Robust Growth-Optimal Portfolios

1. Robust Growth-Optimal Portfolios
N. Rujeerapaiboon1, D. Kuhn1, W. Wiesemann2
1Risk Analytics and Optimization Chair
´Ecole Polytechnique F´ed´erale de Lausanne
2Imperial College Business School

2. 4 Technology Stocks
Ÿ 4 technology companies: Intel, Cisco, Blackberry, and Nokia.
Ÿ Monthly statistics:
µp%q
INTC 0.37
CSCO 0.33
BBRY 0.93
NOK 0.85
Σp%q INTC CSCO BBRY NOK
INTC 0.71 0.33 0.31 0.37
CSCO 0.33 0.91 0.52 0.55
BBRY 0.31 0.52 3.38 0.84
NOK 0.37 0.55 0.84 2.50

3. Markowitz Portfolio Theory

4. Fixed-Mix Strategy
wt pr1, . . . , rt¡1q w
Ÿ Keep portfolio weights constant over time.
Ÿ Memoryless but dynamic.

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

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

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

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

9. All is Lost w.p. 1!

10. Asymptotic Winners Losers
Portfolio’s growth rate = expected logarithmic utility
γVpwq EP log p1 w ˜rq
Ÿ γVpwq ¡ 0: asymptotic winners
Ÿ γVpwq 0: asymptotic losers
Growth optimal portfolio (GOP) is the one that maximizes γVpwq.
wg
€ argmax
w
γVpwq
Kelly Latan´e: GOP outperforms any other causal investment
strategy with probability 1 in the long run.
1
Kelly (1956). Bell System Technical Journal.
2
Latan´e (1959). Journal of Political Economy.

11. Markowitz’ Winners Losers

12. Distributional Assumptions
Ÿ µ and Σ are the only quantities that can be
distilled out of the past.
Ÿ The slightest acquaintance with problems
of analyzing economic time series will
suggest that this assumption is optimistic
rather than unnecessarily restrictive.
— Roy (1952)
1
Roy (1952). Econometrica.

13. Long run may be indeed long . . .
In BS economy, it may take GOP
Ÿ 208 years to beat an all-cash strategy
Ÿ 4,700 years to beat an all-stock strategy
with 95% confidence.
— Rubinstein (1991)
1
Rubinstein (1991). Journal of Portfolio Management.

14. Robust Growth-Optimal Portfolios

15. Robust Measures for Finite Horizons
Performance measures for finite horizons:
Ÿ VaR pwq =
max
γ
5
γ : P
£
1
T
T¸
t1
log p1 w ˜rt q ¥ γ
¥ 1 ¡
C
.
Ÿ WVaR pwq =
max
γ
5
γ : P
£
1
T
T¸
t1
log p1 w ˜rt q ¥ γ
¥ 1 ¡ dP € P
C
.

16. Robust GOP
Performance Guarantees
The fixed-mix strategy w will grow
at least by eT¤WVaR pwq, w.p. 1 ¡
under any distribution P € P.
Robust GOP (R-GOP) is the one that maximizes WVaR pwq.
w¦ € argmax
w
WVaR pwq

17. Calculating WVaR pwq
Ÿ Weak sense white noise ambiguity set
P
2
P : EP p˜rt q µ, EP
˜rs˜rt
¨
δst Σ µµ
@
Ÿ WVaR pwq
max
γ
5
γ : P
£
1
T
T¸
t1
¢
w ˜rt ¡ 1
2
pw ˜rt q2
¥ γ
¥ 1 ¡ dP € P
C
Ÿ Distributionally robust quadratic chance constraint.

