Optimal Portfolio Under Worst-Case Scenarios Carole Bernard (UW), Jit Seng Chen (UW) and Steven Vanduffel (Vrije Universiteit Brussel) Samos, June 2012. Carole Bernard Optimal Portfolio 1/20
Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions Contributions 1 A better understanding of the link between Growth Optimal Portfolio and optimal investment strategies 2 Understanding issues with traditional diversification strategies and how lowest outcomes of optimal strategies always happen in the worse states of the economy . 3 Develop innovative strategies to cope with this observation. 4 Implications in terms of assessing the risk and return of a strategy and in terms of reducing systemic risk Carole Bernard Optimal Portfolio 2/20
Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions Part I: Traditional Diversification Strategies Carole Bernard Optimal Portfolio 3/20
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 4/20
Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions For example, in the Black-Scholes model • A Black-Scholes financial market (mainly for ease of exposition) • 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 (1) , 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 . Carole Bernard Optimal Portfolio 5/20
Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions 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 2 σ 2 � � strategy with X π t = t + σ π W π t , that maximizes π the expected growth rate µ π − 1 2 σ 2 π . It is π ⋆ = Σ − 1 · ( µ − r 1 ) . (2) Carole Bernard Optimal Portfolio 6/20
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 } , (3) 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 7/20
Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions Srategy 1: GOP We invest fully in the GOP. In a crisis (GOP is low), our portfolio is low! Carole Bernard Optimal Portfolio 8/20
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 Srategy 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 10/20
Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions Srategy 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 10/20
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 wealth) • 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 12/20
Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions Optimal Investment Theorem The optimal strategy for U must be cost-efficient . Definition A strategy (or a payoff) is cost-efficient if any other strategy that generates the same distribution under P costs at least as much. 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 13/20
Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions Optimal Investment Theorem The optimal strategy for U must be cost-efficient . Definition A strategy (or a payoff) is cost-efficient if any other strategy that generates the same distribution under P costs at least as much. 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 13/20
Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions Part II: Investment under Worst-Case Scenarios Carole Bernard Optimal Portfolio 14/20
Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions Type of Constraints We are able to find optimal strategies with final payoff V T ◮ 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 , v ∈ R , P ( S ⋆ T � s , V T � v ) = P ( S ⋆ T � s ) P ( V T � v ) 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 when S ⋆ T > q α , 1 − α V ⋆ T = (4) F − 1 ( g ( S ⋆ t , S ⋆ T )) when S ⋆ T � q α , where g ( ., . ) is explicit and t ∈ (0 , T ) can be chosen freely. Carole Bernard Optimal Portfolio 15/20
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 )
Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions Other Types of Dependence Recall that the joint cdf of a couple ( S ⋆ T , V T ) writes as P ( S ⋆ T � s , V T � x ) = C ( H ( s ) , F ( x )) where • The marginal cdf of S ⋆ T : H • The marginal cdf of V T : F • A copula C Independence in the tail (independence copula C ( u , v ) = uv ): ∀ s � q α , v ∈ R , P ( S ⋆ T � s , V T � v ) = P ( S ⋆ T � s ) P ( V T � v ) ◮ We were also able to derive formulas for optimal strategies that generate a Gaussian copula in the tail with a correlation coefficient of -0.5. ◮ Similarly for Clayton or Frank dependence. Carole Bernard Optimal Portfolio 17/20
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 < q α } and a decrease in the market � 0 e rT � D = S ⋆ T < S ⋆ . We further define two events for the portfolio V T < V 0 e rT � V T < 75% V 0 e rT � � � value by A = and B = Cost Sharpe P ( A | C ) P ( A | D ) P ( B | C ) T GOP 5 100 0.266 1.00 1.00 1.00 Buy-and-Hold 5 100 0.239 0.9998 0.965 0.99 Independence 5 101.67 0.214 0.46 0.94 0.13 Gaussian 5 103.40 0.159 0.12 0.90 0.01 Carole Bernard Optimal Portfolio 18/20
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