizer prices varian aff gross val 1 4e6 net val c 1e4 2e4 lower trade long zero upper trade short zero universe trade c long names short names e ntrade 50 forced trade c ABC 25 In this example we are forcing a sell of at least 25 units of asset ABC Mean Variance Optimization To do mean variance optimization you need to give both a variance and ex pected returns and set utility mean variance You will want to specify the risk aversion as well More details of the risk aversion parameter are on page 140 gt opt mvi lt trade optimizer prices varian expected return alphas gross value 1 4e6 net value c 1e4 2e4 ntrade 55 max weight 05 utility mean var risk aversion 3 7 Alternatively you can do mean volatility optimization by using utility mean volatility 7 3 MANAGING CASH FLOW 105 7 3 Managing Cash Flow The optimizer can be used to select trades to take money out of the portfolio or put money into it
174. ting a new one then the command might be gt op te3 lt trade optimizer prices varian utility minimum variance long only TRUE gross val 1 4e6 benchmark spx max weight 0 05 existing cur port ntrade 20 The command creating op te3 is allowing up to 20 assets to be traded It does not control the turnover nor the number of assets in the resulting portfolio Thus op te3 can potentially have up to 75 assets in it if the current portfolio contains 55 assets Suppose that we want to have a portfolio with 45 to 55 assets in it and we want to hold the turnover to 100 000 currency units e g dollars The following command will provide that gt op te4 lt trade optimizer prices varian a utility minimum variance long only TRUE gross value 1 4e6 benchmark spx max weight 0 05 existing cur port turnover le5 port size c 45 55 This command is a realistic possibility The examples above could have been done without specifying the utility argument since there are no expected returns the utility would default to minimum variance but you would get a warning stating that utility was being modified See page 08 for examples of dealing with cash flow Page 150 has an ex ample of minimizing against dual benchmarks for example when you have a prediction of the benchmark after an upcoming rebalancing 6 3 Examples for Active Portfolios Expected returns of the assets are required for optim
175. ting monetary value If you want the file to represent money rather than the number of asset units you can use the multiplier argument gt deport randport2 multiplier prices file randval1 1 randval1 csv Writing compact files If you would like a compact form of the file when there is a large universe of assets you can use the append argument and iteratively write each portfolio to the file gt deport randportBurSt randport2 1 file randcompact gt for i in 2 length randport2 deport randportBurSt randport2 i file randcompact append TRUE We start by first writing just the first portfolio with the default value of append which is FALSE The full name of the deport function needs to be used because the result of the subscripting has lost the class attribute Specifying the file name is a necessity in this case The remaining portfolios are written by a loop with append set to TRUE Note that the loop is assuming that there are at least 2 random portfolios in randport2 S code note If the switch from deport to deport randportBurSt confuses you here is an explanation The deport function is magic in that it is a generic function This means that it looks at the class of its first argument and arranges that the function actually used is the method that matches the class In Portfolio Probe deport has two methods deport randportBurSt which is for objects created by random portfolio and deport portf
176. to form the portfolios with their constraints at the start of 2007 and never change them That is easy to do but probably doesn t answer the question that we really have in mind The second approach forms random portfolios at the start of each window and uses these to calculate the utility that is realized for the window Often the constraints will be static throughout even though the portfolios change with each window In this case imposing a static volatility constraint is a problem we start by looking at what the 8 volatility constraint looks like in the broader context Figure 2 4 shows the minimum ez ante volatility that is achievable through out the time period we are using Clearly we can t always impose a constraint that volatility is no more than 8 since the minimum possible is greater than 8 for most of the period The 8 constraint on the first day is approximately 120 of the minimum volatility So we ll use 120 of the minimum volatility as the upper limit on volatility when it is constrained Figure 2 5 compares the rolling distributions of utility with and without the volatility constraint We see bimodal distributions There is what might be considered a normal mode and a disaster mode In normal mode the volatility constraint reduces the up side about the same amount as the down side In disaster mode the volatility constraint is very useful So this looks quite good for the volatility constraint Note though that
