I stumbled on this not so complicated concept and couldn't figure out what it's called.

I want to buy something that costs $M$ units of money, and have to pay it in $n$ months at a rate of $\frac M n$ every month, with no interest (just to simplify the calculations). I take $M$ units of money and immediately invest it with a compound interest rate of $r$ per month.

At the end of the first month, I'll have $rM - \frac M n$ units of money.

At the end of the second month, I'll have $r(rM - \frac M n) - \frac M n$.

And so on.

This adds up to $M[r^n - \frac 1 n (\sum_{k=1}^n r^{n-k})]$, which simplifies to $M[r^n -\frac 1 n \frac {r^n-1} {r-1}]$.

The idea behind this was to find out if given the choice to pay $m$ units of money upfront (where $m<M$) would be worth it over paying it in $\frac M n$ monthly payments, given I can guarantee the monthly interest rate $r$.

The conclusion I got to (which may be wrong) is that it is worth it if $r^n - \frac 1 n \frac {r^n-1} {r-1} > d$, where $d$ is the discount rate $1-\frac m M$.

For example, if offered a discount rate of $d=3\%$ upfront, over the option of paying the full rate over 12 months, and I have $M$ units of money and can guarantee an interest rate of $r=0.7\%$ a month, I shouldn't take the discount, since at the end of the 12 months I'll have a return of $\approx 4.8\%$ ($>3\%$) over the full amount $M$.

I'm sure this (or something resembling this) is a well known concept/formula, I just couldn't find what it is called, so some guidance would be appreciated.

  • 1
    $\begingroup$ Some form of EMI calculation on a loan maybe? $\endgroup$
    – nimbus3000
    Commented Feb 5, 2021 at 3:31
  • 1
    $\begingroup$ This is just discounting of cashflows, or present value of cashflows, using traditional annuity and or payments of loan with capital and interest. $\endgroup$
    – Attack68
    Commented Feb 5, 2021 at 7:10


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.