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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