Understanding of short question for reserve calculation on page 92

[ActuaryBen]

The short question is extracted from page 92 of SP2 combined material pack, chapter 3 Life Insurance Product (3) under section 1.3 Capital requirement for Immediate Annuity.
Original question is pasted as green while answer is pasted as blue in below. I have change the content to red fond to ease reference for my queries.

Queries:
1) In calculating reserve, why do we need to multiply a factor of (12.85/11.39) whereby 12.85 is mortality rate under valuation basis while 11.39 is mortality rate under premium basis?
With respect to first question above, isn't reserve should be net cashflow*probability of surviving as at end of first year = (100-5)*(1-12.85/100) = 82.7925?

3) the interest rate 5.5% and 4% in pricing and reserve basis were not being used. Is it because the question is about the initial capital strain which is concerning on the first year cashflow only?

4) If question is changed to consider the asset share in second year where renewal expense is included, how should we apply the interest rate, should we use 5.5% or 4%?

Original question
A life insurance company sells an immediate annuity assuming initial commission and expenses of 5% of single premium. (Renewal expenses are ignored.) The interest rate used for pricing is 5.5% pa. At this rate, a life annuity of 1 pa using the premium basis mortality is 11.39 for the policyholder concerned.
The insurance company is required to reserve at a valuation interest rate of 4% pa. The life annuity for the policyholder at this rate of interest and using the valuation mortality basis is 12.85. The insurance company is also required to hold solvency capital of 4% of reserves.
On sale, the commission and expenses were equal to those assumed in the pricing basis. Calculate the capital strain on the sale as a percentage of the single premium.

Answer
Answer: 16.5%
This is derived by considering all of the cashflow components and reserving / solvency capital requirements, as shown below.
For a premium of 100:

Positives:
Premium = 100

Negatives:
Commission and expenses = 5
Setting up reserve = 107.2
Required solvency capital = 4.3

Total = -16.5

where the reserve is derived as (100 – 5) × 12.85 / 11.39 = 107.2
and the solvency capital is derived as 107.2 × 4% = 4.3.
So what does this mean? If the life insurance company wants to sell £100 million of such policies this year, it will require £16.5 million to finance those sales. Therefore the difference between the pricing and the reserving bases costs 12.2% (from 107.2 minus 95) – although we should bear in mind that these numbers are entirely artificial.

 

[Lynn Birchall]

Hi Ben

2) Starting with this one first: 'With respect to first question above, isn't reserve should be net cashflow*probability of surviving as at end of first year = (100-5)*(1-12.85/100) = 82.7925?'

The reserve needs to allow for all future cashflows (and the probability of surviving to them being paid) and then discount them all back to the start of the policy.

The future cashflows once this annuity is in-force are the annuity benefit amounts each year. (We are told renewal expenses are ignored.)

So, we first need to work out what annuity benefit per year would be payable on an annuity with a premium of 100. We can work this out as (Premium - initial commission and expenses) / pricing annuity rate) = (100 - 5) / 11.39.

The reserve is then calculated as using this annuity amount cashflow x reserving basis annuity (12.85). The reserving basis annuity does the allowing for probability of surviving each year and the discounting at the reserving basis rate of 4% for us.

1) So, hopefully the previous reply answers the first part of your 1) as well : the '/11.39' is part of working out the annuity benefit (as we have to calculate it rather than being told the amount in the question), the 'x12.85' is applying the reserving formula for the value of the annuity benefit ( = annuity amount x annuity factor).

3) The 5.5% and 4% are being used. The are used in calculating the 11.39 and 12.85 annuity factors. The initial strain involves the initial reserve. Notice that this reserve includes all the future annuity cashflows (not just those in the first year).

4) I think this question falls away really after the previous ones are sorted? We're ignoring renewal expenses because the question says ignores them. If there were renewal expenses, the would have affected the pricing (so the working out of the original annuity amount) and the reserving in the above. So, we shouldn't now start to include renewal expenses just because we move into the second year.

Hope this helps (and reminds you of some annuity factors and actuarial maths that you've seen in earlier subjects)

 

[ActuaryBen]

Hi Mark,

Thanks for the detailed explanation. It was indeed helpful!