Robust Growth-Optimal Portfolios

Transcription

1 Robust Growth-Optimal Portfolios! Daniel Kuhn! Chair of Risk Analytics and Optimization École Polytechnique Fédérale de Lausanne rao.epfl.ch

2 4 Technology Stocks I 4 technology companies: Intel, Cisco, Blackberry, and Nokia. I Monthly statistics: µ [%] INTC 0.37 CSCO 0.33 BBRY 0.93 NOK 0.85 [%] INTC CSCO BBRY NOK INTC CSCO BBRY NOK

3 Markowitz Portfolio Theory

4 Fixed-Mix Strategy w t (r,...,r t )=w I Keep portfolio weights constant over time. I Memoryless but dynamic.

5 Fixed-Mix Strategy in BS Economy I What to expect:

6 Fixed-Mix Strategy in BS Economy I What really happens:

7 Fixed-Mix Strategy in BS Economy I What really happens:

8 Fixed-Mix Strategy in BS Economy I What really happens:

9 All is Lost w.p.!

10 Asymptotic Winners & Losers Portfolio growth rate = expected log utility (w) =E log ( + w r) CLT: Growth rate determines porfolio performance in the long run. I I (w) > 0: asymptotic winners (w) < 0: asymptotic losers Growth optimal portfolio (GOP) is the one that maximizes (w). w g 2 argmax w (w) Kelly & Latané: GOP outperforms any other non-anticipative investment strategy with probability in the long run. Kelly, Bell Syst. Tech. J Latané, JPE 959

11 Markowitz Winners & Losers

12 Distributional Assumptions I "µ and are the only quantities that can be distilled out of the past." I "The slightest acquaintance with problems of analyzing economic time series will suggest that this assumption is optimistic rather than unnecessarily restrictive." Roy (952) Roy, Econometrica 952

13 The long run may be indeed long... In a Black Scholes economy, it may take the GOP I 208 years to beat an all-cash strategy I 4,700 years to beat an all-stock strategy with 95% confidence. Rubinstein (99) Rubinstein, JPM 99

14 Robust Growth-Optimal Portfolios

15 Finite Investment Horizons Informally, our goal is to maximize terminal wealth TY t= ( + w r t ), which is equivalent to maximizing the growth rate T TX log ( + w r t ). t=

16 Finite Investment Horizons Formally, we can maximize the VaR of the growth rate VaR (w) = max ( : P T TX log ( + w r t )! ). t=

17 Distributional Robustness Stochastic and distributionally robust performance measures: I VaR (w) = max ( : P T TX log ( + w r t )! ). t= I WVaR (w) = max ( : P T TX log ( + w r t )! 8P 2 P ). t=

18 Robust GOP Performance Guarantees The fixed-mix strategy w will grow at least by et WVaR (w ) w.p. under any distribution P 2 P. The robust GOP (RGOP) is the portfolio that maximizes WVaR (w ). w 2 argmax WVaR (w ) w

19 Calculating WVaR (w) I Weak sense white noise ambiguity set I WVaR (w) P = P : E P ( r t )=µ, E P r s r t = st + µµ max ( : P T TX t= w r t 2 (w r! t ) 2 8P 2 P ) I Distributionally robust quadratic chance constraint.

20 Chance Constraints

21 Robust Individual Chance Constraints P(L(x, ) apple 0) 8P 2 P inf P2P m P(L(x, ) apple 0)

22 ASimpleExample Individual chance constraint: P(x ξ + x 2 ξ 2 ) 75% ( ).5 Model : ξ i N (0, ) iid ( ) issocprepresentable: ( ) x 2 /Φ (75%) x x E(ξ )=E(ξ 2 )=0, Var(ξ )=Var(ξ 2 )=, Cov(ξ, ξ 2 )=0 ( )

23 ASimpleExample Individual chance constraint: P(x ξ + x 2 ξ 2 ) 75% ( ).5 Model 2: ξ i 2 δ + 2 δ + iid ( ) x x 2 x + x 2 feasible set is unbounded nonconvex x x E(ξ )=E(ξ 2 )=0, Var(ξ )=Var(ξ 2 )=, Cov(ξ, ξ 2 )=0 ( )

24 ASimpleExample Individual chance constraint: P(x ξ + x 2 ξ 2 ) 75% ( ) Model 3: ξ i κ2 +κ 2 δ κ feasible set changes discontinuously with κ + +κ 2 δ +κ iid x x E(ξ )=E(ξ 2 )=0, Var(ξ )=Var(ξ 2 )=, Cov(ξ, ξ 2 )=0 ( )

