Asse Allocaion wih Higher Order Momens and Facor Models Kris Boud (VU Brussel, Amserdam) Based on join research wih: Wanbo Lu (SWUFE) and Benedic Peeers (Finvex Group) 1
The world of asse reurns is non-normal. Is disribuion ends o be asymmeric and exremes occur oo ofen o be compaible wih he ails of a normal disribuion. 2
If here is no esimaion error, mos invesors would be willing o sacrifice expeced reurn and/or accep a higher volailiy in exchange for a higher skewness and lower kurosis leading o a lower downside risk (e.g. Ang, Chen and Xing, 2006; Harvey and Siddique, 2000, Sco and Horvah, 1980). This rade-off beween posiive preferences for odd momens (mean, skewness) and negaive preferences for even momens (variance, kurosis) can be convenienly summarized ino a single objecive funcion using a Taylor expansion of he expeced uiliy funcion as objecive (Jondeau and Rockinger, 2006; Marellini and Zieman, 2010) or a porfolio downside risk objecive based on he Cornish-Fisher expansion (e.g. Peerson and Boud, 2008; Boud e al., 2013). 3
R package PerformanceAnalyics Higher order comomens are also needed in he porfolio consrains; eg. Downside risk budges ogeher wih consrains on he porfolio urnover, cardinaliy consrains, his makes he porfolio problem complex o solve 4
R packages Deopim and PorfolioAnalyics Differenial Evoluion wih he R package DEopim: Cross-over and muaions gradually improve he iniial populaion and ge close o he global opimum. 5
Smar bea porfolios opimizing risk under various consrains 6
This all leads o a powerful porfolio opimizaion framework, provided ha he opimizaion problem is sensible and he inpu parameers are of good qualiy. 7
The challenge is ha while he sample size in erms of hisory of reurns is ofen limied, many problems require o opimize over a large number of asses: Curse of dimensionaliy in he number of parameers large esimaion errors. The consequences of esimaion error in porfolio opimizaion are well known: Opimized porfolios are ofen no well-diversified (Green and Hollifield, 1993) and behave like error maximizers (Michaud, 1998). 8
The opimizaion framework has o be reliable, sable, and robus wih respec o model and esimaion errors. Impose consrains (weighs and ohers) Clean he daa; Improve he accuracy of he esimaes: Resampling Impose srucure: Facor model; Possibly combined wih shrinkage of he esimaors oward a arge; Large lieraure for he mean and he covariance marix 9
Conribuion Only 1 paper for coskewness and cokurosis: Marellini and Zieman (RFS, 2010): equicorrelaion and single facor model. Single facor assumpion may make sense for equiies bu our applicaion is on asse allocaion (bonds, equiies, commodiies) We hus exend Marellini and Zieman s paper o he mulifacor model. We invesigae he ou-of-sample gains in an asse allocaion framework wih 17 asses. 10
Ouline Why we need higher order comomens for porfolio opimizaion and curse of dimensionaliy in using sample based esimaors; Esimaion of higher order comomens under he mulifacor model; Empirical analysis; Conclusion and furher research. 11
THE NEED FOR HIGHER ORDER COMOMENTS AND THE CURSE OF DIMENSIONALITY 12
Many invesors care abou higher order porfolio momens; When opimizing he porfolio using geneic algorihms like DE, hese momens need o be calculaed housands of ime; Higher order comomen marices allow us o express porfolio momens as explici, rapid o evaluae funcions of he porfolio weighs: 13
