Guaranteed Annuity Rate Options by David O. Forfar

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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 UnitsPrice
• Pension Fund

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• With-profits Policy at Maturity
• (1)Basic Fund (2)Guaranteed Bonuses(3)
  Non-guaranteed BonusesMaturity 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 BonusesGuaranteed
  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 offices Estate).
• (1)Basic Fund(2)Gteed Bonuses(3)Non-gteed
  Bonuses (Terminal Bonus)Full Pension FundPF(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
• MARMarket Annuity Rate (typically now at 65,
  .077.0) GFGuaranteed Fund i.e. excluding
  terminal bonus
• GARGuaranteed Annuity Rate (typically at 65,
  0.111111.11 so GAR1/9)
• 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
• MARMarket Annuity Rate
• GARGuaranteed 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, PFMAR but with minimum of
  the floor pension of GFGAR)
• 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 PFGAR and PFMAR
• 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 PFMAR but with min. of
  GFGAR)
• Turn it into cash terms by valuing the pension
• value of (GFGAR) p.a. GFGAR/MAR
• value of (PF(T)MAR) p.a. PF(T)
• Fund assumed invested in equities
• Guarantee pay-off maximumGFGAR/MAR,PF(T)
• Type 1 GAOmaximum0,GFGAR/MAR-PF(T)

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• Type 2 Annuity Rate Guarantee
• better pension per annum of PFGAR and PFMAR
• 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-offmaximum(GFGAR/MAR-PF),0
• Type of Exchange Option
• Type 2 Guaranteed Annuity Rate Option
• Pay-offmaximum 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 TDeferred annuity rate,
• Value at t of the floor pension is
  GFGARP(t)F(t) D(t)
• GFGAR/MARGFGARF(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-offmaximumD(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 1 GAO Type 1 GAO Type 1 GAO Type 1 GAO
Term Deferred Annuities (PF) 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)GARP(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-offmaximum S(T)(GARF(T)-1),0

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• Pricing Type 2 Annuity Rate Option (Quanto
  option)

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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) Type 2 GAO (borrowing T-bonds) Type 2 GAO (borrowing T-bonds) Type 2 GAO (borrowing T-bonds) 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) Type 2 GAO (borrowing short) Type 2 GAO (borrowing short) Type 2 GAO (borrowing short) 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-off
• e.g. Pay-off for Type 2 GAO was

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• Model 2 (Hull White)
• Complete yield curve driven off the short
  interest rate, r(t) and dr(t)ab-r(t)dtsdW
• Determine x, the rate of interest when the Type 2
  GAO is first in the money
• Determine KN

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

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Type 2 GAO (Model 2) 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