1 guaranteed annuity rate options by david o. forfar international centre for mathematical sciences...

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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 GuaranteePension 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 bonusGAR=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 GuaranteePension 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 TMAR=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 pensionValue 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)=1F(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)

S S

S S

dS(t)=μ (S(t),t)dt+σ dW (t) where dW (t) is the BM driving S(t)

S(t)

F F

F F

dF(t)=μ (F(t),t)dt+σ dW (t) where dW (t) is the BM driving F(t)

F(t)

-(T-t)R(t) R(t) is the redemption yieldP(t)=e .

RR

R R(R(t),t) where dW (t) is the BM driving R(t)dR(t)=μ dt+σ dW (t)

The BMs driving S,F,R are not independent but are correlatedwith correlation coefficients , ,SF SR FR

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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

2 1V(t)=D(t)*N[-d (t)]-S(t)*N[-d (t)]

1|2 *

*1 S(t) 1d (t)= ln( ) σ

σ D(t) 2+

2 2

2 3 2 2

*22

1)

3

* *S F S F SF

R S R SR F R FRttg ttg ttg

ttg ttg ttg

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Type 1 GAO Hedging Strategy

(1) Long on deferred annuities

(2) Short in equities

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Type 1 GAO

Term Deferred Annuities (P*F)

Equities Exchange Option

long short % of Single Premium

30 11 -5 6%

25 12 -6 5%

20 13 -8 5%

15 14 -9 5%

10 16 -12 5%

5 18 -14 4%

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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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Pricing Type 2 Annuity Rate Option (Quanto option)

(t)

1 2(t)*{GAR*F e ]}V(t)=S (t) N[d (t)]-N[d (t)

(t) *1|2 *

1 1ln(F(t)*GAR*e )+ σ

σ 2d (t)=

*2 2 *( )F T t

2

FS F S FR F R

1

2φ(t)=ρ σ σ (T-t)+ ρ σ σ (T-t)

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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 (borrowing T-bonds)

Term Equities DAs T-Bond Quanto

Option

long long short % of Single Premium

30 9 36 -36 9%

25 10 41 -41 10%

20 10 45 -45 10%

15 10 47 -47 10%

10 8 46 -46 8%

5 5 40 -40 5%

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Type 2 GAO (borrowing short)

Term Equities DAs Short bond

Quanto

long long short Option% of Single Premium

30 19 62 -62 19%

25 17 61 -61 17%

20 15 59 -59 15%

15 12 56 -56 12%

10 9 51 -51 9%

5 5 42 -42 5%

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

{ ( )( * ( ) 1) | * ( ) 1 ( ) ( )}

+{ ( )( * ( ) 1) | * ( ) 1 ( ) ( )}

PAY - OFF NOW

GF T GAR F T provided GAR F T and S T GF T

S T GAR F T provided GAR F T and S T GF T

1( )( 1) ( )( * ( ) 1) ( ) as

GAR

MARS T S T GAR F T F T

MAR

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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 KN

2ln ( ) 1( , ){ ( , ) 2( ) }

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( )

( )

T NR

P tB T T N B T T N C T x

T N NN N

T

P tK p e

P t

2ln ( ) 1ω-65 ( , ){ ( , ) 2( ) }2

65

1

( ) 1

( )

T NR

P tB T T N B T T N C T x

T N NN

TN

P tp e

P t GAR

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

ω-65( , )

65 ( ) 65 1 2

1

( )( ). . { [ ( , )] [ ( , )]}

( )

t TT NT t T t N N

TN

P tPF t GAR p p e N d N t K N d N t

P t

( )GAO t

( , )65

1|2

( )1 ln( ) ( , )

* ( , ) ( )

t TN T N

T N

p P t ed N t

N t P t K

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Type 2 GAO (Model 2)

Term Quanto

Option

% Single Premium

30 45%

25 28%

20 17%

15 9%

10 4%

5 1%

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Summary

The hedging strategy works!

(see spreadsheet)

Article in the April issue Actuary Magazine

Full details in the Paper

Copies available