# Find repayment amount on loan repaying a partial amount of the principal in a definite period of time.

I'm trying to calculate the Monthly payment amount on a loan where I want to be able to define the Principal amount that should be repaid and an Outstanding balance after the repayments.

Example:

1749€ is the initial principal amount supposed to be paid in 24 months but I want to take 486,44€ of it and repay that amount in 6 months with a 10% APR compounded monthly. So at the end of the 6 months period, I should have an outstanding balance of 1262,66€.

the amount should be 93,96€ but can't find the perfect formula for it.

Thanks!

• Does the 1749€ not accrue interest as well? Oct 24 '18 at 16:18
• Yep! that's exactly the point! Oct 24 '18 at 16:20
• Your question seems in conflict then, if you pay 486,44€ you'll have more than 1262,66€ left at the end of the six months. So which do you want? To pay 486,44€ or to have 1262,66€ ? Do you want to make these payments in equal proportion every month for six months? Oct 24 '18 at 16:21
• When I say 486,44 it's the principal part that is paid in 6 months. 563,85€ will be the total I'll pay which include the interests during that period (93,98*6). So I should then have a balance of 1262,44€ at the end. Oct 24 '18 at 16:25

I'm not sure I'm interpreting your question correctly. Let me know.

You have a loan for 1749 with annual interest (APR) at 10% compounded monthly. You want to make monthly payments for six months so that the outstanding balance will be 1262,66 at the end.

The formula is this

$$P = \dfrac{r}{(1+r)^n -1}(PV(1+r)^n - OB)$$

where

• $$r$$ is in the interest rate
• $$n$$ is the number of periods (months)
• $$PV$$ is the initial loan amount
• $$OB$$ is the outstanding balance you desire

So with $$r=0,10/12$$, $$n=6$$, $$PV=1749$$, and $$OB=1262,66$$ we can solve for $$P$$ and find $$P=93,96$$. So in order to have an outstanding balance of $$1262,66$$ at the end of the six months, you need to pay $$93,96$$ a month or a total amount of $$93,96*6=563,76$$.

Addendum: I hope I can provide you with some intuition so this isn't just a formula. Hopefully you know

$$PV(1+r)^n$$

gives the total amount due with interest. So

$$PV(1+r)^n - OB$$

is the total amount you'll have to pay in order for $$OB$$ to be left. Say you pay $$P$$ each period. You take out the loan on January 1. You make your first payment on February 1. When the 5 months have gone by, that first payment will have $$P(1+r)^5$$ value. On March 1 you make your second payment $$P$$. When 4 of the remaining months have gone by that second payment will have $$P(1+r)^4$$ value. Continue this for April 1, May 1, and so on. Let's call $$TP$$ the total payment. Hopefully it is clear that

$$TP = P + P(1+r)^1 + P(1+r)^2 + P(1+r)^3 + P(1+r)^4 + P(1+r)^5$$

This can seem very magical, but bear with me. We can simplify this with a clever trick. Multiply by (1+r) and subtract to get

\begin{align*} TP(1+r) &= P(1+r) + P(1+r)^2 + P(1+r)^3 + P(1+r)^4 + P(1+r)^5 + P(1+r)^6\\\\ -\phantom{y=}&\\ TP &= P + P(1+r)^1 + P(1+r)^2 + P(1+r)^3 + P(1+r)^4 + P(1+r)^5\\ \_\_\_\_\_\_\_\_\_\_&\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ TP(1+r) - TP &= P(1+r)^6 - P \end{align*}

Now we can solve for $$TP$$ \begin{align} TP((1+r) - 1) &= P((1+r)^6 - 1 )\\ TP &= P\dfrac{(1+r)^6 - 1 }{(1+r) - 1)}\\ &=P\dfrac{(1+r)^6 - 1}{r} \end{align}

Now we can find the payment amount by setting total payment equal to total balance minus left over balance. \begin{align} TP &= PV(1+r)^6 - OB\\ P\dfrac{(1+r)^6 - 1}{r} &= PV(1+r)^6 - OB\\ P &= \dfrac{r}{(1+r)^6 - 1}(PV(1+r)^6 - OB) \end{align}

This was for $$n=6$$ but you can extrapolate to general $$n$$. I hope this provides some insight.

• Thanks, @Almacomet for your help, I think you understood well my question, and my mistake is on the way I expressed the interest rate, I see that you apply 10%/ month where It should be 0.833% (10%/12). I will test it with the right interest rate... Oct 24 '18 at 17:06
• I tested with the right interest rate and indeed the formula is correct! Thank you very much for the help! Oct 24 '18 at 17:18
• @ALM83 I changed to fit what you have described. As a general note, the phrasing is "10% annual rate compounded monthly" or "10% APR compounded monthly." I also added a derivation. Oct 24 '18 at 18:30
• Great! Yeah, I've seen, I'll adapt my description as well. I see that you might have forgotten a figure from the first answer with the 10% "P=237,93". Thank you for the derivation indeed it will help to understand the formula. Oct 24 '18 at 18:38