Optimal Portfolio Under Worst-Case Scenarios Carole Bernard (UW), Jit Seng Chen (UW) and Steven Vanduffel (Vrije Universiteit Brussel)
SIAM, July 2012.
Carole Bernard
Optimal Portfolio
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Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Contributions 1
Understanding issues with traditional diversification strategies (buy-and-hold, constant mix, Growth Optimal Portfolio) and how lowest outcomes of optimal strategies always happen in the worst states of the economy.
2
Develop innovative strategies to cope with this observation.
3
Implications in terms of assessing risk and return of a strategy and in terms of reducing systemic risk
Carole Bernard
Optimal Portfolio
2/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Part I:
Traditional Diversification Strategies
Carole Bernard
Optimal Portfolio
3/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Growth Optimal Portfolio (GOP) • The Growth Optimal Portfolio (GOP) maximizes expected
logarithmic utility from terminal wealth. • It has the property that it almost surely accumulates more
wealth than any other strictly positive portfolios after a sufficiently long time. • Under general assumptions on the market, the GOP is a
diversified portfolio. • Details in Platen & Heath (2006).
Carole Bernard
Optimal Portfolio
4/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
For example, in the 2-dim Black-Scholes model • Risk-free asset {Bt = B0 e rt , t > 0}
dSt1 St1 dSt2 St2
= µ1 dt + σ1 dWt1 = µ2 dt + σ2 dWt
,
where W 1 and W are two correlated Brownian motions under the physical probability measure P. • Constant-mix strategy: Dynamic rebalancing to preserve the initial target allocation. The payoff of a constant-mix strategy is Stπ = S0π exp(Xtπ ) where Xtπ is normal. • The Growth Optimal Portfolio (GOP) is a constant-mix strategy with Xtπ = µπ − 21 σπ2 t + σπ Wtπ , that maximizes the expected growth rate µπ − 12 σπ2 . It is → → π ? = Σ−1 · µ −r 1 . Carole Bernard
Optimal Portfolio
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Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Market Crisis The growth optimal portfolio S ? can also be interpreted as a major market index. Hence it is intuitive to define a stressed market (or crisis) at time T as an event where the market materialized through S ? - drops below its Value-at-Risk at some high confidence level. The corresponding states of the economy verify Crisis states = {ST? < qα } , (1) where qα is such that P(ST? < qα ) = 1 − α and α is typically high (e.g. α = 0.98).
Carole Bernard
Optimal Portfolio
6/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Strategy 1: GOP We invest fully in the GOP. In a crisis (GOP is low), our portfolio is low!
Carole Bernard
Optimal Portfolio
7/21
Strategy 1 vs the Growth Optimal Portfolio 200
180
Strategy 1
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100
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60 60
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100 120 140 160 Growth Optimal Portfolio, S ∗ (T )
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Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Strategy 2: Buy-and-Hold The buy-and-hold strategy is the simplest investment strategy. An initial amount V0 is used to purchase w0 units of the bank account and wi units of stock S i (i = 1, 2) such that V0 = w0 + w1 S01 + w2 S02 , and no further action is undertaken. Example with 1/3 invested in each asset (bank, S1 and S2 ) on next slide.
Carole Bernard
Optimal Portfolio
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Strategy 2 vs the Growth Optimal Portfolio 220 200 180
Strategy 2
160 140 120 100 80 60 60
80
100 120 140 160 Growth Optimal Portfolio, S ∗ (T )
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Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
I These traditional diversification strategies do not offer protection during a crisis. I In a more general setting, optimal strategies share the same problem... Optimal Portfolio Selection Problem: Consider an investor with fixed investment horizon: max U(XT ) XT
subject to a given “cost of XT ” (equal to initial budget) • Law-invariant preferences XT ∼ YT ⇒ U(XT ) = U(YT ) • Increasing preferences
XT ∼ F , YT ∼ G , ∀x, F (x) 6 G (x) ⇒ U(XT ) > U(YT )
Carole Bernard
Optimal Portfolio
11/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Optimal Investment
max
{XT / initial budget=x0 }
U(XT )
Theorem Optimal strategies for U must be cost-efficient. where we recall the definition of cost-efficiency. Definition - Dybvig (1988) A strategy (or a payoff) is cost-efficient if any other strategy that generates the same distribution F under P costs at least as much. A cost-efficient strategy solves the following optimization problem minXT cost(XT ) . subject to {XT ∼ F } Carole Bernard
Optimal Portfolio
12/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Stochastic Discount Factor and Real-World Pricing: The GOP can be used as numeraire to price under P XT x0 = Cost of −rT = EQ [e XT ] = EP XT at 0 ST? where S0? = 1. Cost-efficiency problem:
minXT EP
h
XT ST?
i
subject to {XT ∼ F }
Theorem A strategy is cost-efficient if and only if its payoff is equal to XT = h(ST? ) where h is non-decreasing.
