Building Consistent Risk Measures into Stochastic Optimization Models

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1 Building Consistent Risk Measures into Stochastic Optimization Models John R. Birge The University of Chicago Graduate School of Business JRBirge Fuqua School, Duke University April 26,

2 Themes Trust Ockham (keep it simple) Different approaches may lead to the same outcomes Consistency is in the eye of the beholder (but not always in the model) Consistency can appear in different ways Actual preferences may support consistent models that violate reasonable axioms JRBirge Fuqua School, Duke University April 26,

3 Outline Probabilistic ( robust ) and expected utility measures Consistent views of alternatives Axioms and principles for model selection and solution Behavioral violations and their implications JRBirge Fuqua School, Duke University April 26,

4 Expected Utility (Recourse Model) Coherent Risk The Debates Old World New World Probabilistic (Chance-constrained) Model Value-at-Risk and Robust Optimization Models JRBirge Fuqua School, Duke University April 26,

5 Mathematical Forms Expected utility: Max x E(f(x,ξ)) Prob. constraints: Max x g(x) s.t. P(h(x,ξ)>=0) α (Coherent) Risk: Max x R(g(x,ξ)) Robust opt.: Max x g(x) s.t. h(x,ξ)>= 0 ξ Ξ JRBirge Fuqua School, Duke University April 26,

6 Issues in Formulations What are the parameters? (f,g,h,r,ξ) Can all the models be consistent? (all right?) Is there a best? How to choose? JRBirge Fuqua School, Duke University April 26,

7 Misinterpretations Objective functions: f, g (h) are the same in each model Probability distribution: P, Ξ must be known with certainty Results are inconsistent with each rationality or behavior JRBirge Fuqua School, Duke University April 26,

8 Forms of Resolution Make models look the same: Robust/Prob => Risk: Let R(g (x,ξ))= g(x)+δ(x h(x,ξ) 0 ξ Ξ) max x X R(g (x,ξ)) max x X g(x) h(x,ξ) 0 ξ Ξ Risk=>Robust: If R is coherent, P(R) domain of P, h(x,g,p)=g-e P (g(x,ξ))=> g(x, P)=E P [f(x,ξ)] max x X R(g(x,ξ)) max x X g h(x,g,p) 0 P P Others: equivalence of solutions (almost always possible) JRBirge Fuqua School, Duke University April 26,

9 How to Choose? Potential criteria: Self-consistency Preference principles (axioms) Tractability Empirical evidence Consensus possible? JRBirge Fuqua School, Duke University April 26,

10 Self-Consistency Principle: the model should have selfconsistent preference rankings Examples: if criteria are expected utilities, then rankings of equivalent outcomes should not vary Market interpretation: prices in the model should be consistent with criteria (e.g., no arbitrage) JRBirge Fuqua School, Duke University April 26,

11 Models and No-Arbitrage Assumption Suppose a future revenue d distributed 0<d i <U< with probability p i (at time t=1 w.l.o.g. for i=1,,n for N future states, riskfree rate r Absence of arbitrage no one in the market can purchase or sell an equivalent cash flow to any share 0 x 1 in such a way that always produces non-negative and, with some positive probability, positive surpluses c L and c U, c L PV(d) c U such that c L >0 and c U <e -r U JRBirge Fuqua School, Duke University April 26,

12 LP for No Arbitrage A buyer cannot, for any c c L, purchase an equivalent cash flow to a share x of d with y in other assets (with current prices s and future prices S i ) and riskfree investing (B) and obtain positive future cash flows, i.e.: 0=max x,y,b i p i (d i x -S it y -e r B) s. t. -cx + s T y + B=0, -d i x+ S it y + e r B 0, i, 0 x 1 JRBirge Fuqua School, Duke University April 26,

