Guaranteed Annuity Options Hansjörg Furrer Market-consistent Actuarial Valuation ETH Zürich, Frühjahrssemester 2008 Guaranteed Annuity Options
Contents A. Guaranteed Annuity Options B. Valuation and Risk Measurement Guaranteed Annuity Options 1
Course material Slides A. Guaranteed Annuity Options Boyle, P. and Hardy, M. (2003). Guaranteed Annuity Options. ASTIN Bulletin, Vol. 33, No. 2, pp. 125-152. Pelsser, A. (2003). Pricing and Hedging Guaranteed Annuity Options via Static Option Replication. Insurance: Mathematics and Economics. Vol. 33, 283-296. The above documents can be downloaded from www.math.ethz.ch/~hjfurrer/teaching/ Guaranteed Annuity Options 2
An Introduction into GAO Definition: Under a guaranteed annuity option, the insurer guarantees to convert the policyholder s accumulated funds into a life annuity at a fixed rate g at the policy maturity date T GAOs provide a minimum return guarantee: - the policyholder has the right to convert the accumulated funds into a life annuity at the better of the market rate prevailing at maturity and the guaranteed rate - if the annuity rates under the guarantee exceed the market annuity rates, then a rational policyholder will exercise the option. In that case, the insurer must cover the difference Guaranteed Annuity Options 3
Origins When many of these guarantees were written in the UK in the 1970ies and 1980ies - long-term interest rates were high - mortality tables did not include an explicit allowance for future mortality improvements (longevity) Ever since, however, long-term interest rates declined and mortality improved significantly for lives on which these policies were sold... To summarize, the guarantee corresponds to a put option on interest rates: - when interest rates rise, the annuity amount per 1000 fund value increases - when interest rates fall, the annuity amount per 1000 fund value decreases Guaranteed Annuity Options 4
Effect of Mortality Improvement The price of the guarantee depends on the mortality: - 13-year annuity certain corresponds to an interest rate of 5.7% since 1000 = 111 1 + 0.057 + 111 (1 + 0.057) 2 + + 111 (1 + 0.057) 13-16-year annuity-certain requires an interest rate of 7.72%: 1000 = 111 1 + 0.0772 + 111 (1 + 0.0772) 2 + + 111 (1 + 0.0772) 16 when mortality improves, the interest rate at which the guarantee becomes effective increases Guaranteed Annuity Options 5
Setting the Scene S = {S(t) : 0 t T } market value of the accumulated funds B = {B(t) : 0 t T } with B(t) = exp{ t 0 r u du} money market account and {r t : 0 t T } instantaneous short rate process P (t, T ) time-t price of a zero-coupon bond with maturity T, t T. P (t, T ) = E Q [B(t)/B(T ) F t ] Hence, a x (T ) : market value of an immediate annuity of 1 p.a. (payable in arrear) to a life aged x at T : a x (T ) = kp x P (T, T + k), (1) k=1 where k p x denotes the conditional probability that a person having attained age x will survive k years (k-year survival probability) Guaranteed Annuity Options 6
The Nature of GAO g : Conversion rate. Determines the guaranteed annuity payment per annum, e.g. if S(T ) = 1000, then g = 9 implies an annuity payment of 1000/9 ( 111) p.a. Y (T ) : payoff from exercising the option at maturity T (= value of the guarantee at maturity T ): S(T ) 0, if g S(T ) a 65 (T ) Y (T ) = ( ) S(T ) S(T ) g a 65 (T ) S(T ), if g > S(T ) a 65 (T ). Conditional on the survival of the policyholder up to time T, one thus has: Y (T ) = S(T ) ( a65 (T ) g 1) +. (2) Guaranteed Annuity Options 7
Valuing the GAO Assumptions: single-premium payments, no expenses financial risk is independent from biometric risk in a first step, it is even assumed that S and {r t : 0 t T } are independent there exists an equivalent martingale measure Q, i.e. an arbitrage-free economy (discounted market prices of tradable securities are Q-martingales) policyholders behave rationally, i.e. policyholders select the highest annuity payout Guaranteed Annuity Options 8