Carole Bernard
Optimal Portfolio
13/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Stochastic Discount Factor and Real-World Pricing: The GOP can be used as numeraire to price under P XT x0 = Cost of −rT = EQ [e XT ] = EP XT at 0 ST? where S0? = 1. Cost-efficiency problem:
minXT EP
h
XT ST?
i
subject to {XT ∼ F }
Theorem A strategy is cost-efficient if and only if its payoff is equal to XT = h(ST? ) where h is non-decreasing.
Carole Bernard
Optimal Portfolio
13/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Idea of the proof (First method) XT min E XT ST? ( XT ∼ F subject to 1 S? ∼ G T
Recall that h i E XT? − E[ 1? ]E[X ] T ST ST 1 corr XT , ? = . 1 ST std( S ? ) std(XT ) T
We can prove that when the distributions for both XT and S1? are T fixed, we have 1 1 XT , ? is anti-monotonic ⇔ corr XT , ? is minimal. ST ST Carole Bernard
Optimal Portfolio
14/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Idea of the proof (Second method) Recall that the joint cdf of a couple (ST? , XT ) writes as P(ST? 6 s, XT 6 x) = C (G (s), F (x)) where G is the marginal cdf of ST? (known: it depends on the financial market), F is the marginal cdf of XT and C denotes the copula for (ST? , X ). XT min E subject to XT ∼ F XT ST? Z Z XT E = (G (1/ξ) − C (G (1/ξ), F (x)))dxdξ, (2) ST? h i The lower bound for E XS T? is derived from the upper bound on C T
C (u, v ) 6 min(u, v ) (where min(u, v ) corresponds to the comonotonic copula). then XT? = F −1 (G (ST? )) has the minimum price for the cdf F . Carole Bernard
Optimal Portfolio
15/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Part II:
Investment under Worst-Case Scenarios
Carole Bernard
Optimal Portfolio
16/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Type of Constraints We find optimal strategies with final payoff XT ∼ F I with a set of probability constraints, for example assuming that the final payoff of the strategy is independent of ST? during a crisis (defined as ST? 6 qα ), ∀s 6 qα , x ∈ R, P(ST? 6 s, XT 6 x) = P(ST? 6 s)P(XT 6 x) Theorem (Optimal Investment with Independence in the Tail) The cheapest path-dependent strategy with cdf F and independent of ST? when ST? 6 qα can be constructed as ? −1 FST? (ST )−α F when ST? > qα , 1−α XT? = (3) F −1 (g (St? , ST? )) when ST? 6 qα , where g (., .) is explicit and t ∈ (0, T ) can be chosen freely. Carole Bernard
Optimal Portfolio
17/21
Strategy 4 vs the Growth Optimal Portfolio 200
180
Strategy 4
160
140
120
100
80
60 60
80
100 120 140 160 Growth Optimal Portfolio, S ∗ (T )
180
200
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Proof & Other Types of Dependence Proof: We make use of our results (JAP, 2012) extending Rachev and R¨ uschendorf (1998) and Tankov (JAP, 2011) to derive improved Fr´echet bounds on copulas when there are constraints on a rectangle. P(ST? 6 s, XT 6 x) = C (G (s), F (x)) where G is the marginal cdf of ST? , F is the marginal cdf of XT and C is a copula. Optimal strategies can be derived explicitly: I Independence in the tail (C (u, v ) = uv ): ∀s 6 qα , x ∈ R, P(ST? 6 s, XT 6 x) = P(ST? 6 s)P(XT 6 x) I Gaussian copula in the tail with correlation -0.5. I Similarly for Clayton or Frank dependence. Carole Bernard
Optimal Portfolio
19/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Proof & Other Types of Dependence Proof: We make use of our results (JAP, 2012) extending Rachev and R¨ uschendorf (1998) and Tankov (JAP, 2011) to derive improved Fr´echet bounds on copulas when there are constraints on a rectangle. P(ST? 6 s, XT 6 x) = C (G (s), F (x)) where G is the marginal cdf of ST? , F is the marginal cdf of XT and C is a copula. Optimal strategies can be derived explicitly: I Independence in the tail (C (u, v ) = uv ): ∀s 6 qα , x ∈ R, P(ST? 6 s, XT 6 x) = P(ST? 6 s)P(XT 6 x) I Gaussian copula in the tail with correlation -0.5. I Similarly for Clayton or Frank dependence. Carole Bernard
