effective rate loan installment

Question

Why is it that the effective rate is based on compounding and it is also paid only on the remaining loan balance in compounding interest we pay interest on interest? Can someone explain in to me? what do we mean that the effective rate is based on compounding.

In this example: 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

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

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

If I used 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.

So my question is since effective rate is based on compounding why didn't we use the formula mentioned above instead we used the EMI for reducing balance with function PMT which equals 30.79.

My question who is it correct that effective is based on compounding effect in which we pay more interest and yet the repayment is based on reducing balance where we pay less interest

Answer 1

With a 7% effective annual rate r = 0.07 and

number of periods per year  n = 12
periodic rate               x = (1 + r)^(1/n) - 1 = 0.00565415
number of years             t = 3
principal                   P = 1000

monthly repayment      d = x P (1 + 1/((1 + x)^(n t) - 1)) = 30.78
total repayment    n t d = 1108.08

If you were to compound £1000 monthly at the periodic rate for one year

A = P (1 + x)^n = 1070

This shows clearly the 7% annual gain.

Over three years

A = P (1 + x)^(n t) = 1225.04

The formula you have: A = P (1 + r/n)^(n t) is OK for use with a nominal rate, but not correct for an effective rate. It seems to be a popular shortcut or error; not accurate when using effective rates.

For a guide to the difference between nominal and effective rates see

https://en.wikipedia.org/wiki/Effective_interest_rate#Calculation

The 7% effective rate can be converted to a nominal annual rate compounded monthly by the following formula

nominal rate compounded monthly = n ((1 + 0.07)^(1/n) - 1) = 6.78497 %

Using this as r, your formula produces the same results as above

For one year

r = 0.0678497

A = P (1 + r/n)^n = 1070

Again, this shows clearly the 7% annual gain.

Over three years, the same result as before.

A = P (1 + r/n)^(n t) = 1225.04

The nominal rate is actually the periodic rate multiplied by the compounding frequency (or the number of periods per year)

0.0678497 = x n

so it would be more accurate to say that the nominal rate is based on compounding. The effective rate is independent of compounding frequency. That is to say, the effective rate does not change according to the number of periods in a year, as does the nominal rate.

E.g. 7% effective annual interest is equivalent to

52 ((1 + 0.07)^(1/52) - 1) = 6.77027 % nominal compounded weekly
12 ((1 + 0.07)^(1/12) - 1) = 6.78497 % nominal compounded monthly
4  ((1 + 0.07)^(1/4)  - 1) = 6.82341 % nominal compounded quarterly

APR in the US is always a nominal rate. In the UK and Europe APR is given as an effective rate.

Finally, there is no reason why the total loan repayments £1108.08 should equal A over 3 years. The loan is being repaid every month so the compounding is operating on a decreasing principal. By contrast, in calculating A there are no repayments being factored in.

Answer 2

Let me offer an example. I lend you $1000 at 6% interest per year. Our agreement is that you will pay me interest only for the first few years. $60 per year is 6%. But you are making monthly payments to me, five dollars per month. You are paying me the 6% but it’s really being paid as 1/2 of 1% per month. As I get that five dollar bill each month I put it in the bank, A generous one that also pays me 6%. Over one full year I have a bit more than $60 saved up from your payments from the effect of “compounding“.

To put it another way, if you paid me $60 at the very end of each year it would be a plain 6% rate. The fact that you were making multiple payments during the year is what creates the compounding effect.

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In response to

A = 1000*(1+0.07/12)^(36) = 1232.92

This would be true for a loan in which I lend you 1000 for 3 years, and you pay me in full at the very end. What a loan calculator does is to account for the declining balance each month. And there's another set of equations to use for loan payment vs a single deposit with compounding.