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

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

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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)

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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 offices Estate). (1)Basic Fund+(2)Gteed Bonuses+(3)Non- gteed Bonuses (Terminal Bonus)=Full Pension Fund=PF(T) =Maturity Value

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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))

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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)

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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).

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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.

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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!

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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)}

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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}

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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

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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)

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14 Model 1 (per WWY 2003)

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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

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

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17 Type 1 GAO TermDeferred Annuities (P*F) EquitiesExchange Option longshort% of Single Premium 3011-56% 2512-65% 2013-85% 1514-95% 1016-125% 518-144%

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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}

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19 Pricing Type 2 Annuity Rate Option (Quanto option)

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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

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21 Type 2 GAO (borrowing T-bonds) TermEquitiesDAsT-Bond Quanto Option long short % of Single Premium 30 936-369% 25 1041-4110% 20 1045-4510% 15 1047-4710% 10 846-468% 5 540-405%

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22 Type 2 GAO (borrowing short) TermEquitiesDAsShort bond Quanto long shortOption % of Single Premium 301962-6219% 251761-6117% 201559-5915% 151256-5612% 10951-519% 5542-425%

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

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24 Model 2 (Hull White) (1)Complete yield curve driven off the short interest rate, r(t) and dr(t)=a*{b-r(t)}dt+σdW (2)Determine x, the rate of interest when the Type 2 GAO is first in the money (3)Determine K N

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

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26 Type 2 GAO (Model 2) Term Quanto Option % Single Premium 3045% 2528% 2017% 159% 104% 51%

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

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