The firs four porfolio momens and how hey relae o comomens E E E E w' R w' w'( R ) w'( R ) 2 3 w' w w' ( w w) 4 w'( R ) w' ( w w w). Needed eg o opimize expeced uiliy under he 4h order aylor expansion; o esimae cornish-fisher VaR and ES, ec. Nx1 NxN NxN 2 NxN 3 E[ R] E E E ( R ( R ( R )( R )( R )( R )' )' ( R )' ( R )' )' ( R )' : Kronec ker Pr oduc 14
Curse of dimensionaliy: Number of unique elemens as a funcion of N Covariance marix Σ: N(N-1)/2 elemens Coskewness marix Φ: N(N+1)(N+2)/6 elemens Cokurosis marix Ψ: N(N+1)(N+2)(N+3)/24 elemens Toal number of unique elemens: N(N+1)/2+ N(N+1)(N+2)/6 + N(N+1)(N+2)(N+3)/24 15
Number of elemens o esimae N=5 120 N=20 10,605 N=100 4,598,025 To be compared wih he number of ime series observaions available in ypical porfolio allocaions wih monhly rebalancing: 5 years: 60 (monhly), 260 (weekly reurns): Large esimaion errors in he sample based esimaors and risk of error maximizing when opimizing ˆ 1 T ˆ ˆ 1 T 1 T T 1 T 1 T 1 ( R ( R ( R ˆ)( R ˆ)( R ˆ)( R ˆ)' ˆ)' ( R ˆ)' ( R ˆ)' ˆ)' ( R ˆ)' 16
FACTOR MODELS AND THE ESTIMATION OF HIGHER ORDER MOMENTS 17
Linear facor model Asse reurns are generaed by a K- dimensional vecor of facors and he remaining idiosyncraic erm is independen: Marix noaion: e Bf a r. 2 1 2 1 2 1 2 22 21 1 12 11 2 1 2 1 N K NK N N K K N N e e e f f f b b b b b b b b b a a a r r r 18
Soluion: Impose srucure o reduce he number of parameers o esimae Covariance marix Σ: N(K+1)+K(K+1)/2 elemens Inpu: Hisory of Reurns K facors Assumpion of a linear facor model Coskewness marix Φ: N(K+1)+ K(K+1)(K+2)/6 elemens Cokurosis marix Ψ: N(K+2)+ K(K+1)/2+K(K+1)(K+2)(K+3)/24 elemens Toal number of unique elemens: N(K+3)+[(1+(K+2)/3+(K+2)(K+3)/12]K(K+1)/2 19
Unresriced One facor Two facors Three facors N=5 120 23 37 61 N=20 10,605 83 112 151 N=100 4,598,025 403 512 631 20
Why? Under he facor model, we can rewrie he higher order comomens as a funcion of he comomens of he facors and a residual marix ha mosly conains 0s (because of he independence assumpion): BSB', BG( B' B'), BP( B' B' B'), Wih: S E ( f )( f )' G E ( f f )( f f )' ( f f )' P E( f )( f )' ( f )' ( f )'. f f f f f f 21
Esimae of exposure marix B: Equaion by equaion leas squares; The residual covariance marix Δ is zero everywhere excep on he diagonal: T The residual coskewness marix Ω is zero everywhere excep for he elemens corresponding o he expeced hird momen of he idiosyncraic erms. 1 T 1 1 T 2 ˆi T 1 ˆ3 i 22
The residual cokurosis marix Υ is zero everywhere excep for: he kurosis of one asse [when i = j = k = l], 2 4 E e ' 6bi Sbi E ei i he cokurosis beween wo asses [when (i=j=k) and l i] ' 2 b Sb E e 3 i l i and he cokurosis beween 3 asses [when (i = j) (k = l)]. b ' i Sb i 2 ' 2 2 2 b Sb E e E e E e E e k k k i i k 23
Code is currenly available from: hps://bibucke.org/rossbenne34/momenes imaion/src Nesed loops: speed gains hrough he use of Rcpp 24
APPLICATION TO THE OPTIMZATION OF A BONDS-EQUITIES-COMMODITIES PORTFOLIO 25
Applicaion: Asse allocaion 17 asses, 2001-2013: four equiy benchmarks (Europe, Norh America, Pacific and Emerging Markes), eigh bond indices (corporae developed high yield index in EUR and USD, corporae developed invesmen grade index in EUR, corporae emerging invesmen grade in USD, sovereign developed invesmen grade in USD, EUR, JPY and sovereign developed and emerging in USD) five commodiy indices (agriculure, energy, indusrial meals, livesock and precious meals). 26
Ou of sample evaluaion period Monhly rebalancings Rolling esimaion samples of 6 years 27