177. to satisfy some of the constraints in par ticular the constraints on the monetary value of portfolios are closely managed 4 18 CONSTRAINT PENALTIES AND SOFT CONSTRAINTS 79 as are threshold constraints and the constraint on the maximum number of assets traded is always obeyed Tools are available in trade optimizer to allow you to adjust the penalties so that most of the constraints those not actively managed can become soft constraints This is done by making the penalties for those constraints small enough so that there will be a trade off between the penalty for the constraint and the utility Consider an example gt soft opl violated 1 variance linear gross value attr linear violations 1 Country Sector There are violations of two linear constraints the variance constraint and the gross value The default value of the penalty constraint argument is 1000 You can change individual elements of the penalty by giving the penalty constraint argument a named vector where the names are the items for which you want to change the penalty You need to give the names exactly they can not be abbreviated The easiest approach is to change the penalty constraint component of the output from the original optimization gt softpen2 lt soft opifpenalty constraint gt softpen2 1 2 lt 1 gt soft op2 lt update soft op1 penalty constraint softpen2 80 CHAPTER 4 CONSTRAINTS Chapter 5 Details of R
178. to what is expected if there were zero skill Figure ZIG shows the outperformance for each of the twenty runs Figure 2 17 plots the mean outperformance over the twenty runs 2 6 Going Farther e Chapter B discusses the function that generates random portfolios 30 Performance percentile Performance percentile 80 60 40 20 100 80 60 40 20 100 CHAPTER 2 RANDOM PORTFOLIO EXAMPLES Figure 2 13 MACD efficacy for Q1 of 2008 Q000 oo0oo0o0o0 100 80 60 40 20 0 Theoretical quantiles Figure 2 14 MACD efficacy for the first half of 2008 o00000000000000 00QO 0O00 100 80 60 40 20 0 Theoretical quantiles 2 6 GOING FARTHER Figure 2 15 MACD efficacy for the year 2008 Performance percentile 60 40 20 80 100 OO0O0O0O0000000000 0 100 80 60 40 20 0 Theoretical quantiles Figure 2 16 Outperformance of MACD for twenty initial portfolios D S LO fa o 2 wn g o 3 S hy D E I E 3 3 e o pe O l o 3 ts 7 0 50 100 150 200 250 Days in 2008 31 32 CHAPTER 2 RANDOM PORTFOLIO EXAMPLES Figure 2 17 Mean outperformance of MACD over twenty initial portfolios T E 8 O WN o 2 g 2 9 5 e Ss 86 E A O Ss D o EN E QO 1 0 50 100 150 200 250 Days in 2008 e Chapter J discusses the constraints that may be imposed e Chapter 5 shows the commands that created the examples that were dis played in this chapter
179. tput can be distinguished from the input An example is gt rep 0 6 1 000000 The user types rep 0 6 followed by return and the next line is the response from S Commands may span more than a single line the second and subsequent lines of a command are introduced by For example gt op lt trade optimizer prices varian gross value 1e6 long only TRUE The second line of the command starts with long only the is not typed by the user but rather by S There is no output in this example S code note The only catch with multi line commands is that it needs to be clear to S that the command is incomplete In this example the command needs a closing parenthesis Occasionally a fragment of code is written in which case there are no intro ductory prompts In addition to S code notes there are boxes which contain cautions and notes 16 Generate Random Optimize CHAPTER 1 ORIENTATION Figure 1 1 Some suggested routes through the document Conceptual Chapter 2 Examples y Chapter 3 Generate Random y Chapter 4 Constraints y Chapter 8 General Use y Chapter 5 Example Code y Chapter 10 Practicalities Chapter 4 Constraints a hapter 6 Long only hapter 7 Long short Chapter 9 Trade Cost y Chapter 13 Utility y Chapter 8 General Use y Chapter 10 Practicalities Chapter 6 Operational Chapter 2 Exampl
