ACTUARIAL RESEARCH CLEARING HOUSE 1993 VOL. 3

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1 ACTUARIAL RESEARCH CLEARING HOUSE 1993 VOL. 3 THE MAGICAL MYSTERY MATRIX Rchard Q. Wendt, F.S.A. Towers Perrn Summary The Cholesk factor matrx has been well known for many years; t s a trangular matrx that satsfes the condton that the factor matrx multpled by ts transpose s equal to a symmetrc matrx. The concept has been used snce the early 1980's to generate asset returns wth specfed means and covarances. Ths paper develops some new uses of the factor matrx, ncludng modelng seral correlaton, addng asset classes wth contngent returns, adjustng prelmnary results to fnal targets and adjustng large, "unfactorable" matrces. Two appendces llustrate a numercal example and nclude lstngs of APL2/PC functons that mplement the methodology. I. INTRODUCTION Gven a symmetrc, postve defnte matrx, M, t s possble to determne an upper trangular factor matrx, F, such that M-F~F where the multplcaton s the matrx nner product and F ~ s the transpose of F. Ths mathematcal defnton plays a valuable role n the generaton of data wth specfed covarances, an ntegral part of many smulaton models. Gven an N row x M colunun ~trx, D, contanng N observatons of M varables; f the mean of each colu~u% s zero, then the covarance matrx for D s defned by (I) COV(D)=DT D (2) Equatons () and (2) lead to the followng well known methodology of generatng random data that satsfes specfed mean and covarance requrements: Gven C, a specfed covarance matrx for M varables F, the Cholesk factor matrx of C R, an N x M matrx of random data, where each column follows a normal dstrbuton wth mean zero and the covarance matrx of R s the dentty matrx 255

2 Then, let R/=R F (3) and t can be easly shown by matrx algebra that the covarance matrx of R' s C. By addng columnar means to R', R' wll be a data matrx wth specfed means and covarances. In summary, the Cholesk factor matrx can generate a data matrx wth specfed means and covarancee, as long as 1. a normal random matrx, R, wth a covarance matrx equal to the dentty matrx can be suppled. Ths s generally done wth a random number generator. A later secton of ths paper wll show how to force the ntal random numbers to exactly satsfy ths requrement. 2. The specfed covarance matrx s factorable. In smple terms, ths means that the specfed covarances and underlyng correlatons must be nternally consstent. For example, f varable A s hghly postvely correlated to varables B and C, then varables B and C would be expected to also have a strong postve correlaton. A specfed covarance matrx wth a strong negatve correlaton between varables B and C would not he factorable. For example, consder varables A, B and C, each wth mean zero and unt standard devaton. If the specfed correlaton matrx for ABC s as follows= A B C A B C Then the oovaranca matrx would be dentcal to the correlaton matrx and t s not factorable wth the Cholesk methodology. In order for the matrx to factor, the correlaton between B and C must be about postve.65 or hgher. II. A SIMPLE ASSET SIMULATION MODEL Some early asset smulaton models that were popular n the early 1980's were based on specfyng means and covarances for each asset class. As long as the specfed covarances were 256

3 factorable,.e., economcally/statstcally feasble, then the model could be developed n the followng process: Gven Then let and M asset classes, N smulatons (scenaros) of each class n each year, Y years for the smulaton tme horzon, R~...~ N x M matrces of normally dstrbuted random varables wth mean zero and covarance matrx - I, C a specfed covarance matrx (M x M), F - the factor matrx of C D,- R, x F, for - I...Y Each D wll satsfy the crtera for specfed covarances; by addng the expected returns of each asset class to the approprate columns, D L wll also have the specfed mean returns for each asset class. By usng smple varatons of ths methodology, both normal and lognormal dstrbutons can be generated. Chart One shows a vsual example of the matrx calculaton, where returns for three asset classes are generated. CREATE RETURN MATRIX WITH SPECIFIED COVARIANCE8 MATRIX D MATRIX R Z! 1 1 z X = I FACTOR MATRIX F M M CHART ONE M The reader wll note that although the smulated asset returns wll exactly satsfy the crtera for means and covarances, each year of the smulaton s ndependently determned and there s 257

