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Distortion Risk Measures and Economic Capital

Explore distortion risk measures in financial pricing and optimal capital levels as discussed by Werner Hurlimann in a paper. Discover coherent distortions and their application, alongside the dialogue by Shaun Wang.

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Distortion Risk Measures and Economic Capital

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  1. Distortion Risk Measures and Economic Capital Discussion ofWerner Hurlimann Paper ---By Shaun Wang

  2. Agenda • Highlights of W. Hurlimann Paper: • 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

  3. 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

  4. 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

  5. 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)

  6. 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

  7. Distortion Risk Measures

  8. Distortion Risk Measures

  9. 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

  10. Hurlimann Result • For the families of bi-atomic risks and 3-parameter Pareto risks, • A specific PH-transform: S*(x)=[S(x)]0.5preserves his order of tail heaviness • Wang transform and TVaR do not preserve his order of tail thickness

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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=0.5if reserve development follows a Brownian motion

  20. Adjustment for Parameter Uncertainty

  21. 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-tQ

  22. A Two-Factor Model • First adjust for parameter uncertainty • F(x)  Normal(0,1)   Student-tQ • Then Apply Wang transform:

  23. Fit 2-factor model to 1999 Cat bondsDate Sources: Lane Financial LLC

  24. Fit 2-factor model to corporate bonds

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