2 hours UNIVERSITY OF MANCHESTER. 8 June :00-16:00. Answer ALL six questions The total number of marks in the paper is 100.

Transcription

1 2 hours UNIVERSITY OF MANCHESTER CONTINGENCIES 1 8 June :00-16:00 Answer ALL six questions The total number of marks in the paper is 100. University approved calculators may be used. 1 of 6 P.T.O.

2 Please note the following before you get started You can find a formula sheet on the last page of this exam paper. The marking is based on your answer booklet only, so make sure you write down your working there. 1. You are given the following mortality data: age x q x Calculate the following: (a) 2 p 51 (b) 2 q 50 (c) 0.5 p 52, using CFM (d) 2.5 p 50.8, using CFM [Total 9 marks] 2. Using the mortality data set out in Question 1 and assuming an interest rate of 3% p.a., calculate: (a) (IA) 50:3 (b) ä 50:3 [6 marks] [4 marks] [Total 10 marks] 2 of 6 P.T.O.

3 3. For a certain mortality model, l x = 100 x, where x [0, 100]. (a) State the value for ω for this model. [1 marks] (b) Prove that µ x+t = x t for x, t > 0 such that x + t < 100 [5 marks] (c) Calculate e 40 [4 marks] [Total 10 marks] 4. A life aged 40 exact takes out a whole life assurance policy with a sum assured of 60,000 payable at the end of the year of death. Premiums are payable annually in advance for the lifetime of the insured life. Assume mortality table AM92 Ult and interest at 4% p.a. throughout this question. (a) Calculate the net premium for the policy. [4 marks] (b) Assume death occurs during the 10 th year of the policy. (i) Calculate the present value of the net loss and (ii) calculate the probability of the net loss taking this value. (c) Write down the general formula for the present value of the net loss for this policy. [7 marks] (d) Using R or otherwise, state in which years death of the policyholder leads to a profit for the company. [6 marks] (e) Therefore calculate the probability of the policy making a profit. [Total 22 marks] 3 of 6 P.T.O.

4 5. A person now aged 60 exact took out a deferred annuity policy 10 years ago. The deferred annuity amount is 20,000 p.a. payable annually in advance from the person s 65 th birthday. Premiums are paid monthly in advance until age 65 or earlier death. Assume interest at 4% p.a. and mortality of AM92 Ultimate. (a) Calculate the net monthly premium for this policy at its commencement. [8 marks] (b) The policy holder has decided to delay his retirement until his 70 th birthday. He is happy to continue paying the net premium calculated in part (a) until his 70 th birthday or earlier death. In return he would like the annual amount of the annuity to be increased to reflect the delay in the annuity starting to be paid and the potentially extra 5 years of premiums that will be paid. The insurer is prepared to do this. They require a fee that amounts to 1% of the reserve, so that the reserve immediately after the alteration is 99% of the reserve for the policy immediately prior to the alteration. (i) Calculate the reserve for the policy when the person is aged 60, on the terms that apply prior to the alteration. [8 marks] (ii) Hence calculate the increased annuity payment for the policyholder. [6 marks] (c) Write down the general formula for the Death Strain at Risk in the t th year of a n years deferred annuity policy for a person aged x. You will need to write a formula for where t < n and for where t n. Assume the policy pays an amount R annually in advance from age x + n. In the formula use the notation t V for the reserve at time t. (d) Calculate the Death Strain at Risk in the year immediately after the policy alteration (Hint: remember that the policy has premiums paid monthly). [5 marks] [Total 29 marks] 4 of 6 P.T.O.

5 6. An insurance company issues an endowment assurance policy to a life aged 45 exact. The policy has a sum assured of 20,000 which is paid at the end of the year of death or on survival to the end of the term of the policy which is 20 years. Premiums are payable annually in advance for 20 years or until earlier death. Expenses are: Initial Expenses of 500 and 10% of the first premium. Renewal expenses, charged on the second and subsequent annual premiums, at 50 increasing each year at 2.5% p.a. plus 2% of premiums. Maturity and claim expenses of 1% of sum assured. (a) Assume interest at 5% p.a. and mortality table AM92 select. Denote the gross annual premium by P. (i) Calculate the EPV of the benefits. (ii) Calculate the EPV of the premiums payable in terms of P. (iii) Calculate the EPV of the expenses in terms of P. (iv) and hence calculate P [5 marks] [7 marks] (b) State whether the premium is expected to be higher or lower if AM92 Ultimate rather than AM92 Select is used. Give a reason for your answer. [Total 20 marks] 5 of 6 P.T.O.

6 Formula sheet Contingencies 1 1. Compound interest i is the interest rate per annum v = (1 + i) 1 δ = log(1 + i) = m((1 + i) 1/m 1) is the nominal rate of interest, convertible m times a year i (m) a n = v + v v n is the present value of an annuity certain in arrears with term n years ä n = 1 + v + v v n 1 is the present value of an annuity certain in advance with term n years 2. Mortality tables and laws l x is the number of lives at age x d x = l x l x+1 is the number of lives who die between ages x and x + 1 tp x is the probability that a life aged x will survive at least t years; p x = 1 p x tq x is the probability that a life aged x will die within t years; q x = 1 q x m n q x is the probability that a life aged x will die between ages x + m and x + m + n K x is the curtate future lifetime for a life aged x; e x = E[K x ] T x is the complete (exact) future lifetime for a life aged x; e x = E[T x ] µ x is the force of mortality at age x: ( tp x = exp t 0 ) µ x+s ds [x] + t denotes a life currently aged x + t who was selected at age x 3. Assurances A x is the EPV of a whole life assurance that pays at the end of the year of death of a life aged x Ā x is the EPV of a whole life assurance that pays at the moment of death of a life aged x E [ v Kx+1] = A x ; Var ( v Kx+1) = A i x (A x ) 2 E [ v Tx] = Āx; Var ( v Tx) = Āi x (Āx) 2 (where i = (1 + i) 2 1. Similar relationships hold for endowment assurances with status x:n, pure endowments with status 1 x:n and term assurances with status x:n 1 ) A (m) x is the EPV of a whole life assurance that pays 1 at the end of that fraction 1/m of a year in which the life aged x dies 4. Annuities a x is the EPV of an annuity for life in arrears for a life aged x ä x = 1 + a x is the EPV of an annuity for life in advance for a life aged x ā x is the EPV of a continuously paying annuity for life for a life aged x E[ä ] = ä Kx+1 x; Var(ä ) = Ai x (A x ) 2 Kx+1 (1 v) 2 E[ā Tx ] = ā x ; Var(ā Tx ) = Āi x (Āx) 2 log(v) 2 (where i = (1 + i) 2 1. Similar relationships hold for temporary annuities with status x:n ) is the EPV of an annuity for life that pays throughout the year m instalments of 1 m each. The first payment is paid at the end of the first 1/m-th of a year a (m) x is identical, except that the payments are made at the beginning of each period rather than the end ä (m) x ä (m) x ä x m 1 2m ; ä (m) x:n ä x:n m 1 2m (1 vn np x ). A x = 1 (1 v)ä x ; Ā x = 1 δā x END OF EXAMINATION PAPER 6 of 6