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

2. 2 Setting the stage 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

3. 3 Aim of This Presentation 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”

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

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

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

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

8. 8 Black-Scholes Economy (2) “Risk-neutral” pricing:  ƒ 0 = e -rT E Q [ ƒ(S T ) ]  Q: “pretend” that S T = exp{ (r -½  2 )T +  z T } Deflator pricing:  ƒ 0 = E P [ ƒ(S T ) R T ]  R T = C (1/S T )  (S large  R small)   =(  -r)/  2 (market price of risk)  P: S T follows “true” process Price ƒ 0 is the same!

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

10. 10 Utility Functions Economic agent has to make decisions under uncertainty Choose “optimal” investment strategy for wealth W 0 Decision today  uncertain wealth W T Make a trade-off between gains and losses

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

12. 12 Optimal Wealth Maximise expected utility E P [U(W T )] By choosing optimal wealth W T First order condition for optimum  U’(W T *) = R T Solution: W T * = (U’) -1 ( R T )  Intuition: buy “cheap” states & spread risk  Solve from budget constraint

13. 13 Optimal Wealth - Example Black-Scholes economy Exp Utility: U(W) = -e -aW / a Condition for optimal wealth:  U’(W T * ) = R T  exp{-aW T * } = C S T -    =(  -r)/  2 (market price of risk) Optimal wealth: W T * = C * +  /a ln(S T )  Solve C * from budget constraint

14. 14 Optimal Wealth - Example (2) 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.

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

16. 16 Optimal Wealth - Example (4) 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)

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

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

19. 19 Principle of Equivalent Utility (2) 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

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

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

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

23. 23 Pricing Insurance Contracts (2) 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)

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

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

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

27. 27 Pricing Insurance Contracts (6) Solution for optimal wealth: Surplus W* BE MVM Price of insurance contract:   0 = e -rT (CNp + ½aC 2 Np(1-p))  “Variance Principle” from actuarial lit.

28. 28 Observations on MVM 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 + ½aC 2 (N+1)p(1-p)) Price for extra contract:  e -rT (Cp + ½aC 2 p(1-p))

29. 29 Observations on MVM (2) 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 + ½aC 2 (N-1)p(1-p)) Price for extra contract:  e -rT (C(1-p) - ½aC 2 p(1-p))  BE + negative MVM!  Give bonus for diversification benefit

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

31. 31 Optimal Wealth with Shortfall Constraint 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 W T < 0  Limit the averaged losses below W T < 0 Optimal investment strategy W* will change due to additional constraint

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

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

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

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

36. 36 Conclusions 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”