Fast gradient descent method for mean-cvar optimization

Transcription

1 Fast gradent descent method for mean-cvar optmzaton Garud Iyengar Alfred Ka Chun Ma February 27, 2009 Abstract We propose an teratve gradent descent procedure for computng approxmate solutons for the scenaro-based mean-cvar portfolo selecton problem. Ths procedure s based on an algorthm proposed by Nesterov [13] for solvng non-smooth convex optmzaton problems. Our procedure does not requre any lnear programmng solver and n many cases the teratve steps can be solved n closed form. We show that ths method s sgnfcantly superor to the lnear programmng approach as the number of scenaros becomes large. 1 Introducton ntro} The goal of portfolo selecton s to dstrbute a fxed amount of captal over a gven set of nvestment opportuntes to maxmze return whle managng the rsk. Although the benefts of dversfyng were well-known, the frst mathematcal model for portfolo selecton was proposed by Markowtz [10]. In the Markowtz model, the return of a portfolo s gven by the expected return of the portfolo and the rsk of the portfolo s measured by the varance of the return of the portfolo. The varance s a good measure of rsk only f the returns are symmetrc. The returns on equty, at least for short tme horzons, can be approxmated by a Normal random varable; consequently, the varance s an adequate measure for the rsk n the portfolo. However, when the dstrbuton of the returns of the underlyng assets s not symmetrc, varance s not an adequate rsk measure. Recently, Condtonal Value-at-Rsk (CVaR [15] has been proposed as a rsk measure for asset classes that have asymmetrc return dstrbutons. CVaR has many nce propertes: t s coherent rsk measure [4], Rockafellar and Uryasev [14] show that the CVaR of Department of Industral Engneerng and Operatons Research, Columba Unversty, New York, NY Emal: [email protected] Department of Industral Engneerng and Operatons Research, Columba Unversty, New York, NY Emal: [email protected] 1

2 a portfolo can be computed from scenaro by solvng a lnear program (LP, usng LP dualty CVaR upper bound constrants can be formulated as lnear constrants, and emprcal studes suggest that the mean-cvar approach where the portfolo return s gven by ts expected return and the portfolo rsk s gven by the CVaR of the portfolo s more approprate than the mean-varance approach f the rsk-return relaton s nonlnear [1]. From the results n Rockafellar and Uryasev [14], t follows that the mean-cvar portfolo selecton problem reduces to an LP. However, the resultng LP s very ll-condtoned and solvng such LP, partcularly when the scenaro sze s large, s very dffcult n practce [2]. We adapt a gradent descent method proposed by Nesterov [13] to solve the mean-cvar optmzaton problem. The method we propose does not requre solvng an LP and therefore t s able to potentally handle a very large number of scenaros. In addton, the method can be easly mplemented. These features mply that a portfolo manager can use our method wthout nstallng any thrd-party LP solvers. We also show how to ncorporate analysts vews nto the mean-cvar portfolo selecton problem [5, 6]. 2 Mean-CVaR optmzaton Suppose there are n assets n the market. Let R R n denote the random returns on the n assets. Let w R n denote the portfolo of the nvestor,.e., 1 T w = n =1 w = 1. The CVaR 1 β ( Rw at the probablty β (0, 1 of the portfolo w s defned as CVaR 1 β ( Rw =E P [ Rw Rw F 1 Rw (β], where Rw denotes the loss on the portfolo w, and F Rw denote the cumulatve densty functon (CDF of the random varable Rw. Thus, the CVaR s condtonal expectaton of the lowest β-quantle of the random portfolo return. The mean-cvar portfolo selecton problem we consder s as follows: mn w W CVaR 1 β( Rw, (1 meancvar} where the set W s the set of all feasble portfolos w. For example, by settng W = } w : E P (R T w = r, 1 T w =1, 2

