JANUARY 2016 EXAMINATIONS. Life Insurance I

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1 PAPER CODE NO. MATH 273 EXAMINER: Dr. C. Boado-Penas TEL.NO DEPARTMENT: Mathematical Sciences JANUARY 2016 EXAMINATIONS Life Insurance I Time allowed: Two and a half hours INSTRUCTIONS TO CANDIDATES: All questions carry equal weight and will be taken into account for the final mark. In addition to this paper you should have available the 2002 edition of the Formulae and Tables. Paper Code MATH 273 Page 1 of 6 CONTINUED

2 1. (a) Given that A 30 = 0.15, p 30 = and i = 4%. Calculate A 31. (b) Give the formula which defines each of the following actuarial symbols. Explain what each symbol measures. (i) n p [x] (ii) m ā x (c) The Thiele s differential equation can be expressed as: d dt t V = δ tt V + P t e t (S t + E t t V )µ [x]+t where P t denotes the annual rate of premium payable at time t. e t denotes the annual rate of premium-related expense payable at time t. E t denotes the expense of paying the sum insured at time t. Interpret the equation together with its individual components. (d) The present value random variable for an insurance policy on (x) is expressed as: v Tx, if T x 10 Z = 2v T x, if 10 < T x 20 3v Tx, if T x > 20 Calculate the expression for E[Z] in terms of whole life insurance benefits. Paper Code MATH 273 Page 2 of 6 CONTINUED

3 2. (a) Under the assumptions of constant force of mortality µ and a constant force of interest δ, evaluate: (i) ā x = E[ā Tx ] (ii) The probability that ā Tx will exceed ā x (b) Let F 0 (t) = 1 e 0.05t, for 0 t 100, where F 0 (t) represents the probability that a newborn does not survive beyond age 0 + t. Calculate: (i) The probability that a life aged 50 dies before age 65. [2 marks] (ii) The probability that a life aged 50 dies between the ages 65 and 80. [2 marks] (iii) The probability that a life aged 60 dies between the ages 75 and 90. [2 marks] (iv) Calculate the expected value of K x, i.e E[K x ], for x = 40 and x = 80. (v) Do you think it is feasible to use this lifetime distribution to model human mortality? Explain your answer. Paper Code MATH 273 Page 3 of 6 CONTINUED

4 3. (a) A 2-year endowment insurance is issued to a life aged 65. The death benefit payable at the end of year of death is constant. In addition, there will be a return of all premiums (without interest) if the insured is alive at time 2. Level premiums are payable annually in advance throughout the duration of the contract. You know that p 65 = 0.9 and p 66 = If the interest rate is 5% for the first year and 10% in the second year and the annual premium is 1,200, calculate: (i) The death benefit using the equivalence principle. (ii) Both the retrospective and prospective reserves at time 2 if the insurance company issued 10,000 identical and independent policies. Explain, with your own words, the relationship between the reserves calculated in these two ways. [8 marks] (b) You are given: 10 q 40 = 0.2 a 50 = 10 i=5% Calculate the expected present value, payable to a life aged 40, of benefit payments set at 1,000 per month, with the first payment at age 50 and continuing for life. [7 marks] Paper Code MATH 273 Page 4 of 6 CONTINUED

5 4. On 1 January 2013 a life insurance company issued a number of whole life insurance policies with sum insured 100,000 to select lives then aged 40. Level premiums are paid annually in advance and the death benefit is paid at the end of the year of death. The premium is calculated using the AM92 select tables and assuming Interest: 4% Initial expenses: 40% of the gross premium Renewal expenses: 5% of gross premium due at the start of each policy year from the second onwards (i) Calculate the gross premium. (ii) Calculate the net premium policy value at t = 2 using the premium basis. (iii) Calculate the gross premium policy value at t = 2 using the premium basis. (iv) Explain why the gross premium policy value is less than the net premium policy value. (v) There were 900 policies in force on 1 January During 2015, there were 3 actual deaths and the actual interest rate earned by the company was 5%. Calculate the profit or loss by the company in respect of these policies for the year (vi) Analyse the profit or loss into components for interest and mortality. Paper Code MATH 273 Page 5 of 6 CONTINUED

6 5. (a) Calculate 3.7 p 30.4 using (i) the method of Uniform Distribution of Deaths (ii) the method of Constant Force of Mortality. Basis: AM92 (b) A life insurance company issues a with profit whole life assurance policy to a life aged 40 exact, under which the sum assured S and any attaching bonuses, are payable immediately on death. Compound bonuses are added annually in advance. Level premiums are payable annually in advance ceasing at exact age 85 or on earlier death. Write down an expression for the net future loss random variable at outset for this policy defining all symbols that are used. [7 marks] (c) A 3-year term life insurance to (x) is defined by the following table: Time Payment p x+t The death benefits are payable at the end of the year of death. Calculate the variance of the present value of the indicated payments given an interest rate of 5%. [7 marks] Paper Code MATH 273 Page 6 of 6 END

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