180. traints cbind Risk1 beta1 gt head cons obj3 lin constraints Risk1 stockA 1 3119461 stockB 0 9886379 stockC 1 1688637 stockD 0 8160228 stockE 1 0312180 stockF 1 2067453 gt cons obj3 bounds lower upper Riski Inf Int gt cons obj3 bounds lt c 0 9 0 95 gt cons obj3 bounds lower upper Risk1 0 9 0 95 62 CHAPTER 4 CONSTRAINTS S code note Two things First the use of cbind in the first line of the example above is merely an easy way to create a one column matrix that has a name for the column Second you might be tempted to do gt cons obj3 bounds lt c 0 9 0 95 instead of gt cons obj3 bounds lt c 0 9 0 95 The difference is that the first of these will result in a plain vector while the second retains the matrix structure of the original object We want that matrix structure We would now use the arguments lin constraints cons obj3 lin constraints lin bounds cons obj3 bounds Constraining more than one risk factor is straightforward gt cons obj4 lt build constraints cbind Riski betal Risk2 beta2 gt head cons obj4 lin constraints Riski Risk2 stockA 1 3119461 0 8972907 stockB 0 9886379 0 9784834 stockC 1 1688637 0 7135961 stockD 0 8160228 1 1195767 stockE 1 0312180 1 0847337 stockF 1 2067453 1 0977745 Now just create the bounds that you want and use the arguments as before Numerical Constraints Market Capitalization If you want to have a portfolio that ha
181. ts with short positions The lin direction argument controls whether you have a long side short side or both side constraint There are three possible values e lin direction 0 means both sides the default e lin direction 1 means long side e lin direction 1 means short side 4 8 LINEAR CONSTRAINTS 65 Of course lin direction gets replicated to be as long as the number of columns in the lin constraints object and can be given as a vector lin direction c 1 1 0 0 When a short side constraint is in effect lin direction 1 then it is the absolute value of the short values that is used your constraints should be non negative Looking at the Effect of the Constraints The constraints realized function identifies the status of linear constraints for a portfolio This function produces the con realized component of objects returned by trade optimizer This component is not a part of the printing of the object but is a part of the summary Using summary can give you information on some of the other constraints as well Here is an example gt opti2 con realized linear lower upper realized nearest violation c tn France 3e 05 2e 05 296632 3368 NA c tn Italy 1e 05 2e 05 180767 19233 NA c tn Spain 1e 05 3e 04 75494 24506 NA c ta France 6e 05 7e 05 600624 624 NA c ta Italy 2e 05 3e 05 206917 6917 NA c ta Spain 4e 05 5e 05 399552 NA 448 c pn France 1e 05 0e 00 24099 24099 NA c
182. turnover for 2007 2008 28 AA 30 O 30 CENAS 31 eee 31 32 edo 107 11 12 LIST OF FIGURES Chapter 1 Orientation This chapter has diverse aims It provides a brief overview of Portfolio Probe functionality e It explains what software you need in order to run Portfolio Probe It suggests a route through the rest of this document given a task and a state of mind It presents the typographic conventions of the document 1 1 Overview of Functionality The two primary aims of Portfolio Probe are e To generate random portfolios the random portfolio function e To optimize a portfolio the trade optimizer function The two functions have the same inputs except for saying how many random portfolios you would like and whether you want random portfolios or random trades All of the rest of Portfolio Probe is support for these two tasks 1 2 Necessary Tools You need to choose a language in which to run Portfolio Probe It can be one of three e R which can be downloaded for free via http www r project org There are some commercial distributions of R as well e S PLUS sold by TIBCO http spotfire tibco com 13 14 CHAPTER 1 ORIENTATION e Cor C You can call Portfolio Probe functionality in a program that you write The last option of using C is not recommended it requires considerable effort and likely has little or no benefit S PLUS and R are versions of the S language and P
183. ue A safer approach would be to give turnover an interval perhaps something like turnover cash required c 1 1 01 6 6 Asset Allocation Asset allocation as we discuss it here means that we are interested in weights rather than prices and asset units like shares or lots We restrict asset alloca tion to long only trade optimizer can handle such problems The key step is to create a price vector that is all ones gt aa prices lt rep 1 length asset names gt names aa prices lt asset names You then select a gross value that is large enough to satisfy the accuracy that you desire For example if getting the weight to the nearest basis point is good enough for you then set the gross value to 10 000 The gross value should be given as an interval that has just one integer in it The command gt aa alti lt trade optimizer aa prices avar expected return aret utility mean var risk aversion 02 long only TRUE gross value 10000 c 5 5 performs a mean variance optimization You can get the weights from the portfolio optimizer object with gt valuation aa alt1 weight A further example of this technique is the multiple scenario analysis on page 150 100 CHAPTER 6 OPTIMIZING LONG ONLY PORTFOLIOS 6 7 Going Farther Tasks that you might want to undertake To add trading costs to your optimization command see Chapter J To add constraints see Chapter Jon page 45 To re