3 where E P [R] denotes the expected returns on the assets, one recovers the canoncal mean-cvar portfolo selecton problem where the goal s to select the mnmum CVaR portfolo that has a target return r. Rockafellar and Uryasev [14, 15] show that CVaR 1 β ( Rw = mn (τ + 1β EP ( Rw τ +, (2 cvar} τ where 1 β s the confdence level and the functon (x + = max(x, 0. It s typcally very hard to explctly characterze the dstrbuton of the returns R, and therefore, n practce, E P ( Rw τ + s approxmated by usng return vectors R generated by some scenaro generator [8]. Let R : =1,..., N} denote N scenaros and let p, =1,..., N, denote the probablty of the -th scenaro. Then the expectaton n (2 can be approxmated as follows. N E P ( Rw τ + p ( R T w τ +. =1 By ntroducng new varables a ( R T w τ+, = 1, =1,..., N, the optmzaton problem (1 can be reformulated nto the lnear program (LP mn τ + 1 β N =1 p a s.t. a R T w τ, =1,..., N, Aw = b, (3 lpmeancva for W n the form of W = a 0, } w : Aw = b. The LP (3 s large t has O(N constrants, and s, often, very ll-condtoned [2]. Thus, solvng the LP (3 as the number of samples N becomes large s very hard. See Secton 4 for further evdence of the numercal nstablty of the LP formulaton. Our soluton method for the optmzaton problem (1 s based on the followng varatonal characterzaton of CVaR [4, 16, 9] CVaR 1 β ( Rw = max Q Q EQ ( Rw, (4 cvardual} where Q denotes a probablty measure on the returns R and the set of measures Q = Q :0 Q P 1 }. β 3

4 Thus, the mean-cvar portfolo selecton problem (1 can be formulated as the followng mn-max problem mn max w W Q Q EQ ( Rw. (5 meancvarg Ths formulaton can be thought of as a game played by the nature and the portfolo manager. It s then natural to consder teratve methods to solve the mean-cvar portfolo selecton problem. When the dstrbuton P s approxmated by N scenaros, the set of measures Q s gven by Q N = q R N : 1 T q =1, 0 q 1 β p }, (6 eq:cq-deg where p =(p 1,..., p N T and the nequaltes are nterpreted as component-wse nequaltes. From now on, we let R T =[R 1,..., R N ] R n N denote the matrx where the -th column s the asset return n the -the scenaro, =1,..., N. Thus, the scenaro-based mean-cvar problem reduces to the saddle-pont problem mn max q T Rw }. (7 meancvarw W q Q N 3 An teratve algorthm We solve the mnmax problem (7 usng a gradent-based procedure proposed by Nesterov [13]. Ths procedure requres that the admssble set of portfolos W be bounded. In practce, there s always margn requrement on the short postons n the portfolo. Such a margn requrement can be modeled as follows. (1 + M ( w + w +, (8 margnreq for some M>0. Snce the portfolo weghts sum to one, we have 1= n w + j n n ( w j + (1 + M ( w j + n n ( w j + = M ( w j +. (9 margnbou Therefore, we have n n w 1 = w + j +( w j + = ( w j (10 normbound M In order to keep the portfolos w bounded, we wll mpose constrants n the form of w /M or w 2 w /M. 4

5 A nave approach to solve the modfed mnmax problem (7 would nvolve generatng terates (w (k, q (k }, where w (k s the best-response to the nature s move q (k 1,.e., } w (k = argmn w W (q (k 1 T Rw, and q (k s the best-response to the nvestor s move w (k 1,.e., q (k = argmn q T Rw (k 1} q Q N The objectve q T Rw s not smooth n (w, q; consequently, ths teratve scheme converges very slowly. Nesterov [13] devsed a procedure that s able to escape ths convergence bottleneck. The Nesterov procedure conssts of two steps. The frst step s smoothng the optmzaton n q:. Let w (k denote the k-th terate. Then the smoothed best response of nature s gven by } q (k = argmax q Q q T Rw (k µd 2 (q, (11 fmu} where µ>0 and d 2 (q s any strongly convex functon. We choose d 2 (q = N q log q +(p /β q log(p /β q. (12 d2} =1 In Appendx A, we show that d 2 (q s strongly convex wth parameter σ 2 = 1 1 β wth respect to the l 1-norm. The Lagrangan functon L for optmzaton n q s gven by N N L(q = q T Rw (k µd 2 (q α(1 T q 1 + µ q ν (q p /β. =1 =1 Settng q L = 0, we have that q (k must satsfy ( R T w (k µ ln q (k p /β q (k α + µ ν =0,.e., q (k p /β q (k = e R T w(k α+µ ν µ. Thus, t follows that for all values of (α, µ, ν, we have that 0 < q (k < p/β. Therefore, complementary 5