18. Distributionally Robust Chance Constraint
Zymler et al.’s results yield an SDP formulation of WVaR pwq.
max γ
s. t. M € SnT 1
, β € R, γ € R
β 1
Ω, M ¤ 0, M © 0
M ©
1
2
°T
t1 Pt ww Pt ¡1
2
°T
t1 Pt w
¡1
2
¡°T
t1 Pt w
©
γT ¡β
'
Ÿ Tractable, but . . .
Ÿ The dimension of the LMIs is pnT 1q¢pnT 1q.
Ÿ n 30, T 120 Ñ #decision variables 6.5M.
1
Zymler, Kuhn Rustem (2013). Math Programming.
2
Ω is the second-order moment matrix of r˜r1 , ˜r2 , . . . , ˜rT s .
3
Pt € Rn¢nT
is a truncation operator: Pt
r1 , . . . , rT
$
rt .

19. Calculating WVaR pwq
Yu et al.’s projection property of distribution families simplifies the
calculation of WVaR pwq.
˜rt € Rn
pµ, Σq è ˜ηt € R pw µ, w Σwq
max γ
s. t. M € ST 1
, β € R, γ € R
β 1
Ωpwq, M ¤ 0, M © 0
M ©
1
2 I ¡1
2 1
¡1
2 1 γT ¡β
Ÿ More tractable.
Ÿ The dimension of the LMIs is pT 1q¢pT 1q.
Ÿ n 30, T 120 Ñ #decision variables 7.4K.
1
Yu, Li, Schuurmans Szepesv`ari (2009). NIPS.

20. Calculating WVaR pwq
Compound symmetric solution
Initial M Compound
symmetric M
Ÿ Highly tractable.
Ÿ The number of decision variables (M, β, γ) reduces to 6!

21. Calculating WVaR pwq
Positive semidefiniteness of any compound symmetric matrix M with
parameters τ1, τ2, τ3, and τ4 can be verified efficiently.
M © 0 ðñ
6
99998
99997
τ1 ¥ τ2
τ4 ¥ 0
τ1 pT ¡1qτ2 ¥ 0
τ4 pτ1 pT ¡1qτ2q ¥ Tτ2
3

22. Calculating WVaR pwq
Ÿ Analytical expression for WVaR pwq
1
2
¤
¥1 ¡
£
1 ¡w µ
™
1 ¡
T
Σ1{2
w
2
¡ T ¡1
T
w Σw
Ÿ Maximizing WVaR pwq is an SOCP whose size is independent
of the horizon T.

23. Remarks
Ÿ Relation to the Markowitz model
R-GOP is an efficient portfolio tailored to T and .
Ÿ Long-term investors
When T Ñ V, WVaR pwq reduces to
1
2
¡ 1
2
p1 ¡w µq2
looooooooooomooooooooooon
nominal growth rate
¡ 1
2
w Σw
loooomoooon
risk premium
Ÿ Relation to El Ghaoui et al.
When T 1, WVaR pwq reduces to
1
2
¤
¦
¦
¥1 ¡
¤
¦
¦
¥1 ¡w µ
™
1 ¡
Σ1{2
w
loooooooooooooooomoooooooooooooooon
formula by El Ghaoui
2
1
El Ghaoui, Oks Outstry (2003). Operations Research.

24. Support Information
Ÿ Weak sense white noise ambiguity set with support Ξ
PΞ P ˆ
2
P : P
r1 , . . . , rT
$
€ Ξ
¨
1
@
Ÿ E.g. ellipsoidal support
Ξ
5
r1 , . . . , rT
$
:
1
T
T¸
t1
prt ¡νq Λ¡1
prt ¡νq ¤ δ
C
leads to a tractable SDP formulation of WVaR pwq.

25. Support Information
WVaR pwq =
max γ
s. t. A € Sn
, B € Sn
, c € Rn
, d € R, α ¥ 0, β ¤ 0, λ ¥ 0, γ € R
β 1
pT A, Σ µµ TpT ¡1q B, µµ 2Tc µ dq ¤ 0
A pT ¡1qB c
c d
T
© α
¡Λ¡1
Λ¡1
ν
ν Λ¡1
δ ¡ν Λ¡1
ν
A ¡B © ¡αΛ¡1
!
A pT ¡1qB λΛ¡1
c ¡λΛ¡1
ν w
c ¡λν Λ¡1 1
2
d β
T
¡γ ¡λpδ ¡ν Λ¡1
νq ¡1
w ¡1 2
(
) © 0
A ¡B λΛ¡1
w
w 2
© 0.
Ÿ The size of this SDP is independent of T.

