Microeconomics questions.answer them well.

A life office issues a three-year unit-linked endowment policy to a life aged exactly 60. The

annual premium is 2,000, payable at the start of each year. The allocation proportion is

90% in year 1 and 97% thereafter. At the end of year of death during the term, the policy

pays the higher of 10,000 and the bid value of units allocated to the policy, after deduction

of the fund management charge. A bonus of 2% of the (bid) value of the unit fund is payable

at maturity. The life office makes the following assumptions in projecting future cash flows:

Mortality A1967-70 ultimate

Initial expenses: 300

Renewal expenses: 50, incurred at the start of the

second and the third years

Fund management charge: 2% per annum, taken at the end of each year

before payment of any benefits

Sterling fund interest rate: 4% per annum

Bid/offer spread: 6%

Unit fund growth rate: 10% per annum.

Construct tables to show the following:

(i) the growth of the unit fund;

(ii) the profit signature, assuming that no sterling reserves are held;

(iii) the profit signature after taking into account sterling reserves, given that the sterling

reserves per policy are to be 36.48 before receipt of the premium due at time 1 year and

78.64 before receipt of the premium due at time 2 years.

In each case, indicate clearly how you calculate your table entries. Ignore the possibility of

surrenders.

(a) In the context of profit-testing of unit-linked business, define the following terms briefly:

(i) the Unit Fund,

(ii) the Sterling Reserves,

(iii) the profit vector of a policy,

(iv) the profit signature of a policy,

(v) the risk discount rate, and

(vi) zeroisation of Sterling Reserves.

(b) A life office issues a large number of identical 4-year annual premium unit-linked endowment assurances to lives aged 65. According to the office's calculations, the profit vector per

policy sold, ignoring withdrawals and assuming that no Sterling reserves are maintained at

the end of each year, is as follows ():

191.12

?111.45

?3.28

10.95

The office's mortality basis is A1967-70 ultimate, and Sterling Reserves earn interest at 5%

per annum.

Calculate

(i) the profit signature per policy sold, ignoring any need to maintain Sterling

given age under the new table, more than double, exactly double, or less than double the mortality rate, q, of the standard table? If u(x) = B c', c > 1, show that the function l, p(x) has its maximum at age No where u(x,) = log c. [Hint: This exercise makes use of Exercise 3.14.] A C* Assume p(x) = 1 + Box - for x > 0. a. Calculate the survival function, s(x). b. Verify that the mode of the distribution of X, the age-at-death, is given by X0 = log(log c) - log A log c 3 10 If p(x) = for 40