Asset share

p8 in chapter 5 of the acted notes gives the following recursive formula for asset share:

AS(t+1) = {(AS(t) + P - E(t))(1+i(t)) - Sq(x+t)}/(1-q(x+t))

where S = death benefit sum assured (and other symbols have their usual meanings)

in the solution to question 4 of the Sept 2008 paper it says:

"the treatment of the cost of the death benefit is inaccurate. it should not be based on the whole guaranteed minimum sum assured, but the excess of the actual death benefit over the asset share"

which is correct?

 

Let's start by rearranging the formula in your post. Multiplying by 1-q(x+t) gives:

AS(t+1)x[1-q(x+t)] = [AS(t) + P - E(t)](1+i(t)) - Sq(x+t)

LHS = AS(t+1) - AS(t+1)q(x+t)

Taking the negative term to the RHS gives:

AS(t+1) = [AS(t) + P - E(t)](1+i(t)) - Sq(x+t) + AS(t+1)q(x+t)
= [AS(t) + P - E(t)](1+i(t)) - [S-AS(t+1)]q(x+t)

So, whether it's correct to use S or S-AS depends on which version of the recursive formula you're using, ie either could be correct, depending on the context.

 

but in the solution to the exam question it says "it should also be divided by (1-q(x))"

 

If we're charging mortality costs based on the full S, then these charges should be sufficient to pay out the full S on death. So the AS's left behind by the dying policyholders are available to spread among the surviving policyholders, ie divide by 1-qx.

However, if we're charging mortality costs based on S-AS, then these charges should be sufficient to pay out only S-AS on death. Since we need to pay out S on death, the AS's of the dying policyholders are needed to make the total payout up to S (ie S-AS + AS = S), so they aren't available to spread among the surviving policyholders, ie don't divide by 1-qx.

So, in the formula, either:

1) base charges on S and divide by 1-q, or
2) base charges on S-AS and don't divide by 1-q

Hopefully this is enough explanation for you to decide whether or not you agree with what's in the Examiners' Report.

 

so is the examiner's report wrong? i had no problem understanding asset shares until i read this exam question, so if the solution is correct maybe i don't understand them afterall.

 

Yes, I think the Examiners' Report is wrong in this instance. You either change the S to S-AS or divide by (1-q) to correct the expression in the question - you don't do both. Let's give them the benefit of the doubt, though, and assume it was just careless wording or poor drafting on their part.

 

okay, thanks for all your replies