25 ASimpleExample Individual chance constraint: P(x ξ + x 2 ξ 2 ) 75% ( ) Model 3: ξ i κ2 +κ 2 δ κ feasible set changes discontinuously with κ + +κ 2 δ +κ iid x x E(ξ )=E(ξ 2 )=0, Var(ξ )=Var(ξ 2 )=, Cov(ξ, ξ 2 )=0 ( )

26 ASimpleExample Individual chance constraint: P(x ξ + x 2 ξ 2 ) 75% ( ).5 Model n: ξ, ξ follow any distribution satisfying ( ) 0.5 x x E(ξ )=E(ξ 2 )=0, Var(ξ )=Var(ξ 2 )=, Cov(ξ, ξ 2 )=0 ( )

27 ASimpleExample Individual chance constraint: P(x ξ + x 2 ξ 2 ) 75% ( ).5 Model n: ξ, ξ follow any distribution satisfying ( ) 0.5 x x E(ξ )=E(ξ 2 )=0, Var(ξ )=Var(ξ 2 )=, Cov(ξ, ξ 2 )=0 ( )

28 ASimpleExample Individual chance constraint: P(x ξ + x 2 ξ 2 ) 75% ( ).5 Model n: ξ, ξ follow any distribution satisfying ( ) 0.5 x x E(ξ )=E(ξ 2 )=0, Var(ξ )=Var(ξ 2 )=, Cov(ξ, ξ 2 )=0 ( )

29 ASimpleExample Individual chance constraint: inf P(x ξ + x 2 ξ 2 ) 75% P P ( ).5 Let P be the set of all distributions satisfying ( ) intersection of feasible sets is convex x x E(ξ )=E(ξ 2 )=0, Var(ξ )=Var(ξ 2 )=, Cov(ξ, ξ 2 )=0 ( )

30 Distributional Robustness Classical chance constraint: P(L(x, ξ) 0) ϵ hedges only against risk sensitive to estimation errors often nonconvex Distributionally robust chance constraint: inf P P P(L(x, ξ) 0) ϵ hedges against risk and ambiguity often convex Calafiore & El Ghaoui, JOTA 2006

31 CVaR Approximation 2 CVaR approximation: P-CVaR ϵ (L(x, ξ)) 0 P-VaR ϵ (L(x, ξ)) 0 P(L(x, ξ) 0) ϵ Thus, the CVaR constraint ϵ VaR ϵ CVaR ϵ 0 Loss P-CVaR ϵ (L(x, ξ)) 0 conservatively approximates the chance constraint P(L(x, ξ) 0) ϵ 2 Nemirovski & Shapiro, SIOPT 2006

32 Exactness of the CVaR Approximation 3 P = {P : E P (ξ) =µ, E P (ξξ T )=Σ + µµ T} Theorem If L(x, ξ) is either concave or quadratic in ξ, then sup P P inf P P P-CVaR ϵ (L(x, ξ)) 0 P (L(x, ξ) 0) ϵ Corollary If L(x, ξ) is either concave or quadratic in ξ, then sup P P = sup P P P-CVaR ϵ (L(x, ξ)) P-VaR ϵ (L(x, ξ)) 3 Zymler, K & Rustem, MPA 202

33 Tractability of the CVaR Approximation P = {P : E P (ξ) =µ, E P (ξξ T )=Σ + µµ T} Theorem If L(ξ) = sup P P min {a m + b m ξ}, then m=,...,l P-CVaR ϵ (L(ξ)) = inf β + ϵ Ω, M s.t. β R, M S k+ +, λ R l + l [ 0 M λ b ] 2 m m 0 2 bt m a m β m= l λ m = m=

34 Tractability of the CVaR Approximation P = {P : E P (ξ) =µ, E P (ξξ T )=Σ + µµ T} Theorem If L(ξ) =ξ Qξ + q ξ + q 0,then sup P P P-CVaR ϵ (L(ξ)) = inf β + ϵ Ω, M s.t. β R, M S k+ + [ Q M q ] 2 2 qt q 0 0 β

35 Robust Growth-Optimal Portfolios

36 Calculating WVaR (w) I Weak sense white noise ambiguity set I WVaR (w) P = P : E P ( r t )=µ, E P r s r t = st + µµ max ( : P T TX t= w r t 2 (w r! t ) 2 8P 2 P ) I Distributionally robust quadratic chance constraint.

37 Calculating WVaR (w) Re-express the distributionally robust chance constraint as an LMI. WVaR (w) = max s. t. M 2 S nt +, 2 R, 2 R + " 2 M h, Mi apple0, M 0 P T P t= P t ww T P t 2 t= P t w 2 PT t= P t w T # I Tractable, but... I The dimension of the LMIs is (nt + ) (nt + ). I n = 30, T = 20! 6.5M. decision variables. is the second-order moment matrix of [ r, r 2,..., r T ]. 2 P t 2 R n nt is a truncation operator: P t [r,...,r T ] = r t.