4 no lnkage between years. In recent years, developng economc theory has lead to a strong ndcatons of lnkages of asset returns between years. More complex models are needed to develop lnkage (.e., seral correlaton) between years; these wll be explored n a later secton of ths paper. III. ADJUSTING SIMULATED RESULTS TO TARGETS In order to ntroduce lnkages of asset returns across years, my frm has developed a sophstcated economc smulaton model that frst smulates yelds and nflaton and secondarly smulates asset returns based on the pattern of smulated yeld and nflaton. Snce the asset class returns have a complex relatonshp wth yelds, nflaton and other asset class returns, there s only ndrect control over the smulated means and covarances of the asset class returns. Where t s mportant to adjust the asset returns to meet specfed targets, there are two possble solutonsz The frst approach s to "tweak" model parameters n an attempt to meet the targets. Even wth expert knowledge of the model, ths can be a dffcult and tme consu~nng process. The second, and more elegant, approach s to apply a matrx "flter" to force the smulated results to ht the specfed targets for means and covarances. As a smple one dmensonal example, f V, a vector of length N, holds randomly smulated data wth mean m and standard devaton s and f the target mean s M and the target standard devaton s ~, then the followng equaton adjusts V to ht the targets: ~= (v-m) x~ M (4) s The adjustment for an N x M data matrx ks analogous to the one dmensonal example, but uses matrx calculatons to control the covarances. Fortunately, the magcal factor matrx can do the job. I For example, see Poterba, James M. and Summers, Lawrence H., 1988, "Mean Reverson n Stock Prces -- Evdence and Implcatons, "Journal of Fnancal Economcs 22,

5 Let D = an N x M matrx of smulated data M~ an N x M matrx, where each column contans the mean of the equvalent column of D M' D an N x M matrx of the target means for each varable C covarance matrx of D, C = (D-Mo) ~ (D-MD) (5) C' F F' target covarance matrx Cholesk factor matrx of C Cholesk factor matrx of C' Then, DI=MIo (D-MD) ( F'xF -I) (6) V' wll have the specfed means and covarences and wll tend to preserve the ntal data patterns for small dfferences between C and C'. One caveat s that the matrx flter works best for small changes n C. In the case where the targets are greatly dfferent from the smulated ovarances, there s frst a queston as to whether C' wll be factorable. If C' s factorable, the means/covarances wll be as desred, but other characterstcs of the data could be dstorted. When usng ths very powerful approach, the results should be carefully nspected to see f there are any undesrable sde effects. An nterestng property of the matrx flter s that t can be set up so that the orgnal pattern of returns for some asset classes can be preserved whle the remanng classes are adjusted. Ths s accomplshed by smply reorderng the data matrx so that the asset class returns to be preserved are n the leftmost columns and the changeable classes are n the rghtmost columns. Ths amazng property wll be used n the next sectons. IV. CONTINGENT RETURNS There are a number of stuatons where we want to add an addtonal asset class to a prevously generated smulaton. All the orgnal data s to be preserved exactly as s end the addtonal asset class s to have a specfed mean and standard devaton and specfed correlatons to the preexstng asset returns. These are called "contngent returns", snce the return patterns of the new asset class depend on the exstng classes. For ths case, we can use a varaton of the matrx flter, usng the prevously mentoned technque to preserve the patterns of the orgnal data and only adjust the added asset class. More 259

6 specfcally, ths technque s defned as follows: Let D O - N x M matrx of smulated data wth exstng classes M~ N x M matrx of means of D O C O covarance matrx for D O R random vector wth mean 0 and standard devaton I D~ D O wth R added as a rghtmost column M~ M 0 wth a column of zeroes added as a rghtmost column CI,,~ covarance matrx for D z C' target covarance matrx, equal to Co, wth new M' F F' Then, covarances appended n last row and column matrx of target means, equal to M0, wth mean of added asset class n last column factor matrx for C~ factor matrx for C DI=,~'+ (DI-M1) x (FIxF "z) Chart Two shows an example of addng contngent returns for an asset class to the returns for three pre-exstng asset classes. ADDING ASSET CLASS WITH CONTINGENT RETURNS z MATRIX D' MATRIX D1 FACTOR MATRIX F' DO I I l I z X 7 = M+I,l,e 4. M4.1 M 4.1 CHART TWO M~I One or more new asset classes can be added to the exstng classes, as long as the covarances of the new classes to all 260

