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