Optimal Portfolio Under Worst-Case Scenarios Carole Bernard (UW), Jit Seng Chen (UW) and Steven Vanduffel (Vrije Universiteit Brussel) Rennes, March 2012. Carole Bernard Optimal Portfolio 1/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs 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 2/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs Part I: Traditional Diversification Strategies Carole Bernard Optimal Portfolio 3/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs 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 (2006). Carole Bernard Optimal Portfolio 4/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs 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 (2006). Carole Bernard Optimal Portfolio 4/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs 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 t S 1 , (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 . W t = ρ W 1 � 1 − ρ 2 W 2 t + t where W 1 and W 2 are independent. Carole Bernard Optimal Portfolio 5/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs 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. • For an initial investment V 0 , V T is given by S π T V T = V 0 , S π 0 where π is the vector of proportions. Carole Bernard Optimal Portfolio 6/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs Growth Optimal Portfolio (GOP) In the 2-dimensional Black-Scholes setting, • The GOP is a constant-mix strategy with µ π − 1 X π � 2 σ 2 � t + σ π W π t = t , that maximizes the expected π growth rate µ π − 1 2 σ 2 π . It is π ⋆ = Σ − 1 · ( µ − r 1 ) . (2) • constant-mix portfolios given by π = απ ⋆ with α > 0 and where π ⋆ is the optimal proportion for the GOP, are optimal strategies for CRRA expected utility maximizers. With a constant relative risk aversion coefficient η > 0, CRRA utility is � x 1 − η when η � = 1 1 − η U ( x ) = log( x ) when η = 1 , and α = 1 /η . Carole Bernard Optimal Portfolio 7/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs 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 8/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs Srategy 1: GOP We invest fully in the GOP. In a crisis (GOP is low), our portfolio is low! Carole Bernard Optimal Portfolio 9/35
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 Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs 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 11/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs 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 11/35
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 Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs Strategy 3: Constant-Mix Strategy Example with 1/3 invested in each asset (bank, S 1 and S 2 ). Carole Bernard Optimal Portfolio 13/35
Strategy 3 vs the Growth Optimal Portfolio 200 180 160 140 Strategy 3 120 100 80 60 60 80 100 120 140 160 180 200 Growth Optimal Portfolio, S ∗ ( T )
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs ◮ These three traditional diversification strategies do not offer protection during a crisis. ◮ In a more general setting, optimal strategies share the same problem... Carole Bernard Optimal Portfolio 15/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs Part II: Optimal portfolio selection for law-invariant preferences Carole Bernard Optimal Portfolio 16/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs Stochastic Discount Factor and Real-World Pricing : The GOP can be used as numeraire to price under P � Price of � � X T � = E Q [ e − rT X T ] = E P [ ξ T X T ] = E P S ⋆ X T at 0 T where S ⋆ 0 = 1. Carole Bernard Optimal Portfolio 17/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs Stochastic Discount Factor and Real-World Pricing : The GOP can be used as numeraire to price under P � Price of � � X T � = E Q [ e − rT X T ] = E P [ ξ T X T ] = E P S ⋆ X T at 0 T where S ⋆ 0 = 1. Carole Bernard Optimal Portfolio 17/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs Cost-efficient strategies (Dybvig (1988)) 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 ) A strategy (or a payoff) is cost-efficient if any other strategy that generates the same distribution under P costs at least as much. The optimal strategy for U must be cost-efficient . Carole Bernard Optimal Portfolio 18/35
Introduction Diversification Strategies Cost-Efficiency Tail Dependence Numerical Example Conclusions Proofs Cost-efficient strategies (Dybvig (1988)) 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 ) A strategy (or a payoff) is cost-efficient if any other strategy that generates the same distribution under P costs at least as much. The optimal strategy for U must be cost-efficient . Carole Bernard Optimal Portfolio 18/35
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