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 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 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 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 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.  (X+ar) =  (X) – a 1.  (X+ar 0 ) =  (X) – a 2. X 2. X  Y   (X)  (Y) 3.  ( X+(1- )Y)  (X)+(1- )  (Y) for in [0,1] 4.  ( X)=  (X) for  0 (In the presence of the other axioms, 3 is equivalent to  (X+Y)   (X)+  (Y).) Theorem: If  is finite,  satisfies 1-4 iff  (X) = -inf{E P (X/r 0 )|P  P } 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” If P gives a single point mass 1, then P can be thought of as a “pure scenario” Other P’s are “random scenarios” Other P’s are “random scenarios” Risk measure arises from “worst scenario” Risk measure arises from “worst scenario” X is “acceptable” if  (X) 0; i.e., no additional capital is required X is “acceptable” if  (X)  0; i.e., no additional capital is required Axiom 4 seems the least defensible Axiom 4 seems the least defensible

Without Axiom 4 Require only: Require only:  1.  (X+ar) =  (X) – a  1.  (X+ar 0 ) =  (X) – a  2. X  2. X  Y   (X)  (Y)   3.  ( X+(1- )Y)  (X)+(1- )  (Y) for all in [0,1] Theorem 1: If  is finite,  satisfies 1-3 iff  (X) = -inf{E P (X/r 0 )-c P | P  P } for some family of probability measures P and constants c P.

Risk measures for investors Suppose: Suppose:  Investor has  Endowment W 0 (describing random end-of-period wealth)  Von Neumann- Morgenstern utility u  Subjective probability P*  Will accept gambles for which E P* (u(X+W 0 )) E P* (u(W 0 ))  Will accept gambles for which E P* (u(X+W 0 ))  E P* (u(W 0 )) or perhaps sup Y Y E P* (u(W 0 +Y)) or perhaps  sup Y  Y E P* (u(W 0 +Y))

How to describe the “acceptable set”? If  is finite, the set A of random variables the investor will accept satisfies: If  is finite, the set A of random variables the investor will accept satisfies:  A is closed  A is convex  X A, Y X Y A  X  A, Y  X  Y  A Theorem 2: There is a risk measure  (satisfying axioms 1. through 3.) for which Theorem 2: There is a risk measure  (satisfying axioms 1. through 3.) for which A = {X |  (X) 0}. A = {X |  (X)  0}.

Remarks  (X) 0 is (by a Theorem 1) the same as E P (X/r 0 ) c P for every  (X)  0 is (by a Theorem 1) the same as E P (X/r 0 )  c P for every P  P 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 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 Wants to sell such a product to investor Must find an X X A Must find an X  X  A Requires finding a solution to system of linear inequalities Requires finding a solution to system of linear inequalities

The sell-side problem,  = {1,2}

“Best” feasible random variable? Barycenter of feasible region? 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? 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 Suppose “investor” is “owner” of a trading firm Investor imposes risk limits on firm via scenarios with loss limits Investor imposes risk limits on firm via scenarios with loss limits Investor asks for firm to achieve maximal (expected) return Investor asks for firm to achieve maximal (expected) return Firm must provide the probability measure Firm must provide the probability measure Given the measure, firm solves LP Given the measure, firm solves LP

Suppose firm has trading desks How to manage? 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 c P to desks: c d,P  Allow desks to trade perturbations to these risk limits at “internal market prices”

With trading of risk limits … Let X d be the random variables available to desk d, for d = 1, 2, … D Let X d be the random variables available to desk d, for d = 1, 2, … D Consistency: Suppose there is a P* F such that Consistency: Suppose there is a P* F such that X X d X  X d  E P* d (X) = E P* 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. 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 = X 1 + X 2 + … + X D is the set of “firm-achievable” random variables), and let Theorem 3: Let X* be the firm-optimal portfolio (where X = X 1 + X 2 + … + X D is the set of “firm-achievable” random variables), and let X d X d be such that X 1 +…+X D =X*. X d  X d be such that X 1 +…+X D =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 X d. (No assumption is needed about the initial allocation of risk limits.) (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 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!) Mean-variance analysis (of course!) Much more … Much more …  Other risk measures  Evaluation of measures of risk  Probability beliefs  Relationship to expected utility maximization

Risk measures considered The standard deviation The standard deviation The semi-variance The semi-variance The expected value of loss The expected value of loss The expected absolute deviation The expected absolute deviation The probability of loss The probability of loss The maximum loss The maximum loss

Connections to expected utility Last chapter of book Last chapter of book  Discusses for which risk measures minimizing risk for a given expected return is consistent with utility maximization  Obtains explicit connections