# under Stochastic Interest Rates Dipartimento di Matematica Applicata alle Scienze University of Trieste Piazzale Europa 1, I Trieste, Italy

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5 Thiele's dierential equation of the actuarial sciences and the famous Black and Scholes equation is developed. 3 The valuation framework The rst two subsections contain a brief overview over the nancial set-up. Details can be looked up in any advanced nancial textbooks, such as Due (1996). The last subsection introduces the insurance factors. 3.1 The nancial assets A time horizon T is xed and the nancial uncertainty is generated by a 2- dimensional standard Brownian motion (W 1 ;W 2 ) dened on a probability space (; F;Q) together with the ltration (F t ; 0 t T ), satisfying the usual conditions and representing the revelation of information. In particular, Q represents the equivalent martingale measure. All trade is assumed to take place in a frictionless market (no transaction costs or taxes, and short-sale allowed). A unit discount bond is a default-free nancial asset that entitles its owner to one unit of account at maturity without any intermediate coupon payments. We denote by B t (s) the market price at time t for a bond maturing at a xed date s t. By denition B s (s) =1. We assume there is a continuum of such bonds maturing at all times s; 0 s T. Furthermore, we assume that a mutual fund is traded and that its market price per unit S t is given by the following stochastic dierential equation under the equivalent martingale measure: ds t = r t S t dt + 1 S t dw 1 t + 2 S t dw 2 t ; (1) where 1 and 2 are constants and the initial value of the process S p 0 is given. Here r t represents the short term interest rate in the economy and may beinterpreted as the instantaneous standard deviation of the rate of return on the mutual fund. As will soon be apparent, the Brownian motion Wt 2 is used to model mutual fund specic risk. 3.2 The Gaussian HJM model The primitives of the HJM-model are the volatility structure and the initial instantaneous forward rates. The volatility structure is given by the function t (u); for t u. We assume it is deterministic, i.e., t (u) is a deterministic function of u and t. Then we denote by f t (u);t u T, the instantaneous forward rates prevailing at time t 0. From the general relationship between the instantaneous forward rates and the bond price, 5

6 B t (s) = exp, Z s t f t (u)du ; and since f 0 (u) for 0 u T is given, the bond prices at time zero are known as well. All relevant quantities are determined by the volatility structure and the initial instantaneous forward rates. For example, the short term interest rate under the equivalent martingale measure is given by r t = f 0 (t)+ Z t 0 v (t) Z t v v (s)dsdv + Z t 0 v (t)dw 1 v : The assumption of deterministic volatility structure implies that r t is Gaussian, hence negative values of r t have positive probability. This fact is a theoretical drawback of Gaussian term structure models, but does not seem to present a problem for reasonable parameter values. The market price of the bond satises the following stochastic dierential equation under the equivalent martingale measure: where db t (s) =r t B t (s)dt + a(t; s)b t (s)dw 1 t ; a(t; s) =, Z s t t (u)du; and the initial value B 0 (t) is determined by the initial instantaneous forward rates. The quantity v(t) = exp, Z t 0 r u du is sometimes called the discount function and represents the stochastic present value at time zero of one unit of account at time t. In this model market prices of nancial assets may be calculated as expectations of discounted cashows under the equivalent martingale measure. In particular, the market price at time t of a unit discount bond expiring at time s may be calculated as v(s) B t (s) =E Q v(t) jf t ; 6

7 where E Q [jf t ] denotes the conditional expectation under the probability measure Q. As a second example consider a European call option on one unit of the mutual fund. The payo of this option is max[s t, G; 0], where the constant G represents the exercise price. Denote the market price at time zero of the described option with expiration at time t by t (G). Given our model of the nancial market it follows that t (G) =S 0 (d 1 t(g)), GB 0 (t)(d 2 t(g)); (2) where d 1 t(g) = 1 1 t 2 2 t +ln S0 ; B 0 (t)g d 2 t(g) =d 1 t(g), t ; t = s Z t 0 a(s; t) 2 ds +( )t, 2 1 Z t 0 a(s; t)ds; and () denotes the cumulative standard normal distribution function. See Amin and Jarrow (1992). The volatility parameter t depends on the volatility structure (a(s; t)) as well as the volatility parameters of the mutual fund ( 1 and 2 ), in addition to time to expiration (t). 3.3 Insurance factors Let C(t) denote an arbitrary insurance benet payable at time t, possibly dependent on the market value at time t of the mutual fund (formally, C(t) is adapted to the ltration (F t ; 0 t T )). Let the random variable T x, dened on another probability space (^; ^F;P), denote the remaining life time of an x-year old person. We assume that the probability density function for T x exists and denote it by f x (). Let t p x = P (T x >t) denote the survival probability ofanx-year old policy buyer. By construction T x is independent ofwt 1 and Wt 2, hence it is independent of all processes reecting nancial quantities. Finally, as indicated and explained in the introduction, we assume risk neutrality with respect to mortality. 4 Single premiums of insurance contracts In this section we derive single premiums at time 0, for a life aged x, of some life insurance contracts of the pure endowment and term insurance types. 7

