Question
(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
(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.
Step by Step Solution
There are 3 Steps involved in it
Step: 1
Get Instant Access to Expert-Tailored Solutions
See step-by-step solutions with expert insights and AI powered tools for academic success
Step: 2
Step: 3
Ace Your Homework with AI
Get the answers you need in no time with our AI-driven, step-by-step assistance
Get Started