7 exstng classes and to each new class can be specfed. Appendx I shows a detaled numercal example of addng a new asset class to three pre-exstng classes. v. CONTROLLING SERIAL CORRELATION The methodology for controllng seral correlaton, the lnkage between years, s very smlar to the approach for creatng contngent returns. There are two possble varatons of ths method - creatng contngent returns wth seral correlaton and adjustng exstng return patterns to specfed seral correlatons. As mentoned n Secton I, early economc smulaton models assumed that return8 were ndependent from year to year. More recent models assume that there s some negatve seral correlaton of returns between years. Seral correlaton allows the modeler to drectly control the shape of compound returns. Chart Three-A shows the results of a smulaton of COMPOUND SIMULATED RETURNS asset class returns wth a WITHOUT SERIAL CORRELATION mean of 13% and a standard devaton of 16%; the graph shows the dstrbuton of compound returns of the "unlnked" yearly returns. By the tenth year, the standard devaton of the annualzed compound return s 4.5%. Chart Three-B shows the results of a smulaton wth negatve seral correlaton; although the mean and standard devaton of all the asset returns s dentcal to the frst smulaton, the standard devaton of compound returns n the tenth year has dropped to 3.0%. Snce the compound return patterns are relevant for forecasts wth long tme horzons, the ablty to control the dstrbuton of compound returns s crtcal to a successful smulaton model. To add contngent returns wth specfed seral correlaton to the pror year's return for the new asset class, generate t s 4 $ o q CHART THREE-A Yf.AR COMPOUND SIMULATED lletulln5 WITH SEll IAL CORllELAT ION :1... t'! 4 S ~ YIr~ CHART THREE-B 261

8 the frst year results by usng the process explaned n secton IV. For the second and later years, use the followng methodology: Let D o R o D, M, C, R D~ M= M r F F" - N x M matrx of smulated data wth exstng classes vector of smulated returns for new asset class n pror year D o wth R 0 added as leftmost column N x M matrx of means of D~ covarance matrx for D, random vector wth mean 0 and standard devaton 1 D I wth R added as a rghtmost column M, wth a column of zeroes added as a rghtmost column covarance matrx for D= target covarance matrx, equal to C,, wth new covarances appended n last row and column; the frst row of the rghtmost column has the covarance of the new asset class between years matrx of target means, equal to M,, wth the mean of added asset class n last column factor matrx for factor matrx for _ DI=M/+ (D=-M=) x (F/xF "z) (8) Snce C and C' only dffer n the last column, all the other columns of rlturns wll be preserved untouched. The last column of D' s the smulated returns for the new asset class, whch wll have the specfed covarance, ncludng the specfed seral correlaton to the pror year returns. 262

9 Chart FOUR gves a vsual example of ths process. For the example of three pre-exstng classes, the pror year's data for the new class s added as the leftmost column and a random vector s added as the rghtmost vector. After the matrx calculaton, the rghtmoet column of the matrx product contans the returns for the new asset class n the current year. These returns are then used to generate the next year's returns. ADDING ASSET CLASS WITH SERIAL CORRELATION FACTOR MATRIX F' Z MATRIX D' MATRIX D2 DO X 04 FACTOR MATRIX F M'~2 "1 M "1 N CHART FOUR The process for adjustng exstng results for seral correlaton s smlar. We want to preserve a11 exstng means and covarances and only change the seral covarance, whch s n the upper rght and lower left cells of the augmented covarance matrx. VI. "PURIFYING" RANDOM DATA It's very easy to generate random data and force the means to zero and the standard devatons to unty; however, those changes do not necessarly remove the correlatons from the data. Fortunately, the magcal Cholesk factor matrx can be used to remove all correlatons. Smply generate a matrx of random data and use the matrx "flter" to force the covarance matrx to be equal to the Identty Matrx. 263

