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Library Introduction to The Economic Theory_6

Đối với một cái nhìn tổng quan về công thức bảo hiểm nhân thọ, Baldwin (2002). Một cuốn sách hữu ích với các derivations toán học chặt chẽ là Gerber (1990). Milevsky (2006) gần đây cuốn sách có chứa nhiều công thức tính toán bảo hiểm hữu ích cho chức năng tử vong cụ thể | CHAPTER 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 TO. 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 e az e aT F z 0 z T 3.1 1 e aT where a 0 is a constant. In the limiting case when T TO this is the well-known exponential function F z eaz see figure 3.1 . Life expectancy denoted z is defined by z zf z dz. J0 Integrating by parts z F z dz. 3.2 J0 For survival function 3.1 z 1 a T eaT 1 . Hence when T TO z 1 a. To obtain some notion about parameter values if life expectancy is 85 then a .012. With this a 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 a 1 ea z T which for any finite T increases with z. When T K the hazard rate is constant equal to a. 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 Ft z i 1 2. Definition Single crossing or stochastic dominance . .