13 Dual for No-Arb LP => EMM Dual: 0=min ρ s.t. i d i p i + λ c + i π i d i -ρ 0, - i S i p i -λ s - i π i S i =0, -e r - λ - i π i e r = 0 For an optimal dual solution, (λ L,π L,ρ L ), π L 0, and ρ L =0: c i d i (p i +π L i)/(-λ L ); s= i (p i + π L i)s i /(-λ L ); -λ=e r (1+ i π L i). At c=c L, the inequality on c must be tight c L = e -r ( i d i [(p i + π L i)/(1+ i π L i)]) =e -r ( i d i [(p i + π L i)/( i (p i + π L i))]) =e -r i d i q L i, where q L i 0 and i q L i= 1. The q L i's define a risk-neutral probability measure (EMM) on d i for the lower bound price c L (that is also consistent with the other assets so that s=e -r ( i S i q L i)). JRBirge Fuqua School, Duke University April 26,

14 Seller s Problem For any c c U, a sale of a share of an equivalent cash flow to d cannot produce an arbitrage: 0=max x,y,b i p i (-d i x+s it y+e r B) s. t. cx - s T y - B=0, d i x -S it y - e r B 0, i, 0 x 1 The dual solution then yields, c U =e -r i d i q U i and s=e -r i S i q U i for an equivalent risk-neutral measure q U. JRBirge Fuqua School, Duke University April 26,

15 General EMM s By the overall absence of arbitrage assumption, c U c L. Both q L and q U provide risk-neutral prices for the market assets with price s any convex combination of q L and q U is a consistent risk-neutral probability measure for any c L c c U, q c consistent with all market prices s = e -r i S i q c i and c = e -r i d i q c i. JRBirge Fuqua School, Duke University April 26,

16 Complete Market Implications If the market is complete, some combination of market assets can reproduce d c L =c U and EMM is unique If not complete, bounds possible; values outside bounds may lead to extremal solutions (inconsistent with market prices). JRBirge Fuqua School, Duke University April 26,

17 Further Implications and Limitations In models: Adjust all probabilities to consistent EMM Optimize with risk-neutral expectations What if market is not complete? Choice of EMM s Alternative decisions and values consistent with market (in some range) Which one to pick? Be consistent with preference of decision maker Generate alternatives Find range of consistent decisions and objectives JRBirge Fuqua School, Duke University April 26,

18 Finding a Unique Choice Possible situations: Easier: decision-maker (investor) preference depends only on the random quantities (not on decisions) Can pick an EMM before solving the optimization model Harder: preference depends on decisions and the random outcomes Model must adjust preference mapping depending on the decisions JRBirge Fuqua School, Duke University April 26,

19 Principles/Axioms: Coherent and Rational Risk Measures R is a coherent (negative) risk measure if R is concave and increasing (monotonic) R(g(x,ξ)+a)=R(g(x,ξ)) + a, a < (translation invariant) R(λ g(x,ξ)) = λ R(g(x,ξ)) λ 0 (positive homogeneous) Von Neumann-Morgenstern (rational) utility: Complete, Transitive, Continuous, Independent JRBirge Fuqua School, Duke University April 26,

20 Axiom Implications Coherent risk measures: R(g(x,ξ))= min P P E P [g(x,ξ)] Similar to Maxmin Expected Utility May preserve higher orders of stochastic dominance. vnm utility: R(g(x,ξ))=E P [f(x,ξ)] JRBirge Fuqua School, Duke University April 26,

21 What does not fit axioms? Mean-variance or mean-standard deviation Not monotonic E(g(x,ξ)) - γ (Var(g(x,ξ)) 0.5 Value-at-Risk Not convex α-var(g(x,ξ))=min{t P(g(x,ξ) t) α} (e.g., X/Y=0 w.p. 0.95, -1 w.p , P(X=Y=-1)= Var(X)=0.05-VaR(Y)=0; 0.05-VaR(0.5(X+Y))=-0.5) JRBirge Fuqua School, Duke University April 26,

22 What does fit? Coherent objectives: Semi-deviations: R(X)=E[((t-E(X)) - ) p ] 1/p Conditional tail expectation/cvar: R(X)=E[t t α-var(x))] LP form: R(X)=min t {t-(1/α)e[(t-x) - ]} - Other tail expectations JRBirge Fuqua School, Duke University April 26,

