Pricing insurance contracts: an incomplete market approach

1. Pricing insurance contracts: an incomplete market approach ACFI seminar, 2 November 2006 Antoon Pelsser University of Amsterdam & ING Group

2. Arbitrage-free pricing: standard method for financial markets Assume complete market Every payoff can be replicated Every risk is hedgeable Insurance “markets” are incomplete Insurance risks are un-hedgeable But, financial risks are hedgeable Setting the stage 2

3. Find pricing rule for insurance contracts consistent with arbitrage-free pricing “Market-consistent valuation” Basic workhorse: utility functions Price = BE Repl. Portfolio + MVM A new interpretation of Profit-Sharing as “smart EC” Aim of This Presentation 3

4. Recap of Arbitrage-free pricing Utility Functions & Optimal Wealth Principle of Equivalent Utility Pricing Insurance Contracts Optimal Wealth with Shortfall Constraint Outline 4

5. Recap of Arbitrage-free pricing Utility Functions & Optimal Wealth Principle of Equivalent Utility Pricing Insurance Contracts Optimal Wealth with Shortfall Constraint Outline 5

6. Arb-free pricing = pricing by replication Price of contract = price of replicating strategy Cash flows  portf of bonds Option  delta-hedging strategy Arbitrage-free pricing 6

7. Two assets: stock S and bond B S0=100, B0=1 Bond price BT=erT const. interest rate r (e.g. 5%) Stock price ST = 100 exp{ ( -½2)T + zT } zT ~ n(0,T) (st.dev = T)  is growth rate of stock (e.g. 8%)  is volatility of stock (e.g. 20%) Black-Scholes Economy 7

8. “Risk-neutral” pricing: ƒ0 = e-rT EQ[ƒ(ST) ] Q: “pretend” that ST = exp{ (r -½2)T + zT} Deflator pricing: ƒ0 = EP[ ƒ(ST)RT ] RT = C (1/ST) (S large  R small) =(-r)/2 (market price of risk) P: ST follows “true” process Price ƒ0 is the same! Black-Scholes Economy (2) 8

9. Recap of Arbitrage-free pricing Utility Functions & Optimal Wealth Principle of Equivalent Utility Pricing Insurance Contracts Optimal Wealth with Shortfall Constraint Outline 9

10. Economic agent has to make decisions under uncertainty Choose “optimal” investment strategy for wealth W0 Decision today  uncertain wealth WT Make a trade-off between gains and losses Utility Functions 10

11. Exponential utility: U(W) = -e-aW / a RA = a Power utility: U(W) = W(1-b) / (1-b) RA = b/W Utility Functions - Examples 11

12. Maximise expected utility EP[U(WT)] By choosing optimal wealth WT First order condition for optimum U’(WT*) = RT Solution: WT* = (U’)-1(RT) Intuition: buy “cheap” states & spread risk Solve  from budget constraint Optimal Wealth 12

13. Black-Scholes economy Exp Utility: U(W) =-e-aW / a Condition for optimal wealth: U’(WT*) = RT exp{-aWT*} =  CST- =(-r)/2 (market price of risk) Optimal wealth: WT* = C* + /a ln(ST) Solve C* from budget constraint Optimal Wealth - Example 13

14. Optimal investment strategy: invest amount (e-r(T-t)/a) in stocks borrow amount (e-r(T-t)/a -Wt) in bonds usually leveraged position! Intuition: More risk averse (a)  less stocks (/a) We cannot observe utility function We can observe “leverage” of Insurance Company Use leverage to “calibrate” utility fct. Optimal Wealth - Example (2) 14

15. Optimal Wealth - Example (3) Default for ST<75 Optimal Surplus 15

16. Risk-free rate r =5% Growth rate stock  =8% Volatility stock  =20% Hence:  = (-r)/2 = 0.75 Amount stocks: e-rT/a = e-0.05T 0.75/a Calibrate a from leverage (e.g. the  or D/E ratio) Optimal Wealth - Example (4) 16

17. Recap of Arbitrage-free pricing Utility Functions & Optimal Wealth Principle of Equivalent Utility Pricing Insurance Contracts Optimal Wealth with Shortfall Constraint Outline 17

18. Economic agent thinks about selling a (hedgeable) financial risk HT Wealth at time T = WT - HT What price 0 should agent ask? Agent will be indifferent if expected utility is unchanged: Principle of Equivalent Utility 18

