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Risk Management with Coherent Measures of Risk IPAM Conference on Financial Mathematics: Risk Management, Modeling and Numerical Methods January 2001 ADEH axioms for regulatory risk measures Definition: A risk measure is a mapping from random variables to real numbers The random variable is the net worth of the firm if forced to liquidate at the end of a holding period Regulators are concerned about this random variable taking on negative values The value of the risk measure is the amount of additional capital (invested in a “riskless instrument”) required to hold the portfolio The axioms (regulatory measures) 1. r(X+ar0) = r(X) – a 2. X Y r(X) r(Y) 3. r(lX+(1-l)Y) lr(X)+(1-l)r(Y) for l in [0,1] 4. r(lX)=lr(X) for l 0 (In the presence of the other axioms, 3 is equivalent to r(X+Y) r(X)+r(Y).) Theorem: If W is finite, r satisfies 1-4 iff r(X) = -inf{EP(X/r0)|PP} for some family of probability measures P. If P gives a single point mass 1, then P can be thought of as a “pure scenario” Other P’s are “random scenarios” Risk measure arises from “worst scenario” X is “acceptable” if r(X) 0; i.e., no additional capital is required Axiom 4 seems the least defensible Without Axiom 4 Require only: 1. r(X+ar0) = r(X) – a 2. X Y r(X) r(Y) 3. r(lX+(1-l)Y) lr(X)+(1-l)r(Y) for all l in [0,1] Theorem 1: If W is finite, r satisfies 1-3 iff r(X) = -inf{EP(X/r0)-cP | PP} for some family of probability measures P and constants cP. Risk measures for investors Suppose: Investor has Endowment W0 (describing random end-of-period wealth) Von Neumann- Morgenstern utility u Subjective probability P* Will accept gambles for which EP*(u(X+W0)) EP*(u(W0)) or perhaps supYY EP*(u(W0+Y)) How to describe the “acceptable set”? If W is finite, the set A of random variables the investor will accept satisfies: A is closed A is convex XA, Y X YA Theorem 2: There is a risk measure r (satisfying axioms 1. through 3.) for which A = {X | r(X) 0}. Remarks r(X) 0 is (by a Theorem 1) the same as EP(X/r0) cP for every PP Investor can describe set of acceptable random variables by giving loss limits for a set of “generalized scenarios”. (Sometimes used in practice – without the benefit of theory!) The “sell side” problem Seller of financial instruments can offer net (random) payments from some set X (In simplest case X is a linear space) Wants to sell such a product to investor Must find an X X A Requires finding a solution to system of linear inequalities The sell-side problem, W = {1,2} “Best” feasible random variable? Barycenter of feasible region? If u is quadratic, this maximizes investor’s expected utility; if “locally nearly quadratic” it nearly does so The value maximizing expected value for some probability? Perhaps investor trusts seller to have a better estimate of true probabilities More like Markowitz – maximize expected return subject to a risk limit Gives rise to a standard LP Another situation Suppose “investor” is “owner” of a trading firm Investor imposes risk limits on firm via scenarios with loss limits Investor asks for firm to achieve maximal (expected) return Firm must provide the probability measure Given the measure, firm solves LP Suppose firm has trading desks How to manage? Each desk may have its own probability P*d (for expected value computations) Assign risk limits to desks? How to distribute risk limits? Allow desks to trade limits? Initially allocate cP to desks: cd,P Allow desks to trade perturbations to these risk limits at “internal market prices” With trading of risk limits … Let Xd be the random variables available to desk d, for d = 1, 2, … D Consistency: Suppose there is a P*F such that XXd EP*d(X) = EP*F(X) Suppose each desk tries to maximize its expected return, taking into account the costs (or profits) from trading risk limits, choosing its portfolio to satisfy its resulting trading limits. Theorem 3: Let X* be the firm-optimal portfolio (where X = X1 + X2 + … + XD is the set of “firm-achievable” random variables), and let XdXd be such that X1+…+XD=X*. Then there is an equilibrium for the internal market for risk limits (with prices equal to the dual variables for the firm’s optimal solution) for which each desk d holds Xd. (No assumption is needed about the initial allocation of risk limits.) Summary Control of risk based on scenarios and scenario risk limits has the potential to Allow investors to describe their preferences in an intuitively appealing way Allow portfolio-choosers to use tools from linear programming to select portfolios Allow firms to achive firm-wide optimal portfolios without having to do firmwide optimization. Back to Markowitz (book, 1959) Mean-variance analysis (of course!) Much more … Other risk measures Evaluation of measures of risk Probability beliefs Relationship to expected utility maximization Risk measures considered The standard deviation The semi-variance The expected value of loss The expected absolute deviation The probability of loss The maximum loss Connections to expected utility Last chapter of book Discusses for which risk measures minimizing risk for a given expected return is consistent with utility maximization Obtains explicit connections