11 Generalization of Redington's Theory of Immunization Elias 5. W. Shiu ABSTRACT In the classical model of immunization, there is no distinction between short term and long term interest rates, i.e., all yield curves are assumed to be flat. This is a violation of the no-arbitrage principle. Moreover, the interest rate shocks are assumed to be small. In this paper we generalize the theory to the case where the interest rate is a function of time and the shock is of arbitrary magnitude. 1. INTRODUCTION Consider a block of insurance business and its associated assets. For t ~ 0, let At denote the asset cash flow expected at time t , i.e. the investment income and capital maturities expected at that time. Let Lt denote the liability cash flow expected at time t , i.e. the policy claims plus policy surrenders plus expenses minus premium income expected at that time. Define the net cash flow at time t, t ~ 0, as Nt=At-Lt · The theory of immunization is concerned with two kinds of risk that may occur due to interest rate fluctuations: (I) positive net cash flows may have to be reinvested at lower interest rates, (2) negative net cash flows may involve the liquidations of assets at depreciated values because of then higher interest rates. -69- 2. REDINGTON'S THEORY OF IMM~IZATION Let 5(&) denote the surplus of the blodc of business evaluated at a given force of interest &. Thus 5(&) is the present value of the net cash flows, 5(&) = 2: Nt e-&t t~O If the force of interest changes from & to & + f, the surplus value changes from 5(&) to 5(& + f). What are the conditions on the cash flows such that 5(& + E) ~ 5(&) ? By Taylor's formula with remainder, . 5(& + £) = 5(&) + £5'(&) + £25"(& + ~)12 , (2.1) where ~ is some point between 0 and £. Hence, if 5'(&) =0 (2.2) and 5"(&) > 0 , (2.3) we have 5(& + £) > 5(&) provided that If I is sufficiently small. In this paper the cash flows are always assumed to be independent of interest rate fluctuations. Thus 5'(&) = - 2: t Nt e-&t t~O and 5"(&) = 2: t2 Nt e-&t . t~O Hence conditions (2.2) and (2.3) becmle 2: t Nt e-&t = 0 (2.'1) t~O 2: t2 Nt e-&t > 0 , (2.5) t~O respectively. In terms of the asset and liability cash flows, these two conditions can be formulated as -70- (2.6) L t 2At e-&t > L t2Lt e-&t . (2.7) t~O t~O 3. A GENERALIZATION OF REDINGTON'S THEORY It haS been pointed oot by FiSher and Wei! (3, p.417) that Redington's model shoold be generalized to the case where both the force of interest & and the shoc:Ic E are functions of time. Thus we consider S(&) = Nt exp(-J t &(s) dS) L o t~O 5(& + E) =L Nt exp( - Jot (&(s) + E(S» ds) . t~O In the fill2lnce literature, the force-of-interest function &0 is called the instllntllneous forWllrd-r8te function. Again, we want to know whether there are cooditlons on the net cash flows (Nt) such that S(& + E) ~ s(&) . Note that formula (2.1) is not applicable here, since the argument of S is a function, not a scalar. Moreover, we do not require the interest rate shock E(·) to be of small magnitude. Define rlt = Nt exp( - Jot &(s) ds) and f(t) =exp(-J t E(S) dS) . o Thus tnt) are the present values of the net cash flows evaluated with the force-of-Interest function &0. Clearly, S(& + E) - S(&) = L rlt(r(t) - J) . t~O By Taylor's formula with integral remainder, -71- f(t) = f(a) + u'(a) + f I (t - w) f"(w) dw o = I - t£(a) + I (t - w) f"(w) dw . f o Hence, 5(& + £) - 5(&) = -£(a)'2: tilt + 2: Ilt'f l (t - w)f"(w)dw . o t~a t~a \ Interchanging the order of summation and integration yields ~ Ilt'f l (t - w) f"(w) dw = fooo (~ nt (t - w)) f"(w) dw . o t~a t~w Now, suppose that the cash flows {Nt} satisfy either ~ Ilt (t - w) ~ a for all positive w (3.1) t~w or 2: Ilt (t - w) ~ a for all positive w ; (3.2) t~w then, by the weighted mean value theorem for integrals, there exists a positive number ~ such that fooo (~Ilt (t - w)) f"(w) dw = f"(~) fooo (~ Ilt (t - w))dw. t~w t~w Reversing the order of integration and summation, we have fooo (2: Ilt (t - w))dw = 2: Ilt <fot(t - w)dw) t~w t~a = 2: Ilt (t212) . (3.3) t~a Thus, subject to (3. I) or (3.2), 5(& + £) - 5(&) = -£(a)·~ tilt + 1/2'f"(~)'~ t21lt . t~a t~a This is an extension of equation (2.1). If, in addition to (3.1) or (3.2), we also assume that the first moment of the present values of the net cash flows is zero, i.e. 2: tilt =a , (3.4) t~a then 5(& + £) - 5(&) = 1/2'f"(~)'2: t21lt. (3.5) t~a -72- We rv:JW make three observations: (i) Equation (3.4) is a generalization of (2.4). (ii) Because of equation (3.3), the term L t 21\ (3.6) t~O is ranegative if (3.1) holdS anCI nonpositive if (3.2) holds. (iii) Since f·(s) = f(s)[£(s»)2 - £'(s») for all s • the sign of f·(~) is the same as that of (£(t»2 - £,(~) . Note that, if the interest rate shock function £(.) is constant as in Redington's maCIel, by (iii) f·(t) ~ 0 . (3.7) It follows from (3.5) that to secure S(& + £) ~ S(&) for all shocks £(.) satisfying (3.7), we should structure the cash flows (Nt) such that (3.4) anCI (3.1) hold. Hence we have a generalization of Redington's maCIel. Equation (3.5) implies that, subject to (3.4) anCI (3.1) or (3.2), the term (3.6) or its absolute value may be used as a measure of the interest rate risk of the cash flows. For the case where there is only one positive cash outflow, an expression similar to (3.6) has been derived by Fang anCI Vasicek (6) anCI is described as a measure of risk for an immunized portfolio. For a simple derivation of the Fang-Vasicek result, see [Ill. Note that, according to (3.5), if (3.8) we have 5(&+ £) = 5(&) (3.9) for each interest rate shock EO. However, if (3.4), (3.8) anCI (3'1) or (3.2) hold, then, for each t > 0, Nt =0 anCI thus we automatically have (3.9). -73- 1 Equation (3.5) suggests a strategy, conservative with respect to interest rate fluctuations, for structuring the cash flows of a block of business: Solve for the optimal (Nt} by minimizing expression (3.6) sUbject to (3. I), (3.'1) and a~ other necessary constraints or by maximizing (3.6) sub ject to (3.2), (3.4) and a~ other necessary constraints. (Since the cash flows (Nt} appear linearly in (3.6), (3.1), (3.2) and (3.4), we may use the method of linear programming.) However, we remarlc that, since the point ~ depends on the initial present values of the net cash flOws (nt} (and the interest rate shoclc EO), minimizing I ~ t2ntl t~O does not necessarily Imply that I f"(O ~ t2"t I t~O is minimized even for a fixed function f(')' 4. MEAN ABSOLUTE DEVIATION CONSTRAINT What is the meaning of conditions (3.1) and (3.2)7 To appreciate their meaning, we shall derive some equivalent conditions in this section and the next. Consider the function x. = maximum(O, x) . Then ~ nt (t - w) ~ nt (t - w). t~w t~O Since (t - w). =(I t - wi' t - w)12 , ~ nr(t - w). = (~nr I t - wi' ~ nrt - w·~"t )/2 . If we assume that (4.1 ) and (3.4) -74- then if and only if i.e., if and only if t t l: It - wi At exp(-Jo &(s)ds) ~ l:lt - wi Lt exp(-Jo &(S)dS). t~O t~O The last inequality has been called the mean absolute deviation (MAD) constraint by Fang and Vasicek (5, p.232J. 5. J. KARAMAT A'S THEOREM Consider the equality x+ =(-x)+ + X • Thus l: nr(t - w)+ = l: nr(w - t)+ + l: ot·t - w·l: nt If we again assume (4.1) and (3.4), then l: nt (t - w) ~ 0 t~w if and only if l: ot (w - t)+ ~ 0 . (5. I) t~O Another way to write (5.1) is l: ot (w - t) ~ 0 . (5.2) w>t~O Inequality (5.2) can be obtained from a classical result of J. Karamata (8) in convex analysis. Let me row explain. Define «P(t) = f(t) - I . -75- Then (5.3) S(c • f) - s(c) = l: nt CP(t) . t~O A natural question one might asle is whether there are theorems in mathematics that provide, for a given class of functions cP, sufficient conditions on tnt} such that (5.3) is nomegative. Motivated by (3.1), we shall only consider convex 'P's. Note that the functions we encountered above, 'P(t) = (t - w). , CP(t) = It - wi and CP(t) = (w - t)+ , are convex functions. The following is Karamata's Theorem [8; 10, p.