Question

A company that sells annuities must base the annual payout on the probability distribution of the length of life of the participants in the plan. Suppose the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and a standard deviation of 3.5 years (Show your workings clearly).

a) What proportion of the plan recipients would receive payments beyond age 75?

b) What proportion of the plan recipients die before they reach the standard retirement age of 65?

c) Find the age at which payments have ceased for approximately 86% of the plan participants.

Answer 1

Explanation:

Let X be length of life of the participants in the plan.

Given that X is N(68,3.5)

We convert this to standard normal score z using

`z=(x-68)/(3.5)`

a) proportion of the plan recipients that would receive payments beyond age 75=
`P(X\geq 75) = P(Z\geq 2)\\= 0.025`

b) proportion of the plan recipients die before they reach the standard retirement age of 65=
`P(X\leq 65) = P(z\leq -0.86)\\=0.5-0.2764\\=0.2236`

c) x for 86% ceased

`P(Z<c) = 0.14\\c=-1.08\\X=68-1.08*3.5\\x=64.22`