Interest rate question

Question

I'm having a little trouble figuring this out. Suppose I took out a loan with the following terms:

Loan Amount: 1,000
Interest Rate (APR): 10%
Compound Frequency: Monthly (12 compounding periods)
Number of Payments: 12
Number of Years: 1.0

Compound loan total: 1,104.71
Monthly Payment: 87.92
Expected Loan Cost (monthly payment * num payments): 87.92*12 = 1,055.04

Plugging in the above numbers in an online payment calculator results in the same monthly payment of $87.92, so now I'm confused: why is the final amount different?

I tried another compound interest calculator online and I got the same result: $1,055.04

The compound loan total seems correct: if I'm getting charged 10% per year, at the end of a 1 year loan I should have paid 10% + compound interest which adds up to $1,104.71. The actual interest rate will be about 5.5% if I'm making monthly payments of $87.92. What am I missing here?

Update: This compound interest calculator results in the $1,104.71.

Answer 1

Each monthly payment you make reduces your principal. So the loan interest calculated every month is different. On average, you have approximately half of the original principal every months - hence the $54 interest when you have 10% APR.

The compound interest calculator calculates the interest accumulated, but it doesn't take into the account the reduction of principal because of the monthly payments. You should use the amortization calculator for that, and that would give you the total of $1054.99 (depending on rounding etc, its close to the $1055.04 you got).

Answer 2

The number of payments is the source of confusion here.

The interest that's typically charged in a monthly-payment scenario is on the principal remaining after each payment. With each monthly payment, the principal balance owing is reduced. So, each subsequent month, the amount of interest charged decreases in line with the principal.

You'd only pay $104.71 total interest if the $1000 loan was outstanding for the entire year. This kind of loan would be one with a single "balloon payment" due at the end of the loan term, not where the payments are spread out over the life of the loan.

In a situation with 12 monthly payments and the loan paid off by the final payment, you are borrowing – on average – a bit over half of that amount over the course of the year. Hence why the interest is ~$55 as opposed to ~$100+.

Answer 3

Here are some step-by-step calculations so you can see fairly clearly what's going on:-

The effective annual interest rate is given by

ear = (1 + i/n)^n - 1

where i is the nominal interest rate and n is the number of compounding periods.

ear = (1 + 0.1/12)^12 - 1 = 0.104713 = 10.4713 %

The monthly rate is

r = (ear + 1)^(1/n) - 1 = 0.00833333 = 0.833333 %

The monthly repayment is given by the formula

p = r*pv/(1 - (1 + r)^-n)

where pv is the present value of the loan

p = 0.00833333*1000/(1 - (1 + 0.00833333)^-12) = 87.9159

This gives a total repayment of p*12 = 1054.99

You expected the loan to cost (1 + ear)*pv = (1 + 0.104713)*1000 = 1104.71 but the repayments progressively reduce the amount owed so the total repayment is only 1054.99.