Coherent Measures of Risk

David Heath

Carnegie Mellon University

Joint work with Philippe Artzner, Freddy Delbaen, Jean-Marc Eber; research partially funded by Société Generale

Measuring Risk

Purpose:– Manage and control risk– Make good risk/return tradeoff– “Risk adjust” traders’ profits

To help with:– Regulation of traders and banks– Portfolio selection– Motivating traders to reduce risk

How should a risk measure behave?

Should provide a basis for setting “capital requirements”

Should be “reasonable”– Encourage diversification– Should respect “more is better”

Should be useable as a management tool– Should be compatible with allocation of risk limits to

desks– Should provide sensible way to “risk-adjust” gains of

different investment strategies (desks)

The basic model

For now, think only of “market risk” For now, assume liquid markets A “state of the market” is then a set of prices

for all securities. (i.e., a copy of WSJ) For a given portfolio and a given state , set

X() = market value of in state .

A risk measure assigns a number (X) to each such (random variable) X.

More generally ... Notice maps X’s (not ’s) into numbers. More complexity can be introduced through X

– X should give the value of the firm if required to liquidate at the end of period, for every possible state of the world

– State can specify amount of liquidity– Can consider “active” management over period

» must describe evolution of markets over period

» instead of portfolio , must consider strategy (e.g., rebalance each day using futures to stay hedged)

Let’s focus on Want to provide capital requirements.

– Suppose firm is required to allocate additional capital - what do they do with it

» Riskless investment (which, and how riskless)?» Risky investment?

– We assume: some particular instrument is specified. It’s price today is 1, and at end is r0(). (Might be pdb, money market, S&P)

– (X) tells the number of shares of this security which must be added to the portfolio to make it “safe enough”.

Axioms for “coherent” Units:

– (X+r0) = (X) - (for all )

Diversification: – ((X+Y)/2) ((X)+(Y))/2

More is better:– If X Y then (Y) (X)

Scale invariance:– (X) = (X) (for all 0)

An aside ...

In the presence of the linearity axiom, the diversification axiom can be written (X+Y) (X) + (Y)

This means that a risk limit can be “allocated” to desks

If the inequality failed for a firm desiring to hold X+Y, firm could reduce capital requirement by setting up two subsidiaries, one to hold X and the other Y.

Do any such exist?

Do we want one? (Maybe not!) There are many such ’s:

– Take any set A of outcomes

Set (X) = - inf{X()/r0() | A}» Think of A as set of scenarios; gives worst case

– Take any set of probabilities P

Set p(X) = - inf{EP(X/r0) | PP}» Think of each P as a “generalized scenario”

Are there any more?

Theorem: If is a finite set, then every coherent risk measure can be obtained from generalized scenarios.

So: specifying a coherent risk measure is the same as specifying a set of generalized scenarios.

How can (or does) one pick generalized scenarios?

SPAN uses generalized scenarios:– To set margin on a portfolio consisting of shares of

some futures contract and options on that contract, consider prices (scenarios) by:

– Let the futures price change by -3/3, -2/3, -1/3, 0, 1/3, 2/3, 3/3 of some “range”, and vols either move up or move down. (These are scenarios.)

– Let the futures go up or down by an “extreme” move, vols stay the same. Need cover only 35% of the loss. (These are generalized …)

Another method Let each desk generate relevant scenarios for

instruments it trades; pass these to firm’s risk manager

Risk manager takes all combinations of these scenarios and may add some more

Resulting set of scenarios is given back to each desk, which must value its portfolio for each

Results are combined by firm risk manager

What about VaR?

VaR specifies a risk measure VaR

VaR is computed for an X as follows: For a given probability P (the best guess at the “true” (physical or martingale?) probability)Compute the .01 quantile of the distribution of X

under P

The negative of this quantile = VaR(X)

(implicitly assumes r0 = 1.)

VaR is not coherent!

VaR satisfies all axioms except diversification (and it uses r0 = 1).

This means VaR limits can’t be allocated to desks: each desk might satisfy limit but total portfolio might not.

Firms avoid VaR restrictions by setting up subsidiaries

VaR says: don’t diversify!

Consider a CCC bond. Suppose:– Probability of default over a week is .005– Value after default is 0– Yield spread is .26/yr or .005/week

Consider the portfolio:– Borrow $300,000 at risk-free rate– Purchase $300,000 of this bond

Value at end if no default is $1500 Probability of default is .005, so VaR says OK!

– In fact, can do this to any scale!

If you diversify: If there are 3 independent bonds like this

Consider borrowing $300,000 and purchasing $100,000 of each bond

Probability distribution of worth at end: (Let’s pretend interest rate = 0)

Probability Value

0.985075 1500

0.01485 -99000 VaR requires 99000 7.46E-05 -199500

1.25E-07 -300000

Even scarier

Most firms want to “get the highest return per unit of risk.”

If they use VaR to measure risk, they’ll be led to pile up the losses on a “small” set of scenarios (a set with probability less than .01)

If they use “black box” approach to reducing VaR they’ll do the same, probably without realizing it!

Does anything like VaR work? Suppose we have chosen a P which we’d use

to compute VaR Suppose X has a continuous distribution

(under P) Then set (X) = -EP(X | X -VaR(X))

This is coherent! (requires a proof) It’s the smallest coherent which depends

only on the P-distribution of X’s and which is bigger than VaR.

