tailieunhanh - The annuity puzzle economic of annuities_5

Tham khảo tài liệu 'the annuity puzzle economic of annuities_5', khoa học xã hội, kinh tế chính trị phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | CHAPTER 3 Survival Functions Stochastic Dominance and Changes in Longevity 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 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 . Life expectancy denoted z is defined by z zf z dz. J0 Integrating by parts z F z dz. J0 For survival function 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 . 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 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 Consider two survival functions Ft z i 1 2. Definition Single crossing or stochastic dominance . .