Geometric progression in annuity

Question

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

$$\left(1+\frac{6\%}{12}\right)=1+i$$

Hence,

$$1+i = 1.005$$

The present value at the moment of the first deposit is

$$2,000\left(1+\frac{1+r}{1+i}+\left(\frac{1+r}{1+i}\right)^2+ \dots + \left(\frac{1+r}{1+i}\right)^{239}\right)$$

(Note: there are 240 conversions in 20 years)

so,

$$2,000 \frac{1-\left(\frac{1+r}{1+i}\right)^{-240}}{1-\frac{1+r}{1+i}}$$

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.

Answer 1

What I think you are looking for is the present value of a growing annuity.

1. Your problem says that your nominal interest rate is 6%, you will have to find the effective interest rate:

$$r=(1+\frac{r_n}{12})^{12}-1$$

$$0.061678=(1+\frac{0.06}{12})^{12}-1$$

2. Then you can find the present value of the annuity:

$$PV=C\left(\frac{1}{r-g}\right)\left(1-\left(\frac{1+g}{1+r}\right)^N\right)$$

$$429607.68=2000\left(\frac{1}{(\frac{0.061678}{12})-(\frac{0.05}{12})}\right)\left(1-\left(\frac{1+\frac{0.05}{12}}{1+\frac{0.061678}{12}}\right)^{240}\right)$$

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.

Answer 2

I think that you're looking to simplify this expression. $$\begin{array}{lll} PV&=&2000\bigg[(1.005)+(1.005)^2+\dots+(1.005)^{12}\bigg]\\ &&+2000(1.05)^1\bigg[(1.005)^{12\cdot 1+1}+(1.005)^{12\cdot 1+2}+\dots+(1.005)^{12\cdot 1+12}\bigg]\\ &&+2000(1.05)^2\bigg[(1.005)^{12\cdot 2+1}+(1.005)^{12\cdot 2+2}+\dots+(1.005)^{12\cdot 2+12}\bigg]\\ &&\dots\\ &&+2000(1.05)^{19}\bigg[(1.005)^{12\cdot 19+1}+(1.005)^{12\cdot 19+2}+\dots+(1.005)^{12\cdot 19+12}\bigg]\\ \end{array}$$