Loan repayment calculation and monthly compounding interest problem

Question

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!

Answer 1

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*

Answer 2

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

Answer 3

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