Microeconomics questions

An explorer aged exactly 57 has just made a proposal to a life office for a whole life assurance

with a sum assured of 10,000 payable at the end of the year of death. For lives accepted at

normal rate, level annual premiums are payable until death under this policy.

The explorer is about to undertake a hazardous expedition which will last three years. The life

office estimates that during these three years the explorer will experience a constant addition of

0.02871 to the normal force of mortality, but after three years will experience normal mortality.

The life office quotes a level extra premium payable for the first three years.

Calculate this level extra premium on the following basis:

normal mortality: A1967-70 ultimate

interest: 3% per annum

expenses: none

An impaired life aged exactly 55 wishes to effect a without profit endowment assurance for a

sum assured of 1,000 payable at the end of 10 years or at the end of the year of earlier death.

Level annual premiums are payable throughout the term of the policy.

Special terms are offered on the assumption that the life will experience mortality which can

be represented by:

(a) for the first five years, a constant addition of 0.009569 to the normal force of mortality,

and

(b) for the remaining five years, the mortality of a life 8 years older.

The life office quotes a level extra premium payable throughout the term. Calculate this level

extra premium. Basis:

normal mortality: A1967-70 ultimate

interest: 3% per annum

expenses: none

A group of impaired lives now aged 40 experience mortality according to A1967-70 ultimate

with an addition to the force of mortality. The addition is 0.0005 at age 40, increasing linearly

to 0.0025 at age 60, at which level the addition remains constant.

Find the probability that an impaired life aged exactly 40

(i) will die within 20 years;

(ii) will die within 30 years;

(iii) will die between 20 and 30 years from the present time.

(i) Can you envisage circumstances under which an office could offer an impaired life who

wishes to pay the "normal" premiums a level debt but not a diminishing debt?

Hint Consider a life who is very severely impaired, and think of an approximate relationship between

(a) the level debt and

(b) the initial debt when debts decrease linearly to zero.

It is thought that a plumber charges f22 per hour plus an administrative charge of f15 per call- out. A sample of eight invoices was obtained corresponding to jobs with durations of 1 hour, 2 hours, ..., 8 hours. For each invoice the total cost of the job was noted with the following results: Time x (hours): 1 2 3 4 5 6 7 8 Cost y (f): 40 50 81 89 122 128 151 179 [(x-x)2=42 [(y-v)?=16,492 _(x-x)(y-y)=826 The following model is used to represent the data: Y, =a+bx; +e; where Y, (i =1, 2,...,8) are the costs, x; (i =1,2,...,8) are the fixed times and e; (/ =1,2,...,8) are independent errors with a /(0, of) distribution. (i) (a) Derive formulae for the least squares estimators of a and b. (b) Explain how your answer to part (i)(a) would have differed if you had been asked to calculate the maximum likelihood estimators and justify your answer. (ii) Calculate the regression coefficients a and b. (iii) Carry out a test to establish whether or not the slope in the model agrees with the suggested f22 per hour. (iv) Calculate a 90% confidence interval for the: (a) average cost of a job lasting 4 hours (b) cost of an individual job lasting 6 hours. 16 (v) Comment on relative widths of the two intervals calculated in part (iv). [Total 19