Distortion Risk Measures and Economic Capital Discussion of Werner Hurlimann Paper --- By Shaun Wang April 11, 2003

Agenda Highlights of W. Hurlimann Paper: Discussion by S. Wang: Search for distortion measures that preserve an order of tail heaviness Optimal level of capital Discussion by S. Wang: Link distortion measures to financial pricing theories Empirical studies in Cat-bond, corporate bond April 11, 2003

Assumptions We know the dist’n F(x) for financial losses In real-life this may be the hardest part Risks are compared solely based on F(x) Correlation implicitly reflected in the aggregate risk distribution April 11, 2003

Axioms for Coherent Measures Axiom 1. If X  Y  (X)  (Y). Axiom 2. (X+Y)  (X)+ (Y) Axiom 3. X and Y are co-monotone  (X+Y) = (X)+ (Y) Axiom 4. Continuity Wang/Young/Panjer (1997) April 11, 2003

Representation for Coherent Measures of Risk Given Axioms 1-4, there is distortion g:[0,1][0,1] increasing concave with g(0)=0 and g(1)=1, such that F*(x) = g[F(x)] and (X) = E*[X] Alternatively, S*(x) = h[S(x)], with S(x)=1F(x) and h(u)=1 g(1u) Wang/Young/Panjer (1997) April 11, 2003

Some Coherent Distortions TVaR or CTE: g(u) = max{0, (u)/(1)} PH-transform: S*(x) = [S(x)]^, for  <1 Wang transform: g(u) = [1(u)+], where  is the Normal(0,1) distribution April 11, 2003

Distortion Risk Measures April 11, 2003

Distortion Risk Measures April 11, 2003

Ordering of Tail Heaviness Hurlimann compares risks X and Y with equal mean and equal variance If E[(X c)+^2]  E[(Y c)+^2] for all c, Y has a heavier tail than risk X He tries to find “distortion measures” that preserve his order of tail heaviness April 11, 2003

Hurlimann Result For the families of bi-atomic risks and 3-parameter Pareto risks, A specific PH-transform: S*(x)=[S(x)]0.5 preserves his order of tail heaviness Wang transform and TVaR do not preserve his order of tail thickness Lookback: S*(x)=[S(x)]^  [1  ln(S(x)) ], see Hurlimann (19989) April 11, 2003

Optimal Risk Capital Definitions : Economic Risk Capital: Amount of capital required as cushion against potential unexpected losses Cost of capital: Interest cost of financing Excess return over risk-free rate demanded by investors April 11, 2003

Optimal Risk Capital: Notations X: financial loss in 1-year C = C[X]: economic risk capital i borrowing interest rate r < i risk-free interest rate April 11, 2003

Dilemma of Capital Requirement Net interest on capital (i  r)C  small C Solvency risk X  C(1+r)   large C Let R[.] be a risk measure to price insolvency See guarantee fund premium by David Cummins Minimize total cost: R[max{X  C(1+r),0}] + (i  r)C April 11, 2003

Optimal Risk Capital: Result Optimal Capital (Dhane and Goovaerts, 2002): C[X] = VaR(X)/(1+r) with  =1 g1[(i  r)/(1+r)] When (i  r) increases, optimal capital decreases! Eg. XNormal(,), i=7.5%, r=3.75%, and g(u)=u^0.5,  C[X]=[+3]/1.0375 April 11, 2003

Remarks In standalone risk evaluation, distortion measures may or may not preserve Hurlimann’s order of tail heaviness However, individual risk distribution tails can shrink within portfolio diversification We need to reflect the portfolio effect and link with financial pricing theories April 11, 2003

Properties of Wang transform If the asset return R has a normal distribution F(x), transformed F*(x) is also normal with E*[R] = E[R]   [R] = r (risk-free rate)  = { E[R] r }/[R] is the “market price of risk”, also called the Sharpe ratio April 11, 2003

Link to Financial Theories Market portfolio Z has market price of risk 0 corr(X,Z) =  Buhlmann 1980 economic model  It recovers CAPM for assets, and Black-Scholes formula for Options April 11, 2003

Unified Treatment of Asset / Loss The gain X for one party is the loss for the counter party: Y = X We should use opposite signs of , and we get the same price for both sides of the transaction How good is it for pricing insurance and credit risks? Reality check  we need a second factor for parameter uncertainty & skew / kurtosis April 11, 2003

Risk Adjustment for Long-Tailed Liabilities The Sharpe Ratio  can adjust for the time horizon: (T) = (1) * (T)b, where 0.5 b 1 where T is the average duration of loss payout patterns b=0.5 if reserve development follows a Brownian motion April 11, 2003

Adjustment for Parameter Uncertainty We have m observations from normal(,2). Not knowing the true parameters, we have to estimate  and  by sample mean & variance. When assessing the probability of future outcomes, we effectively need to use Student-t with k=m  2 degrees-of-freedom April 11, 2003

Adjust for Parameter Uncertainty Baseline: For normal distributions, Student-t properly reflects the parameter uncertainty Generalization: For arbitrary F(x), we propose the following adjustment: F(x)  Normal(0,1)   Student-t Q April 11, 2003

A Two-Factor Model First adjust for parameter uncertainty F(x)  Normal(0,1)   Student-t Q Then Apply Wang transform: April 11, 2003

Fit 2-factor model to 1999 Cat bonds Date Sources: Lane Financial LLC April 11, 2003

Fit 2-factor model to corporate bonds April 11, 2003