Coherent Risk Measures. Acceptance Sets. L = {X G : X(ω) < 0, ω Ω}.

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1 So far in this course we have used several different mathematical expressions to quantify risk, without a deeper discussion of their properties. Coherent Risk Measures Lecture 11, Optimisation in Finance The choice of the right measure to quantify risk generally depends strongly on the specific risk management decision at hand. Examples of situations with different needs are the following: MSc in Mathematical Finance Oxford University, TT 2009 Dr Raphael Hauser (hauser@comlab.ox.ac.uk) an insurance company needs to decide on the premiums to charge so as to keep the risk of bankruptcy low, a financial regulator needs to impose capital requirements on banks so as to keep the risk of systemic failure of the financial system low, a broker needs to decide when and how much margin to call on a portfolio that contains derivatives, pensioners (or their pension funds) need to decide how to invest their savings so as to preserve capital and generate a consumable return, a young person is investing money for their pension that they will not consume for several decades. In this lecture we will discuss the mathematical properties of risk measures that are useful to the first four situations, but less relevant to the last. Acceptance Sets We consider the question of which investments are acceptable over a single investment period [0,T] for a given investor or regulator. Let (Ω, F,P) be a probability space describing the set of states of nature at time T, where P may be known or unknown. We write G for the set of measurable functions X : Ω R, and L + = {X G : X(ω) 0, ω Ω}, L = {X G : X(ω) 0, ω Ω}, L = {X G : X(ω) < 0, ω Ω}.

2 Definition 1. A is an acceptance set if the following axioms are satisfied, Any function X G is called a risk. This is thought of as the final net worth of a risky position (which may be an entire portfolio of investments). A G the set of risks that are acceptable to the investor or regulator (this may be a broker or some other overseer). 1. L + A. 2. A L =. 3. A is a convex cone convexity implies that diversification is more acceptable than the individual parts positive homogeneity implies that acceptance is scaleinvariant. Risk Measures Associated with an acceptance set of risks is a risk-measure that is defined as follows, ρ : G R ρ A (X) = min{α : X + αr A }, where α is a cash postition with risk-free rate r. That is, ρ A (X) is the minimum requirement of extra capital to make X an acceptable risk. Note that α can be both positive (for nonacceptable risks) or negative (for acceptable ones). Definition 2. A risk measure is coherent if it satisfies the following axioms, 1. ρ(x + αr) = ρ(x) α (transitivity) 2. ρ(x 1 +X 2 ) ρ(x 1 )+ρ(x 2 ) (subadditivity, based on the idea that a merger does not create additional risk; this property makes decentralised risk management usable), 3. ρ(τx) = τρ(x) for all τ 0, X G (positive homogeneity), Risk measure can also be defined independently as a function ρ : G R. 4. X(ω) Y (ω) ω Ω implies ρ(y ) ρ(x) (monotonicity).

3 Theorem 3 (Artzner, Delbaen, Eber, Heath 1998). Counterexample 1. Value-at-risk is not subadditive and hence not coherent. i) If A is an acceptance set of risks, then ρ(x) = min{α : X + αr A } is a coherent risk measure. ii) Conversely, if ρ is a coherent risk measure, then A ρ = {X G : ρ(x) 0} is an acceptance set. To see this, consider a portfolio consisting of an underwritten put and an underwritten call that are each out of the money with probability 0.96 at maturity time T. Individually, each asset s loss has a VaR 0.95 of zero, but combined they have a loss function whose VaR 0.95 is nonzero. As a result, the use of VaR can discourage diversification of investments and fail to detect concentration of risks. To see this, we follow an example of Albanese: Consider the situation in which the base rate be zero and the spread for all corporate bonds is 2%. These bonds default independently of one another with probability An amount of 1M is borrowed at the base rate and invested in a single corporate bond. The payoff of this investment is as follows, 20K with probability 0.99, X = 1000K with probability Therefore, ρ(x) = VaR 0.95 ( X) = 20K, which is an acceptable investment. Compare this to the situation where 1M is borrowed and invested in a diversified portfolio of 100 different corporate bonds, at equal amounts of 10K. The payoff of this investment is as follows, 20K with probability =.366, 9.8K with probability 100 (0.99) =.370, X = 0.4K with probability ( ) (.99) 98 (0.01)2 = 0.185,. Now we have ρ(x) > 0, and the investment is not acceptable. Thus, by diversifying, the risk has gone up, not down!

