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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:

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

1 Answer

+1 vote
by (360 points)

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. 

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