More about this VaR-like To compute a 1% VaR by simulation, one

might generate 10,000 random scenarios (using P) and use -the 100th worst one.

The corresponding estimate of our would be the negative of the average of the 100 worst ones

If X is normally distributed, this (X) is very close to VaR

This may be a good first step toward coherence

What’s next?

What are the consequences of trying to maximize return per unit of risk when using a coherent risk measure? – We think that something like that does make

sense Could a bank perform well if each desk

used such a measure?– We think so.

Conclusions (to part 1 of talk)

Good risk management requires the use of coherent risk measures

VaR is not a coherent risk measure– Can induce firms to arrange portfolio so that when

the fail, they fail big– Discourages diversification

There is a substitute for VaR which is more conservative than VaR, is about as easy to compute, and is coherent

Ongoing research (results tentative!!)

Can coherent risk measures be used for– Firm-wide risk management?– In portfolio selection?

What criteria make one coherent risk measure (or one set of generalized scenarios) better than another?

Can such measures help with– Decentralized portfolio optimization?– “Risk adjusting” trading profits?

Maxing expected return per unit risk

Using VaR, problem is:– Maximize E(X)

– subject to VaR(X) K Problem is (usually) unbounded

– It is if there’s any X with E(X)>0 and VaR(X) 0 (like being short a far out-of- the-money put)

VaR constraint is satisfied for arbitrarily large position size!

With a coherent risk measure

We’ll see that – Firms can achieve “economically optimal”

portfolios– Decision problem can be allocated to desks

– Desks can each have their own PDesk

– If these aren’t too inconsistent, still works! But first -- in addition to regulators we need

the firm’s owners

Meeting goals of shareholders

So far, risk measures were for regulation Shareholders have a different point of view

– Solvency isn’t enough– Don’t want too much risk of loss of investment

Shareholders may have different risk preferences than regulators

Firm must respect both regulators’ and shareholders’ demands

A “shareholders’” risk measure Require firm to count shareholder’s investment

as liability This “desired shareholder value” may be

– Fixed $

– Some index

– In general, some random variable, say T (target)

– Risk is the risk of missing target Apply coherent risk measure to X-T. Shareholders have risk measure SH

The optimization problem

Let Reg denote the regulator’s risk measure

Let P be some given probability measure Let T be the “investor’s target” Let SH be the shareholders’ risk measure

Problem: Choose available X to maximize EP(X) subject to: Reg(X) 0 and

SH(X-T) 0.

In liquid markets: Linear Program

In liquid markets the initial price of X, 0(X) is a linear function of X.

Traded X’s form a linear space Available X’s satisfy 0(X) = K (capital)

Objective function (EP(X)) is linear in X

Constraints, written properly, are linear:– Reg(X) 0 is same as EQ(X) 0 for all Q QReg

– SH(X-T) 0 is same as EQ(X) EQ(T) for all Q QSH

Is the resulting portfolio optimal?

Can firm get to shareholder’s optimal X? Suppose:

– Shareholders (or managers) have a utility function u, strictly increasing

– Desired portfolio is solution X* to:» Maximize EP(u(X)) over all available X satisfying regulator’s

constraints

– Suppose such an X* exists

– Can managers specify T and SH so that X* is the solution to the above LP?

Forcing optimality Theorem:

Let T = X* and QSH = set of all probability measures. Then the only feasible solution to the LP is X*.

Proof: If X is feasible, then shareholder constraints require X T (= X*). But if any available X X* were actually larger (on a set with positive P-measure), EP(u(X)) would be bigger than EP(u(X*)), so X* wouldn’t have maximized expected utility

If the firm has trading desks Let X1, X2, …, XD the spaces of random

terminal worths available to desks 1, 2, …D Then random variables available to firm are

elements of X = X1+ X2+ … + XD .

Suppose target T* is allocated arbitrarily to desks so that T* = T1 + T2 + … + TD.

Suppose initial capital is arbitrarily allocated to desks: K = K1 + … + KD

and Regulator’s risk is assigned (for each regulator probability Q QReg) to desks: rQ,1, rQ,2, …, rQ,D summing to 0.

Let desk d try to solve

Choose Xd* Xd to maximize EP(Xd) subject to:– 0(Xd) = Kd

– EQ(Xd) EQ(Td) for every Q QSH

– EQ(Xd) rQ,d for every Q QReg

Clearly ...

X1* + X2* + … + XD* is feasible for the firm’s problem, so EP(X1*+…+XD*) is EP(X*).

– i.e., desks can’t get better total solution than firm could get

Since X* can be decomposed as X1 + X2 + …

+ XD where Xd Xd, with appropriate “splitting of resources” as above desks will achieve optimal portfolio for the firm

How can firm do this allocation?

Set up an internal market for “perturbations” of all of the arbitrary allocations. Desks can trade such perturbations; i.e., can agree that one desk will lower the rhs of one of its constraints and the other will increase its. But this agreement has a price (to be set internally by this market). (Value of each desk’s objective function is lowered by the amount of its payments in this internal market.)

Market equilibrium

The only equilibrium for this market produces the optimal portfolio for the firm.– (Look at the firm’s dual problem; this tells the

equilibrium internal prices associated with each constraint.)

What if each desk has its own Pd?

If there is some P such that EPd(X) = EP(X)

for all X Xd then any market equilibrium solves the firm’s LP for this measure P.

If there isn’t then there is “internal arbitrage” and no market equilibrium exists.