A 20-year with-profits endowment with a basic sum assured of $15,000 was issued to a

man aged 45 exact at entry. The policy has been in force for 7 years.

The policy is subject to simple reversionary bonuses that vest at the start of each year,

and death benefits are paid at the end of the policy year of death. In addition, a terminal

bonus equal to 25% of the sum assured plus existing bonus, is payable on a death or

maturity claim.

Simple bonuses were declared at the rate of 4% for the first year of the policy, and

3.75% pa thereafter to date. The company expects to pay future bonuses at a rate of

3.5% pa from now on.

Calculate the net premium reserve for this policy as at the present time. Assume AM92

Ultimate mortality and 4% pa interest.

{i} Explain with the aid of formulae what is meant by the term 'Standardised Mortality Ratio '. {ii} For a certain year. the estimated mid-year population of women in England and Wales and the number of deaths from cancer were as follows: Exact Estimated Number of ages population deaths 9-3 5 1 2.939.599 939 35-59 4.241.199 3.925 59-95 4.192.499 15.549 95-39 3.4 1'1 .999 29.33 'F over 39 1.143.599 1T.544 For the same year. the corresponding data for women in a certain district were as follows: Exact Estimated Number of ages population deaths 9-3 5 33 .992 5 35-59 29 .9 T9 3 1 59-95 39.1 92 93 95-39 19.2 29 1 49 over 39 5.?99 3 2 Calculate the 3tandardised Mortality Ratio with rmpect to cancer for women in this district for the year in question. using the national rates as the standard. The following data refers to permanent assurances. males. in 193339 at durations 2 years and over. Actual 1991-1 E Actual 199.4 E deaths by 311439 deaths by AM39 ultimate ultimate 113 34 52 132 115 Source G M. 1. Bureau {i} Suppose that. at all ages. the force of mortality of non-smokers is cw times the force of mortality of M39 ultimate. whilst that of smokers is :13 times the force of mortality of M39 ultimate. Find approximate 95% condence intervals for cm; and :13. {ii} Suppose that. at all ages. the force of mortality of the combined group (of smokers and non-smokers together} is car;- times the force of mortality of M39 ultimate. Find an approximate 95% condence interval for :13. Let 9 m Poisson(mE}. where E 'E a constant and where 111 takes the {unknown} value '31. Show that ii}. = 9 f E '13 the maximmn likelihood mtimator of 131. and also that it is an unbiased and efcient estimator of #31