Risk Management with Coherent Measures of Risk

IPAM Conference on Financial IPAM Conference on Financial Mathematics: Risk Management, Mathematics: Risk Management,

Modeling and Numerical MethodsModeling and Numerical Methods

January 2001January 2001

ADEH axioms for regulatory risk measures Definition: A risk measure is a mapping Definition: A risk measure is a mapping

from random variables to real numbersfrom random variables to real numbers The random variable is the net worth of the The random variable is the net worth of the

firm if forced to liquidate at the end of a firm if forced to liquidate at the end of a holding periodholding period

Regulators are concerned about this random Regulators are concerned about this random variable taking on negative valuesvariable taking on negative values

The value of the risk measure is the amount The value of the risk measure is the amount of additional capital (invested in a “riskless of additional capital (invested in a “riskless instrument”) required to hold the portfolioinstrument”) required to hold the portfolio

The axioms (regulatory measures) 1. 1. (X+ar(X+ar0) = ) = (X) – a(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{EP(X/r0)|PP} for some family of probability measures P.

If P gives a single point mass 1, then P can If P gives a single point mass 1, then P can be thought of as a “pure scenario”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 is “acceptable” if (X) (X) 0; i.e., no 0; i.e., no additional capital is requiredadditional capital is required

Axiom 4 seems the least defensibleAxiom 4 seems the least defensible

Without Axiom 4 Require only:Require only:

1. 1. (X+ar(X+ar0) = ) = (X) – a(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{EP(X/r0)-cP | PP} for some family of probability measures P and constants cP.

Risk measures for investors Suppose: Suppose:

Investor hasInvestor has Endowment WEndowment W0 0 (describing random (describing random

end-of-period wealth)end-of-period wealth)Von Neumann- Morgenstern utility uVon Neumann- Morgenstern utility uSubjective probability P*Subjective probability P*

Will accept gambles for which Will accept gambles for which EEP*P*(u(X+W(u(X+W00)) )) E EP*P*(u(W(u(W00))))

or perhaps or perhaps sup supYYYY E EP*P*(u(W(u(W00+Y))+Y))

How to describe the “acceptable set”? If If is finite, the set is finite, the set AA of random variables of random variables

the investor will accept satisfies:the investor will accept satisfies: AA is closed is closed AA is convex is convex XXAA, Y , Y X X Y YAA

Theorem 2: There is a risk measure Theorem 2: There is a risk measure (satisfying axioms 1. through 3.) for which(satisfying axioms 1. through 3.) for which

AA = {X | = {X | (X) (X) 0}. 0}.

Remarks

(X) (X) 0 is (by a Theorem 1) the same as 0 is (by a Theorem 1) the same as EEPP(X/r(X/r00) ) c cPP for every 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 Seller of financial instruments can offer net (random) payments from some set (random) payments from some set XX

(In simplest case (In simplest case XX is a linear space) is a linear space) Wants to sell such a product to investorWants to sell such a product to investor Must find an X Must find an X XX AA Requires finding a solution to system of Requires finding a solution to system of

linear inequalities linear inequalities

The sell-side problem, = {1,2}

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

-2 -1 0 1 2 3 4 5 6 7

X(1)

“Best” feasible random variable? Barycenter of feasible region?Barycenter of feasible region?

If u is quadratic, this maximizes If u is quadratic, this maximizes investor’s expected utility; if “locally investor’s expected utility; if “locally nearly quadratic” it nearly does sonearly quadratic” it nearly does so

The value maximizing expected value for The value maximizing expected value for some probability?some probability? Perhaps investor trusts seller to have a Perhaps investor trusts seller to have a

better estimate of true probabilitiesbetter estimate of true probabilities More like Markowitz – maximize More like Markowitz – maximize

expected return subject to a risk limitexpected return subject to a risk limit Gives rise to a standard LPGives rise to a standard LP

Another situation Suppose “investor” is “owner” of a trading Suppose “investor” is “owner” of a trading

firmfirm Investor imposes risk limits on firm via Investor imposes risk limits on firm via

scenarios with loss limitsscenarios with loss limits Investor asks for firm to achieve maximal Investor asks for firm to achieve maximal

(expected) return (expected) return Firm must provide the probability measureFirm must provide the probability measure Given the measure, firm solves LPGiven the measure, firm solves LP

Suppose firm has trading desks How to manage?How to manage?

Each desk may have its own probability Each desk may have its own probability P*P*d d (for expected value computations) (for expected value computations)

Assign risk limits to desks?Assign risk limits to desks?How to distribute risk limits?How to distribute risk limits?

Allow desks to trade limits?Allow desks to trade limits?Initially allocate cInitially allocate cPP to desks: c to desks: cd,Pd,P

Allow desks to trade perturbations to Allow desks to trade perturbations to these risk limits at “internal market these risk limits at “internal market prices”prices”

With trading of risk limits … Let Let XXdd be the random variables available to be the random variables available to

desk d, for d = 1, 2, … Ddesk d, for d = 1, 2, … D Consistency: Suppose there is a P*Consistency: Suppose there is a P*FF such that such that

XXXXd d EP*d(X) = EP*F

(X) Suppose each desk tries to maximize its Suppose each desk tries to maximize its

expected return, taking into account the costs expected return, taking into account the costs (or profits) from trading risk limits, choosing (or profits) from trading risk limits, choosing its portfolio to satisfy its resulting trading its portfolio to satisfy its resulting trading limits.limits.

Theorem 3: Let X* be the firm-optimal Theorem 3: Let X* be the firm-optimal portfolio (where portfolio (where XX = = XX11 + + XX22 + … + + … + XXDD is the is the set of “firm-achievable” random variables), set of “firm-achievable” random variables), and let and let

XXddXXdd be such that X be such that X11+…+X+…+XDD=X*. =X*.

Then there is an equilibrium for the internal Then there is an equilibrium for the internal market for risk limits (with prices equal to market for risk limits (with prices equal to the dual variables for the firm’s optimal the dual variables for the firm’s optimal solution) for which each desk d holds Xsolution) for which each desk d holds Xd.d.

(No assumption is needed about the initial (No assumption is needed about the initial allocation of risk limits.)allocation of risk limits.)

Summary Control of risk based on scenarios and Control of risk based on scenarios and

scenario risk limits has the potential toscenario risk limits has the potential to Allow investors to describe their Allow investors to describe their

preferences in an intuitively appealing waypreferences in an intuitively appealing way Allow portfolio-choosers to use tools from Allow portfolio-choosers to use tools from

linear programming to select portfolioslinear programming to select portfolios Allow firms to achive firm-wide optimal Allow firms to achive firm-wide optimal

portfolios without having to do firmwide portfolios without having to do firmwide optimization.optimization.

Back to Markowitz (book, 1959)

Mean-variance analysis (of course!)Mean-variance analysis (of course!) Much more …Much more …

Other risk measuresOther risk measures Evaluation of measures of riskEvaluation of measures of risk Probability beliefsProbability beliefs Relationship to expected utility Relationship to expected utility

maximizationmaximization

Risk measures considered

The standard deviation The standard deviation The semi-varianceThe semi-variance The expected value of lossThe expected value of loss The expected absolute deviationThe expected absolute deviation The probability of lossThe probability of loss The maximum lossThe maximum loss

Connections to expected utility

Last chapter of bookLast chapter of book Discusses for which risk measures Discusses for which risk measures

minimizing risk for a given expected minimizing risk for a given expected return is consistent with utility return is consistent with utility maximizationmaximization

Obtains explicit connectionsObtains explicit connections