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

2. 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

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
April 11, 2003

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 Continuity
Wang/Young/Panjer (1997)
April 11, 2003

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)
Wang/Young/Panjer (1997)
April 11, 2003

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
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7. Distortion Risk Measures
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8. Distortion Risk Measures
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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
April 11, 2003

10. 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

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
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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
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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
April 11, 2003

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
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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
April 11, 2003

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
April 11, 2003

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
April 11, 2003

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
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

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=0.5 if reserve development follows a Brownian motion
April 11, 2003

20. 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

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-t Q
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22. A Two-Factor Model First adjust for parameter uncertainty
F(x)  Normal(0,1)   Student-t Q
Then Apply Wang transform:
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23. Fit 2-factor model to 1999 Cat bonds Date Sources: Lane Financial LLC
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24. Fit 2-factor model to corporate bonds
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