Optimal Portfolio
19/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Some numerical results We define two events related to the market, i.e. the market crisis C = {S?T < qα }. Define A = XT < x0 erT . GOP Buy-and-Hold Independence Gaussian
Carole Bernard
T 5 5 5 5
Cost 100 100 101.67 103.40
Sharpe 0.266 0.239 0.214 0.159
P(A|C) 1.00 0.9998 0.46 0.12
Optimal Portfolio
20/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
Some Implications I Trade-off between losing “utility” and gaining protection during a “crisis”: the new strategies do not incur their biggest losses in the worst states in the economy. I This can be used to reduce systemic risk. • the idea of assessing risk and performance of a portfolio not
only by looking at its final distribution but also by looking at its interaction with the economic conditions is indeed related to the increasing concern to evaluate systemic risk. • Acharya (2009) explains that regulators should “be regulating each bank as a function of both its joint (correlated) risk with other banks as well as its individual (bank-specific) risk”. • An insight of this work is that if all institutional investors implement strategies that are resilient against crisis regimes, as we propose, then systemic risk can be diminished.
Do not hesitate to contact me to get updated working papers! Carole Bernard
Optimal Portfolio
21/21
Introduction
Diversification Strategies
Tail Dependence
Numerical Example
Conclusions
References I Bernard, C., Boyle P., Vanduffel S., 2011, “Explicit Representation of Cost-efficient Strategies”, available on SSRN. I Bernard, C., Jiang, X., Vanduffel, S., 2012. “Note on Improved Fr´ echet bounds and model-free pricing of multi-asset options”, Journal of Applied Probability. I Bernard, C., Maj, M., Vanduffel, S., 2011. “Improving the Design of Financial Products in a Multidimensional Black-Scholes Market,”, North American Actuarial Journal. I Bernard, C., Vanduffel, S., 2011. “Optimal Investment under Probability Constraints,” AfMath Proceedings. I Bernard, C., Vanduffel, S., 2012. “Financial Bounds for Insurance Prices,”Journal of Risk Insurance. I Cox, J.C., Leland, H., 1982. “On Dynamic Investment Strategies,” Proceedings of the seminar on the Analysis of Security Prices,(published in 2000 in JEDC). I Dybvig, P., 1988a. “Distributional Analysis of Portfolio Choice,” Journal of Business. I Dybvig, P., 1988b. “Inefficient Dynamic Portfolio Strategies or How to Throw Away a Million Dollars in the Stock Market,” Review of Financial Studies. I Goldstein, D.G., Johnson, E.J., Sharpe, W.F., 2008. “Choosing Outcomes versus Choosing Products: Consumer-focused Retirement Investment Advice,” Journal of Consumer Research. I Jin, H., Zhou, X.Y., 2008. “Behavioral Portfolio Selection in Continuous Time,” Mathematical Finance. I Nelsen, R., 2006. “An Introduction to Copulas”, Second edition, Springer. I Pelsser, A., Vorst, T., 1996. “Transaction Costs and Efficiency of Portfolio Strategies,” European Journal of Operational Research. I Platen, E., 2005. “A benchmark approach to quantitative finance,” Springer finance. I Tankov, P., 2011. “Improved Fr´ echet bounds and model-free pricing of multi-asset options,” Journal of Applied Probability, forthcoming. I Vanduffel, S., Chernih, A., Maj, M., Schoutens, W. 2009. “On the Suboptimality of Path-dependent Pay-offs in L´ evy markets”, Applied Mathematical Finance.
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Optimal Portfolio
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