23 Information Issues with VaR Value of Information: Blau s dilemma Suppose demand=b=0 w.p. 0.9 and 1 w.p. 0.1 Problem: min x s.t. 0.9-VaR(b) x or P[x b] 0.9 Solution: x*=0 With perfect information: x P =0 w.p. 0.9 and 1 w.p. 0.1 EVPI = Exp. Value without Perfect Information Exp. Value with Perfect Information = = -0.1 < 0 (Same may be true with EVSampleInformation) For RO, let Ξ = {b P[b] 0.9} = {0} JRBirge Fuqua School, Duke University April 26,

24 Problems with Paradox Utility may depend on information level With no information, 0.9 may be acceptable but not the same with more information Cannot make direct comparisons in information value Not including role of competitor Competitor may gain information as well In this case, more information may not always be beneficial JRBirge Fuqua School, Duke University April 26,

25 Reasons for Choice: Tractability Choose model so that we can solve it Computational problems: Probabilistic constraints may give nonconvexity Non-coherent risk measures may be nonconvex Non-convexity makes solutions difficult JRBirge Fuqua School, Duke University April 26,

26 Coherent Risk and Tractability Coherent risk measures are convexity-preserving (concave and increasing over g concave in x) in decision variables x Overall, coherent risk min-max form may have convex subproblems With finite (nice) moments, inner moment problem is tractable: R(g(x,ξ))= min P P E P [g(x,ξ)] =min p{i} i p i g(x,ξ i ) i p i h(ξ i )=h, i p i =1 JRBirge Fuqua School, Duke University April 26,

27 Toward a Consistent View: Competition Suppose (a) competitor(s) choose(s) y(x,ξ) to maximize c(x,y,ξ) Formulation: min x X E P [f(x,y,ξ) y argmax c(x,y,ξ)] y fixed (or f independent of y) => EU y=ξ Ξ, f(x,y,ξ)=c(x,y,ξ)=g(x,ξ) => RO EU assumes irrelevant adversary RO assumes perfect adversary Coherent risk (MEU) assume constrained adversary JRBirge Fuqua School, Duke University April 26,

28 Evidence on Preferences? Expected utility maximizing? (Maybe) Stochastic order on outcomes? (Usually) Probabilistic measure on outcomes (Maybe) Worst-case (or worst with some probability) measure on outcomes (robust measures) (Maybe) JRBirge Fuqua School, Duke University April 26,

29 All True? What is observed? (e.g., Kahnemann- Tversky prospect theory) Targets define utility Preference depends on closeness to targets Local properties as in models Too far away Close to 2nd goal Close to 1 st goal Satisfied Traditional EU applies Small prob. Form for RO/Prob constrained JRBirge Fuqua School, Duke University April 26,

30 Converging Models EU, Coherent Risk, and RO models can apply for observed preferences Interpretation of a competitor can bring them together Paradoxes generally concern misinterpretations What about methods? JRBirge Fuqua School, Duke University April 26,

31 Convergent Methods Bounding methods for EU/Coherent Risk: Find P* s.t. E P* [f(x,ξ)] ( ) E P [f(x,ξ)] Equivalent to Max(Min) P P E P [f(x,ξ)] Procedures: Generalized programming (subproblems to generate weights on ξ Ξ) Use of convexity properties Finite support (but often non-convex subproblems) Direct interpretation for RO JRBirge Fuqua School, Duke University April 26,

32 Combining: When to Use What? Risk-neutral expectation Complete markets (after transformation) and discounting Traditional expected utility Can define function, incomplete market (keep consistent) Worst-case robust or given probability Little information, only survivability counts Competition and distribution domains Allows consistent view from risk-neutral to worst case ; can be tractable JRBirge Fuqua School, Duke University April 26,

33 Conclusions Risk-neutral equivalence (or at least marking to market) for consistency Utility forms enter in incomplete markets and can take different forms Adversary view makes many forms consistent View of prospect theory (+) can drive further consistency with local properties for tractability JRBirge Fuqua School, Duke University April 26,