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
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.
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.
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.
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!
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.
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).