CT5 September 2012, Q13

[Aditi Vaidya]

Hey, I'm finding it a little difficult to understand the monthly premium equation that is used in the Examiner's report.
The way I worked the three elements out is as follows:
Benefits:
73,500 * A:[55] * (1.04^0.5) + 1500 * I(A):[55] * (1.04^0.5)
= 42,280.74857

Premiums:
12 * P * adue(12):[55]:<30> = 12 * P * ( adue:[55]:<30> - (11/24)*(1-D85/D[55]))
12 * P * ( adue:[55] – adue:85 * (D85/D[55]) - (11/24)*(1-D85/D[55]))
= 14.89970076 * 12P = 178.7964091 * P

Expenses:
275 + 65 * ( adue:[55] - 1) + 0.725 * P * 12(adue(12):[55]:<1>) + 0.025 * P * 12(adue(12):[55]:<30>) + 200 * A:[55] * (1.04^0.5)

However, the manner in which commission is calculated in the Examiner's report, is very different from this method. They haven't considered that commission is deducted with every premium. (i.e monthly.) and not sure how renewal commission is calculated in their method..
It would be great if someone could help me understand the examiner's report.
Thank you!

 

[Aditi Vaidya]

Equation according to examiner's report:-

If the monthly premium and sum assured are denoted by P and S respectively then:
0.975×12Padue(12):[55]:<30> + 0.025P
= (0.98S + 200)A:[55] * (1.04^0.5) + 0.02S(IA):[55] * (1.04^0.5) +275+ 65(adue:[55] −1) + 0.75×12P

 

[Lucy England]

Initial commission is 75% of the total premiums payable in the first policy year. Initial expenses/commission are typically incurred at t=0 of a policy (you can assume this is the case unless the question specifies otherwise). This indicates that the insurer has to pay 0.75*12P at policy outset, ie 75% of the twelve monthly premiums payable in year 1. That explains the last term in the expression given in the Examiners' Report.

Renewal commission is then 2.5% of the second and subsequent monthly premiums. So from the second month of the policy onwards, the monthly expense is 0.025*P. We can write this as 0.025*12P(adue(12):[55]:<30> - 1/12). The -1/12 in the brackets is used to exclude the first premium. The Examiners' Report brings this part of the expression over to the first line of workings straight away, which is slightly confusing! They've combined the expression for the premium with the renewal commission, giving:

12P*adue(12):[55]:<30> - 0.025*12P(adue(12):[55]:<30> - 1/12) = 0.975*12P*adue(12):[55]:<30> + 0.025P

It looks like you've assumed that the initial commission is charged on each of the first 12 premiums, which isn't the case - the full expense is paid at policy outset, ie the start of month 1. The regular commission is then charged from the start of month 2 onwards, rather than the start of year 2 onwards.

BTW, great work typing up the notation! It's all really clear.

 

[Aditi Vaidya]

Thank you very much, Lucy! this was super helpful