Transcript Selling an Idea or a Product

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” w is then a set of
prices for all securities. (i.e., a copy of WSJ)
 For a given portfolio p and a given state w,
set Xp(w) = market value of p in state w.
 A risk measure r assigns a number r(X) to
each such (random variable) X.
More generally ...
r maps X’s (not p’s) into numbers.
 More complexity can be introduced through X
 Notice
– 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 w can specify amount of liquidity
– Can consider “active” management over period
» w must describe evolution of markets over period
» instead of portfolio p, must consider strategy (e.g.,
rebalance each day using futures to stay hedged)
Let’s focus on r
 Want
r 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(w). (Might be pdb, money market, S&P)
– r(X) tells the number of shares of this security
which must be added to the portfolio to make it
“safe enough”.
Axioms for “coherent” r
 Units:
– r(X+ar0) = r(X) - a
(for all a)
 Diversification:
– r((X+Y)/2)  (r(X)+r(Y))/2
 More
is better:
– If X  Y then r(Y)  r(X)
 Scale
invariance:
– r(aX) = a r(X)
(for all a  0)
An aside ...
 In
the presence of the linearity axiom, the
diversification axiom can be written
r(X+Y)  r(X) + r(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 r exist?
 Do
we want one? (Maybe not!)
 There are many such r’s:
– Take any set A of outcomes
Set rA(X) = - inf{X(w)/r0(w) | wA}
» Think of A as set of scenarios; r gives worst case
– Take any set of probabilities P
Set rp(X) = - inf{EP(X/r0) | PP}
» Think of each P as a “generalized scenario”
Are there any more?
If W is a finite set, then every
coherent risk measure can be obtained from
generalized scenarios.
 Theorem:
 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?
specifies a risk measure rVaR
 rVaR is computed for an X as follows: For a
given probability P (the best guess at the
“true” (physical or martingale?) probability)
 VaR
Compute the .01 quantile of the distribution of X
under P
The negative of this quantile = rVaR(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
7.46E-05
-199500
1.25E-07
-300000
VaR requires 99000
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 r(X) = -EP(X | X  -VaR(X))
 This r is coherent! (requires a proof)
 It’s the smallest coherent r which depends
only on the P-distribution of X’s and which
is bigger than VaR.
More about this VaR-like r
 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 r would
be the negative of the average of the 100
worst ones
 If X is normally distributed, this r(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-ofthe-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 rSH
The optimization problem
rReg denote the regulator’s risk measure
 Let P be some given probability measure
 Let T be the “investor’s target”
 Let rSH be the shareholders’ risk measure
 Problem: Choose available X to maximize
EP(X) subject to: rReg(X)  0 and
rSH(X-T)  0.
 Let
In liquid markets: Linear Program
liquid markets the initial price of X, p0(X) is
a linear function of X.
 Traded X’s form a linear space
 Available X’s satisfy p0(X) = K (capital)
 Objective function (EP(X)) is linear in X
 Constraints, written properly, are linear:
– rReg(X) 0 is same as EQ(X)  0 for all Q  QReg
– rSH(X-T) 0 is same as EQ(X)  EQ(T) for all Q  QSH
 In
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 rSH 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
Xd*  Xd to maximize EP(Xd)
subject to:
 Choose
– p0(Xd) = Kd
– EQ(Xd)  EQ(Td) for every Q  QSH
– EQ(Xd)  rQ,d for every Q  QReg
Clearly ...
+ X2* + … + XD* is feasible for the
firm’s problem, so EP(X1*+…+XD*) is 
EP(X*).
 X1*
– 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.