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If I have a payment amount, interest rate, and term can I determine the Present Value?

for example:

Payment: $631.35
Rate: 8.50% 
Term: 3 years
num pay per year: 12
Present Value: ?  ($20,000)

Thanks for your help!

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up vote 2 down vote accepted

The problem is, details of the problem are missing. Are the payments made at the beginning or end of each month? Is the interest rate the annual effective rate, or is it the nominal rate convertible 12 times per year? Also, you haven't told us your background. Do you know geometric sum formulas? Do you have a financial calculator?

I will assume the payments are made at the end of each month and that the interest rate is the nominal rate convertible 12 times per year (or compounded 12 times a year). This means that the actual interest rate is 8.5% / 12 per year. Let's call this rate j.

Now, if the first payment is 1 month from now, its present value is $631.35 (1 + j)^{-1}$, since $j$ is the monthly interest rate. The second payment is 2 months from now, so its PV is $631.35 (1 + j)^{-2}$, and so on, until the 36th payment, which happens 36 months from now, has PV $631.35 (1 + j)^{-36}$. Therefore, the PV of all the payments is the sum of the PV of each payment (this involves a geometric series):

$\begin{align*} PV &= 631.35 (1 + j)^{-1} + 631.35 (1 + j)^{-2} + 631.35 (1 + j)^{-3} + \cdots + 631.35 (1 + j)^{-36} \\ &= 631.35 \cdot \frac{(1+j)^{-1} + (1+j)^{-37}}{1 - (1 + j)^{-1}} \\ &= 631.35 \cdot \frac{1 - (1+j)^{-36}}{j} \end{align*}$

where in the second step I used the sum of a geometric sum formula, and in the third step I multiplied through by $1+j$ to simplify things.

Plugging in $j$, as above, I get \$19999.98. I did not round early throughout this, so this answer is probably more correct than \$20000, but maybe your teacher doesn't care. And, it looks like I made the correct assumptions, since I got almost exactly the answer you were expecting.

By the way, in general, the PV of an annuity that pays P at the end of each period for $n$ periods, where the interest rate is rate $i$ per payment period, is

$$PV = P \cdot\frac{1 - (1+i)^{-n}}{i}$$

If you have that memorized, you don't have to bother with the geometric series any more. Note, $i$ is the rate per payment period, so if you're given an interest rate that doesn't match up with the payment period, as in this problem, you will need to figure that out.

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