# Loan repayment calculation and monthly compounding interest problem

I need to write a mock up application that returns a quote to potential borrowers. The specification says that "The monthly and total repayment should use monthly compounding interest".

Program input: Requested Amount, Rate, Loan length in months

Program output: monthly repayment, total repayment amount

This is the example they give:

``````Input:
Requested amount: £1000
Rate: 7.0%
Months: 36

Output:
Monthly repayment: £30.78
Total repayment: £1108.10
``````

The problem is that I don't know how they arrived at this result. After consulting some websites, for example here. I found that the formula for calculating the compound interest rate is

``````A = P (1 + r/n) ^ nt
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual interest rate (decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for
``````

Using this on our example we get A = 1000*(1+0.07/12)^(36) = 1232.92, which is not 1108.10 as they say in their example.

I am wondering if their example is wrong or am I missing something here

Thanks!

• Total repayment is not the future value. Total repayment is n t P which is 36*30.78 = 1108.08 . But it is not very important, the main thing is to compute P. Oct 10, 2015 at 18:43
• Duplicate of money.stackexchange.com/questions/54655 Oct 11, 2015 at 16:33

The calculation can be made on the basis that the loan is equal to the sum of the repayments discounted to present value. (For more information see Calculating the Present Value of an Ordinary Annuity.)

With

``````s = value of loan
d = periodic repayment
r = periodic interest rate
n = number of periods
``````

Deriving the loan formula from the simple discount summation. ``````d = (r s)/(1 - (1 + r)^-n)
``````

As you can see, this is the same as the loan formula given here.

In the UK and Europe APR is usually quoted as the effective interest rate while in the US it is quoted as a nominal rate. (Also, in the US the effective APR is usually called the annual percentage yield, APY, not APR.)

Using the effective interest rate finds the expected answer.

``````Requested amount, s = £1000
Effective Rate: 7.0%, ∴ monthly rate, r = (1 + 0.07)^(1/12) - 1
Months, n = 36

d = (r s)/(1 - (1 + r)^-n) = 30.7789
``````

The total repayment is `£30.78 * n = £1108.08`

Using a nominal interest rate does not give the expected answer.

``````Requested amount, s = £1000
Nominal Rate: 7.0% compounded monthly, ∴ monthly r = 0.07/12
Months, n = 36

d = (r s)/(1 - (1 + r)^-n) = 30.8771   *incorrect*
``````
• Have an idea how to calculate rate if we have requestedAmount & total repayment, please ? Mar 7, 2019 at 13:28
• @aguetat With `s = requestedAmount` and `n*d = total repayment` you can solve `s - (d - d (1 + r)^-n)/r = 0` for the periodic rate `r`, as shown here, solved graphically. Then convert the periodic rate to an effective or nominal annual rate as required. You could use an online solver, e.g. using the above figures and this website type in `1000-(30.7789-30.7789*(1+r)^-36)/r=0` and solve for variable `r`. Mar 7, 2019 at 15:11
• Have your the solution of this equation (i am looking for the mathematical formula), please ? Mar 7, 2019 at 15:12
• @aguetat It doesn't have an algebraic solution. It has to be solved numerically. Mar 7, 2019 at 15:14

You need the Present Value, not Future Value formula for this. The loan amount or 1000 is paid/received now (not in the future). The formula is \$ PMT = PV (r/n)(1+r/n)^{nt} / [(1+r/n)^{nt} - 1] \$ See for example http://www.calculatorsoup.com/calculators/financial/loan-calculator.php

With PV = 1000, r=0.07, n=12, t=3 we get PMT = 30.877 per month

• Present Value is £1000 and he already had it. The PMT part is correct though. Oct 11, 2015 at 3:33

Well, what you are asking is EMI, which comes to 30.78 in your case. The formula you are applying is of compounding a value, which is completely different.

In EMI, person keeps paying money every month or any other period as specified. This amount is firstly allocated towards the interest for the period and the balance for principal amount. So, in effect principal keeps decreasing and subsequently interest thereon. Also, since, interest is getting paid every time it becomes due, compounding actually do not happen at all.

In the case of compounding, interest gets applied at certain interval, but do not get paid. So, in effect every time when interest gets applied, it applies on complete Principal outstanding as well as interest unpaid. Hence, this complete amount gets payable at the end. In this case, total amount payable is obviously high, because of 2 reasons: 1. Since, Principal gets unpaid during whole period, you are paying interest on complete amount for complete period. 2. You will be paying interest on interest (compounding of interest) since you are not paying it as it is becoming due.

Hence, both are different. You need to find EMI calculator or EMI formula, to achieve your purpose.

EDIT: The formula for calculating EMI:

``````EMI = (L × I) × [(1 + I)^ N ÷ {(1 + I)^ N } -1]

Where
L = loan amount
l = interest rate per annum divided by 12
^ = to the power of
N = loan period in months
``````

Assuming a loan of Rs. 1 lakh at 9 % per annum, repayable in 15 years, the EMI calculation using the formula will be: EMI = (1,00,000 × 0.0075) × [(1 + 0.0075) 180 ÷ {(1+0.0075) 180 } - 1] = 750 × [3.838 ÷ 2.838] = 750 × 1.35236 = 1,014

• How does ths answer the question? Oct 11, 2015 at 14:12
• It indeed tells what is the difference between the two, which OP don't seems to know. It also tells when to use when, why and how. Since, it describes the whole logic, one would be grasp and the question. If you need just a formula, then wait a few minutes while I may post it. Oct 11, 2015 at 14:20