# What is the name of this concept/formula?

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 $$m) would be worth it over paying it in $$\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.

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