184. uld be gt op iri utility value sqrt 252 The negative sign is because the optimizer always minimizes When quantities are being maximized information ratio or utility then it really minimizes the negative of the quantity Note that this is the information ratio you would get if all of your inputs were exact The actual information ratio that you achieve is likely to be worse We now turn to the case of maximizing the information ratio when there is a benchmark involved The Information Ratio with a Tracking Error Constraint It is common to perform a mean variance optimization relative to the bench mark The risk aversion is adjusted so that the tracking error is near its desired level It makes more financial sense to maximize the information ratio in abso lute terms as in the command that created op ir1 while having a constraint that the tracking error is no larger than a given value The mechanism in trade optimizer that allows a constraint relative to a benchmark is the bench constraint argument Here is a command to constrain the tracking error to be no more than 4 while maximizing the information ratio gt op ir2 lt trade optimizer prices varian F long only TRUE expected return alphas gross value 1 4e6 bench constraint c spx 0 0472 252 The value of the bench constraint argument needs to be named so that it knows what benchmark you want and the value is in the scale of the variance Using the c functio
185. umn which is numeric and the asset names are used as the row names This is then converted from a data frame into a matrix Finally the drop function changes the object from being a one column matrix into an ordinary vector The reason that there is the intermediate step of converting to a matrix is that the row names are not preserved when a data frame is dropped to a vector There is a reason for this you can just think of it as an unfortunate fact Other data will be imported in a similar fashion Using as matrix is fairly likely drop is only appropriate when there is a single column or row of data and a simple vector is desired Variance Matrix A variance matrix is often used in Portfolio Probe If your problem is an asset allocation with at most a few dozen assets then a sample variance matrix as computed by the var or cov wt S functions should do though there may be better methods For a large number of assets a factor model or a shrinkage model will probably be more appropriate You can use any variance matrix that you want It can come from a com mercial source or you can compute it yourself If you need to compute the variance matrix yourself there are functions in the BurStFin package that will do that If you are using R on Windows then you can get BurStFin with gt install packages BurStFin repos http www burns stat com R You can get the source with gt download packages BurStFin some_direct
186. ut note that this is the zero based location of the distance for columns corresponding to distance utilities The second row is similar to the first except that it is columns of the vtable The third row is the zero based index of the destination for the utility with this combination of alpha and variance The utility for the combination is computed multiplied by its weight in the destination the sixth row of the utility table and added to whatever is there That is a destination may hold a weighted average of utilities If a destination is a negative number the utility is not placed into a destination these are combinations that are presumably to be used in a constraint There must be at least one zero in the destination row and the destinations must be numbered from zero upwards with no integers skipped The fourth row is an indicator for the type of utility to be performed The allowable codes are given in Table 14 2 150 CHAPTER 14 ADVANCED FEATURES Table 14 2 Utilities codes for the utility table The fifth row holds the risk aversion parameter for the cases where it is used the value is ignored in the other cases When the utable argument is given then the risk aversion in the utility table is used rather than using the value of the risk aversion argument unless force risk aver is set to TRUE The example utility table above specifies that six utilities are to be computed and each put into a different destination Each