Change of Measure Technique For valuing the GAO, a switch from the spot martingale measure Q to the T -forward measure Q T turns out to be appropriate. Let X be a contingent claim that settles at T. Then Spot martingale measure T -forward measure Notation Numéraire Pricing formula State price density Q B(t) π t (X) = B(t) E Q [ X B(T ) ] Ft Q T P (t, T ) π t (X) = P (t, T ) E QT [X F t ] dq dp dq T dq = E[ζ F t ] Ft Ft = P (t,t ) P (0,T )B(t) Guaranteed Annuity Options 9
Main Steps (1/2) Let X = Y (T ) = ( ) S(T ) (a x (T ) g) + g It follows from the above assumptions that the time-t value of the GAO is given by: π t (X) = P (t, T ) E QT [ S(T ) Ft ] EQT [ (ax (T ) g ) + F t ] T t p x (T t) g By the martingale condition, we have that P (t, T )E QT [S(T ) F t ] = S(t). Hence, π t (X) = S(t) E QT [ (ax (T ) g ) + Ft ] T t p x (T t) g (3) Guaranteed Annuity Options 10
Main Steps (2/2) Using the definition of a x (T ) from (1), one obtains π t (X) = S(t) E QT [ ( J k=1 ) + ] kp x P (T, T + k) g F t T tp x (T t) g (4) The expression inside the expectation in (4) corresponds to a call option on a coupon paying bond. The coupon payments at the time instants T + k are k p x. Jamshidian [3] showed that if the short rate follows a one-factor process, then the option price on a coupon paying bond equals the price of a portfolio of options on the individual zero-coupon bonds (Brigo and Mercurio [2], p. 68): CBO(t, T, τ, c, K) = n c i ZBO(t, T, T i, K i ) (5) Guaranteed Annuity Options 11 i=1
Applying the Hull-White Short Rate Model We now specify the term structure of interest rates via the following short rate dynamics (Hull-White one-factor model): dr t = κ ( θ(t) r t ) dt + σ dw (t) (t 0) In the Hull-White context, bond options can be calculated explicitly, see for instance Brigo and Mercurio [2], p. 65. Example: European call option with strike K, maturity S written on a zero-bond maturing at time T > S: ZBC(t, S, T, K) = P (t, T ) Φ(h) K P (t, S) Φ(h σ p ) (6) Guaranteed Annuity Options 12
with σ p = σ κ 1 exp{ 2κ(S t)} 2κ ( 1 e κ(s t)), h = 1 ( P (t, T ) ) log + σ p σ p P (t, S)K 2 Combining (4), (5) and (6), one obtains an explicit formula for the price of the GAO: π t (X) = S(t) J k=1 kp x ZBO(t, T, T + k, K k ) P (t, T ) T tp x (T t) g (7) Guaranteed Annuity Options 13
Discussion the market value for the GAO is proportional to S(t) the simple pricing formula (7) relies on strong assumptions such as - equity returns are independent of interest rates - term structure of interest rates is given by a one-factor Gaussian short rate model Formula (7) can be generalized by assuming that S(T ) and P (T, T + k) are jointly lognormally distributed. In that case, a closed-form solution for the value of the GAO can still be derived. Guaranteed Annuity Options 14
The Need for Multi-Factor Models Example: Hull-White two-factor model: dr t = κ ( θ(t) + u(t) r t ) dt + σ1 dw 1 (t) du(t) = b u(t) dt + σ 2 dw 2 (t) with u(0) = 0 and dw 1 (t)dw 2 (t) = ρdt. Note: the value of a swaption depends on the joint distribution of the forward rates (F (t; T 0, T 1 ), F (t; T 1, T 2 ),..., F (t; T n 1, T n )). The payoff can thus not be additively separated as in the case of e.g. a cap Later we will show that the payoff of GAOs can be (statically) replicated by a portfolio of receiver swaptions Guaranteed Annuity Options 15
Correlation among the forward rates has an impact on the contract value Multi-factor models allow for more general correlation patterns than one-factor models Thus: simple one-factor models usually give reasonable prices for instruments, but good hedging schemes will assume many factors Guaranteed Annuity Options 16
Interest Rate Swaps (IRS) Definition (Interest rate swap): contract that exchanges fixed payments for floating payments, starting at a future time instant Tenor structure: reset dates: T α, T α+1,..., T β 1 payment dates: T α+1,..., T β 1, T β fixed-leg payments: N τ i K (N: notional amount, τ i : year-fraction from T i 1 to T i ) floating-leg payments: N τ i F (T i 1 ; T i 1, T i ) Guaranteed Annuity Options 17
