Hi All, There is a question on pg4 of ch5, which discusses that the cost of cover lie between 0 and SA. I didn't understand much the wording of its solution, but continuing reading, my understanding is that: 1. On aggregate basis : the cost of cover must lie between 0 and SA. Because if a policy have negative asset share, other policies may have positive, so the aggregate will be positive. 2. On individual basis : it is not the case. The cost of cover can lie outside the range of 0-SA. Because, for a single policy, if the asset share is negative, then the cost of cover will be more than SA. Please confirm if my understanding is correct? But still I am not able to understand the reasoning behind it. Why it is different in individual and aggregate basis? Furthermore, on pg5, why it is that we deduct the death benefit in excess of AS on individual basis but only death benefit on aggregate basis?
Hi Bharti The word 'cost' can mean many things. In this question we were just taking a very simple view (and then showing it was wrong). To a policyholder they may consider that there is no cost for life cover if they have not died - they didn't receive a death benefit and so there has been no cost to the insurer. If they did die then the sum assured is the cost. Some policyholders do think this and ask for their money back from a term assurance if they haven't died. This view is wrong though, as some policyholders will have died and this cost is spread over all the policies. The question doesn't go any deeper than that. It is just there to show that the cost of cover when calculating a per policy asset share can't be zero or the sum assured, it must be something else. Now let's look at how asset shares should be calculated. First of all, let's consider the aggregate asset share. This is just the accumulated value of all the cashflows in and out for the cohort of policies. Imagine a big pot of money that is initially empty. Premiums come into the pot and claims and expenses go out. So if there is a claim, we deduct the full value of the sum assured, as that is what leaves the pot (and it is this cashflow that we refer to as the cost). The asset share could be negative early on - this doesn't change the fact that the cost is equal to the sum assured paid out. Note that we are not using cost to mean profit - profit would depend on how big the pot is. Now let's look at individual (or per policy) asset shares. We can actually calculate these two ways. Firstly we could take the aggregate asset share and divide it by the number of survivors (assuming all policies were the same size). In this case we still deduct the full sum assured paid out for each death. So we calculate the asset share as (assuming expenses and premiums at the start of year and claims paid at the end): AS_t = [(AS_t-1 + P - E)(1+i) - S q] / (1-q) So in this first case we have deducted the full value of the sum assured for each death (with mortality rate of q). But we can rearrange this formula to get AS_t (1-q) = (AS_t-1 + P - E)(1+i) - S q AS_t = (AS_t-1 + P - E)(1+i) - (S - AS_t) q This leads to the second way to calculate per policy asset shares. In this case we have deducted the sum assured in excess of the asset share. It doesn't matter which way you calculate the individual asset share. You can remember one of these formula and then work out the other by rearranging it. Note that the two formulae use cost in a different sense. The first formula uses cost as the cashflow paid out, ie the sum assured. The second formula uses cost in the sense of profit, ie we make a loss if the sum assured paid out is more than the asset share that we have. Sadly, many words have several potential meanings. The important thing though is to understand how asset shares work and use them correctly whatever definition of cost that we have. Best wishes Mark
