Microeconomics questions

Your office is considering the issue of 3-year annual-premium endowment assurance policies

without profits to lives aged 62. In respect of a policy with sum assured 10, 000, payable at

the end of the year of death (if within 3 years) or on maturity, calculate the net present value

of the profit signature on the following assumptions:

premium basis: mortality: A1967-70 ultimate

interest: 6% p.a.

expenses: 3% of all premiums.

reserve basis: net premium method using A1967-70 ultimate,

4% p.a. interest

rate of interest to

be earned in life fund: 8% p.a.

expenses: 3% of office premiums

mortality: A1967-70 ultimate

risk discount rate: 10% p.a.

A life office issues 3-year term assurance policy to a man aged exactly 59. The sum assured

is 15, 000, payable at the end of the year of death. Level premiums are payable annually in

advance. Expenses are expected to be as follows:

initial expenses: 10

renewal expenses: 2 incurred at the beginning of the 2nd and each subsequent policy year.

It is assumed that interest of 7% per annum will be earned on the life funds, and that mortality

follows the A1967-70 ultimate table. The risk discount rate used by the office is 15% per annum.

The office calculates the annual premium by requiring that the net present value of the expected

profit on each policy is equal to 20% of one office premium. Calculate the office premium on

each of the following reserving bases:

(i) The office holds zero reserves at each year-end.

(ii) The office sets up a reserve at each year-end (except the last) equal to 80% of one office

The number of claims per year from individual policies in a portfolio is believed to follow a Poi(1) distribution. The Gamma(5,2) distribution is chosen as a prior distribution for 2. A random sample of 10 policies is observed over the last year, and the numbers of claims are found to be as follows: 4, 1, 0, 0, 2, 0, 0, 1, 3, 1 (i) Derive the posterior distribution for 1. [3 (ii) Hence, determine the Bayesian estimate for 1: (a) under squared error loss (b) under zero-one error loss (c) under absolute error loss. [5 [Total 8 Arandom variable Y has a gamma distribution with PDF: fly) =. U" [(a) (i) (a) Show that f(y) can be written in the form of a member of the exponential family and identify the natural parameter and the dispersion parameter. (b) Use the properties of exponential families to obtain the mean and variance of this distribution. (c) Hence, or otherwise, determine the variance function. [7 (ii) Another random variable, X , has a negative binomial distribution with PF: P(X = x) =- I(k+ x) [(x +1)[(k)" p*(1-p)* x =0,1,2,... (k #1) Explain clearly why this negative binomial distribution is not a member of the exponential family. [2