Coherent Measures of Risk

1
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

2
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

3
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)

4
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.

5
More generally ...

• Notice r maps Xs (not ps) 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 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)

6
Lets 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. Its price today is 1, and at end is
  r0(w). (Might be pdb, money market, SP)
• r(X) tells the number of shares of this security
  which must be added to the portfolio to make it
  safe enough.

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Axioms for coherent r

• Units
• r(Xar0) r(X) - a (for all a)
• Diversification
• r((XY)/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 ³ 0)

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An aside ...

• In the presence of the linearity axiom, the
  diversification axiom can be written
• r(XY) 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 XY, firm could reduce capital requirement
  by setting up two subsidiaries, one to hold X and
  the other Y.

9
Do any such r exist?

• Do we want one? (Maybe not!)
• There are many such rs
• Take any set A of outcomes
• Set rA(X) - infX(w)/r0(w) wÎA
• Think of A as set of scenarios r gives worst
  case
• Take any set of probabilities P
• Set rp(X) - infEP(X/r0) PÎP
• Think of each P as a generalized scenario

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Are there any more?

• Theorem If W 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.

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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 )

12
Another method

• Let each desk generate relevant scenarios for
  instruments it trades pass these to firms 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

13
What about VaR?

• 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)
• Compute the .01 quantile of the distribution of X
  under P
• The negative of this quantile rVaR(X)
• (implicitly assumes r0 1.)

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VaR is not coherent!

• VaR satisfies all axioms except diversification
  (and it uses r0 1).
• This means VaR limits cant be allocated to
  desks each desk might satisfy limit but total
  portfolio might not.
• Firms avoid VaR restrictions by setting up
  subsidiaries

15
VaR says dont 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!

16
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
• (Lets pretend interest rate
  0)
• Probability Value
• 0.985075 1500
• 0.01485 -99000 VaR requires
  99000
• 7.46E-05 -199500
• 1.25E-07 -300000

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Even scarier

• Most firms want to get the highest return per
  unit of risk.
• If they use VaR to measure risk, theyll 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
  theyll do the same, probably without realizing
  it!

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Does anything like VaR work?

• Suppose we have chosen a P which wed 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)
• Its the smallest coherent r which depends only
  on the P-distribution of Xs and which is bigger
  than VaR.

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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

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Whats 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.

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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

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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?

23
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 theres any X with E(X)gt0 and
  VaR(X) 0 (like being short a far out-of-
  the-money put)
• VaR constraint is satisfied for arbitrarily large
  position size!

24
With a coherent risk measure

• Well see that
• Firms can achieve economically optimal
  portfolios
• Decision problem can be allocated to desks
• Desks can each have their own PDesk
• If these arent too inconsistent, still works!
• But first -- in addition to regulators we need
  the firms owners

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Meeting goals of shareholders

• So far, risk measures were for regulation
• Shareholders have a different point of view
• Solvency isnt enough
• Dont want too much risk of loss of investment
• Shareholders may have different risk preferences
  than regulators
• Firm must respect both regulators and
  shareholders demands

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A shareholders risk measure

• Require firm to count shareholders 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

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The optimization problem

• Let rReg denote the regulators risk measure
• Let P be some given probability measure
• Let T be the investors 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.

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In liquid markets Linear Program

• In liquid markets the initial price of X, p0(X)
  is a linear function of X.
• Traded Xs form a linear space
• Available Xs 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) ³ 0 for all Q Î QReg
• rSH(X-T) 0 is same as EQ(X) ³ EQ(T) for all Q Î
  QSH

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Is the resulting portfolio optimal?

• Can firm get to shareholders 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
  regulators constraints
• Suppose such an X exists
• Can managers specify T and rSH so that X is the
  solution to the above LP?

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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 ³ T ( X). But if any
  available X ³ X were actually larger (on a set
  with positive P-measure), EP(u(X)) would be
  bigger than EP(u(X)), so X wouldnt have
  maximized expected utility

31
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

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• and Regulators risk is assigned (for each
  regulator probability Q Î QReg) to desks rQ,1,
  rQ,2, , rQ,D summing to 0.

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Let desk d try to solve

• Choose Xd Î Xd to maximize EP(Xd) subject to
• p0(Xd) Kd
• EQ(Xd) ³ EQ(Td) for every Q Î QSH
• EQ(Xd) ³ rQ,d for every Q Î QReg

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Clearly ...

• X1 X2 XD is feasible for the firms
  problem, so EP(X1XD) is EP(X).
• i.e., desks cant get better total solution than
  firm could get
• Since X can be decomposed as X1 X2 XD
  where Xd Î Xd, with appropriate splitting of
  resources as above desks will achieve optimal
  portfolio for the firm

35
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 desks objective
  function is lowered by the amount of its payments
  in this internal market.)

36
Market equilibrium

• The only equilibrium for this market produces the
  optimal portfolio for the firm.
• (Look at the firms dual problem this tells the
  equilibrium internal prices associated with each
  constraint.)

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What if each desk has its own Pd?

• If there is some P such that EPd(X) EP(X) for
  all X Î Xd then any market equilibrium solves the
  firms LP for this measure P.
• If there isnt then there is internal arbitrage
  and no market equilibrium exists.