MATH39522 CONTINGENCIES 2
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MATH39522
CONTINGENCIES 2 - COURSEWORK
2022
1. Consider a life assurance policy for two lives y and 夕 which pays a sum assured at the end of the year of death of y, in the event that either y dies before 夕 or both y and 夕 die in the same year.
(a) Write down an expression for the expected present value EPV of this policy in the form of a summation.
[2 marks]
(b) Assuming the following:
· y is a male aged 50 and 夕 is a female aged 55 and
· an interest rate of 4% a year and mortality table PA92C20
calculate the EPV of this assurance.
[4 marks]
[Total 6 marks]
2. An insurance policy provides an annual income to a person y who has just retired. We call this income the pension amount.
The pension amount is paid for life with the first payment made immediately on retirement and subsequent payments made on the anniversary of retirement if the member y is still alive. Assume y retires on 1 January 2022, so that payments are made on each 1 January until y dies.
The initial pension amount payable on retirement is Pó and more generally the payment made at time t is Ph for t = 0. 1. 2. ●●●●●● .
The pension amount is increased each year at a compound rate of interest of 3% a year with the first increase made on 1 January 2023.
The insurance company also provides a reversionary annuity payable annually to a life 夕 who is the partner of y. The amount that the reversionary annuity pays is 60% of the amount Ph that would have been paid if y had remained alive (hint: so for any payment of the reversionary annuity, work out how much the pension payable to y would have paid at that point and take 60% of that ).
The first payment of the reversionary annuity will be made on the 1 January immediately following the death of y (assuming the partner 夕 is alive).
Now consider the following data and assumptions to be used in making calculations in this question:
· y is a female aged 65 exact and 夕 is a female aged 62 exact at 1 January 2022, · the amount of pension Pó = £20.000 payable on 1 January 2022, · an interest rate of 5% a year and mortality table PA92C20.
(a) Write down the formula for Ph
[1 marks]
(b) Calculate the EPV of the reversionary annuity at 1 January 2022.
[4 marks]
(c) Calculate what the reserve for this policy would be as at 1 January 2025, assuming that y has died, but 夕 is alive.
[3 marks]
(d) In addition to the above benefits, there is a lump sum payment at the end of the year in which y dies so long as this is before the 10h9 payment of pension, i.e. P0 is made (due on 1 January 2031).
The lump sum payment at time t is equal to:
(10 _ t) . 0 ●4 Ph
if 夕 is still alive at t, or
(10 _ t) . Ph
if 夕 has already died before t.
Calculate the EPV of this lump sum death benefit.
[6 marks]
Aside : the payments from the lump sum and reversionary pension, together with payments to y while they are both alive, is such that the combination of all three broadly means that if y dies within 10 years of retirement, that a total of 10 years’ pension is paid by the insurer in all circumstances (we might call that a 10 year guarantee). The payments fall short in two areas - (i) there is no allowance for future 3% increases to the pension within the lump sum and also (ii), if 夕 dies after y but before 10 years then the total payments by the insurer will be less than 10 years’ pension as the reversionary annuity stops payment on 夕t s death.
[Total 14 marks]
2022-03-26