(i) Define the following symbol in words, and give a formula in terms of an integral for each

of them:?

(a) A

x

1

y

(b) A2

xy

(c) ay|x

(ii) Consider the following sets of payments:

(1) 1 immediately on the death of (y) if (y) dies before (x), and

(2) an income of ? p.a. payable continuously to (x) after the death of (y), plus 1 immediately on the death of (x) if this occurs after that of (y).

Prove that the present values (at force of interest ? p.a.) of (1) and (2) are equal. Hence write

down a relationship involving A

x

1

y

, A2

xy

and ay|x.

A special life policy on 2 lives aged x and y respectively provides cash sums of 10,000 and

20,000 immediately on the first and second deaths respectively. In addition, an annuity

at the rate of 1,000 per annum will be paid continuously, commencing immediately on the

first death and ceasing immediately on the second death. Obtain an expression for the mean

present value of the benefits in terms of joint-life and single- life annuity functions and the

force of interest. Ignore expenses.

An office issues a policy on the lives of a woman aged 60 and her husband aged 64. Under this

policy, level premiums are payable annually in advance for 20 years or until the first death of

the couple, if earlier.

On the first death of the couple, the survivor will receive an annuity of 10,000 per annum,

payable weekly, beginning immediately on the first death.

Calculate the annual premium if the office uses the basis given below:

Mortality males: a(55) males ultimate

females: a(55) females ultimate

Expenses: 20% of the first premium

5% of each premium after the first

Interest: 6% per annum.

A special annuity, payable yearly in arrear, is effected on the lives of a man aged x and his

wife aged y. The conditions of payment are:

(a) so long as both survive the rate of payment will be 3,000 per annum;

(b) if the wife dies first, the rate of payment will be 2,000 per annum until the man's death;

(c) payments at the rate of 3,000 per annum will continue for six years certain after the

death of the husband, the first payment being at the end of the year of his death, and will be

reduced thereafter to 1,500 per annum during the lifetime of the wife.

Obtain an expression for the present value of this annuity in terms of single and joint-life

annuity factors, life table and compound interest functions. Assume that the same (nonselect) table of mortality is appropriate for the two lives.

Find 11(0), c(x}f 6K1), and dujx) if \"(17) = Pr[X = x] where (3.5.20) is to be used to produce a table of the p.f. of the random variable X when it has a a. A Poisson distribution with parameter in b. A binomial distribution with parameters a and 30. Formula (3.5.20) is to be used to produce tables of compound interest func- tions. Find 24(1), c(r) ,l cilia), and 1 f d(x) when a. u(x) = a}, b. u(x) = ea. or: 3.6 Verify the entries for the constant force of mortality and the hyperbolic as- sumption in Table 3.61. Note that the entry for spa: in the hyperbolic column provides justication for the hyperbolic name. Graph p.{x + t), 0 <: r for each of the three assumptions in table also graph survival function assumption. using ix column compute as- sumptions use and an assumption uniform distribution deaths year age to find median future lifetime a pawn a. b. if qm="{1.04" q calculate probability that will die between ages under are uniformly distributed within hyperbolic age. ii lim m60 h .11 c. p. . constant force is adopted show _ f etc as log :3.>