Guaranteed Annuity Options

Transcription

1 Guaranteed Annuity Options Hansjörg Furrer Market-consistent Actuarial Valuation ETH Zürich, Frühjahrssemester 2008 Guaranteed Annuity Options

2 Contents A. Guaranteed Annuity Options B. Valuation and Risk Measurement Guaranteed Annuity Options 1

3 Course material Slides A. Guaranteed Annuity Options Boyle, P. and Hardy, M. (2003). Guaranteed Annuity Options. ASTIN Bulletin, Vol. 33, No. 2, pp Pelsser, A. (2003). Pricing and Hedging Guaranteed Annuity Options via Static Option Replication. Insurance: Mathematics and Economics. Vol. 33, The above documents can be downloaded from Guaranteed Annuity Options 2

4 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

5 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

6 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 = ( ) ( ) year annuity-certain requires an interest rate of 7.72%: 1000 = ( ) ( ) 16 when mortality improves, the interest rate at which the guarantee becomes effective increases Guaranteed Annuity Options 5

7 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

8 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

9 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

10 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

11 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

12 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

13 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

14 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

15 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

16 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

17 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

18 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

19 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

20 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

21 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

22 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

23 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

24 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

25 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

26 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

27 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

28 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

29 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

30 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

31 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

32 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

33 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

34 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

35 References [1] Boyle, P. and Hardy, M. (2003). Guaranteed Annuity Options. ASTIN Bulletin, Vol. 33, No. 2, [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, [4] Pelsser, A. (2003). Pricing and Hedging Guaranteed Annuity Options via Static Option Replication. Insurance: Mathematics and Economics. Vol. 33, [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, Guaranteed Annuity Options 34