6 slackness mples that µ = ν = 0, and q (k = where α s the soluton of the equaton β 1 p 1+e 1 µ (RT w(k +α, =1,..., N, (13 q-opt} p β 1 1+e 1 µ (RT w(k +α =1. (14 sumofq} The second-step n the Nesterov procedure s to compute the update w (k usng a convex combnaton of two updates z (k and y (k defned as follows. y (k = argmn q (k 1 Ry + Ω } y w (k 1 2 2, (15 ykdef} y W 2µσ 2 z (k = argmn z W w (k = ( 1 k +3 k 1 ( t +1 2 t=0 ( k +1 k +3 z (k + ( Ω q (t Rz + z 2}, (16 zkdef} 2µσ 2 y (k, (17 where Ω = max max q 1 1 w 2 1 ( q T Rw 2 = max R 2 2 and σ 2 s the convexty parameter for the strongly convex functon d 2 (q. The terate y (k s a modfed best-response where one penalzes large movements from the last response w (k 1. The terate z (k n (16 consders all the prevous responses q (t : t =0,..., k 1 } to compute the response. The weght on y (k ncreases as the teraton count k ncreases. When the set W s descrbed by lnear equaltes,.e., W = w : Aw = b }, we add the addtonal constrant w 2 1+2/M, and n ths case t s easy to show that (15 and (16 can be solved n closed form. When the set W s descrbed by lnear nequalty constrants, we mpose the constrant w 1 1+2/M. Then (15 and (16 are quadratc programs that can, n practce, be solved very effcently usng actve set methods. Note that each quadratc problem encountered n the course of our proposed teratve procedure has n varables and O(m constrants, where m denotes the number of components n b. Nesterov [13] proves that after N steps the output (ŵ, q of the algorthm dsplayed n Fgure 1 satsfes } } ( mn q T Rŵ max q T D1 D 2 Ω 1 2 Rw < δ N = 1, (18 gap} q Q N w W σ 2 K 6

7 Nesterov Procedure D (1 + ( 2 M 2, D 2 1 β β ln(β (1 β ln(1 β, σ2 1 Ω max R 2 2, K 1 ΩD 1D 2 ε σ 2, µ ε 2D 2, w (0 1 n 1 for k 0 to K do } q (k argmax q Q q T Rw (k µd 2 (q } y (k+1 argmn y W q (k Ry + Ω 2µσ 2 y w (k 2 2 z (k+1 argmn z W ( ( k t+1 t=0 2 q (t Ω Rz + ( ( w (k+1 z (k+1 + y (k+1 1 k+3 return ŵ = y (K, q = K k=0 ( k+1 k+3 2(+1 (N+1(N+2 q (k. 2µσ 2 } 1 β Fgure 1: Nesterov Procedure fg:neste.e., after K teratons the algorthm produces a par (ŵ, q that are δ N -optmal polces for nature and the nvestor. One can, therefore, termnate the algorthm once we are satsfed wth the qualty of the portfolo. ( 1 2 D Moreover, choosng K 1D 2Ω σ 2 1 ε can ensure that the output of the algorthm s ε-optmal. In our numercal calculatons we found that usng the gap n (18 termnates the algorthm much qucker than usng the upper bound. The man features of ths algorthm are as follows. (a The modfed best-response y (k and z (k of the nvestor are computed by solvng a separable quadratc optmzaton problem that s smlar to the mean-varance portfolo selecton wth uncorrelated assets. Ths mples that the technology for mean-varance optmzaton can be used to solve the mean-cvar problem. (b The terates ( w, q are at least δ N -optmal, and often, the qualty of the soluton s sgnfcantly superor to that mpled by the bound. Thus, one can termnate the algorthm at any stage where one obtans a soluton of suffcent qualty. (c In Secton 4, we show that ths algorthm converges to a reasonably accurate soluton wth the error ε = 10 3 very quckly even when the number of scenaros N = Snce the scenaro-based mean- CVaR problem s tself an approxmaton to the orgnal problem, solvng the scenaro-based CVaR very accurately does not serve any purpose. 7

