# Annuity formula proof $\frac{a_{\overline{n}|}}{a_{\overline{k}|}}$

I have the actuarial exam FM in 2 days and there is one more thing that I would like to understand.

I cam across a problem having to do with identities and this is the following.

A perpetuity paying $50 on the last day of each year was purchased on January 1, 1928. On January 1, 1978, the perpetuity was exchanged for a 15-year annuity-due with semi-annual payments of amount X. The interest rate is 6 percent, convertible monthly. Find X. When there are $$n$$ payments of $$1$$ with $$k$$ conversion periods between each payments at the end of each $$n$$ conversion period, I can see why the present value of those payments will be $$\frac{a_{\overline{n}|}}{s_{\overline{k}|}}$$ I think the argument is that if a payment of $$1$$ is to be made in one payment period at the end, the equivalent payments that we can pretend that there is is $$1 \over {s_{\overline{k}|}}$$. Thus calculating the present value is straight forward. However, if the payments are made at the beginning I am thinking that each payment of $$1$$ is equivalent to $$1 \over {a_{\overline{k}|}}$$ but the numerator should be an annuity-due rather than an annuity-immediate which is expressed as $$\frac{a_{\overline{n}|}}{a_{\overline{k}|}}$$ Either I am not even deriving the first formula or there is something missing in my argument... can someone help me out? ## 2 Answers Let be$i^{(12)}=6\%$the nominal interest convertible monthly, so that the effective annual interest rate is$i=\left(1+\frac{i^{(12)}}{12}\right)^{12}=6.17\%$. The value of the perpetuity at the time of the exchange is $$50 a_{\overline{\infty}\!|i}=\frac{50}{i}=\frac{50}{0.0617}= 810.66 \tag 1$$ The semiannual interest rate is$i^{(2)}=6.08\%$found by $$\left(1+\frac{i^{(2)}}{2}\right)^{2}=\left(1+\frac{i^{(12)}}{12}\right)^{12}\Longrightarrow i^{(2)}=2\left[\left(1+\frac{i^{(12)}}{12}\right)^{6}-1\right]=6.08\%$$ For the 15-year annuity-due with semi-annual payments we have 30 payments at the beginning of each period. The present value is $$X\,\ddot a_{\overline{30\,}\!|i^{(2)}}=X\,\frac{1-v^n}{1-v}=X\, \times 14.4839\tag 2$$ where$v=\frac{1}{1+i^{(2)}}$. Putting$(1)=(2)$, we find $$50 a_{\overline{\infty}\!|i}=X\,\ddot a_{\overline{30\,}\!|i^{(2)}}\Longrightarrow \boxed{X= 55.97 }$$ There are two small errors in alexjo's answer. First, the equation for the the calculation of the effective annual interest rate should be$i = (1 +\frac{i^{(12)}}{12})^{12} - 1$. Second, you do not use$i^{(2)}$in the 15-year annuity-due but$\frac{i^{(2)}}{2}$. The quantity$i^{(2)}$is an annual nominal rate of interest, compounded semi-annually, which yields an effective six-month rate of$0.005$. Because the period we are dealing with is half a year, we need to use the effective six-month rate.$$810.66 = X\,\ddot a_{\overline{30\,}\!|0.005}$$ $$X = 40.336\tag1$$ However, the multiple choice only offers less straightforward answers, all in symbolic form. We can still follow the path alexjo outlined for us, but we must be aware of an added twist while we do it$-$do not convert the interest rate of 6%, convertible monthly. Therefore, our annuities' time periods will in the unit of months. A payment of 50 at the end of the year is equivalent to$50/{s_{\overline{12\,}\!|0.005}}$. And so, the present value of the perpetuity paying \$50 on the last day of each year at the time of the exchange is $$\left(\frac{50}{s_{\overline{12\,}\!|0.005}}\right)\left(\frac{1}{0.005}\right).$$

Now for the exchange. There are 180 months in 15 years, which gives us $\ddot a_{\overline{180\,}\!|0.005}$ for an annuity-due, but we must account for the fact that payments are made on a semiannual basis; so we must divide by $\ddot a_{\overline{6\,}\!|0.005}$. The result is $$\frac{10000}{s_{\overline{12\,}\!|0.005}} = X\frac{\ddot a_{\overline{180\,}\!|0.005}}{\ddot a_{\overline{6\,}\!|0.005}},$$

which, when simplified, gives $$X = \frac{10000a_{\overline{6\,}\!|0.005}}{s_{\overline{12\,}\!|0.005}a_{\overline{180\,}\!|0.005}},$$

which agrees with (1). Remember, the dots on the annuity-due cancel.