# loan repayment- find the loan and interest paid

I have a loan at a nominal annual interest rate compounded monthly $$i^{(12)} = 12$$% and is repaid with $$120$$ monthly payments starting one month after the loan. The monthly payments are $$600$$.

I am asked to find the i) loan first, ii) find the total interest paid, iii) find the interest paid in 10th payment and iv) the principal repaid in the 20th payment.

I was not sure if I calculated and used the right formulas. I calculated the loan by doing

$$n = 120/12 = 10$$ years

since the nominal interest $$i^{(12)} = 12%$$ then $$i = 0.12/12 = 0.01$$

Since there are $$120$$ monthly payments of $$600$$ dollars each, I did

\require{enclose} \begin{align} L = 600 a_{\enclose{actuarial}{10} i} = 600(1-(1/1.12)^{10}/0.01 = 40681.60 \end{align}

For the total interest: I took the monthly payments of $$600$$ dollars multiplied by the $$120$$ and subtract by the total loan found above to get $$31318.40$$

For finding the interest paid in 10th payment and the principal repaid in the 20th payment I wasn't sure if I did it correctly using the table form, i did

$$\begin{array}{c|c|c|c|c} \text{Period} & \text{End Payment} & \text{Interest repaid} & \text{Priniple repaid} & \text{Outstanding balance} \\ \hline 0 & 0 & 0 & 0 & 40681.60 \\ 1 & 600 & 406.81 & 193.184 & 40488.42 \\ 2 & 600 & 404.88 & 195.11 & 40293.30 \\ 3 & 600 & 402.93 & 197.06 & 40096.23 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 10 & 600 & 386.60 & 213.39 & 38447.07 \\ \end{array}$$

I'm assuming towards the end of the 120th payment, the outstanding should be zero. I don't have a special program other than excel... and calculating a specific number of payments can be tedious. Is there a quicker way to calculate?

• What is your loan? Mar 27, 2021 at 21:49
• I can't figure out how to type the annuity symbol in MathJax. Anyway, you need the $a^{(12)}$ version. Basically, you've got $120$ payments at $1\%$ a month, so forget about the $10$ years and the $12\%$ Mar 27, 2021 at 21:55

## 1 Answer

As I said in a comment, the amount of the loan should be calculated as $$\require{enclose} L = 600 a^{(12)}_{\enclose{actuarial}{10} i} = 600\left(1-(1/1.01)^{120}\right)/0.01 = 41820.31$$

You calculation of the total interest is correct. It's just the total payments less the amount of the loan.

The interest paid in the $$10$$ payment is $$1\%$$ of the amount of the loan outstanding after $$9$$ payments. Since there are then $$111$$ payments left, you do it it just as you calculated the amount of the loan, but with $$111$$ in the exponent instead of $$120$$.

For the amount of principal paid in the $$20$$ payment, calculate the amount of interest in the payment as above. The amount of principal in the payment is $$600$$ minus the amount of interest in the payment.

• ah, I forgot I am not using years but months. I also noticed that it is one month after so there shouldn't be any period 0. so for the 20th payment it will be 31514.43, but i want the principal repaid, so the 19th payment gives 31697.46. I take the difference to get 183.02. Mar 27, 2021 at 23:11
• @comp890 Yes, that's another way to do it, though I think the way I suggested is easier. Mar 27, 2021 at 23:50