1. Finance for Actuaries Interest Rate Sensitive Insurance Products 2000 Investment Conference Jeroen van Bezooyen Shyam Mehta

2. Finance for Actuaries2 Agenda Product Description Valuation Methodology Interest Rate Model Example Hedging Strategies Model Extensions

3. Finance for Actuaries3 Product Description Based on French interest rate products Single premium Guaranteed minimum annual bonus Discretionary additional bonus based on portfolio yield Maturity: 8 years with option to extend Early surrender option (subject to surrender and/or tax penalties)

4. Finance for Actuaries4 Product Description Bonus policy is not hard-coded Awarded bonus credits driven by –Guarantee –Competitor bonuses –Portfolio yield Modelling approach: specify functional form that captures the above effects

5. Finance for Actuaries5 Valuation Methodology Traditional actuarial approach ‘Single scenario’ Specify discount rate and bonus credit assumptions Specify lapse rates Project cash flows Value product as present value

6. Finance for Actuaries6 Valuation Methodology Drawback of actuarial approach No explicit allowance for optionality (actuarial judgement?) No risk management/hedging policy (what to do when interest rates go to 2%?)

7. Finance for Actuaries7 Valuation Methodology Financial economic approach Specify stochastic interest rate model Calibrate model to current market conditions Specify bonus rate function Specify lapse rates Solve model ‘backwards’ (numerically)

8. Finance for Actuaries8 Valuation Methodology Advantages of financial economic approach Allows for optionality Consistency with market prices Specifies risk management/hedging policy Possible to ‘hedge’ model assumptions

9. Finance for Actuaries9 Interest Rate Model Hull-White (1-factor) interest rate model, i.e. model short rate as –Mean-reverting and –Normal process Parameters –Volatility –Mean reversion rate –Mean reversion level (time-dependent for calibration) Implement as trinomial tree

10. Finance for Actuaries10 Trinomial Tree  tttt dBdtrdr 

11. Finance for Actuaries11 Trinomial Tree Backward solution method Set value (N,j) at final time layer equal to 1 Product value at node (i,j) is product of –exp [ -interest rate (i,j) ]* –[ 1 + bonus(i,j) ]* –Probability up-move * up value (i+1) + Probability middle-move * middle value (i+1) + Probability down move * down value (i+1)

12. Finance for Actuaries12 Example Bonus 3.5% guarantee, plus fraction of difference (if positive) between –weighted average of 8-t year spot rate and fixed 5% rate, and –3.5% guarantee

13. Finance for Actuaries13 Example Early surrender –Sliding scale of surrender penalties –Fraction of investors withdraws rationally –Withdrawal behaviour of rest is driven by a lapse rate function, where surrender rate depends on difference between yield and bonus Option to extend at maturity

14. Finance for Actuaries14 Example - Lapse Rate Function

15. Finance for Actuaries15 Example - Valuation Results Euro term structure as at 31 st May 2000 Lump sum investment of €100, 5% initial charge. Product value to investor: –Bonus only: €97.26 / 92.40 –Including early surrender: €100.92 / 95.88 –Including extension option €102.31 / 97.20

16. Finance for Actuaries16 Example Product Value as Function of Yield Level

17. Finance for Actuaries17 Example Product Value as Function of Yield Curve Tilt

18. Finance for Actuaries18 Example Product Value as Function of Volatility

19. Finance for Actuaries19 Hedging Valuation method is based on replicating portfolio Consequently, outcome is product value as well as hedging strategy Hedging strategy is dynamic, i.e. depends on interest rate level

20. Finance for Actuaries20 Hedging - Example Consider two assets –8-year bond with 6% coupon –Cash 31 st May 2000 hedge strategy is to invest –84% in bond, and –16% in cash

21. Finance for Actuaries21 Hedging - Dynamic Strategy Hedge Ratio as Function of Yield Level

22. Finance for Actuaries22 Hedging - Model Assumptions Hedging strategies based on model parameters Parameters can be calibrated against market prices (term structure, options, etc.) However, parameters can change over time! (e.g. volatility, yield curve slope) These modelling assumptions can also be hedged

23. Finance for Actuaries23 Hedging - Model Assumptions Volatility parameter can be hedged with an option position Introduce third asset: 8-year, 3.5% floor contract Hedge portfolio is to invest –55% in 8-year bond –43% in cash –2% in floor (total notional € 198)

24. Finance for Actuaries24 Model Extensions Additional behavioural analysis –Bonus declarations –Lapse rate function –Extension rationality Two-factor Hull-White model Calibration to other instruments

25. Finance for Actuaries25 Conclusions Advantages of using a market value model –Allows explicitly for optionality –Calibration to market prices –Delivers value as well hedge strategy This should be in every Actuary’s toolkit! Actuarial judgement on specification of lapse rate and bonus credit functions

26. Finance for Actuaries Interest Rate Sensitive Insurance Products 2000 Investment Conference Jeroen van Bezooyen Shyam Mehta