Project L3 Scenario 3

[ActStudent1405]

The method I used to calculate the expected present value of premiums for Scenario 3 is different from that in the model answer. As an illustration I used the EPV Premiums for age 50.

(1) Before option date – all policies

The period before the option is equivalent to a term policy. The EPV of premiums for the first ten years for all policies is:

äx:10 * P x:10 * Sum Assured - using standard mortality rates

Answer I got is: £337.34

(2) After option date – policyholders taking up option – Basic Sum Assured

This is equivalent to a new whole of life policy at standard premium rates

ä60 * P60 * v10 * 10px * 0.7 * Sum Assured - using standard mortality rates

Answer: £1,928.67

(3) After option date – policyholders taking up option – Basic Sum Assured

This is equivalent to a new whole of life policy at standard premium rates

ä60 (at higher mortality rates) * P60 (standard mortality rates) * v10 * 10px (standard mortality rates) * 0.3 * Sum Assured

Answer: £683.58


(4) After option date – policyholders taking up option – Additional Sum Assured

This is equivalent to a new whole of life policy at standard premium rates

ä60 (at higher mortality rates) * P 60 (standard mortality rates) * v10 * 10px (standard mortality rates) * 0.3 * Sum Assured

Answer: £683.58

Summing these values gives an answer of £3,633.16, This is different from the model answer. For policyholder taking up the options: adding £337.34 * 0.7 + £1,928.67 = £2,164.82 which is part of the answer in cell R11 of the worksheet titled Rated in the model solutions. The premium paid by policyholders taking up the option for the additional sum assured in (4) tallies with the model answer. My guess is that something is not correct in part (3). Any help is appreciated.

 

[ActStudent1405]

Just a polite reminder to my earlier query. Thanks.

 

[Steve Hales]

Hi - thanks for the reminder

I think that your P60's should use the higher rate of mortality in items (3) and (4). That should do it.

 

[ActStudent1405]

Thank you Steve for your hint. I tried the P60 at the higher mortality rates for parts (3) and (4) but did not get anywhere close to the answer. For example, in the answer, the value of extra premium which is part (4) in my computation, the P60 at standard rates (column T in the worksheet titled: "Rated") is used.

 

[Steve Hales]

This is what I think we're aiming for; an EPV of premiums (for policyholder aged 50) of 4,287.81. This comes from the model solution via

=RATED!R11 + RATED!T11 + RATED!V11 = 4,287.81

This is equal to the [Value of the basic premiums] + [Value of the extra premiums] + [Value of the option premiums].

Here's what you need in your four steps to reproduce this.

(1) Before option date – all policies

= 10,000 * A(x:10)
= 10,000 * BASE!L11
= 337.34 - using standard mortality rates

(2) After option date – policyholders not taking up option – Basic Sum Assured

= 10,000 * A(60) * 0.7 * \(v^{10}\) * \({}_{10}p_{50}\)
= 10,000 * BASE!F21 * 0.7 / (1 + Interest)^10 * BASE!J11
= 1,928.67 - using standard mortality rates

(3) After option date – policyholders taking up option – Basic Sum Assured

= 10,000 * A(60) * 0.3 * \(v^{10}\)* \({}_{10}p_{50}\)
= 10,000 * RATED!F21 * 0.3 / (1 + Interest)^10 * BASE!J11
= 1,010.90 - using higher mortality rates for the A(60) but standard mortality rates for \({}_{10}p_{50}\)

(4) After option date – policyholders taking up option – Additional Sum Assured

= 10,000 * A(60) * 0.3 * \(v^{10}\) * \({}_{10}p_{50}\)
= 10,000 * RATED!F21 * 0.3 / (1 + Interest)^10 * BASE!J11
= 1,010.90 - using higher mortality rates for the A(60) but standard mortality rates for \({}_{10}p_{50}\)

Then summing these we have 337.34 + 1,928.67 + 1,010.90 + 1,010.90 = 4,287.81.

Are you sure your formulae are pointing to the right cells? Where I use RATED!F21 in items (3) and (4), you should have RATED!G21*RATED!E21.

 

[ActStudent1405]

Thank you for the step-by-step answer. I did alter the premium rates using the higher mortality rates. However, as you pointed out, this leads to an EPV(Benefits) = EPV(Premiums).

If I amend the calculation as you suggest, the value of the option premium would be zero. My model uses a four tiered premium calculation (explained earlier) to determine to come to a value of EPV(Premium) which should yield an answer equal to the summation of Rated!R11 and Rated!T11 in the examiner’s solution.

My formula for the option premium is similar to the examiner’s solution EPV(Benefits) – EPV(Premium). Although, I fully understand your calculation, I am struggling to try to get an option premium of £594.2 (Rated!V11) using my assumptions. Is there a way of doing it, or should I simply scrap my assumptions and follow the exam solutions?

Thanks

 

[Steve Hales]

Firstly, EPV(Total Benefits) = EPV(Total Premiums) is exactly what you're aiming for so that's not a problem. The sample solution gives EPV(Option premium) = EPV(Total benefits) - EPV(Some of the premium).

Secondly, that £594.2 (Rated!V11) is the amount that the option is worth to the policyholder, and so they have to pay an additional £72.18 in premiums for it. By using your approach it's as if you've already allowed for that optionality in the setting up of the formulae. This means that you'll need to find an expression for what the option value represents - maybe something like [Value of the policy with the option] - [Value of the policy without the option]. This is only a suggestion and not the solution!

I hate to give up on a good challenge, but we're well beyond the amount of time it's wise to spend on this in terms of CA2! Remember that CA2 is about model documentation and reporting rather than having a perfect (and sometimes even a correct) spreadsheet.