Calculate the following probabilities on the basis of English Life Table No. 12-Males.

(i) 40p25,

(ii) 60|10q25,

(iii) the probability that (30) survives for at least 10 years,

(iv) the probability that (40) survives to age 65,

(v) the probability that (50) dies within 10 years,

(vi) the probability that (50) fails to reach age 70,

(vii) the probability that (60) dies between ages 80 and 85,

(viii) the probability that (60) dies within the first five years after retiring at age 65.

In addition, express (i) and (ii) in words.

On the basis of E.L.T. No. 12 - Males, find the probability that a life aged 30 will

(i) survive to age 40,

(ii) die before reaching age 50,

(iii) die in his 50th year of age, i.e. between ages 49 and 50,

(iv) die between his 40th birthday and his 50th birthday,

(v) die either between exact ages 35 and 45 or between exact ages 70 and 80.

A man aged 50 has just retired because of ill health. Up to exact age 58 he will be subject to

a constant force of mortality of 0.019803 p.a., after which his mortality will be that of E.L.T.

No. 12 - Males. Find the probability that he will

(i) die before age 55,

(ii) live to age 65,

(iii) die between ages 55 and 60.

For the first 5 years after arrival in a certain country, lives are subject to a constant force of

mortality of 0.005. Thereafter lives are subject to mortality according to English Life Table

No. 12- Males with an addition of 0.039221 to the force of mortality.

(i) A life aged exactly 30 has just arrived in the country.

(a) Show that the probability that the life will survive to age 35 is 0.97531.

(b) Find the probability that the life will survive to age 60.

(ii) What is the probability that a life aged exactly 33 who has been in the country for 3

years will die between ages 50 and 51?

(Assume that these lives will remain in the given country.)

1.5 For a certain animal population,

lx =

l0

(1 + x)

2

(x ? 0)

Calculate

(i) the complete expectation of life at birth,

(ii) the force of mortality at age 1 year,

(iii) the chance that a newly-born animal will die between ages 1 and 2 years.

Let A be a random variable representing the present value of the benefits of a whole of life assurance, and Y be a random variable representing the present value of the benefits of a temporary assurance with a term of n years. Both assurances have a sum assured of 1 payable at the end of the year of death and were issued to the same life aged x. (1) Describe the benefits provided by the contract which has a present value represented by the random variable X - Y. [1] (ii) Show that: cov(X, n) = 'A- AA and hence or otherwise that: var(X - Y) = Ax - (HA. ) - 2AL where the functions A are determined using an interest rate of i, and functions 2 A are determined using an interest rate of i + 2i. [7] [Total 81