Value of a (payer) IRS [ β ( V (t) = E Q P (t, T i ) τ i F (Ti 1, T i ) K ) ] F t =... i=α+1 = β i=α+1 ( ) P (t, T i 1 ) (1 + τ i K)P (t, T i ) Forward swap rate: value of the fixed-leg rate K that makes the present value of the contract equal to zero: S α,β (t) = P (t, T α) P (t, T β ) β i=α+1 τ i P (t, T i ) Guaranteed Annuity Options 18
Swap options, Swaptions Definition: A European payer swap option where the holder has the right to pay fixed and receive floating, is an option on the swap rate S α,β (t). A European receiver swap option where the holder has the right to pay floating and receive fix, is an option on the swap rate S α,β (t). The swaption maturity often coincides with the first reset date of the underlying IRS Example: receiver swaption provides payments of the form ( K S α,β (T α ) ) +. If K = 7% and S α,β (T α ) = 5%, it is optimal to exercise the option and receive fixed payments of S α,β (T α ) + (K S α,β (T α )) + = K By entering a receiver swaption, the holder protects itself against the risk that interest rates will have fallen when the swaption matures. Guaranteed Annuity Options 19
IRS and Swaptions in a Nutshell Type Discounted payoff at T α Price (time-t value) Payer IRS Payer Swaption Receiver IRS Receiver Swaption β i=α+1 ( β i=α+1 β i=α+1 ( β i=α+1 P (T α, T i ) τ i ( F (Ti 1, T i ) K ) = ( Sα,β (T α ) K ) A α,β (T α ) P (T α, T i ) τ i ( F (Ti 1, T i ) K )) + = ( Sα,β (T α ) K ) + Aα,β (T α ) P (T α, T i ) τ i ( K F (Ti 1, T i ) ) = ( K Sα,β (T α ) ) A α,β (T α ) P (T α, T i ) τ i ( K F (Ti 1, T i ) )) + = ( K Sα,β (T α ) ) + Aα,β (T α ) β i=α+1 ( ) P (t, T i 1 ) (1 + τ i K)P (t, T i ) A α,β (t) E QA [ (Sα,β (T α ) K ) + Ft ] β i=α+1 ( ) P (t, T i 1 ) + (1 + τ i K)P (t, T i ) A α,β (t) E QA [ (K Sα,β (T α ) ) + F t ] Guaranteed Annuity Options 20
Hedging the Interest Rate Risk of a GAO The quantity A α,β (t) is given by β i=α+1 τ i P (t, T i ) and defines the change of measure from the spot martingale measure Q to the measure Q A : dq A dq = A α,β(t β )/A α,β (0) B(T β )/B(0) = A α,β(t β ) A α,β (0) B(T β ). Recall that the GAO gives the right to obtain a series of cash payments n p x g at different dates T 1, T 2,.... Hence, the interest rate exposure in a GAO is similar to that under a swaption. Pelsser [4] advocates the usage of long-dated receiver swaptions for dealing with the interest rate risk under a GAO (static replicating portfolio approach) Price of the GAO value of a portfolio of long-dated receiver swaptions Guaranteed Annuity Options 21
Static Replicating Portfolio Recall that in the case of GAOs the expresssion ( a(t ) g 1) + = ( J k=1 kp x g P (T, T + k) 1 ) + gives the right to receive a series of cash payments ( k p x /g) for a price of 1 Cash flows from GAO are gradually decreasing over time (due to the decreasing survival probabilities), whereas cash flows associated with an N-year swap are constant over time Idea: Combine positions in receiver swap contracts all starting at time T, but with different maturities T + k Guaranteed Annuity Options 22
Construction of the Hedge Portfolio Aim: determine the amount to be invested in each swap Let ω be the limiting age of the mortality table (e.g. ω = 120) At time T + (ω x): - cash flow to be replicated: ( ω x p x /g) - cash flow of a swap with fixed leg K ω x and length ω x : 1 + K ω x - amount H ω x to be invested at time t: H ω x := ω xp x g(1 + K ω x ) (8) Guaranteed Annuity Options 23
Note: equation (8) can be rewritten as H ω x K ω x = ω x p x g H ω x (9) At time T + (ω x) 1: - cash flow to be replicated: ( ω x 1 p x /g) - cash flow from swap with fixed leg K ω x and length ω x : K ω x - cash flow from swap with fixed leg K ω x 1 and length ω x 1 : 1+K ω x 1 - amount H ω x 1 to be invested at time t: H ω x 1 := ( ω x 1p x ω x p x ) /g + Hω x 1 + K ω x 1 (10) Observe that H ω x K ω x + H ω x 1 (1 + K ω x 1 ) = ω x 1 p x /g Guaranteed Annuity Options 24