4 Counterexample 2. Let X = R T x be the return of a portfolio with weights x. Then Var(X) = x T Qx is not a coherent risk measure, since it violates the homogeneity condition ρ(τ X) = τ ρ(x). Note that it also violates subadditivity, as (x + x) T Q(x + x) = 4x T Qx > x T Qx + x T Qx. Thus, while subadditivity and homogeneity imply convexity, a convex function may be neither subadditive nor homogeneous! Further, Var(X) violates the translation condtion 1, as Var(X + αr) = Var(X) Var(X) α. Thus, this risk-measure has no interpretation in terms of extra margin required to make the investment acceptable. Counterexample 3. Also not coherent are the related risk measures ρ(x) = E[X]+λVar(X) (violates homogeneity and subadditivity), ρ(x) = E[X] + λvar(x) 1/2 (violates monotonicity), ρ(x) = E[X] + Var(max(0,E[X] X)) (violates subadditivity), etc. Let us now study positive examples: Example 1. ρ(x) = CVaR q ( X) is coherent for all quantiles q. Proof. The only nontrivial property is subadditivity. Let us also assume for simplicity that the cumulative distribution function F(x) of the random variable X has no discontinuities, so that VaR q ( X) = F 1 (q) =: x q. Let 1 X xq be the indicator function of the event {ω Ω : X(ω) x q } that the loss of incurred by the risk X exceeds the value-at-risk at level q. Then CVaR q (X) = E[ X1 X xq ]. Now, if Z = X + Y is the sum of two risks, then CVaR q (Z) CVaR q (X) CVaR q (Y ) = E[ Z1 Z zq + X1 X xq + Y 1 Y yq ] = E[X(1 X xq 1 Z zq )] + E[Y (1 Y xq 1 Z zq )] x q E[1 X xq 1 Z zq ] y q E[1 Y yq 1 Z zq ] = x q (q q) y q (q q) = 0, where the inequality holds because 0, if X x q, 1 X xq 1 Z zq 0, if X > x q.

5 The next result shows that any coherent risk measure has a dual descriptiion: Mathematical Properties of Coherent Risk Measures Lemma 4.Let {ρ τ : τ T } be a family of coherent risk measures. Then ρ(x) = sup τ T ρ τ (X) is a coherent risk measure. Proof. Since for all τ, ρ τ (X + αr) = ρ τ (X) α, we have ρ(x + αr) = sup(ρ τ (X) α) = ρ(x) α. Therefore, ρ is transitive. Furthermore, subadditivity, positive homogeneity and monotonicity are properties preserved under taking supremums. Theorem 5 (Hu, Artzner-Delbaen-Eber-Heath). ρ is a coherent risk measure if and only if there exists a family P of probability measures on (Ω, F) such that for all X G, ρ(x) = sup{e P [ X/r] : P P}. (1) Proof. For the if part, note that for any probability measure P, the linearity of E P implies homogeneity additivity (and hence subadditivity), and monotonicity. Further, E P [ (X + αr)/r] = E P [ X/r] α. Therefore, for each P P, ρ P (X) = E P [ X/r] is a coherent risk measure, and the if part follows from Lemma 4. Note that P is a convex set of probability measures, which has the following consequences: To sketch a proof of the only if part, construct a set of probability measures P = {P : P is a probability measure s.t. E P [ Y/r] ρ(y ) Y G }. It is then clear that ρ(x) sup{e P [ X/r] : P P}, by construction of P. It can be shown that the supremum is actually achieved. To prove the achievement of the sup in the last part of the proof of Theorem 5, convex separation can be used. If ρ is defined by (1) for some set P of probability measures, P can be replaced by its convex hull P = conv(p). Equation (1) has the interpretation of a robust insurance premium over the convex uncertainty set P of probability measures that describe the state of nature at time T.

6 The last result we will discuss shows that VaR has an interpretation as a pointwise best-case estimate of risk: Theorem 6 (Artzner-Delbaen-Eber-Heath). For every risk X G it is true that where VaR α (X) = inf{ρ(x) : ρ coherent and ρ VaR α }, VaR α (X) = sup{x : P[ X xr] < α} and P is the physical probability measure as usual, and where ρ VaR α means ρ(y ) VaR α (Y ), Y G.