10 VII. FACTORING "UNFACTORABLE" MATRICES As dscussed above, some specfed covarance matrces may not be factorable by the Cholesk methodology. Ths may occur for two possble reasons. Frst, the specfed correlatons may not be economcally or statstcally sensble. (Note that a covarance matrx calculated from actual data wll always be factorable, snce the pattern of correlatons of actual data s, by defnton, sensble.) Second, the factor methodology may run nto computatonal problems wth large matrces, due to roundng error. Practtoners generally fnd that, as the number of asset classes ncreases, t becomes more dffcult to factor the specfed covarance matrx. A possble soluton to ths problem s a specal process to apply the matrx flter to data matrces wth large numbers of varables. Ths process chooses random subsets of the varables and attempts to adjust each subset to the specfed covarances for that subset. The next teraton chooses a dfferent random subset from all the varables (ncludng the results of all pror adjustments) and adjusts that subset to ts specfed covarances. If the specfed covarances are sensble, ths process wll converge farly quckly to the desred results. VIII. CONCLUSION The factor matrx approach has been used n economc models snce the early 1980'8. Ths paper presented some new uses of the factor matrx for smulatng more complex asset models, controllng compound returns wth specfed seral correlatons and for adjustng prevously smulated results to meet specfed means and covarance8. 264

11 APPENDIX I Numercal Example Ths appendx gves a numercal example of the calculatons to add an asset class wth contngent returns to three pre-exstng asset classes. For ths smplfed example, we wll use 25 observatons of each varable. The methodology follows the process descrbed n Secton IV of the paper. For purposes of presentaton, the data has been truncated to sx decmals, or less. The actual calculatons n APL2/PC use up to 12 dgts of accuracy. Snce a hgh degree of accuracy s needed to factor the covarance matrces, t s recommended that extended precson calculatons be used. 265

12 The pre-exstng data s as follows: I Observaton CLASS 1 CLASS Mean Std Dev ~ [ I CLASS 3, ! The standard devaton used here g the populaton standard devaton rather than the sample standard devaton (.e., dvson by N rather than N-l}. The populaton standard devaton s equal to the square rqot of the dagonals of the covarance matrx. 266

13 The correlaton matrx for the data s the followng: CLASS CLASS 1 I CLASS 2 [ CLASS 3 CLASS CLASS Consstent wth the standard devatons and correlatons s the followng covarance matrxz CLASS CLASS 1 CLASS 2 CLASS 3 CLASS CLASS I For the purposes of ths example, we wlll add a fourth asset class, wth the followng characterstcs: Mean.13 Std Dev..16 Correlatons to Classes 1-3: 0,.2,.1 Based on these characterstcs, n combnaton wth the characterstcs of the pre-exstng classes, the target covarance matrx s the same as before, but wth a row and column added. The reader can easly verfy that the target covarance matrx s the followngz CLASS I CLASS 2 CLASS 3 CLASS 4 CLASS CLASS 2 CLASS 3 CLASS

14 We wll now append a column of 25 random numbers selected from a lognormal dstrbuton wth a mean and standard devaton of unty. The actual mean and standard devaton of the random sample wll vary wth sample error. The followng table dsplays the numbers whch were used for ths example: Random Numbers I-5, F I , , I " l J I I I ! t , The Cholesk factor matrx of the augmented data matrx n our example s shown n the followng table: CLASS 1 CLASS 2 CLASS 3 CLASS 4 CLASS CLASS CLASS 3 CLASS The Cholesk factor matrx of the target covarance matrx s dentcal for the frst three columns, correspondng to the preexstng data, whch s to be preserved. The target factor matrx s shown n the followng table: CLASS I CLASS 2 CLASS 3 CLASS 4 CLASS CLASS CLASS CLASS I /

15 Then, f we multply the target factor matrx by the nverse of the exstng factor matrx, we get the followng matrx: Jl CLASS I CLASS 2 CLASS 3 CLASS I I m l u I III CLASS 4 CLASS m m u CLASS I CLASS Note that the frst three columns and rows of ths matrx form the Identty Matrx; ths s the mechansm that preserves the orgnal data. 269

16 After multplyng the augmented data matrx by the last matrx, we obtan the new data matrx, as shown n the followng table: MEAN STD DEV CLASS CLASS CLASS CLASS , , I I , ,0499 0, ,1291! , I

17 Note that the frst three columns are dentcal to the startng data; the covarance matrx of the generated data s exactly equal to the target covarance matrx. Therefore, the generated asset class satsfes the requrements for mean, standard devaton and correlatons. 271