8 4.1 Pure endowment and term insurance types of contracts From the assumed risk neutrality with respect to mortality and independence between mortality and nancial risk, it follows that the market price at time 0 of a pure endowment insurance contract with benet C(T )payable at time T if the insured is alive is = T p x E Q [v(t )C(T )] : (3) Similarly, the market price at time 0 of a term insurance with benet C(t) payable upon death at time t T is 1 = Z T 0 E Q [v(t)c(t)] f x (t)dt: (4) In the remainder of this section we consider three dierent kinds of benets: C (1) (t) = 1, C (2) (t) = max[s t ;G t ], and C (3) (t) = max[min[s t ;K t ];G t ]. In the rst example the benet is deterministic, as in traditional life insurance contracts. This example is included to isolate the eect of the stochastic interest rate. In the second example the benet is the maximum of the value of one unit of the fund and a guaranteed amount G t. In principle the amount the insurance company is obliged to pay under this contract has no upper bound. Therefore in the last contract a maximum amount, a cap, is included. 4.2 Deterministic benets We will calculate the single premiums of the policies at time zero. First we turn to the rst example and calculate the market premiums of the pure endowment insurance and the term insurance. From the formulas of the previous subsection we obtain and (1) = T p x B 0 (T ) Z T 1 = (1) B 0 (t)f x (t)dt: 0 For the pure endowment insurance the single premium is the market price at time zero of one unit of account payable at time T multiplied by the probability of payment. For the term insurance B 0 (t)f x (t)dt can, similarly, beinterpreted as the market value at time zero of the expected payo in the time interval (t; t + dt). The single premium is then the sum of these expected payos over the whole term of the contract. 8

9 These formulas resemble the corresponding classical formulas. However, note carefully an important dierence: The usual present values of future payos are replaced with market values of future payos. Persson (1998) derives similar results for traditional life insurance based on the Vasicek (1977) model of the term structure. 4.3 Equity-Linked Policies with Guarantees For the second example let U 1 (t) =E Q [v(t) max[s t ;G t ]]: Here U 1 (t) can be interpreted as the market value at time zero of a benet which expires at time t with probability 1. We observe that the payo max[s t ;G t ] resembles the structure of the benet in the BS-model. The calculation of the above expectation is presented in the following proposition. Proposition 1 The market value at time zero of the benet max[s t ;G t ] payable at time t is U 1 (t) =S 0 (d 1 t(g t )) + G t B 0 (t)(,d 2 t (G t )): Proof 1 From equation (2) we know the time zero value of the claim max[s t, G t ; 0]. Observe that max[s t ;G t ] = max[s t, G t ; 0] + G t. The market price at time zero of the last term is B 0 (t)g t. The formula of Proposition 1 is then the sum of the two time zero market values. The resulting formula depends on the initial forward rates (B 0 (t)) and ve parameters: the parameters of the mutual fund price process (S 0, 1, 2 ), the guarantee (G t ), and time to expiration (t). Incorporating the insurance aspects, the single premiums of the two contracts can now be expressed by exploiting relations (3) and (4) as and (2) = T p x [S 0 (d 1 T (G T )) + G T B 0 (T )(,d 2 T (G T ))] Z T 1 = (2) [S 0 (d 1 (G t t)) + G t B 0 (t)(,d 2 (G t t))]f x (t)dt: 0 For constant interest rate these formulas reduce to the results in Theorem 1 and 2 of Aase and Persson (1994). 9

