Comparing an Upfront Discount with Investing Installment Payments
Summary
The document compares paying a purchase price in equal, interest-free monthly installments with paying a discounted amount upfront while investing the full price at a monthly compound rate. It derives the account balance after the final installment by repeatedly applying the investment return and subtracting each payment, then expresses the result as a geometric sum.
The author proposes comparing that ending balance with the upfront discount to decide which payment option is financially preferable. The example considers a 12-month schedule, a monthly return of 0.7%, and a 3% discount, and concludes that the investment route yields about 4.8% relative to the full price. This is a useful time-value-of-money comparison, but the post is a question rather than a validated derivation. Its compounding convention and the timing of the monthly payments need to be specified carefully, and the assumption that the return is guaranteed is central to the conclusion.
Key ideas
- Equal installment payments can be compared with an upfront discount by accounting for investment growth over the payment period.
- The balance recurrence compounds the remaining funds and subtracts each installment.
- The proposed closed form uses a geometric series of monthly returns.
- The example favors installments under its stated return and discount assumptions.
- Payment timing and the reliability of the assumed investment return affect the comparison.
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# What is the name of this concept/formula?
# 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 $m<M$) 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.