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Guaranteed Annuity Rate Options by David O. Forfar. International Centre for Mathematical Sciences and Isaac Newton Institute. Unit-Linked Policy at Maturity Value of units =Number of Units*Price =Pension Fund. With-profits Policy at Maturity

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Guaranteed annuity rate options by david o forfar l.jpg

Guaranteed Annuity Rate Optionsby David O. Forfar

International Centre for Mathematical Sciences and Isaac Newton Institute


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Unit-Linked Policy at Maturity

Value of units

=Number of Units*Price

=Pension Fund


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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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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


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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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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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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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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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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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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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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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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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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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Model 1(per WWY 2003)


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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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Type 1 GAO Hedging Strategy

(1) Long on deferred annuities

(2) Short in equities


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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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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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Type 2 GAO : Guaranteed Sum at Maturity, modifies the pay-off

e.g. Pay-off for Type 2 GAO was


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Model 2 (Hull White) pay-off

  • 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



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Summary pay-off

The hedging strategy works!

(see spreadsheet)

Article in the April issue Actuary Magazine

Full details in the Paper

Copies available


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