X4 assignment Ques 13

[Eleanor Cawston]

Question 13 from 2020 CM1 assignment X4
I don't understand the timing of the addition of the bonuses is this question. The policy is sold on 1/7/15. Bonuses (if I have correctly understood) are declared on 1st January each year. So in the first half of each policy year, the bonus for that year has not yet been declared. Given that the benefit is payable immediately on death, I made the usual approximation that it is paid in the middle of the year of the (policy) year of death. To me it would seem natural then that the number of bonuses declared is one less than the policy year.

That would mean the benefit is 100000(1+i)^0.5(0.97*WL_[20] +0.03(IA)_[20] ), rather than the (1+i)^0.5(100000*WL_[20]+3000(IA)_[20]) given.

Using the latter, if the policy holder dies in the first half of the first policy year, his sum assured is increased by the as-yet-undeclared bonus.

Any comments on how to justify this would be appreciated.
Thank you.

 

[Lucy England]

The bonuses are declared at the start of each policy year in this question, not each calendar year. The wording of this question is a little vague, but the wording of part (ii) gives an extra hint by saying "assume bonus entitlement is earned immediately on payment of premium". Since premiums are paid annually in advance from the start of the policy, this indicates that the first bonus is earned at policy time 0, ie on 1 July 2015.

The formula you gave for the EPV of the benefits doesn't quite allow for the bonus being awarded 6 months into the policy. It allows for payment of benefits halfway through each policy year (on average) and the first bonus vesting at the end of the first policy year. The 97,000 * A:[20] + 3,000 * (IA):[20] represents an annual benefit starting at 100,000 and increasing by 3,000 each year (on each policy anniversary). Multiplying this by (1+i)^0.5 represents the payment of benefits being made immediately on death (or halfway through each policy year on average), but you're only bringing forward the payment timing and not the application of the bonuses.

Having a change in benefit after the first half year of the policy would get really complicated as probabilities of death / survival would cause problems. I think you'd have to have something like a half-year term assurance followed by another expression to account for the benefit increasing annually thereafter. It's definitely too complicated for the 4 marks on offer for part (ii), anyway!

 

[Eleanor Cawston]

The formula you gave for the EPV of the benefits doesn't quite allow for the bonus being awarded 6 months into the policy. It allows for payment of benefits halfway through each policy year (on average) and the first bonus vesting at the end of the first policy year. The 97,000 * A:[20] + 3,000 * (IA):[20] represents an annual benefit starting at 100,000 and increasing by 3,000 each year (on each policy anniversary). Multiplying this by (1+i)^0.5 represents the payment of benefits being made immediately on death (or halfway through each policy year on average), but you're only bringing forward the payment timing and not the application of the bonuses.

Yes, I thought this was more correct than having it vest at the beginning, since it wasn't declared then (in my understanding). Not perfect still, but I couldn't see a way to make it be added to the SA in the middle of the policy year (without maybe having a half year assurance function with no bonus and then defining the rest of it in terms of policy year, but that sounded overly complicated). Essentially the problem was, I accredited relevance to the information that it started in July, and since many questions say "policy year", I thought it was deliberately making the point that the bonus declaration happened at a different time! Perhaps over-thinking the information given.

Thank you for your very prompt answer!
Eleanor

 

[Lucy England]

No problem

I don't think you were necessarily overthinking things. I'll request that we tighten up this question wording next time we update the assignments to avoid confusion in future.