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Distortion Risk Measures and Economic Capital

Explore distortion measures preserving tail heaviness order, optimal capital levels, and the link with financial pricing theories discussed in Werner Hurlimann's paper by Shaun Wang. Learn about coherent risk measures and distortion transformations in financial risk management.

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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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