18 APPENDIX II Ths appendx contans source lstngs for several APL2/PC programs that facltate the processes descrbed n ths paper. These functons are: FACTOR CHANGE CHANGEX Calculate Cholesk factor matrx for a ~ven data matrx Flter a gven data matrx to meet specfed covarances Flter a gven data matrx wth a large number of varables to meet specfed covarances 272

19 [ o~ [ 1] [ z] [ 3] -[ 4] [ 5] [ 5] [ 7] [ S] [ 9] [1o] [I] [12] [13] [14] [15] [Is] B~FACTOR A;I;J;X A FUNCTION TO CREATE CHOLESKI FACTOR MATRIX A A IS SQUAP~E INPUT MATRIX n B IS OUTPUT FACTOR MATRIX B~(~A)p0 B[I;]~A[;I]+A[1;I]*0.5 I~J~2 START:4(J=I)/DIAG B[J;I]~(A[I;J]-(B[xI-I;I]+.xB[~I-I;J]))+B[J;J] ~END. " DIAG:~(0>X~A[I;J]-+/B[xI-I;I]*2)/ERROR B[J;I]~X*0.5 END:~(I~J~J+I)/START J~2 ~((It~A)~I+I+I)/START 40 ERROR:B~I;I AERROR; RETURN ROW NUMBER OF ERROR [ 0] B ~WANT CHANGE A;M [ ] n FUNCTION TO CHANGE COVARIANCES OF A TO SPECIFIED COVARIANCES [ 2] A A IS INPUT DATA MATRIX [ 3] n WANT IS SPECIFIED COVARIANCE MATRIX [ 4] n B IS OUTPUT DATA MATRIX [ 5] A B WILL HAVE SPECIFIED COVARIANCE AND COLUMNAR MEANS OF A [ 6] B~FACTOR WANT [ 7] ~(I:;,B)/NO A TEST FOR FACTORABLE B MATRIX [ 8] M ~(;A)~(+~A) It~A [ 9] B~M+(A-M)+.xBDFACTOR COVM A [10] NO:~0 AIF ERROR, RETURN ROW WITH ERROR 273

20 .[ o] [ I] [ 2.] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [o] [zz] [12] [13] [14] [15] [16] [17] [18] [s] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] B~WANT CHANGEX A;AA;BBB;COLS;EPS;LOW;MASK;MAXLOOP;NCOLS;NLOOP;R;T;W R FUNCTION TO CHANGE LARGE DATA MATRICES TO R SPECIFIED COVARIANCES R A IS INPUT DATA MATRIX R WANT IS SPECIFIED COVARIANCE MATRIX A B IS OUTPUT DATA MATRIX R B WILL HAVE SPECIFIED COVARIANCES AND COLUMNAR MEANS OF A R THIS FUNCTION MAY WORK WHEN WANT WILL NOT FACTOR DUE TO ITS SIZE R CHANGEX CALLS CHANGE ON SUBSETS OF THE DATA UNTIL CONVERGENCE MAXLOOP~50 EPS~ IE- 6 LOW+ lo00oo0 MASK~ (~R)o.~R MASK~ MASK+ +/, MASK NLOOP~0 LOOP:NLOOP~NLOOP+I ~(NLOOP>MAXLOOP)/END NCOLS~I+?-2+R ~ GET RANDOM NUMBER OF COLUMNS COLS~(xR)[NCOLS?R] ~ GET UNIQUE SAMPLE OF COLUMNS COLS~COLS[4COLS] R SORT INTO ORIGINAL ORDER W*WANT[COLS;COLS] BBB~W CHANGE A[;COLS] ~ ITERATE THROUGH CHANGE ~(I=~,BBB)/LOOP ~DIDN'T FACTOR, TRY AGAIN A[;COLS]~BBB ~(LOW~T~(+/,(M~SK~WANT-COVM A)*2)*0.5)/NEXT RRMSE ERROR AA~A ASAVE LOWEST RESULT AALTHOUGH ALGORITHM LETS ERROR GET HIGHER RCONVERGENCE IS ACTUALLY BETTER LOW~T NEXT:~(EPS~'T)/OUT BSTOP IF ERROR ~ EPS ~LOOP END:D~'DOES NOT CONVERGE' AMAX LOOPS OUT:B+AA ~GET BEST RESULT 274

21 ACTUARIAL EDUCATION AND RESEARCH FUND 1993 PRACTITIONERS AWARD PAPERS 275

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