Methods of solving this exam FM problem with geometric-investments.

Question

The problem I am working on is as follows.

Matthew makes a series of payments at the beginning of each year for $20$ years. The first payment is $100$. Each subsequent payment through the tenth year increases by $5\%$ from the previous payment. After the tenth payment, each payment decreases by $5\%$ from the previous payment. Calculate the present value of these payments at the time the first payment is made using an annual effective rate of $7\%$.

In this exam, time is of essence and the way I tried to solve this problem took much longer and I wanted to and what is worse, did not get the answer.

What I tried was the following.

$$\begin{align} PV &= 100(1+(1.05v)+(1.05v)^2+ \cdots +(1.05v)^9)+100(1.05^9v^{10})(.95+.95^2v+ \cdots +.95^{10}v^{9}) \\ &=\frac{100}{105v}(\alpha+\alpha^2+ \cdots +\alpha^{10})+100(1.05v)^9(\beta + \beta^2+ \cdots +\beta^{10})\\ &=\frac{100}{105v}a_{\overline{10}\rceil j}+100(1.05v)^9a_{\overline{10}\rceil k}\\ & \approx 1308.4 \end{align}$$

where $\alpha=1.05v, \ \beta=.95v$ and $j=\alpha^{-1}-1, \ k=\beta^{-1}-1$.

In retrospect, I feel as though using the geometric sum would have been a safer route to take rather than the annuity immediate or due. But since my calculations will end up being the same I am thinking that I missed something (the answer is supposedly 1385)

Can someone help me out? I did not quite understand the solution that the book provides for it uses some formula that I am not fully familiar with.

Answer 1

For an annuity of $n$ periods (starting a period from now, thus $n$ payments) with initial payment $A$ that grows at $g$ per period we have that PV of this annuity with effective rate $r$

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

Note: This essentially finds present value of payments one period before first. With this it should be much easier to find answer

Answer 2

Your cash flow, written out, is correct. However, your simplification and expression in terms of actuarial notation is not quite optimal because the formula reverts to the usual finite geometric series sum anyway: Let $\alpha = 1.05v = 1/(j+1)$, $\beta = 0.95v = 1/(k+1)$ be the equivalent present value discount factors. Then $$\begin{align*} PV &= 100(1 + \alpha + \cdots + \alpha^9) + 100 (1.05)^9 (0.95) v^{10} (1 + \beta + \cdots + \beta^9) \\ &= 100 a_{\overline{10}\rceil j} + 95 (1.05)^9 v^{10} a_{\overline{10}\rceil k} \\ &= 100 \frac{1 - \alpha^{10}}{1 - \alpha} + 95 (1.05)^9 v^{10} \frac{1 - \beta^{10}}{1- \beta}. \end{align*}$$ Sure, you could use a BA II financial calculator to calculate the annuities-immediate, but why in this case?

The answer, to more significant digits, is $1384.645\ldots$.