We will consider wo objecives: Minimize variance Maximize expeced uiliy (CRRA, seing mean o 0) ( 1) ( 1)( 2) EU w) m 2) ( w) m(3) ( w) m 2 6 24 ( ( (4) ( w Consrains: Long only, fully invesed; Each asse class has o conribue equally o he porfolio risk Variance Or expeced shorfall. Esimaion of porfolio momens using he sample esimaor, single facor or 3-facor model (saisical). Rolling esimaion sample of 6 years. Opimizaion wih DEopim. ). 28
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Main resul: Three acions seem o increase he ou-of-sample reurn and reduce he porfolio downside risk: (i) imposing more srucure on he esimaes (moving from he sample esimaor o he facor model based esimaes); (ii) swiching from an equal variance conribuion consrain o an equal expeced shorfall consrain; (iii) swiching from a minimum variance objecive o a CRRA expeced uiliy objecive. 30
This is for one sample More research is needed in erms of sensiiviy o he sample sudied, urnover analysis, risk budges, he reamen of currency effecs, he handling of ouliers and, imporanly, he choice of facors. 31
CONCLUSION 32
Conclusion Higher order momens are imporan in porfolio allocaion; Nooriously difficul o esimae already in moderae dimension because of curse of dimensionaliy; Srucure needs o be imposed in order o reduce he number of parameers o esimae; This paper sudies he use of he mulifacor model: many parameers are zero a huge reducion in number of parameers o esimae. 33
Unresriced One facor Two facors Three facors N=5 120 23 37 61 N=20 10,605 83 112 151 N=100 4,598,025 403 512 631 34
Curren research The mulifacor model assumpion solves problems, bu raises ohers: How o selec facors; especially hose ha are informaive abou he higher order comomens (no jus PCA explaining he covariance); Impac of exremes: Winsorizaion accouning for he non-normaliy; Dynamics; Model diagnosics: 35
Model diagnosics: Are he elemens really 0? The fac ha many of hese elemens are 0 is a model assumpion (idiosyncraic erms are independen). In curren research we develop a es for his. Good news is: asympoic disribuion of es saisics based on producs of residuals of differen asses are independen of he equaion by equaion regression esimaion needed o compue hose residuals (Randles, 1997) 1 T i T 1 1 ˆ ˆ i j; T j; i j k; i 1 ˆ k; T j k l Challenge is: so many elemens ha can be esed; individual es (many spurious deecions); join hypohesis es (curse of dimensionaliy). T 1 ˆ ˆ i j T 1 ˆ ˆ i j ˆ k ˆ l 36
References R packages Deopim, PerformanceAnalyics, PorfolioAnalyics, xs, Rcpp, + GSoC 2014 [Ross Benne] Marellini, Lionel and Volker Ziemann. 2010. Improved esimaes of higher-order comomens and implicaions for porfolio selecion. Review of Financial Sudies 23, 1467-1502. Zivo, E. 2011. Facor model risk analysis. Presenaion a R/Finance 2011. Jondeau, Eric, Ser-Huang Poon and Michael Rockinger. 2007. Financial modelling under non-gaussian disribuions. Springer. Boud, Kris, Peerson, Brian and Chrisophe Croux. 2008. Esimaion and decomposiion of downside risk for porfolios wih non-normal reurns. Journal of Risk 11, 79-103. Boud, Kris and Benedic Peeers. 2013. Asse allocaion wih risk facors. Quaniaive Finance Leers 1, 60-65. Boud Kris, Carl, Peer and Brian Peerson. 2013. Asse allocaion wih Condiional Value-a-Risk Budges. Journal of Risk 15, 39-68. This paper: Boud, Kris, Wanbo, Lu and Benedic Peeers. 2013. Asse allocaion wih higher order momens. Available on SSRN. 37