11 of the premium, denoted by d t, deemed to be invested in the mutual fund. Without guarantees, the number of units acquired at time t should therefore be equal to d t =S t, but at this pointweintroduce the minimum guarantee provision, expressed by a minimum number of units guaranteed at time t. Let g t represent this guarantee, and n t denote the actual number of units deemed to be invested in the mutual fund at time t. Thus, n t = max g t ; d t ;t=0; 1; :::; T, 1: S t The market value at time t of the periodical premium P t must be equal to the value of n t units at time t, i.e., P t = n t S t = d t + g t max[s t, k t ; 0]; (5) with k t = d t =g t. The time t payo of the minimum guarantee provision, P t,d t, is then equal to the payo of g t call options on (units of) the mutual fund with exercise price k t and maturity t. Observe that the amount of periodical premium depends on the time t value of the mutual fund and, thus, is stochastic. 5.2 The benet If death occurs at time between t and t + 1, with t =0; 1; :::; T, 1, the benet C() is simply the market value at time of the accumulated investments in the mutual fund, i.e., C() =S tx j=0 whereas the benet at maturity T, due if the insured is alive, is T X,1 C(T )=S T This contract thus merely represents a way of saving, and does not include any additional coverage against unfavorable events such as death or disability. j=0 5.3 Constant periodical premium The life insurance policy just described is a pure nancial instrument, in which the mortality risk is completely absent from the insurance company's point of view. The mortality risk, indeed, determines only the time to expiration of the policy. n j ; n j : 11

15 death. The relations of subsection 5.2 are modied in the following way: t,1 X C(t) = max gs t ; d S t ;t=1;:::;t: S j j=0 Observe that 2 C(t) max4 gtst ; Xt,1 j=0 3 d S t 5 ; S j so that there is an implicit minimum benet guaranteed at time t, given by the market value of g t units of the mutual fund. We recall that in the BS-contract the benet, that we denote by C (t), is instead given by 2 C (t) = max4 Gt ; t,1 X j=0 3 d S t 5 ; S j where G t represents the minimum amount guaranteed at time t, expressed in the usual unit of account. It is also interesting to compare the periodical premium for the minimum guarantee provision in both models. As already said, in our model this market price is proportional to the time 0 value of a portfolio of European call options on one unit of the mutual fund. We recall that for the BS-contract the periodical premium, denoted by P, is instead given by where h PT E Q P t=1 tv(t) max = d + t = ( h G t, P t,1 j=0 d S t=s j ; 0 P T,1 t=0 B 0(t) t p x ; (7) t,1p x (1, 1 p x+t,1 ) ; t =1; :::; T, 1 T,1p x ; t = T represents the probability that the policy expires at time t. The periodical premium for the guarantee, P, d, is then proportional to the value at time 0 of a portfolio of European put options on the accumulated investments in the mutual fund, each one with maturity t and exercise price G t, see Bacinello and Ortu (1994). We observe, however, that for the BS-contract the minimum guarantee G t can be xed in such away to supply the insured with an adequate coverage against early death. In our model, instead, the minimum guarantee could reach an adequate level only in the long run so that, even in the case of constant premiums, our policy may represent, from the insured's point of view, mainly an appealing way ofinvesting money, but not a suitable coverage in the case of death during the rst years of contract (since the mortality component has ii 15

16 the only function of levelling the premium). Anyway, the goal of getting, in the same time, an interesting nancial and insurance product can be easily attained if the insured buys, in addition to the policy here described, a standard term insurance contract. 5.6 Numerical results In this subsection we present some numerical results for the constant premiums P and P dened in expressions (6) and (7) respectively, all obtained under the assumption of a constant volatility structure, i.e. t (u) for all t; u. It is easy to check analytically the behavior of P with respect to the parameters on which it depends by the sign of its partial derivatives, all in closed form. In particular this premium is increasing with respect to the initial unit value of the mutual fund S 0, the minimum number of units guaranteed at each premium payment date g, the amount d deemed to be periodically invested in the fund, the instantaneous forward rates f 0 (t) prevailing at time 0, the volatility parameters ; 1 ; 2 (at least when they are positive), while it is decreasing with respect to the time 0 prices of unit discount bonds B 0 (t). It is not a priori clear the behavior of P with respect to the maturity T and to the survival probabilities t p x (or, alternatively, to the age x of the insured at the inception of the contract). To study this behavior, to get a numerical intuition for the price of the minimum guarantee provision in our contract, P, d, and to compare it with the corresponding price in the BS-model, P, d (not in closed form), some numerical examples are presented below. For comparison, we have xed the BS-parameter G t = gts 0 =B 0 (t). This quantity can be interpreted as the riskless return at time t of the amount gts 0 invested at time 0 in unit discount bonds with maturity t. If the same amount were invested in the mutual fund, its stochastic return at time t, gts t,would give exactly the implicit minimum guaranteed benet in our model, as shown in the previous subsection. To evaluate the expectation in expression (7) Monte Carlo simulations are employed. To this end we have simulated 1; 000; 000 trajectories for the standard Brownian motions W 1 t and W 2 t in the time interval (0;T] and used them for building corresponding trajectories of and r t = f 0 (t)+ 2 t 2 + W 1 t 2 ; v(t) = exp, Z t 0 r u du ; S t = S 0, v(t) exp t 2 ( )+ 2 1W 1 + t 2W 2 t : 16