187. view common mistakes see Section 0 1 To improve your solution or examine the quality of solutions you are get ting go to Chapter 2 on page 133 To export your solution to a file see Section 8 3 To perform optimizations with multiple variances expected returns and or benchmarks go to Chapter 4 Chapter 7 Optimizing Long Short Portfolios This chapter discusses commands to optimize long short portfolios The examples in this chapter are just sketches of what you are likely to want to do in practice Chapter 4 discusses all the constraints that may be imposed Some of the examples limit the number of assets that are held in the portfolio or traded These limits are convenient but threshold constraints page D2 may be better to use Chapter O covers trading costs it is usually a good idea to constrain trading in some way whether with trading costs or a turnover constraint 7 1 Required Inputs The trade optimizer function needs a price vector The prices argument needs to be a vector of prices with names that are the asset names The assets in prices control the assets that are used in the problem Other inputs such as the variance and linear constraints may contain additional assets which will be ignored Other than prices there is no single argument that is required However there are two concepts that are required the amount of money in the portfolio has to be constrained somehow and there has to be so
188. without them the step regarding benchmarks can go wrong When iterations max is set to zero then a minimal amount of optimization is performed You will get a warning that this is not the same as no optimization We don t care about optimization in this instance we only care about the setup for optimization Check for Benchmark If you are not using a benchmark you don t need to be concerned with this step If you are using a benchmark then the default behavior makes the variances relative to the benchmark This is seen in the vtable component of the output of our dummy run gt qc temp vtable 4 1 23 E 3 variance 0 1 2 benchmark 500 500 500 utility only 1 0 0 attr benchmarks 1 Spx spx spx In our example we want the last two columns to not have a benchmark Hence we would do an operation like gt qcvtab lt qc temp vtable gt attr qcvtab benchmarks lt c spx If you are generating random portfolios and you didn t include the variance con straints in the dummy run then you will also want to change the utility only row to be zero in the columns that pertain to your constraints The argument vtable qcvtab should be given in the actual computation You can learn more about the vtable argument on page 44 78 CHAPTER 4 CONSTRAINTS Constraints out of Utility This step does not pertain to random portfolios unless a utility constraint is used This step tells the o
189. y the number of factors and Y is a diagonal matrix containing the specific variances This will most likely be the result of a statistical factor model e full factor models These have formula ADA Y where A is a matrix of loadings with dimensions that are the number of assets by the number of factors is the variance matrix of the factors among themselves and Y is a diagonal matrix containing the specific variances e vech This is the lower triangle of the variance matrix stacked by column So there is the variance of the first asset and the covariances of the first asset with all of the other assets then the variance of the second asset and the covariances of the second asset with assets 3 and onward etc The Variance List When at least one variance is not full the variance argument to the optimizers needs to be a list with special components The components of the list are e vardata Always required This is a vector of the actual numbers for the variance representations all concatenated together The order of the data in the representations is full matrix The first column the second column etc This is of course the same as the first row the second row etc simple factor model The loadings for the first asset the loadings for the second asset etc Then the specific variances full factor model Each column of the factor variance matrix in turn Then the loadings for the first asset the loadings
