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 u Purpose: –Manage and control risk –Make good risk/return tradeoff –“Risk adjust” traders’ profits u To help with: –Regulation of traders and banks –Portfolio selection –Motivating traders to reduce risk

How should a risk measure behave? u Should provide a basis for setting “capital requirements” u Should be “reasonable” –Encourage diversification –Should respect “more is better” u 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 u For now, think only of “market risk” u 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. u 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 r 0 (  ). (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”  u Units: –  (X+  r 0 ) =  (X) -  (for all  ) u Diversification: –  ((X+Y)/2)  (  (X)+  (Y))/2 u More is better: – If X  Y then  (Y)   (X) u Scale invariance: –  (  X) =   (X) (for all   0)

An aside... u In the presence of the linearity axiom, the diversification axiom can be written  (X+Y)   (X) +  (Y) u This means that a risk limit can be “allocated” to desks u 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? u Do we want one? (Maybe not!)  There are many such  ’s: –Take any set A of outcomes Set   (X) = - inf{X(  )/r 0 (  ) |  A} »Think of A as set of scenarios;  gives worst case –Take any set of probabilities P Set  p (X) = - inf{E P (X/r 0 ) | P  P} »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. u So: specifying a coherent risk measure is the same as specifying a set of generalized scenarios.

How can (or does) one pick generalized scenarios? u 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 u Let each desk generate relevant scenarios for instruments it trades; pass these to firm’s risk manager u Risk manager takes all combinations of these scenarios and may add some more u Resulting set of scenarios is given back to each desk, which must value its portfolio for each u 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) u (implicitly assumes r 0 = 1.)

VaR is not coherent! u VaR satisfies all axioms except diversification (and it uses r 0 = 1). u This means VaR limits can’t be allocated to desks: each desk might satisfy limit but total portfolio might not. u Firms avoid VaR restrictions by setting up subsidiaries

VaR says: don’t diversify! u 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 u Consider the portfolio: –Borrow $300,000 at risk-free rate –Purchase $300,000 of this bond u Value at end if no default is $1500 u Probability of default is.005, so VaR says OK! –In fact, can do this to any scale!

If you diversify: u If there are 3 independent bonds like this Consider borrowing $300,000 and purchasing $100,000 of each bond u Probability distribution of worth at end: (Let’s pretend interest rate = 0) Probability Value VaR requires E E

Even scarier u Most firms want to “get the highest return per unit of risk.” u 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) u If they use “black box” approach to reducing VaR they’ll do the same, probably without realizing it!

Does anything like VaR work? u Suppose we have chosen a P which we’d use to compute VaR u Suppose X has a continuous distribution (under P)  Then set  (X) = -E P (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  u 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 u This may be a good first step toward coherence

What’s next? u 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 u Could a bank perform well if each desk used such a measure? –We think so.

Conclusions (to part 1 of talk) u Good risk management requires the use of coherent risk measures u VaR is not a coherent risk measure –Can induce firms to arrange portfolio so that when the fail, they fail big –Discourages diversification u 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!!) u Can coherent risk measures be used for –Firm-wide risk management? –In portfolio selection? u What criteria make one coherent risk measure (or one set of generalized scenarios) better than another? u Can such measures help with –Decentralized portfolio optimization? –“Risk adjusting” trading profits?

Maxing expected return per unit risk u Using VaR, problem is: –Maximize E(X) –subject to VaR(X)  K u 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) u VaR constraint is satisfied for arbitrarily large position size!

With a coherent risk measure u We’ll see that –Firms can achieve “economically optimal” portfolios –Decision problem can be allocated to desks –Desks can each have their own P Desk –If these aren’t too inconsistent, still works! u But first -- in addition to regulators we need the firm’s owners

Meeting goals of shareholders u So far, risk measures were for regulation u Shareholders have a different point of view –Solvency isn’t enough –Don’t want too much risk of loss of investment u Shareholders may have different risk preferences than regulators u Firm must respect both regulators’ and shareholders’ demands

A “shareholders’” risk measure u Require firm to count shareholder’s investment as liability u This “desired shareholder value” may be –Fixed $ –Some index –In general, some random variable, say T (target) –Risk is the risk of missing target u Apply coherent risk measure to X-T.  Shareholders have risk measure  SH

The optimization problem  Let  Reg denote the regulator’s risk measure u Let P be some given probability measure u Let T be the “investor’s target”  Let  SH be the shareholders’ risk measure  Problem: Choose available X to maximize E P (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. u Traded X’s form a linear space  Available X’s satisfy  0 (X) = K (capital) u Objective function (E P (X)) is linear in X u Constraints, written properly, are linear: –  Reg (X)  0 is same as E Q (X)  0 for all Q  Q Reg –  SH (X-T)  0 is same as E Q (X)  E Q (T) for all Q  Q SH

Is the resulting portfolio optimal? u Can firm get to shareholder’s optimal X? u Suppose: –Shareholders (or managers) have a utility function u, strictly increasing –Desired portfolio is solution X* to: »Maximize E P (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 u Theorem: Let T = X* and Q SH = 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), E P (u(X)) would be bigger than E P (u(X*)), so X* wouldn’t have maximized expected utility

If the firm has trading desks  Let X 1, X 2, …, X D the spaces of random terminal worths available to desks 1, 2, …D  Then random variables available to firm are elements of X = X 1 + X 2 + … + X D. u Suppose target T* is allocated arbitrarily to desks so that T* = T 1 + T 2 + … + T D. u Suppose initial capital is arbitrarily allocated to desks: K = K 1 + … + K D

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

Let desk d try to solve  Choose X d *  X d to maximize E P (X d ) subject to: –  0 (X d ) = K d –E Q (X d )  E Q (T d ) for every Q  Q SH –EQ(X d )  r Q,d for every Q  Q Reg

Clearly...  X 1 * + X 2 * + … + X D * is feasible for the firm’s problem, so E P (X 1 *+…+X D *) is  E P (X*). –i.e., desks can’t get better total solution than firm could get  Since X* can be decomposed as X 1 + X 2 + … + X D where X d  X d, with appropriate “splitting of resources” as above desks will achieve optimal portfolio for the firm

How can firm do this allocation? u 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 u 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 P d ?  If there is some P such that E P d (X) = E P (X) for all X  X d then any market equilibrium solves the firm’s LP for this measure P. u If there isn’t then there is “internal arbitrage” and no market equilibrium exists.