19. Principle of Equivalent Utility is consistent with arbitrage-free pricing for hedgeable claims Principle of Equivalent Utility can also be applied to insurance claims Mixture of financial & insurance risk Mixture of hedgeable & un-hedgeable risk Literature: Musiela & Zariphopoulou V. Young Principle of Equivalent Utility (2) 19

20. Recap of Arbitrage-free pricing Utility Functions & Optimal Wealth Principle of Equivalent Utility Pricing Insurance Contracts Optimal Wealth with Shortfall Constraint Outline 20

21. Assume insurance claim: HT IT, Hedgeable risk HT cash amount C stock-price ST Insurance risk IT number of policyholders alive at time T 1 / 0 if car-accident does (not) happen salary development of employee Pricing Insurance Contracts 21

22. Apply principle of Equivalent Utility Find price 0 via solving “right-hand” optimisation problem Pricing Insurance Contracts 22

23. Consider specific example: life insurance contract Portfolio of N policyholders Survival probability p until time T Pay each survivor the cash amount C Payoff at time T = Cn n is (multinomial) random variable E[n] = Np Var[n] = Np(1-p) Pricing Insurance Contracts (2) 23

24. Solve optimisation problem: Mixture of random variables W,R are “financial” risks n is “insurance” risk “Adjusted” F.O. condition Pricing Insurance Contracts (3) 24

25. Necessary condition for optimal wealth: EPU[ ] does not affect “financial” WT EPU[ ] only affects “insurance” risk n Exponential utility: U’(W-Cn) = e-a(W-Cn) = e-aW eaCn Pricing Insurance Contracts (4) 25

26. For large N: n  n( Np , Np(1-p) ) MGF of z~n(m,V): E[etz] = exp{ t m + ½t2V } Hence: EPU[ eaCn ] = exp{ aC Np + ½(aC)2Np(1-p) } Equation for optimal wealth: Pricing Insurance Contracts (5) 26

27. Solution for optimal wealth: Surplus W* BE MVM Price of insurance contract: 0 = e-rT(CNp + ½aC2Np(1-p)) “Variance Principle” from actuarial lit. Pricing Insurance Contracts (6) 27

28. MVM  ½ Risk-av * Var(Unhedg. Risk) Note: MVM for “binomial” risk! “Diversified” variance of all unhedgeable risks Price for N+1 contracts: e-rT(C(N+1)p + ½aC2(N+1)p(1-p)) Price for extra contract: e-rT(Cp + ½aC2p(1-p)) Observations on MVM 28

29. Sell 1 contract that pays C if policyholder N dies Death benefit to N’s widow Certain payment C + (N-1) uncertain Price: e-rT(C + C(N-1)p + ½aC2(N-1)p(1-p)) Price for extra contract: e-rT(C(1-p) - ½aC2p(1-p)) BE + negative MVM! Give bonus for diversification benefit Observations on MVM (2) 29

30. Recap of Arbitrage-free pricing Utility Functions & Optimal Wealth Principle of Equivalent Utility Pricing Insurance Contracts Optimal Wealth with Shortfall Constraint Outline 30

31. Regulator (or Risk Manager) imposes additional shortfall constraints e.g. Insurance Company wants to maintain single-A rating “CVaR” constraint Protect policyholder Limit Probability of WT < 0 Limit the averaged losses below WT < 0 Optimal investment strategy W* will change due to additional constraint Optimal Wealth with Shortfall Constraint 31

32. Maximise expected utility EP[U(WT)] By choosing optimal wealth WT with CVaR constraint:  denotes a confidence level V denotes the -VaR (auxiliary decision variable) This formulation of CVaR is due to Uryasev Optimal Wealth with Shortfall Constraint (2) 32

33. Optimal Wealth with Shortfall Constraint (3) Less Shortfall VaR = 0.28 33

34. Optimal Wealth with Shortfall Constraint (4) EC “Contingent Capital” 34

35. Finance EC by selling part of “upside” to policyholders: Profit-Sharing EC -/- Profit Sharing Optimal Wealth with Shortfall Constraint (5) 35

36. Pricing insurance contracts in “incomplete” markets Principle of Equivalent utility is useful setting Decompose price as: BE Repl. Portfolio + MVM MVM can be positive or negative Profit-Sharing can be interpreted as holding “smart EC” Conclusions 36