How do I calculate fixed amount of monthly payments to pay off a debt with interest?

Question

If my balance owed is 10,000 and my APR is 27%, if I wanted to pay the debt in full in 4 months, what would be my payments per month?

Here is the website where it takes inputs: Debt Repayment Calculator

According to the site, my payments would be 2,642 per month.

How does it calculate the answer?

Note: The company uses 'daily average of balance' and uses compound interest, the billing cycles are every 30 days.

Here is a model to better understand the question:

Debt: 1000
Billing cycle 1 has 30 days, at the end of this cycle, we were charged: (1000 * .27 (APR) * 30 (billing cycle days)) / 365 (365 calendar days) = ~22.1912 dollars of interest.

Debt: 1022.1912
Billing cycle 2: has 30 days, at the end of this cycle, we were charged: ~22.6842 (using the previous formula)

Debt 1044.8754
Billing cycle 3: .....

So what amount per month in 4 months would pay the full balance considering interest?

What we know is that $1000/4 = 250$, so we know that it has to be greater than 250 considering we get interest charges per month.

Answer 1

It should be obvious the answer will be a fourth of the principal plus the interest accrued over the 4 months. We have that the annual percentage rate is 27%, so dividing by twelve, we have a monthly percentage rate of $\frac{9}{4}$%. Substituting into the compound interest formula, we have $10000*1.0225^4$ as the final balance. That is around 10930 dollars, or 2732 dollars a month. Notice that this is if you saved that amount each month, and paid at the end of the 4 months. If you paid 2642.19$ a month, you would pay the debt off exactly in the 4th payment. Of course, to minimize interest, you should pay as much as possible each month.

Answer 2

I will answer a slightly more general question, and then apply it to your question.

Suppose the balance (in dollars) = $B$, the interest (as a percentage) = $I$, and you make constant monthly payments of $m$ dollars. Let us write a function $f_n$ that gives the balance left after $n$ months. After putting some thought into it, we can agree as a recursive function we have

\begin{align*} f_0 &= B,\\ f_n &= (f_{n-1}-m) + I \, (f_{n-1}-m),\\ &= (1+I)f_{n-1} - (1+I)m. \end{align*}

Solving this recurrence relation is not too hard. You can see that

\begin{align*} f_n &= (1+I)f_{n-1} - (1+I)m\\ &= (1+I)^2 f_{n-2} - 2(1+I)m\\ &\hspace{2mm}\vdots\\ &= (1+I)^n B - n(1+I)m. \end{align*}

Now that we have this function, we want to know what $m$ needs to be so that $f_4$ = 0. Solving $f_4 = 0$ for $m$ yields $$(1+I)^4 B = 4(1+I)m \implies m = \frac{(1+I)^3 B}{4}.$$

Now, plug in $B = 10000$, $I = \frac{(.27)*(30)}{365} \approx .0222$ and see that this yields $m = 2670.2237$ which is very close to the answer that website gave you and the discrepancy likely has to do with the fact that there is not 30 days in every month, or something to do with the APR. To be honest I don't know anything about how APR's work, or most other things from the real world either :P