8 4 Numercal results results} We tested our procedure on the example n [12]. Our asset unverse conssted of Treasury bonds wth 2, 5, 10, and 30 years to maturty. As n the example n [12], we approxmated the returns on the assets a Delta-Gamma approxmaton usng the yelds on bonds wth 6 month, 2 years, 5 years, 10 years, 20 years, and 30 years to maturty as the rsk-factors. We smulated N scenaros for the rsk factors and then used the Delta-Gamma approxmaton to compute N return scenaros. We refer the reader to [11] for a detaled dscusson of the smulaton procedure. In Table 1, we dsplay the optmal soluton to the LP formulaton for the mean-cvar problem (3 wth β =0.05 and N = We use MOSEK [3] to solve these LPs. Table 2 shows the optmal portfolo computed by our proposed algorthm wth the error tolerance ε = The portfolos produced by our algorthm and the LP formulaton (3 are qute dfferent; although the CVaR values are close. These results only mply that the LP approach and the our proposed teratve approach are consstent,.e., both approaches are able to solve the mean-cvar problem; these results are not able to dfferentate between the two approaches. The most mportant results of ths secton are n Tables 3 and 4. In Table 3 we dsplay the CPU tme for solvng the LP formulaton usng ILOG CPLEX [7] and MOSEK, and the CPU tme for computng an ε =0.001 optmal soluton usng our algorthm as a functon of the number of scenaros N. It s clear that the ndustry leader LP solver CPLEX performs very poorly on ths problem. MOSEK performs much better but the run tmes for ths commercal solver s an order of magntude hgher than that of our MATLAB-based code. Table 4 dsplays the run tmes and the number of teratons requred by our algorthm as a functon of the accuracy ε. The performance of our algorthm degrades very quckly as ε decreases. Therefore, ths algorthm s only suted for applcatons where one wants to compute a reasonably accurate soluton very quckly. An example of such an applcaton s hgh-frequency tradng. The data n hgh-frequency tradng s typcally very nosy; therefore, t s pontless to compute a very accurate soluton. Note that the LP approach does not allow any flexblty n settng the accuracy level. Next, we show how to use analysts vews to bas the sample probablty mass functon p. We restrct ourselves to vews of the form: ν T R g, where ν R n s a vector that determnes the partcular lnear combnaton of the return vector R, and g s a probablty densty on R. We convert ths vew on the dstrbuton of the random return R to a vew on 8

9 dstrbuton of the N sample returns R, t =1,..., n, by defnng a vew probablty vector p = g(ν T R N, =1,..., N. k=1 g(νt R k Suppose we have m dfferent vews,.e., there are m dfferent vew probablty vectors p (j, j =1,..., m. We combne these vectors nto a sngle sample probablty vector p as follows: m p = u j p (j + u 0 p (0, (19 probabl where p (0 = 1 N 1 denotes the emprcal measure, and u j denotes the confdence weght on vew p (j. Snce p s a probablty vector, we requre that N j=0 u j = 1. Next, we solve the mean-cvar problem wth scenaro probablty vector p. Our algorthm also works wth other technques for combnng vews, see, for example [5, 6, 12]. For our numercal experments, we set m = 2. The two vews were chosen to be [ T ν (1 = ], g (1 = unf[0, 0.001], [ T ν (2 = ], g (2 = unf[0, ]. The weght vector was set such that u 0 =0.9, u 1 = u 2 =0.05,.e., we assumed that we had 90% confdence n the emprcal dstrbuton and 5% confdence n each of the two vews. Table 5 shows the optmal portfolo computed usng the LP formulaton (3. As n the prevous case, the LP was solved usng MOSEK. Table 6 shows the results computed usng our algorthm wth ε = Concluson In ths paper, we propose an effcent algorthm for solvng mean-cvar portfolo selecton problem wthout usng an LP solver. As shown n the numercal experments, the algorthm s a useful alternatve to the LP approach when one wants a very fast solver that guarantees an accuracy algorthm wth ε Ths technque can also be extended to solve many other types of portfolo selecton problems. 9

10 bond/target return (r y y y y CVaR Table 1: Optmal portfolo and CVaR for the Mean-CVaR problem solved by LP approach. tab:1} bond/target return (r y y y y CVaR Error Table 2: Optmal portfolo and CVaR for the Mean-CVaR problem solved by our algorthm, and the absolute error compared wth LP approach. tab:1b} References [1] V. Agarwal and N.Y. Nak. Rsks and portfolo decsons nvolvng hedge funds. Revew of Fnancal Studes, 17(1:63 98, Sprng [2] S. Alexander, T.F. Coleman, and Y. L. Mnmzng CVaR and VaR for a portfolo of dervatves. Journal Bankng and Fnance, 30(2: , February [3] E. D. Andersen and K. D. Andersen. The MOSEK optmzaton toolbox for MATLAB manual Verson /tools/help/ndex.html, [4] P. Artzner, F. Delbean, J.M. Eber, and D. Heath. Coherent measure of rsks. Mathematcal Fnance, 9(3: , July [5] F. Black and R. Ltterman. Asset allocaton: combnng nvestor vews wth market equlbrum. Goldman Sachs Fxed Income Research, [6] F. Black and R. Ltterman. Asset allocaton: combnng nvestor vews wth market expectatons. Journal of Fxed Income, 1(1:7 18, September [7] ILOG. ILOG CPLEX [8] Y.K. Koskosds and A.M. Duarte Jr. A scenaro-based approach to actve asset allocaton. Journal of Portfolo Management, 23:74 85, Wnter

