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Question 1

Question 4.19 An office uses the multiple state sickness model to calculate premiums for a three-year combined sickness and endowment assurance policy issued to a healthy policyholder aged 57 at inception. Premiums are payable annually in advance for 3 years and are waived during periods of sickness. At the end of each of the first 2 years a benefit of [10,000 is payable if the life is then sick. A sum assured of f1 5,000 is payable at the end of the year of death if this occurs during the term of the policy or at the end of the three years if the life is alive and has never claimed any sickness benefit. The benefit payable at the end of year 3 is $10,000 if the life has previously claimed sickness benefit. The transition probabilities are as follows (for /= 0, 1 ,2): P (dead at / + 1 | healthy at () = 0.02 P( dead at / + 1 | sick at () = 0.05 P ( sick at / + 1 | healthy at /) =0.10 P ( sick at / +1 | sick at /) = 0.09 (i) Calculate the probabilities of being in each state at times / = 0, 1, 2, 3. [3] (ii) Calculate the profit margin for the contract by carrying out a profit test using the following assumptions: Annual premium: $5,300 Risk discount rate: 10% pa Interest on cash flows and reserves: 8% pa Initial expenses: (200 incurred on payment of the first premium Renewal expenses: f40 at times / = 1, 2 whether healthy or sick Claim expense: E30 at the date of claim of any benefit Reserves: OV = V =0 , VH =5, 000 if the policyholder is healthy at time t ( ( = 1, 2) , V = 4,000 if the policyholder is sick at time ? (/ = 1, 2) [16] [Total 19]10.3 For many years a company has recruited, uniformly over each year, 200 employees on their 20th birthdays and a fixed number of additional employees on their 25th birthdays. Mor- tality has followed English Life Table No. 12 - Males. Employees may retire on their 60th or 65th birthdays, and one third of employees reaching their 60th birthdays retire on that date. Employees leave the company only through death or retirement, and the total number of employees is 10.000. Find the total number of new recruits each year. 10.4 The male population of a certain country has been stationary for many years, there being 100,000 male births per annum, spread uniformly over the year; the mortality of males follows English Life Table No.12 - Males, and migration may be ignored. (a) Calculate the size of the country's male population at any time. (b) The government of the country has decided to introduce a social security plan, under which all employed men between ages 15 and 65 must contribute a fixed sum every week, the same sum being also payable by their employers. Men over age 65 will receive a pension of 100 units of currency per week, and those sick or unemployed between ages 15 and 65 also receive this amount. If it may be assumed that at any time 95% of men between ages 15 and 65 are employed, while the remaining 5% are sick or unemployed, calculate the weekly contribution payable by each employed worker. The scheme is to be financed on a pay-as-you-go basis (no fund is built up) and administrative costs are to be ignored. 10.5 A large manufacturing company has for many years staffed one of its divisions by the recruit- ment, uniformly over each year, of 1.000 staff at exact age 20. At the end of one year in the job, new staff are examined for suitability, and 20% are dismissed. All employees are assessed at age 35, and 50% are immediately moved out of the division. At age 40, all remaining em- ployers are moved out of the division. Death is the only other reason for leaving the division. Staff experience mortality according to English Life Table No. 12 - Males. Calculate the number of staff in the division