Ambiguity in portfolio optimization

Transcription

1 May/June 2006

2 Introduction: Risk and Ambiguity Frank Knight Risk, Uncertainty and Profit (1920) Risk: the decision-maker can assign mathematical probabilities to random phenomena Uncertainty: randomness cannot be expressed in terms of specific mathematical models. Ellsberg (1961) Uncertainty: the probabilistic model is known, but the realizations of the random variables are unknown ( aleatoric uncertainty ) Ambiguity: the probability model itself is unknown ( epistemic uncertainty ).

3 It has frequently been observed that the solutions of portfolio optimization problems are much sensitive w.r.t. the parameters of the distributions (Klein and Bawa (1976), Chopra and Ziemba (1993)). However, the distributions are estimated from data and therefore contain some estimation error and model ambiguity. Therefore we need to combine Statistics ( how to estimate the probability model and its parameters and to assess confidence ) and Optimization ( how to find the best decision ) in one approach.

4 Problem formulation (ξ 1,..., ξ M ) random returns for M asset categories (x 1,..., x M ) portfolio weights Y x = M m=1 x mξ m portfolio return A(Y x ) acceptability functional Let ˆP be a baseline probability measure for (ξ 1,..., ξ M ). If we fully trust in the validity of this model, we solve the non-ambiguous portfolio optimization problem Maximize (in x) : EˆP (Y x) subject to AˆP (Y x) q x 1l = 1 x 0

5 Acceptability functionals Required properties: (i) translation equivariance (ii) concavity A(Y + c) = A(Y ) + c A(λY 1 + (1 λ)y 2 ) λa(y 1 ) + (1 λ)a(y 2 ). (iii) continuity Y A(Y ) continuous (iv) version independence: If Y 1 and Y 2 have the same distribution, then A(Y 1 ) = A(Y 2 )

6 s for such acceptability functionals A are the expectation A(Y ) = E(Y ) the average-value-at risk (conditional value-at-risk, expected shortfall) A(Y ) = AV@R α (Y ) = 1 α α G 1 Y 0 (p) dp, where G 1 Y (p) = inf{v : P{Y v} p} the (lower) standard deviation corrected expectation (0 < λ < 1) A(Y ) = E(Y ) λstd(y ) or E(Y ) λstd (Y ), Std (Y ) = E[[Y EY ] ) 2

7 Stress testing We solve the portfolio problem for the baseline probability model, but check its performance under new probability models. : 500 historic weakly returns from NYSE model composition optimal for weakly return acceptabilty AV@R 0.1 (ˆP) (ˆP) 0.8% 0.92 (P ) (ˆP) -0.13% 0.90 (P + ) (ˆP) 1.17% 0.93 (P ) (P ) -0.07% 0.92 (P + ) (P + ) 1.9% 0.92 (ˆP) : historic data (P ) : 5% higher prob. for bad scenarios (P + ) : 5% lower prob. for bad scenarios

8 The ambiguity problem - maximin formulation Let P be an ambiguity set, i.e. the set of probability models, to which the modeler in indifferent. The portfolio selection model under P-ambiguity is of maximin type and reads Maximize (in x) : min{e P (Y x ) : P P} subject to A P (Y x ) q for all P P x 1l = 1 x 0

9 Dealing with uncertainty: Deterministic optimization: max{f (x) : F i (x) 0} F is the objective, F i are the constraint functions. Robust optimization (maximin approach): A set Ξ of possible parameters ξ is given. max{min{f (x, ξ) : ξ Ξ} : F i (x, ξ) 0; ξ Ξ} Robust optimization produces very conservative portfolios.

10 Stochastic optimization: (Ξ, A, P) is a probability space, typical element is called ξ. or more generally max{e P [F (x, ξ)]; E P [F i (x, ξ)] 0} max{a P [F (x, ξ)]; A (i) P [F i(x, ξ)] 0} where A ( ) P are aggregating probability functionals, mapping the distribution of F (x, ξ) under P to the (extended) real line. Stochastic optimization produces well diversified portfolios.

11 Robust Opt. Stochastic Opt. Robust Stoch. Opt. Bayesian Opt.??

12 The ambiguity problem - Bayesian formulation Let λ be a probability on the ambiguity set P. λ is called the Bayesian prior on the set of probability models. The Bayesian formulation is Maximize (in x) : P A P[F (x, ξ)] dλ(p) subject to P A(i) P [F i(x, ξ)] dλ(p) 0 x 1l = 1 x 0

13 The ambiguity problem as robust stochastic optimization We repeat the maximin formulation of the problem: Maximize (in x) : min{e P (Y x ) : P P} subject to A P (Y x ) q for all P P x 1l = 1 x 0

14 for the maximin portfolio problem Let X = {x : x 1l = 1, x 0, A P (Y x ) q for all P P}, then the ambiguity problem reads max min E P[Y x ]. x X P P By continuity and concavity of A, X is a compact convex set. Moreover, (P, x) E P [Y x ] is bilinear in P and x and hence convex-concave. Therefore x X is a solution of if and only if there is a P P such that (P, x ) is a saddle point, i.e. E P [Y x ] E P [Y x ] E P [Y x ] for all (P, x) P X. Our problem is semi-infinite. Direct saddle point methods were used by Rockafellar (1976), Nemirovskii and Yudin (1978).

15 Successive convex programming (SCP) 1. Set n = 0 and P 0 = {ˆP} with ˆP P. 2. Solve the outer problem Maximize (in x, t) : t subject to t E P (Y x ) for all P P n A P (Y x ) q for all P P n x 1l = 1; x 0 and call the solution (x n, t n ). 3. Solve the first inner problem Minimize (in P) : E P (Y xn ) subject to P P and call the solution P (1) n.

