For regular expense: If aged \(x\) have an expense paid in annuity starting in the second year, is it right to write it as \( a_{x+1} \) and if it's for temporary annuity for \(n\) years, then \(a_{x+1:\overline{n-1]}}\)? Is this correct?

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Hi @Junlin, welcome to the online hotseat. If I'm interpreting your question correctly, you're asking: what is the "default" assumption (and formula) for valuing a renewal expense that is incurred every year beginning one year after the inception of the policy.

I would think that, usually, the expense would only be incurred if the life has survived until the time at which it is to be paid (i.e. alive at time 1). In this case, it should be valued with the formula \( a_x \), which happens to be equal to \( _1 p_{x} v \ddot{a}_{x+1} \). Note that this is *not* the same as \(a_{x+1} \).

In the case of a term annuity, we would write it as \(a_{x:\overline{n}]} = {_1}p_{x}v\ddot{a}_{x+1:\overline{n}]} \).

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Using such expressions is fine, provided you also multiply the annuity factors with the respective probabilities of the life surviving to the start of the second year i.e. probability of the life surviving the first year $$p_x$$.

But there is an easier (slightly more intuitive) way to handle such a stream of expenses:

Suppose there are initial expenses of R100 per policy, and then the regular expenses of R10 starting in the second year, payable annually in advance thereafter. So the R10 is payable at times 1,2,3 ...,n.

This can be treated as initial expenses of R90, plus a stream of R10 payable at time 0,1,2,3,...,n. So you could use a normal annuity factor 10*a[x] (in advance) + 90 to give the expected present value of this stream of expenses.

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I've just edited the typesetting - hope I didn't change the meaning of your question!