Q11 September 2021 CM1A

[Ray K]

Hi,

I have two questions regarding this question:

1) In part i) you attempt to set an annuity payable monthly in advance for 15 years or until earlier death equal to an endowment assurance payable at the end of year of death or at the end of term if earlier.

For a term assurance or a whole life assurance it is seen that you can simply multiply by 1/(1+b) and set the interest rate to j where j = (1+i)/(1+b) - 1

Why can you not do 1/(1+b) * the endowment assurance at 4%?

2) Somewhat related - how would I attempt a similar question with compound increasing assurances payable immediately on death? How would you adjust say, a term assurance increasing compound?

3) In part ii it asks for the effective rate of return if they survive to the end of the contract. An equation of value is set out setting the EPV of premiums to the EPV of benefits. I do not understand why the benefits do not include the reversionary bonus. I was under the impression the bonus is the minimum you would achieve - so would be included in the return?

Many thanks.

 

[Joe Hook]

Hi,

1) With bonuses vesting at the end of the year there's a disconnect between the amount paid on death and survival in the final year. On death the sum assured would have 14 bonuses included whereas the life gets the full 15 years on survival. Since the amounts differ we have to split up death and survival benefits.

2) When payments are immediately on death our first principles expression for a term assurance with compound increasing (vesting at the end of the year) benefits would look something like (approximating for payments on average half way through the year):

(TA):x:<n> = v^0.5 * qx + (1+b) * v^1.5 * px * qx+1 + ...

We need the first term in the form (1+b)*v*prob so we need to pull out a factor of v^-0.5 / (1+b) = (1+i)^0.5 / (1+b)

That (1+i)^0.5 factor should look familiar as our claims acceleration factor.

3) This is a with-profits contract so bonuses are typically not guaranteed although there can be a guaranteed element to them. This means in the very worst case scenario there are no bonuses awarded. This situation would give the minimum return.

Hope this helps.
Joe