$n = 36$
$FV = 20000$
$r = 0.20$ per year or $\approx$ $0.0153$ per month
$PMT =$ ?
Question: What type of cash flow is this?
I know the first payment starts in $t = 0$, so, I think the formula I should apply to this is:
$$FV =PMT \times (1+r) \times \frac{(1+ r)^n - 1}{r}$$
I understand that $\frac{(1+ r)^n - 1}{r}$ in the above formula is the future value factor (English ins't my first language, so I don't know if I translated it correctly), but I can't understand the $(1+r)$ after the $PMT$.
I think this $1+r$ is related with the $PMT$ made in $t = 0$ (the first payment made TODAY), but this is still not very clear for me.
Can anyone explain me how to get to the formula that solves the problem and why you get this?
Maybe you are used to the formula of FV if the $n$ payments are made at the end of each of the $n$ periods?
$$FV \text{ for payment at the END of periods} = PMT \cdot \frac{(1+r)^n - 1}{r}$$
But now every payment is made at the beginning of each period, so each payment is compounded for one more period:
$$FV = PMT (1+r) \cdot \frac{(1+r)^n-1}{r}$$
The first PMT today will be compounded for $n$ periods, so the FV of this PMT alone is $PMT(1+r)^n$.
The second PMT one period later will be compounded for $n-1$ periods, so the FV of this PMT alone is $PMT(1+r)^{n-1}$.
Then the final PMT $n-1$ periods later will be compounded for $1$ period, so the FV of that PMT alone is $PMT(1+r)$.
By considering the total FV as a geometric series, the total FV of all PMTs is
$$\begin{align*} FV &= PMT(1+r)^n + PMT (1+r)^{n-1} + \cdots + PMT(1+r)\\ &= PMT(1+r)^n \cdot \frac{1-(1+r)^{-n}}{1-(1+r)^{-1}}\\ &= PMT(1+r) \cdot \frac{(1+r)^n-1}{(1+r)-1} \end{align*}$$