16 4. Solve the second inner problem Minimize (in P) : A P (Y xn ) subject to P P call the solution P (2) n and let P n+1 = P n {P (1) n } {P (2) n }. 5. If P n+1 = P n then stop. Otherwise set n := n + 1 and goto 2.

17 Convergence Proposition. Assume that P is compact and convex and that (P, x) E P [Y x ] as well as (P, x) A P [Y x ] are jointly continuous. Then every cluster point of (x n ) is a solution of the minimax problem. If the saddle point is unique, then the algorithm converges to the optimal solution.

18 ˆP

19 ˆP P 1

20 ˆP P 1 P 2

21 ˆP P 3 P 1 P 2

22 Ambiguity sets as balls - distances of probability measures We want to use an ambiguity sets of the form P = {P : d(p, P ) ɛ} as ambiguity sets. But which distance d to choose? Transportation distance, Wasserstein distance d T (P 1, P 2 ) = sup{ h(u) dp 1 (u) h(u) dp 2 (u) : h(u) h(v) u v }

23 Kolmogorov-Smirnov distance d KS (P 1, P 2 ) = sup{ P 1 ((, a]) P 2 ((, a]) : a R} Variational distance d V (P 1, P 2 ) = sup{ P 1 (A) P 2 (A) : A mesurable } If densities exist, this is equivalent to the L 1 distance: dp 1 d L1 (P 1, P 2 ) = dµ dp 2 dµ dµ.

24 The ambiguity set must reflect our current information about P If our information is based on statistical estimation, the ambiguity set must coincide with a confidence set If the distance is too fine, no statistical confidence set can be constructed without further assumptions (e.g. for the variational distance) If we base our estimate for P on some historical data ξ 1, ξ 2,..., ξ n, we use the empirical distribution ˆP n {ξ i } = 1 n

25 Confidence sets P{d KS (P, ˆP n ) M/ n} 58 exp( 2M 2 ) (Dvoretzky, Kiefer, Wolfowitz inequality) E P [d T (P, ˆP n )] Cn 1/M for some constant C (Dudley (1969)). By Markov s inequality P{d T (P, ˆP n ) ɛ} E[d(P, ˆP n )]/ɛ n 1/M C/ɛ. Under smoothness conditions on P, the confidence sets may improved (Kersting (1978)).

26 6 assets IBM - International Business Machines Corporation PRG - Procter & Gamble Corporation ATT - AT&T Corporation VER - Verizon Communications Inc INT - Intel Corporation EXX - Exxon Mobil Corporation Risk constraint: AV@R 0,1 0.9 Ambiguity set: P = {P : d T (P, ˆP) ɛ}. In order to make the ambiguity sets more interpretable, we define a robustness parameter γ as the maximal relative change of the expected returns and relate this to ɛ by ɛ = max{η : sup E P (ξ (i) ) (1 + γ)eˆp (ξ(i) ) : for all i}. d T (P,ˆP) η

27 Working with Transportation distances By the well known theorem of Kantorovich-Rubinstein, the Transportation ambiguity set can be represented as {P : d T (P, ˆP) ɛ} = {P : there is a bivariate probability K(, ) s.t. K(u, dv) = P(u); K(du, v) = ˆP(v); u v 1 K(du, dv) ɛ}. v u If the probability space Ω is finite, Ω = {x (1),..., x (n) }, the ambiguity set is a polyhedral set u v {P : d T (P, ˆP) ɛ} = {P = (P 1,..., P S ) : P j = i K ij ; j K ij = ˆP i ; K ij 0; i,j x (i) x (j) 1 K ij ɛ}.

28 Illustration of the Transportation distance P 1 P 2 d T (P 1, P 2 ) = minimal E ( transported mass distance ).

29 The bivariate probability K has the interpretation as the solution of Monge s mass transportation problem. The Transportation distance describes the minimal effort (in terms of expected transportation distances), to change the mass distribution ˆP into the new mass distribution P (Rachev and Rüschendorf (1998)). In the case of a finite probability space Ω it is not difficult to find a solution for the inner problems, i.e. to determine and by a special algorithm. inf{e P (Y x ) : d T (P, ˆP) ɛ} inf{a P (Y x ) : d T (P, ˆP) ɛ}

30 The solution of the Return Basic Probability Model Worst Case Probability Model Robustness Risk Basic Probability Model Worst Case Probability Model Robustness Portfolio Composition Robustness Expected returns, risks and portfolio composition. The assets from top to bottom are: EXX, VER, ATT, PRG, INT, IBM.

31 1.025 Basic Return Robustness Risk Efficient frontiers in dependence of the robustness parameter γ. Risk and return are calculated w.r.t. the basic model ˆP.

32 1.03 Worts Case Return Robustness Risk Efficient frontiers in dependence of the robustness parameter γ. Risk and return are calculated w.r.t. the worst case model.

33 Types of problems in Finance and Risk management Estimation Modeling Pricing (Equation solving): F (x) = y Optimization: max F (x) Game Theory (Equilibrium): max min F (x, y)

34 In order to capture scenario uncertainty (aleatoric uncertainty) and probability ambiguity (epistemic uncertainty) we use a probabilistic maximin approach. The ambiguity neighborhood should be chosen in such a way that it corresponds to a probabilistic confidence regions for which bounds for the covering probability are available. The result of model ambiguity is a further diversification of he portfolio It turns out that often the price to be paid for including ambiguity in the optimization problem is very little. Minimax models are also relevant for the pricing of swing options in Electricity markets.