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$n = 36$

$FV = 20000$

$r = 0.20$ 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?

Corvo
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  • Uh, where do you get interest rates like that? I'm game! – Oscar Lanzi Nov 07 '21 at 19:27
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    Hahaha just in the textbook of the crap college I do – Corvo Nov 07 '21 at 19:29
  • It's not clear to me how often the interest is compounded. We have it as an Annual Percentage Rate (APR), but idk how often interest is actually accrued. – Justin Benfield Nov 07 '21 at 19:48
  • @JustinBenfield 20% per year is the same as 1,53% per year. The 20% is an APR, but I converted this to a mensal percentage rate (1,53%) because the interest rate and PMT must be in the same base unit of time. Since the PMT is monthly, I converted the interest rate from annual to monthly.

    I use this formula to convert from anual to monthly:

    monthly interest rate $= (1+r)^{1/q}-1$ where $q$ is the number of months in a year, so:

    monthly rate $= (1+0,20) ^{1/12}-1$

     – Corvo Nov 07 '21 at 20:01
  • Ok so the interest is compounded monthly, I didn't see where in the problem statement that was communicated. – Justin Benfield Nov 07 '21 at 20:08
  • Where do you get 20%? – markvs Nov 07 '21 at 20:19
  • @markvs The 20% per year was given by the problem of the textbook. If I want to discover the PMT, I should convert the annual interest rate to monthly interest rate or convert 36 months to 3 years. – Corvo Nov 07 '21 at 20:44
  • @JustinBenfield If you use the equivalent interest rate formula, you found out that 20% per year is the same as 1,53% per month, right? At least that's how I learned... I also have the option to convert $n = 36$ months to years (3 years) and keep the annual interest rate (the 20%). – Corvo Nov 07 '21 at 20:46
  • It is not a realistic problem. I do not think there was time after 1980 when you could get 20%. It is more like 1% now. The thing is usually compounded daily (weekly, monthly ...). – markvs Nov 07 '21 at 21:01
  • @markvs I know, you're right. I'm studying at a low quality college just for formality. Just for this exercise you can already see the low level of this course I do... I learn more about finance from outside sources than from this ridiculous college, with this kind of stupid and useless question lol – Corvo Nov 07 '21 at 21:37

1 Answers1

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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

$$\begin{align*} FV &= PMT(1+r)^n + PMT (1+r)^{n-1} + \cdots + PMT(1+r)\\ &= PMT(1+r)^n \cdot \frac{1-(1+r)^{-n}}{1-(1+r)^{-1}}\\ &= PMT(1+r) \cdot \frac{(1+r)^n-1}{(1+r)-1} \end{align*}$$

peterwhy
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