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.

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? 0.1957 using Excel or 0.1949 using Table E.2

c.Find the age at which payments have ceased for approximately 90% of the plan participants.

Answer 1

Given mean = 68 years

Standard deviation = 3.5 years

Question (a)

Proportion of the plan recipients that would receive payments beyond age 75 will be the ones that will be alive after 75 years

we need to calculate Z-score and then find the respective proportion value from it

Z-score = ( X - ) /

= (75 - 68) / 3.5

= 2

The corresponding value from the positive Z-table attached below is 0.97725. But in the Z-table it represents area to the left of Z-score. here we need the area to the right of it. (Since we are looking for recipients that would receive payments beyond 75)

which implies that proportion of plan recipients that would receive payments beyond age 75 = 1 - 0.97725

= 0.02275 = 2.275%

Question (b)

Proportion of the plan recipients that die before they reach the standard retirement age of 65

Here X = 65,   = 68, = 3,5

So Z-score = ( X - ) /

= (65 -68) / 3.5

= -0.8571

  Z-score = -0.86 (rounded to 2 decimals)

The corresponding value from the negative Z-table attached below is 0.19489. In the Z-table it represents area to the left of Z-score.

So Proportion of the plan recipients that die before they reach the standard retirement age of 65 = 0.19489 = 19.489%

Question (c)

The age at which payments will be ceased for approximately 90% of plan participants

So we would find out the age by which 90% of the plan recipients would die

So, in the positive Z-table attached below we would look for 0.9 value and find the corresponding Z-score

for Z-score of 1.28 the corresponding value is 0.89973 which is approximately 0.9 the value we are looking for

We know that

Z-score = ( X - ) /

1.28 = (X - 68) / 3.5

X - 68 = 1.28 * 3.5

X - 68 = 4.48

X = 72.48

So at the age of 72.48 years, payments for approximately 90% of the plan participants will cease