I am working on the following problem that involves annuity which deposits form a geometric progression.
Stan elects to receive his retirement benefit over $20$ years at the rate of $2,000$ per month beginning one month from now. The monthly benefit increases by $5\%$ each year. At a nominal interest rate of $6\%$ convertible monthly, calculate the present value of the retirement benefit.
I understand that when the deposits are forming a geometric progression, the present value at the time of the first deposit will be an annuity-due with the appropriate interest.
So, since the deposits increase with an annual rate of $5\%$, each month the deposits increase by $r\%$ which can be calculated from
$$(1+r)^{12}=1.05$$
Hence,
$$1+r=\sqrt[12]{1.05}-1$$
Also, since the account adds a nominal interest rate of $6\%$ convertible monthly, the deposit will accumulate $i\%$ each month which can be calculated from
$$(1+\frac{6\%}{12})=1+i$$
Hence,
$$1+i = 1.005$$
The present value at the moment of the first deposit is
$$2,000(1+\frac{1+r}{1+i}+(\frac{1+r}{1+i})^2+ \dots + (\frac{1+r}{1+i})^{239})$$
(Note: there are 240 conversions in 20 years)
so,
$$2,000 \frac{1-(\frac{1+r}{1+i})^{-240}}{1-\frac{1+r}{1+i}}$$
Using the present value factor $(1+j)^{-1}=\frac{1+r}{1+i}$ we can see that the above expression is equivalent to
$$2,000 \ddot{a}_{\overline{240}\rceil j}$$
So, according to my calculation the present value at the time of the first deposit $X$ is equal to
$$X=2,000 \ddot{a}_{\overline{240}\rceil j} \approx 430,816.22$$
Since this value is one conversion after the the very first month, I want to say that the answer to this problem must be
$$(1.005)^{-1}X \approx 428,627.86$$
However, the answer in the book is supposedly $419,253$.
I thought that I counted the number of conversions wrong, and I tried it a couple of times but it still did not give me the right answer. Can I have some help?
Thank you.
Best Answer
What I think you are looking for is the present value of a growing annuity.
$$r=(1+\frac{r_n}{12})^{12}-1$$
$$0.061678=(1+\frac{0.06}{12})^{12}-1$$
$$PV=C\left(\frac{1}{r-g}\right)\left(1-\left(\frac{1+g}{1+r}\right)^N\right)$$
$$429607.68=2000\left(\frac{1}{(\frac{0.061678}{12})-(\frac{0.05}{12})}\right)\left(1-\left(\frac{1+\frac{0.05}{12}}{1+\frac{0.061678}{12}}\right)^{240}\right)$$
I still don't get the same answer than your book. However, I didn't try converting everything to a yearly payment. Calculate it rather than on 240 \$2000 payments, on 20 \$x payments.