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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
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)|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”
 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 | PP} for some
family of probability measures P and
constants cP.
Risk measures for investors
 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  supYY EP*(u(W0+Y))
How to describe the “acceptable
If W 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
 Theorem 2: There is a risk measure r
(satisfying axioms 1. through 3.) for which
A = {X | r(X)  0}.
r(X)  0 is (by a Theorem 1) the same as
EP(X/r0)  cP 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
(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
 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
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
XXd 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
XdXd 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.)
Control of risk based on scenarios and
scenario risk limits has the potential to
 Allow investors to describe their
preferences in an intuitively appealing
 Allow portfolio-choosers to use tools
from linear programming to select
 Allow firms to achive firm-wide optimal
portfolios without having to do firmwide
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
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
 Obtains explicit connections