Sample chapter for the economic theory of annuities_6

C H A P T E R 3 Survival Functions, Stochastic Dominance, and Changes in Longevity 3.1 Survival Functions As in chapter 2, age is taken to be a continuous variable, denoted z, whose range is from 0 to maximum lifetime, denoted T. Formally, it is possible to allow T = ∞. When considering individual decisions, age 0 should be interpreted as the earliest age at which decisions are undertaken. Uncertainty about longevity, that is, the age of death, is represented by a survival distribution function, F(z), which is the probability of survival to age z. The function F(z) satisfies F(0) = 1, F(T) = 0, and F(z) strictly decreases in z. We shall assume that F(z) is differentiable and hence that the probability of death at age z, which is the density function of 1−F(z), exists for all z, f(z) = −dF(z)/dz > 0, 0 ≤ z ≤ T. A commonly used survival function is −αz −αT F(z) = 1 −e−αT ,0 ≤ z ≤ T, (3.1) where α > 0 is a constant. In the limiting case, when T = ∞, this is the well-known exponential function F(z) = e−αz (see figure 3.1). Life expectancy, denoted z, is defined by Z T z = zf(z)dz. 0 Integrating by parts, Z T z = F(z)dz. (3.2) 0 For survival function (3.1), z = (1/α) − (T/(eαT −1)). Hence, when T = ∞, z = 1/α. To obtain some notion about parameter values, if life expectancy is 85, then α = .012. With this α, the probability of survival to age 100 is e−1.2 = .031, somewhat higher than the current fraction of surviving 100-year-olds in developed countries. 16 • Chapter 3 Figure 3.1. Survival functions. The conditional probability of dying at age z, f(z)/F(z), is termed the hazard rate of survival function F(z). For function (3.1), for example, the hazard rate is equal to α/(1 −eα(z−T)), which for any finite T increases with z. When T = ∞, the hazard rate is constant, equal to α. It will be useful to formalize the notion that one survival function has a “shorter life span” or “is more risky” than another. The following is a direct application of the theory of stochastic dominance in investment decisions.1 Consider two survival functions, Fi(z), i = 1,2. Definition (Single crossing or stochastic dominance). The function F1(z) is said to (strictly) stochastically dominate F2(z) if the hazard rates satisfy f2(z) f1(z) F2(z) F1(z) 0 ≤ z ≤ T. (3.3) In words, the rate of decrease of survival probabilities, dln F(z) f(z) dz F(z) 1 See, for example, Levy (1998) and the references therein. Survival Functions • 17 Figure 3.2. F1(z) stochastically dominates F2(z). is smaller at all ages with survival function 1 than with survival function 2. Two implications of this definition are important. First, consider the functions Fi(z) Fi(z) zi 0 Fi(z)dz 0 ≤ z ≤ T,i = 1,2. Being positive and with their integral over (0,T) equal to 1, they must intersect (cross) at least once over this range. At any such crossing, when F1(z) F2(z) 0 F1(z)dz 0 F2(z)dz condition (3.3) implies that ! d F1(z) d dz 0 F1(z)dz dz ! F2(z) 0 F2(z)dz Hence, there can only be a single crossing. That is, there exists an age zc, 0 < zc < T, such that (figure 3.2) F1(z) F2(z) 0 F1(z)dz 0 F2(z)dz as z S zc. (3.4) 18 • Chapter 3 Intuitively, (3.4) means that the dominant (dominated) distribution has higher (lower) survival rates, relative to life expectancy, at older (younger) ages. Second, since Fi(0) = 1, i = 1,2, it follows from (3.4) that Z T Z T z1 = F1(z)dz > F2(z)dz = z2; (3.5) 0 0 that is, stochastic dominance implies higher life expectancy. 3.2 Changes in Longevity It will be useful in later chapters to study the effects of changes in longevity. Thus, suppose that survival functions are a function of age and, in addition, a parameter, denoted α, that represents longevity, F(z,α). We take an increase in α to (weakly) decrease (in analogy to the function e−αz) survival probabilities at all ages: ∂F(z,α)/∂α ≤ 0 (with strict inequality for some z) for all 0 ≤ z ≤ T. How does a change in α affect the hazard rate? Using the previous definitions, ∂ f(z,α) ∂µ(z,α) ∂α F(z,α) ∂z where 1 ∂F(z,α) F(z,α) ∂α is the relative change in F(z,α) due to a small change in α. It is seen that a decrease in α (increasing survival rates) reduces the hazard rate when it has a proportionately larger effect on survival probabilities at older ages, and vice versa2 (figure 3.3). This observation will be important when we discuss the effects of changes in longevity on individuals’ behavior. A special case of a change in longevity is when lifetime is finite and known with certainty. Thus, let ( F(z,α) = 1, 0 ≤ z ≤ T, (3.7) 2 A sufficient condition for (3.6) to be positive is that ∂2F(z,α) < 0. For F(z,α) = e−αz, ∂ ∂α(∂z ) R 0 as αz R 1. However, ∂α F(z,α) = 1 for all z. Survival Functions • 19 Figure 3.3. An increase in longevity reduces the hazard rate. where T = T(α) depends negatively on α. Survival is certain until age T. An increase in longevity means in this case simply a lengthening of lifetime, T. The condition in figure 3.3 is satisfied in a discontinuous form: ∂F(z,α)/∂α = 0 for 0 ≤ z < T and ∂F(T,α)/∂α < 0. Function (3.1) has two parameters, α and T, that affect longevity in different ways: −αz −αT F(z,α,T) = 1 −e−αT . We can examine separately the effects of a change in α and a change in T (figure 3.4): ∂F(z,α,T) 1 −e−αz T z ∂α 1 −e−αT eαT −1 eαz −1 0 < z < T = 0, z = 0, T, (3.8) and ∂F(z,α,T) α 1 −e−αz ∂T eαT −1 1 −e−αT 0 < z ≤ T = 0, z = 0. (3.9) ... - tailieumienphi.vn