17 As for the term structure of interest rates at time 0, in all our numerical examples we have set f 0 (t) =r 0 + qt (so that B 0 (t) = exp,,r 0 t, qt 2 =2 ), and although in most cases we have xed q =0(at term structure), we have also considered increasing (q >0) and decreasing (q <0) term structures. Moreover, we have xed d = g = S 0 = 1. Finally, wehave constructed the probabilities tp x and t from the Italian Statistics for Males Mortality in Table 1 reports some results obtained when the maturity T varies between 5 and 15 while the other parameters are xed. TABLE 1 x =40;f 0 (t) =r 0 =0:04;=0:06; 1 =0:03; 2 =0:2 T P P P, P From Table 1 one can see that both P and P are increasing with respect to the maturity T. The price for the minimum guarantee provision is never negligible, and for the BS-contract it is on average 514 basis points (bp) higher than for our contract. In Table 2 we show the behavior of the premiums with respect to an age of the insured at time 0 between 30 and

18 TABLE 2 T =10;f 0 (t) =r 0 =0:04;=0:06; 1 =0:03; 2 =0:2 x P P P, P From Table 2 one can observe that both P and P are decreasing with respect to x. However, the absolute dierences are small and we are tempted to conclude that the insurer's age has only an imperceptible inuence on the premiums. In this connection, we point out that also the use of dierent mortality tables proved to be almost irrelevant in the premium calculation. To interpret this fact recall that, at least in our model, the mortality component has the only function of levelling the premium. Also in these examples the premium P is higher than P, 586 bp on average. The results reported in Tables 3 to 7 show the behavior of P and P with respect to the initial term structure. More precisely, in Table 3 we consider the case of at term structures, and report the premiums corresponding to dierent values of the initial spot rate r 0.InTables 4 and 5 we consider linearly increasing term structures corresponding to two dierent slopes and to various levels of the initial spot rate, while in Tables 6 and 7 we show similar results obtained when the initial forward rates f 0 (t) linearly decrease with respect to their time to maturity t. 18

19 TABLE 3 x =40;T =10;f 0 (t) =r 0 ;=0:06; 1 =0:03; 2 =0:2 r 0 P P P, P TABLE 4 x =40;T =10;f 0 (t) =r 0 +0:001t; =0:06; 1 =0:03; 2 =0:2 r 0 P P P, P TABLE 5 x =40;T =10;f 0 (t) =r 0 +0:002t; =0:06; 1 =0:03; 2 =0:2 r 0 P P P, P

20 TABLE 6 x =40;T =10;f 0 (t) =r 0, 0:001t; =0:06; 1 =0:03; 2 =0:2 r 0 P P P, P TABLE 7 x =40;T =10;f 0 (t) =r 0, 0:002t; =0:06; 1 =0:03; 2 =0:2 r 0 P P P, P As expected, from Tables 3 to 7 one can notice that the initial spot rate r 0 has a strong inuence on the premiums, no matter if the term structure is at, increasing, or decreasing. Moreover, as analytically established under the assumption B 0 (t) = exp,,r 0 t, qt 2 =2, our premium is an increasing function both with respect to r 0 and with respect to q, and the same behavior can also be numerically veried for the corresponding premium for the BS-contract P, which isontheaverage 605 bp higher than P. Table 8 reports some results obtained when the parameter, characterizing the volatility structure in the Gaussian HJM model, varies between 0 and 0:2 with step 0:01. 20

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