Guaranteed Annuity Rate Options by David O. Forfar

1. Guaranteed Annuity Rate Optionsby David O. Forfar International Centre for Mathematical Sciences and Isaac Newton Institute

2. Unit-Linked Policy at Maturity Value of units =Number of Units*Price =Pension Fund

3. With-profits Policy at Maturity (1)Basic Fund +(2)Guaranteed Bonuses+(3) Non-guaranteed Bonuses=Maturity Value of the Pension Fund =PF(T) (1)=Basic Fund, set at policy outset (2)=Guaranteed bonuses, declared every year by the life office and are guaranteed (3)=Non-guaranteed Bonuses =Terminal Bonus, decided only at policy maturity and are non-guaranteed (1)Basic Fund+(2)Guaranteed Bonuses=Guaranteed Fund (GF)

4. Annuity Rate Guarantees • Expenses assumed to be % of the premium, • Premium accumulated at investment return achieved, • The terminal bonus determined after smoothing of investment return, • Any guarantee/option paid for from outside the policy (i.e. by the life office’s Estate). • (1)Basic Fund+(2)G’teed Bonuses+(3)Non-g’teed Bonuses (Terminal Bonus)=Full Pension Fund=PF(T) =Maturity Value

5. Annuity Rate Guarantees Two quite distinct types of annuity rate guarantee depending on:- Type 1: the annuity rate guarantee applies only to the guaranteed fund (GF(T)=(1)+(2)) Type 2: the annuity rate guarantee applies to the full pension fund (PF(T)=(1)+(2)+(3))

6. Type 1 Annuity Rate Guarantee Pension pay-off per annum at Maturity Maximum(PF(T)*MAR,GF(T)*GAR) per annum PF(T)=Full Pension Fund at maturity MAR=Market Annuity Rate (typically now at 65, .07=7.0%) GF=Guaranteed Fund i.e. excluding terminal bonus GAR=Guaranteed Annuity Rate (typically at 65, 0.1111=11.11% so GAR=1/9) In words: there is a ‘floor pension’ (GF(T)*GAR) below which a life office cannot go, no matter what happens to the stock-market or how expensive market annuity rates become. The annuity rate guarantee (GAR) applies only to the guaranteed fund - GF(T)

7. Type 2 Annuity Rate Guarantee Pension payoff per annum at Maturity Maximum(PF(T)*MAR,PF(T)*GAR) per annum =PF(T)*Maximum(GAR,MAR) per annum PF(T)=Total Pension Fund at T MAR=Market Annuity Rate GAR=Guaranteed Annuity Rate In words: the total pension fund - PF(T) - is applied at whichever is the better of the market annuity rate (MAR) or the guaranteed annuity rate (GAR). The guarantee applies to the full fund (PF).

8. Type 1 Annuity Rate Guarantee (pension per annum, PF*MAR but with minimum of the ‘floor pension’ of GF*GAR) Risks Exposed to:- • Interest rate risk (MAR low) • Longevity risk (MAR low) • Equity risk (on GF only, not the PF) • If decade of retirement 60-70 (European option is in fact a Bermudan Option) Control available : through not making the guaranteed fund (GF) too large i.e. not making the guaranteed bonuses, declared every year, too large.

9. Type 2 Annuity Rate Guarantee pension per annum, better of PF*GAR and PF*MAR Risks Exposed to:- • Interest rate risk (MAR low) • Longevity risk (MAR low) • Equity risk (PF high) • If decade of retirement 60-70 (European option is in fact a Bermudan Option) No control available!

10. Type 1 Annuity Rate Guarantee (pension p. a. of PF*MAR but with min. of GF*GAR) Turn it into cash terms by valuing the pension value of £(GF*GAR) p.a.= GF*GAR/MAR value of £(PF(T)*MAR) p.a. =PF(T) Fund assumed invested in equities Guarantee pay-off =maximum{GF*GAR/MAR,PF(T)} Type 1 GAO=maximum{0,GF*GAR/MAR-PF(T)}

11. Type 2 Annuity Rate Guarantee better pension per annum of PF*GAR and PF*MAR Turn it into cash terms by valuing the pension Value of PF(T)*GAR p.a.=PF(T)*GAR/MAR Value of PF(T)*MAR p.a.=PF(T) Guarantee Pay-off =Maximum(PF(T)*GAR/MAR,PF(T)) Type 2 GAO =PF(T)*maximum{(GAR/MAR-1),0}

12. Type 1 Guaranteed Annuity Rate Option Pay-off=maximum{(GF*GAR/MAR-PF),0} =Type of Exchange Option Type 2 Guaranteed Annuity Rate Option Pay-off=maximum PF*{(GAR/MAR-1),0} =Type of Quanto option

13. Type 1 GAO P(t)=T-bond price, P(T)=1 F(t)=Annuity of £1 p.a. commencing at T (age 65) but bought forward i.e. price agreed at t but not paid until T F(T)=1/MAR F(t)*P(t)= Value at t of a pension of £1 p.a. commencing at T=Deferred annuity rate, Value at t of the ‘floor pension’ is GF*GAR*P(t)*F(t) =D(t) GF*GAR/MAR=GF*GAR*F(T)=D(T) Value of PF at time t =PF(t) assumed to be all shares so replace PF(t) by S(t)

14. Model 1(per WWY 2003)

15. Pricing Type 1 GAO (Exchange option) Option pay-off=maximum{D(T)-S(T),0} V(t)=Value of Type 1 GAO at t

16. Type 1 GAO Hedging Strategy (1) Long on deferred annuities (2) Short in equities

18. Type 2 GAO Value at t of PF(t)*GAR p.a.=S(t)*GAR*P(t)*F(t) P(t)=value at t of T-bond (zero-coupon bond redeeming at T) F(t)= forward annuity at t, annuity of £1 p.a. commencing at T, price paid at T but agreed at t, F(T)=1/MAR Value of PF at time t =S(t) Pay-off=maximum S(T)*{(GAR*F(T)-1),0}

19. Pricing Type 2 Annuity Rate Option (Quanto option)

20. Type 2 GAO Hedging (1) Invest all the option premium in shares, (2) Long in deferred annuities, financed by, (3) Short in T-bonds (zero-coupon bonds redeeming at T). If the borrowings are not in the T-bond but are short makes great difference to price

23. Type 2 GAO : Guaranteed Sum at Maturity, modifies the pay-off e.g. Pay-off for Type 2 GAO was

24. Model 2 (Hull White) • Complete yield curve driven off the short interest rate, r(t) and dr(t)=a*{b-r(t)}dt+σdW • Determine x, the rate of interest when the Type 2 GAO is first in the money • Determine KN

25. Formula under the Hull-White Model for a Type 2 GAO

27. Summary The hedging strategy works! (see spreadsheet) Article in the April issue Actuary Magazine Full details in the Paper Copies available