# annuity expense starting second year

+1 vote
74 views
asked Aug 11, 2016
edited Aug 11, 2016

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?

commented Aug 11, 2016 by (4,220 points)

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

## 2 Answers

+2 votes
answered Aug 11, 2016 by (4,220 points)
selected Aug 12, 2016

Best answer

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 $$a_{x:\overline{n}]} = {_1}p_{x}v\ddot{a}_{x+1:\overline{n}]}$$.

commented Aug 12, 2016 by (200 points)
@Simon, yes, you interpretation was perfect. Thank you so much, it helps a lot.
commented Aug 12, 2016 by (200 points)

@simon, I'm just a bit confused why is n not decrease by 1 in the temporary annuity? Shouldn't it be one less payment because you only start paying in the second year?

commented Aug 12, 2016 by (4,220 points)

Usually I think there would be the same number of payments as the term of the policy (occuring at the end of years $$1, 2, \dots, n$$) $$-$$ unless it is stated otherwise in the question.

0 votes
answered Aug 11, 2016 by (1,350 points)

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.

commented Aug 12, 2016 by (200 points)

Ohh, this makes so much sense! Thank you so much!