Risk Management with Coherent Measures of Risk

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 Þr(X) ³ 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 l³ 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)|PÎP} 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 Þr(X) ³ 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 | PÎP} 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)) ³ EP*(u(W0)) or perhaps ³ supYÎY 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 • XÎA, Y ³ X Þ YÎA • 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) ³ cP for every PÎP • 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 ÎXÇ 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 XÎXd Þ 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 XdÎXd 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

Risk Management with Coherent Measures of Risk