Continously paid annuity (until n years after death)

Question

BUS3024S Test 1 2005.pdf (0,1 MB)

In the above paper, Q4 asks us to write down an expression for an annuity which will be 1 p.a. from now until n years after the death of \((x)\). I understand the way the memo proceeds to do the question, however, I cannot understand why my way is incorrect:

\[\int_0^\infty e^{-\delta t} ~_tp_x\mu_{x+t}~ \overline{a}_{\overline{n}|}dt+\overline{a}_x=\overline{A}_x \overline{a}_{\overline{n}|} +\overline{a}_x\]

I am not sure why the above is wrong. The first part represents the probability that (x) lives till t then dies, upon which the annuity (of length n) begins. The second term is simply the payment while (x) is alive. However, the memo's answer is:

$$\overline{a}_{\overline{n}|}+v^n\overline{a}_x$$

Answer 1

Your answer is not incorrect. There are often different ways of representing the same benefit. Yours is just another way. If you use the fact that \( \bar{A_{x}}=1-\delta{\bar{a_{x}}}\) you should be able to derive the memo's result.