A Universal Framework For Pricing Financial and Insurance Risks

1. Shaun Wang, FCAS, Ph.D. SCOR Reinsurance Co. Shaun Wang, 2001 A Universal Framework For Pricing Financial and Insurance Risks Presentation at the ASTIN Colloquium July 2001, Washington DC

2. CAPM Price Data ? Black-Scholes Outline: A Puzzle Game • Present a new formula to connect CAPM with Black-Scholes • Piece together with actuarial axioms • Empirical findings • Capital Allocations

3. Market Price of Risk • Asset return R has normal distribution • r --- the risk-free rate • ={ E[R] r }/[R] is “the market price of risk” or excess return per unit of volatility.

4. Capital Asset Pricing Model Let Ri and RM be the return for asset i and market portfolio M.

5. The New Transform • extends the “market price of risk” in CAPM to risks with non-normal distributions is the standard normal cdf.

6. If FX is normal(), FX* is another normal( )  E*[X] =   • If FX is lognormal( ), FX* is another lognormal( )

7. Correlation Measure • Risks X and Y can be transformed to normal variables: Define New Correlation

8. Why New Correlation ? • Let X ~ lognormal(0,1) • Let Y=X^b (deterministic) • For the traditional correlation: (X,Y)  0 as b  + • For the new correlation: *(X,Y)=1 for all b

9. Extending CAPM • The transformrecoversCAPM for riskswith normal distributions •  extends the traditional meaning of { E[R] r }/[R] • New transformextendsCAPM to risks with non-normal distributions:

10. Brownian Motion • To reproduce stock’s current value: • Stock price Ai(T) ~ lognormal Ai(0) = E*[ Ai(T)] exp(rT) • Implies

11. Co-monotone Derivatives • For non-decreasing f, Y=f(X) is co-monotone derivative of X. • e.g. Y=call option, X=underlying stock • Y and X have the samecorrelation *with the market portfolio • Same should be used for pricingtheunderlyingand itsderivative

12. Commutable Pricing • Co-monotone derivative Y=f(X) • Equivalent methods: • Applytransformto FX to get FX*, then derive FY* from FX* • Derive FY from FX, then applytransform to FY to get FY*

13. Recover Black-Scholes • Applytransformwith same i from underlying stock to price options • Both i and the expected return i drop out from the risk-adjusted stock price distribution!! • We’ve just reproduced the B-S price!!

14. Option Pricing Example A stock’s current price = $1326.03. Projection of 3-month price: 20 outcomes: 1218.71, 1309.51, 1287.08, 1352.47, 1518.84, 1239.06, 1415.00, 1387.64, 1602.70, 1189.37, 1364.62, 1505.44, 1358.41, 1419.09, 1550.21, 1355.32, 1429.04, 1359.02, 1377.62, 1363.84. The 3-month risk-free rate = 1.5%. How to price a 3-month European call option with a strike price of $1375 ?

15. Computation • Sample data: =4.08%, =8.07% • Use=(r)/ =0.320 as “starter” • The transform yields a price =1328.14, differing from current price=1326.03 • Solve to match current price. We get=0.342 • Use the true to price options

16. Using New Transform (=0.342)

17. Loss vs Asset • Loss is negative asset: X= – A • New transform applicable to both assets and losses, with opposite signs in  • Alternatively, …

18. Use the same  without changing sign: apply transform to FA for assets, but apply transform to SX=1– FX for losses. Loss vs Asset

19. Actuarial World • Loss X with tail prob: SX(t) = Pr{ X>t }. • Layer X(a, a+h)=min[ max(Xa,0), h ]

20. Loss Distribution

21. Venter 1991 ASTIN Paper • Insurance prices by layer imply a transformed distribution • layer (t, t+dt) loss: SX(t) dt • layer (t, t+dt) price: SX*(t) dt • implied transform: SX(t)  SX*(t)

22. Graphic Intuition

23. Theoretical Choice • extends classic CAPM and Black-Scholes, • equilibrium price under more relaxed distributional assumptions than CAPM, and • unified treatment of assets & losses

24. Reality Check • Evidence for 3-moment CAPM which accounts for skewness [Kozik/Larson paper] • “Volatility smile” in option prices • Empirical risk premiums for tail events (CAT insurance and bond default) are higher than implied by the transform.

25. 2-Factor Model • 1/b is a multiple factor to the normal volatility • b<1, depends on F(x), with smaller values at tails (higher adjustment) • b adjusts for skewness & parameter uncertainty

26. Calibrate the b-function • Let Q be a symmetric distribution with fatter tails than Normal(0,1): • Normal-Lognormal Mixture • Student-t • Two calibrations lead to similar b-functions at the tails

27. 2-Factor Model: Normal-Lognormal Calibration

28. Theoretical insights of b-function • Relates closely to 3-moment CAPM. • Explains better investor behavior: distortion by greed and fear • Explains “volatility smile” in option prices • Quantifies increased cost-of-capital for gearing, non-liquidity markets, “stochastic volatility”, information asymmetry, and parameter uncertainty

29. Fit 2-factor model to 1999 transactionsDate Sources: Lane Financial LLC Publications

30. Use 1999 parameters to price 2000 transactions

31. 2-factor model for corporate bonds: same lambda but lower gamma than CAT-bond

32. Cross Industry Comparison  and  by industry: equity, credit, CAT-bond, weather and insurance Cross Time-horizon comparison Term-structure of  and  Universal Pricing

33. Capital Allocation • The pricing formula can serve as a bridge linking risk, capital and return. • Pricing parameters are readily comparable to other industries. • A more robust method than many current ERM practices