This yields a recursive relation for the amounts to be invested in swaps with tenor length n: ( ) np x n+1 p x /g + Hn+1 H n = 1 + K n Guaranteed Annuity Options 25
Price of the GAO With the portfolio of swaps ω x n=1 H n V swap (T, K n ) all cash flows of the GAO can be replicated Value of the GAO: ( ω x n=1 H n V swap (T, K n )) + ω x n=1 H n (V swap (T, K n )) + = ω x n=1 H n V swapt (T, K n ) Guaranteed Annuity Options 26
Discussion of the Static Replicating Portfolio Approach Pros and cons: + no need for dynamic hedging (no further buying and selling until maturity) + based on the right type of interest rate options + swap market is more liquid than bond market + cheaper and better protection than reserving (reserving at 99%-level may be insufficient) hedge against the interest rate risk only (hedging mortality risk by selling more life insurance?) in a period of rising stock returns, insurer must keep purchasing swaptions Guaranteed Annuity Options 27
B. Valuation and Risk Measurement Pricing Derivative Securities Consider an economy of d + 1 assets (S 0, S 1,..., S d ) Trading strategy: H = (H 0, H 1,..., H d ) with H i (t) denoting the number of units held of the ith asset at time t Value of the portfolio at time t: V (t; H) = d H i (t) S i (t) i=0 Guaranteed Annuity Options 28
The strategy H is self-financing if V (t) V (0) = d t i=0 0 H i (u) ds i (u) When pricing a derivative, the drift parameters µ i in the dynamics of S i do not appear: one does not need to know anything about an investor s attitude towards risk Rationale: risk preferences are irrelevant because contingent claims can be replicated by trading in the underlying assets Price of the derivative: minimal investment to implement the trading strategy Guaranteed Annuity Options 29
The Role of the Measures P and Q Real-world measure P asset returns vary by asset class the measure P describes the empirical dynamics of asset prices Risk-neutral measure Q the rate of return on any risky asset is the same as the risk-free rate in E Q [ ]-expectation, the risky assets behave like the money market account Guaranteed Annuity Options 30
Risk measurement poses the question: Risk Measurement How does the portfolio value V change in response to changes in the underlying risk factors? Z = (Z 1, Z 2,..., Z m ) vector of risk factors Z = Z(t + t) Z(t) : change in Z over t Portfolio loss: L = V (t; Z) V (t + t; Z(t) + Z) Guaranteed Annuity Options 31
Calculation of the loss distribution function F L If we were to determine the loss distribution function F L by means of Monte Carlo simulation, we would have to proceed as follows: For each of n replications, (i) generate a scenario under P, i.e. a vector of risk factor changes Z i (ii) re-value the portfolio V (t + t; Z(t) + Z i ) at time t + t under Q, given the outcome of Z i (iii) compute the loss L i = V (t; Z) V (t + t; Z(t) + Z i ) Estimate P[L x] using 1 n n i=1 1 {Li x} Guaranteed Annuity Options 32
Estimation of a conditional expectation The bottleneck in the above recipe is the portfolio revaluation of step (ii). This means computing a conditional expectation (or an estimate thereof) Guaranteed Annuity Options 33
References [1] Boyle, P. and Hardy, M. (2003). Guaranteed Annuity Options. ASTIN Bulletin, Vol. 33, No. 2, 125-152. [2] Brigo, D. and Mercurio, F. (2001). Interest Rate Models. Theory and Practice. Springer, Berlin. [3] Jamshidian, F. (1989). An exact bond option formula. Journal of Finance, 44, 205-209. [4] Pelsser, A. (2003). Pricing and Hedging Guaranteed Annuity Options via Static Option Replication. Insurance: Mathematics and Economics. Vol. 33, 283-296. [5] Wilkie, A. D., Waters, H. R., and Yang, S. (2003). Reserving, Pricing and Hedging for Policies with Guaranteed Annuity Options. British Actuarial Journal, Vol. 9, No II, 263-425. Guaranteed Annuity Options 34