190. ying or selling More detail on costs due to taxes can be found in and its references Several optimizers implement piecewise linear cost functions While this choice is likely to be due to mathematical tractability costs and benefits from taxes really are piecewise linear in some circumstances The tax liability often depends on the length of time an asset has been held and there may be several holding periods for a particular asset Portfolio Probe does not have piecewise linear costs but you can approximate them with nonlinear costs Taxes also have a sort of time decay similar to options Their effect changes as the end of the tax period gets closer 9 6 Going Farther Additional steps might be e To look for trouble spots with Chapter OJon the facing page Chapter 10 Practicalities and Troubleshooting The best question to ask after a computation has been performed is Does it make sense The purpose of this chapter is to help with that question to give hints about how it might not make sense and to give possible solutions when it doesn t Though there will be a tendency to come to this chapter only when you think something is wrong the best time to come to it is when you think everything is right While some of the problems listed in this chapter are specific to optimization there are many that will affect the generation of random portfolios also The sections of this chapter are 1 Easy ways to be wrong 2 Supp
191. you get with utility information ratio The second approach is to have a separate term for costs T ar w Clu vwTVw You get this second version with utility exocost information ratio 13 2 Mean Variance Utility Mean variance utility is invoked with utility mean variance With mean variance utility we maximize aT w yw Vw C w where y is the risk aversion parameter and w satisfies all of the constraints Many implementations have a one half in the variance term meaning that the risk aversion parameter in those cases is a factor of two different than here Also in some optimizers the parameter used divides the variance rather than multiplies it in which case it is a risk tolerance parameter The mean variance risk aversion parameter is invariant to annualization As long as both expected return and variance are for the same time scale it doesn t matter what time scale it is However risk aversion is affected when expected returnand variance are for returns that are put into percent When percent returns are used the risk aversion is divided by 100 Kallberg and Ziemba 1983 classify risk aversion greater than 3 as very risk averse 1 to 2 as moderate risk aversion and less than 1 as risky these num bers would be divided by 100 for data representing percent returns Some additional sense of suitable values for the risk aversion parameter may be found in Burns 20030 All of the above are focused
192. zationf 0 000004 6 5 Managing Cash Flow Injecting Money into a Portfolio Extracting Money out of a Portfolio 6 6_ _Asset_Allocation o 6 7 Going Farthed e 74 75 75 76 76 76 77 77 78 78 78 81 83 84 84 85 85 86 87 88 89 90 90 7_ Optimizing Long Short Portfolios 7 1 Required Inputs 7 3 9 Trading Cost 9 1 Background 0 0 0 0 0 0 0 000000000002 008 9 2 Specifying Cost 7 5 Real Time Monitoring 7 6 Going Farthe 8 General Use 8 1 Setting Up Data 2 ee 8 2 The Random Generation or Optimizatio 8 3_ Post Optimizatio 10 5 S Language Problems and Solutions CONTENTS Monetary Value i A EATE ETE E E aa ae Managing Cash Flow 2 ee Injecting Money into a Portfolio 4 Extracting Money out of a Portfolio 0 7 4 Money Constraints o e a ee Adding a Benchmark to the Variancd Explore the Trade Export the Trade A Gindicn uy eae ewe A Polynomial Costs O Re a aera ee so oo AS 10 Practicalities and Troubleshooting Input Manel 10 2 Suppressing Warning 3S OB an fan SL Gt See a a a ta T 10 3 Cheatshest ia Se Are the hn hy hy He Oe Se Se dv Ah a Implied Ranged 1 a a Threshold Input Utility Problemy 2
193. zero There will be a warning if you do this 10 2 Suppressing Warning Messages While warning messages help avoid mistakes they can be annoying and counter productive if they warn about something that you know you are doing The do warn argument of random portfolio and trade optimizer can be used to suppress most warnings Here is an example of its use do warn c utility switch FALSE value range FALSE Note that abbreviation of the names is allowed as long as the abbreviation is unique among the choices Some warnings are never suppressed these are more likely to be mistakes You should reformulate the command to avoid such warnings whether or not the original is a mistake The default is TRUE for all entries except converged which defaults to FALSE since non convergence of the optimization is seldom of concern Here are the possible entries for do warn e benchmark long short A benchmark is used in a long short portfolio While this can be correct it often is not See Section 12 e bounds missing There are rows missing from the lin bounds object that should logically be there The missing bounds are taken to be infinite This is almost certainly caused by something going wrong or at least sloppy use 10 2 SUPPRESSING WARNING MESSAGES 125 converged optimization only Convergence is declared if more than fail iter consecutive iterations fail to improve the solution If the max imum number of iterations is perf

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