11 N CPLEX MOSEK Our algorthm (Iteratons ( ( ( ( (1 Table 3: CPU tme for both methods n second and number of teratons requred for our algorthm. tab:3} ε η CPU tme Iteratons CVaR Error Table 4: CPU tme and teraton counts for our algorthm. tab:4} [9] H. Lüth and J. Doege. Convex rsk measures for portfolo optmzaton and concepts of flexblty. Mathematcal Programmng, 104(2: , November [10] H.M. Markowtz. Portfolo selecton. Journal of Fnance, 7(1:77 91, March [11] A. Meucc. Rsk and asset allocaton. Sprnger, [12] A. Meucc. Beyond black-ltterman: Vews on non-normal markets. Rsk Magazne, 19:87 92, [13] Y. Nesterov. Smooth mnmzaton of non-smooth functons. Mathematcal Programmng, 103(1: , May [14] R.T. Rockafellar and S. Uryasev. Optmzaton of condtonal value-at-rsk. Journal of Rsk, 2(3:21 41, [15] R.T. Rockafellar and S. Uryasev. Condtonal value-at-rsk for general loss dstrbutons. Journal Bankng and Fnance, 26(7: , July [16] R.T. Rockafellar, S. Uryasev, and M. Zabarankn. Devaton measures n rsk analyss and optmzaton. Techncal report, Department of Industral and System Engneerng, Unversty of Florda, Appendx A Detals of the parameters n the Nesterov algorthm The Hessan 2 d 2 (q of the smoothng functon d 2 (q = N by q log q +(β 1 p q log(β 1 p q s gven 2 (d 2 (q = dag([q 1 1,..., q 1 N ] + dag([β 1 p 1 q 1 1,..., (β 1 p N q N 1 ]. compmaxt 11

12 r Bond New Change New Change New Change New Change 2y y y y CVaR Table 5: Optmal portfolo and CVaR for the Mean-CVaR problem solved by our algorthm wth weghts on vews u 0 =0.9, u 1 = u 2 =0.05 by LP approach. tab:2a} r Bond New Change New Change New Change New Change 2y y y y CVaR Error Table 6: Optmal portfolo and CVaR for the Mean-CVaR problem solved by our algorthm wth weghts on vews u 0 =0.9, u 1 = u 2 =0.05 by our algorthm. tab:2b} Therefore, h T 2 (d 2 (qh = = = N =1 ( N =1 ( N =1 h 2 q + N =1 h 2 q h 2 (β 1 p q ( N =1 q 1 ( N + =1 h q q β β h 2 1, h 2 ( N (β 1 p q ( N =1 =1 (β 1 p q β 1 1 (20 h β 1 p q β 1 p q 2 (21 where (20 follows from the fact that q = 1 and (β 1 p q =β 1 1, and (21 follows from the Cauchy-Schwatrz nequalty. By settng w (k = 0 n (11, t follows that q mn = argmn q QN d 2 (q} satsfes q mn = β 1 p, =1,..., N, 1+eα/µ 12

13 where α s chosen to ensure that 1 T q mn = 1. Therefore, t follows that q mn = p, and mn q Q N d 2 (q = p log p + ( p (β 1 1 log p + log(β 1 1. Snce d 2 (q s a convex functon, max q QN d 2 (q occurs at extreme ponts of the polytope Q N. The extreme ponts of the polytope Q N are of the form: β 1 p, π(1,..., π(k 1}, q = 0, π(k + 2,..., π(n} where π s a permutaton of the set 1,..., N} and q π(k+1 [0, β 1 p π(k+1 ] s chosen to ensure that N =1 q = 1. The value d 2 (q = β 1 p π( ln(β 1 p π( : π(k+1 + q π(k+1 ln(q π(k+1 +(β 1 p π(k+1 q π(k+1 ln(β 1 p π(k+1 q π(k+1 β 1 p π( ln(β 1 p π(, where the last nequalty follows from q π(k+1 ln(q π(k+1 + (β 1 p π(k+1 q π(k+1 ln(β 1 p π(k+1 q π(k+1 (β 1 p π(k+1 ln(β 1 p π(k+1. Thus, D 2 = max q Q d 2(q mn q Q d 2(q β 1 p log(β 1 p p log p = β 1( β log β + (1 β log(1 β. ( p (β 1 1 log p + log(β 1 1 (22 D2_q} 13