38 Calculating WVaR (w) Simplify the SDP using a projection property of distribution families. r t 2 R n (µ, ) t 2 R (w µ, w w) WVaR (w) = max s. t. M 2 S T +, 2 R, 2 R + M I More tractable. h (w), Mi apple0, M 0 apple 2 I 2 2 T I The dimension of the LMIs is (T + ) (T + ). I n = 30, T = 20! 7.4K decision variables. Yu et al., NIPS 2009

39 Calculating WVaR (w) Compound symmetric solution Initial M Compound symmetric M I Highly tractable. I The number of decision variables (M,, ) reduces to 6!

40 Calculating WVaR (w) Positive semidefiniteness of any compound symmetric matrix M with parameters, 2, 3, and 4 can be verified very efficiently. M 0 () 8 2 >< 4 0 +(T ) 2 0 >: 4 ( +(T ) 2 ) T 3 2

41 Calculating WVaR (w) WVaR (w) = max s. t. 2 R 4, 2 R, 2 R h i T 2p + µ p 2 + T (T )µ 2 p 2 + 2T µ p apple (T ) ( +(T ) 2 ) T T (T ) 2 0 ( 4 T + ) 2 +(T ) 2 T I Nonlinear program with 6 variables and 9 constraints!

42 Calculating WVaR (w) I Analytical solution: WVaR (w) = 2 w µ + r T /2 w! 2 T T w wa I Maximizing WVaR (w) gives rise to an SOCP whose size is independent of T.

43 Discussion I Relation to Markowitz RGOP is Markowitz-efficient, tailored to T and. I Long-term investors When T!, WVaR (w) reduces to 2 2 ( w µ) 2 {z } nominal growth rate 2 w w {z } risk premium I Relation to El Ghaoui et al. When T =, WVaR (w) reduces to w µ + r /2 w {z } formula by El Ghaoui 2 C A C A El Ghaoui et al., OR 2003

44 More Information: Support I Weak sense white noise ambiguity set with support I E.g. ellipsoidal support P = P \ P : P r,...,r T 2 = = ( r,...,r T : T TX t= (r t ) (r t ) apple )

45 More Information: Support WVaR (w) = max s. t. A 2 S n, B 2 S n, c 2 R n, d 2 R, 0, apple 0, 0, 2 R + (T ha, + µµ i + T (T ) hb, µµ i + 2T c µ + d) apple 0 apple apple A +(T )B c c d T A B 2 A +(T )B + c w 4 c + d+ ( ) 2 T w 2 apple A B + w w I The size of this SDP is independent of T.

46 Less Information: Moment Uncertainty I Moment estimates can differ greatly from the true values. I 2 nd -layer of robustness: confidence region. n (µ, ) :(µ ˆµ) ˆ (µ ˆµ) apple, ˆ o 2 I Analytical expression for WVaR (w) w ˆµ + p r! ( ) 2 + T ˆ /2 w! 2 2 (T ) w ˆ w T

47 Synthetic Experiment: Horizon Effects I Black Scholes economy I Calculate µ and from 0 Industry Portfolios [] I Compare VaRs and Sharpe Ratios of RGOP and GOP VaR: Break-even point 70 years SR: Always 4.24% better on avg. Data Source: Fama French Online Data Library

48 Synthetic Experiment: Ambiguity Effects I Calculate µ and from 0 Industry Portfolios I T = 360 months, = 5% I Compare VaRs of RGOP and GOP under 2 distributions in P outperformance % Lognormal WC-Dist for GOP WC-Dist for RGOP

49 Out-of-Sample Backtests Dataset ˆr p ˆp 0IND 2IND ishares DJIA csr c TR c NR [MDD RGOP /n GOP RGOP /n GOP RGOP /n GOP RGOP /n GOP winners

50 References I Kelly, J. L. A new interpretation of information rate. Bell System Technical Journal 35, 4 (956), I Latanè, H. A. Criteria for choice among risky ventures. Journal of Political Economy 67, 2 (959), I Roy, A. D. Safety first and the holding of assets. Econometrica 20, 3 (952), I Rubinstein., M. Continuously rebalanced investment strategies. Journal of Portfolio Management 8, (99), I Rujeerapaiboon, N., Kuhn, D., and Wiesemann, W. Robust growth-optimal portfolios. Working Paper (204). I Yu, Y., Li, Y., Schuurmans, D., and Szepesvári, C. A general projection property for distribution families. Advances in Neural Information Processing Systems 22 (2009), I Zymler, S., Kuhn, D., and Rustem, B. Distributionally robust joint chance constraints with second-order moment information. Mathematical Programming 37, -2 (203), I Zymler, S., Kuhn, D., and Rustem, B. Worst-case value-at-risk of nonlinear portfolios. Management Science 59, (203),