26. Moment Uncertainty
Ÿ Moment estimates can differ greatly from the true values.
Ÿ 2nd
-layer of robustness: confidence region.
3
pµ, Σq : pµ ¡ ˆµq ˆΣ¡1
pµ ¡ ˆµq ¤ δ1, Σ ¨ δ2
ˆΣ
A
Ÿ Analytical expression for WVaR pwq
£
1 ¡w ˆµ
£
—
δ1
™
p1 ¡ qδ2
T
ˆΣ1{2
w
2
¡ δ2 pT ¡1q
T
w ˆΣw

27. Synthetic Experiment: Horizon T
Ÿ Calculate µ and Σ from 10 Industry Portfolios
Ÿ BS economy simulation
Ÿ Compare VaRs and SRs of GOP and R-GOP
VaR: Break-even point 170 years SR: Always 14.24% better on avg.

28. Synthetic Experiment: Distributional Ambiguity
Ÿ Calculate µ and Σ from 10 Industry Portfolios
Ÿ T 360 months, 5%
Ÿ Compare VaRs of GOP and R-GOP under 21 distributions in P
0.0 0.2 0.4 0.6 0.8 1.0
0.0 11.60 11.53 11.58 11.90 13.27 81.01
0.2 11.42 11.49 12.12 12.05 65.81
0.4 11.37 11.76 13.15 65.79
0.6 12.07 12.61 60.05
0.8 12.78 58.52
1.0 14.75 outperformance %
Lognormal WC-Dist for GOP WC-Dist for R-GOP
1
Available from Fama-French Data Library.

29. Out-of-Sample Test
10IND 12IND iShares
Mean return
RGOP
MV
GOP
1/n
10IND 12IND iShares
Standard deviation
RGOP
MV
GOP
1/n
10IND 12IND iShares
Sharpe ratio
RGOP
MV
GOP
1/n
10IND 12IND iShares
INIT
Terminal wealth
RGOP
MV
GOP
1/n

30. References
Ÿ Kelly, J. L. A new interpretation of information rate. Bell System
Technical Journal 35, 4 (1956).
Ÿ Latan`e, H. A. Criteria for choice among risky ventures. Journal of
Political Economy 67, 2 (1959).
Ÿ Roy, A. D. Safety first and the holding of assets. Econometrica 20, 3
(1952), 431-449.
Ÿ Rubinstein, M. Continuously rebalanced investment strategies. Journal
of Portfolio Management 18, 1 (1991).
Ÿ Rujeerapaiboon, N., Kuhn, D., and Wiesemann, W. Robust
growth-optimal portfolios. Management Science 62, 7 (2016).
Ÿ Yu, Y., Li, Y., Schuurmans, D., and Szepesv´ari, C. A general projection
property for distribution families. Advances in Neural Information
Processing Systems 22 (2009).
Ÿ Zymler, S., Kuhn, D., and Rustem, B. Distributionally robust joint chance
constraints with second-order moment information. Mathematical
Programming 137, 1-2 (2013).

31. Calculating WVaR pwq
WVaR pwq =
max γ
s. t. τ € R4
, β € R, γ € R
β 1
T
¡
σ2
p µ2
p
©
τ1 TpT ¡1qµ2
pτ2 2Tµpτ3 τ4
%
¤ 0
τ1 ¥ τ2
τ4 ¥ 0
τ1 pT ¡1qτ2 ¥ 0
τ4 pτ1 pT ¡1qτ2q ¥ Tτ2
3
τ1 ¡ 1
2
¨
¥ τ2
τ4 ¡γT β ¥ 0
τ1 ¡ 1
2
¨
pT ¡1qτ2 ¥ 0
pτ4 ¡γT βq
τ1 ¡ 1
2
¨
pT ¡1qτ2
¨
¥ T
τ3 1
2
¨2