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 1/21
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 { B t = B 0 e rt , t � 0 } dS 1 t = µ 1 dt + σ 1 dW 1 t S 1 t , dS 2 t = µ 2 dt + σ 2 dW t t S 2 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 S π t = S π 0 exp( X π t ) where X π t is normal. • The Growth Optimal Portfolio (GOP) is a constant-mix µ π − 1 strategy with X π � 2 σ 2 � t + σ π W π t = t , that maximizes π the expected growth rate µ π − 1 2 σ 2 π . It is � → → π ⋆ = Σ − 1 · � µ − r . 1 Carole Bernard Optimal Portfolio 5/21
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 = { S ⋆ T < q α } , (1) where q α is such that P ( S ⋆ T < q α ) = 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 200 180 160 140 Strategy 1 120 100 80 60 60 80 100 120 140 160 180 200 Growth Optimal Portfolio, S ∗ ( T )
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 V 0 is used to purchase w 0 units of the bank account and w i units of stock S i ( i = 1 , 2) such that V 0 = w 0 + w 1 S 1 0 + w 2 S 2 0 , and no further action is undertaken. Example with 1/3 invested in each asset (bank, S 1 and S 2 ) on next slide. Carole Bernard Optimal Portfolio 9/21
Strategy 2 vs the Growth Optimal Portfolio 220 200 180 160 Strategy 2 140 120 100 80 60 60 80 100 120 140 160 180 200 Growth Optimal Portfolio, S ∗ ( T )
Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions ◮ These traditional diversification strategies do not offer protection during a crisis. ◮ In a more general setting, optimal strategies share the same problem... Optimal Portfolio Selection Problem: Consider an investor with fixed investment horizon : max U ( X T ) X T subject to a given “cost of X T ” (equal to initial budget) • Law-invariant preferences X T ∼ Y T ⇒ U ( X T ) = U ( Y T ) • Increasing preferences X T ∼ F , Y T ∼ G , ∀ x , F ( x ) � G ( x ) ⇒ U ( X T ) � U ( Y T ) Carole Bernard Optimal Portfolio 11/21
Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions Optimal Investment / initial budget = x 0 } U ( X T ) max { X T 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 min X T cost ( X T ) { X T ∼ F } . subject to 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 � x 0 = Cost of � � X T � = E Q [ e − rT X T ] = E P S ⋆ X T at 0 T where S ⋆ 0 = 1. � � X T min X T E P S ⋆ Cost-efficiency problem: T subject to { X T ∼ F } Theorem A strategy is cost-efficient if and only if its payoff is equal to X T = h ( S ⋆ T ) 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 � x 0 = Cost of � � X T � = E Q [ e − rT X T ] = E P S ⋆ X T at 0 T where S ⋆ 0 = 1. � � X T min X T E P S ⋆ Cost-efficiency problem: T subject to { X T ∼ F } Theorem A strategy is cost-efficient if and only if its payoff is equal to X T = h ( S ⋆ T ) 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) � X T � min E S ⋆ X T T � X T ∼ F subject to 1 T ∼ G S ⋆ Recall that � � X T − E [ 1 T ] E [ X T ] E � X T , 1 � S ⋆ S ⋆ T corr = . S ⋆ std( 1 T ) std( X T ) T S ⋆ 1 We can prove that when the distributions for both X T and T are S ⋆ fixed, we have � X T , 1 � � X T , 1 � is anti-monotonic ⇔ corr is minimal. S ⋆ S ⋆ T T 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 ( S ⋆ T , X T ) writes as P ( S ⋆ T � s , X T � x ) = C ( G ( s ) , F ( x )) where G is the marginal cdf of S ⋆ T (known: it depends on the financial market), F is the marginal cdf of X T and C denotes the copula for ( S ⋆ T , X ) . � X T � min E subject to X T ∼ F S ⋆ X T T � X T � � � = ( G (1 /ξ ) − C ( G (1 /ξ ) , F ( x ))) dxd ξ, (2) E S ⋆ T � � X T The lower bound for E is derived from the upper bound on C S ⋆ T C ( u , v ) � min( u , v ) (where min( u , v ) corresponds to the comonotonic copula). then X ⋆ T = F − 1 ( G ( S ⋆ T )) 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 X T ∼ F ◮ with a set of probability constraints, for example assuming that the final payoff of the strategy is independent of S ⋆ T during a crisis (defined as S ⋆ T � q α ), ∀ s � q α , x ∈ R , P ( S ⋆ T � s , X T � x ) = P ( S ⋆ T � s ) P ( X T � x ) Theorem ( Optimal Investment with Independence in the Tail) The cheapest path-dependent strategy with cdf F and independent of S ⋆ T when S ⋆ T � q α can be constructed as � � F S ⋆ T ( S ⋆ T ) − α F − 1 S ⋆ T > q α , when X ⋆ 1 − α T = (3) F − 1 ( g ( S ⋆ t , S ⋆ S ⋆ T )) when T � q α , 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 200 180 160 140 Strategy 4 120 100 80 60 60 80 100 120 140 160 180 200 Growth Optimal Portfolio, S ∗ ( T )
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