APR – the interpretation from non business degree guy

APR is probably the most commonly used financial term that everyone have access to.
When you apply for credit card, different services offer different APRs, when you apply for auto loans, they offer you APR, when you apply for property loans everyone is talking about APR. To fully understand what is APR will let you not only understand why you are paying that much interest but also help you avoid the traps that sales rep set up for you.

Look at the image below, if you don’t fully understand why the total amount you pay 8.56 * 12 = 102.72 doesn’t equal to 105, (100 + 100 * 5%). Then this article will be helpful to you.

First of first, people knows the interest is calcuated based on the rate times the loan. However, to reduce the risk of one amount huge loan payback, lenders tend to require the borrow to pay back at periodic time like yearly, monthly or even daily(for certain credit card). In that case, the interest is not as straight forward as you might imagine. On the other hand, APR(annual interest rate) need to be used to calculate the monthly interest rate to suffice the granulairty.

Say if we borrow $100 and the APR is 5%. If we pay at the end of the year, indeed, we need to pay $105 following basic math. However, if we pay every half year. Then we know the half year interest rate is 5%/2 = 2.5%.

Then the payment will be like:
at first payment, we pay half the money we borrow plus the interest for $100:
$50 + $100 * (5%/2) = 52.5
Then by the end of the year, we pay the rest of the money and the interest for the other half.
$50 + $50 * (5%/2) = 51.25
In total, we payed only $103.75.

Say we pay by monthly… the payments will look like this:

Month 1: (100/12) + (100) * (5%/12) = 8.75
Month 2: (100/12) + (100-100/12) * (5%/12) = 8.715278

Month N: (100/12) + (100-(N-1)*100/12) * (5%/12) = …
Month 12: (100/12) + (100-11*100/12) * (5%/12) = 8.368056

When you add up all the months, it is 102.7083, if you want to pay fixed amount of money every month, then 102.7083/12 = 8.559025 ~ $8.56.
Clearly, there are usually two parts in your monthly payment, the principle and the interest. Clearly, you have to pay your interest in full first, and then the rest of your montly payment will go to your principle.

I hope the explainations above is making sense to you, as you can see, there is one part that I am using 5% and simply divide by 12 to calculate the monthly interest rate. Am I doing it right? or am I doing to accurately enough?

Then we need to talk about compound interest, which basically means, if your monthly interest rate is m, your first month payment due should be 100(1+m), then if the loan carrier forward without payment, you need to pay the interest you owe the first month, then the due amount next month will be 100(1+m)^2… so on and so forth. The most scientific expression between the (MIR)monthly interest rate and APR should be

APR = (1+MIR)^12
my version: APR=MIR*12
Are they really that different?

I guess if you have taken college math, you probably can recall Taylor Series some point some where in your curriculum, it basically says a guys called Brook Taylor claims that any function could be expressed by a series of terms related with derivatives… anyway, somewhere it says.

(1+x)^n = 1 + x n + x^2 n(n-1)/2 + …

Let’s correlate this to APR…
n is kind of like the payment number, 12 in this case.
x is like the MIR(montly interest rate), about 0.4%…

the first term is about xn… and then the second term is about (xn)^2.. so on and so forth…

xn is about 0.048
(xn)^2 is about 0.002304, which is only less than 5% of the second term… and the rest of the other terms will keep decreasing, so if you don’t have a calculator in front of you to help solve

(1+m)^12=(1+0.05), which m = (1+0.05)^(1/12)-1 = 0.004074124

You can simply use
m = 1 + 0.05/12 -1 = 0.004166667

As you can see, the approximation will give you a slightly higher interest rate, I guess bank will be more than happy to use the higher and simpler approach to make their life easier.

In the end, if we generalize the monthly calculation, say we use C is represent the total amount of money we borrowed, N represent the total number of payments, and R to represent the APR..

Then the whole thing will be like:

SUM(C/N + (C – (n-1)C/N) * R/12) where n changes from 1 to N.

The final result for the total interest will be like: CR*(N+1)/24

Using this equation, we can quickly estimate that the interest for a 30 year loan a 5% APR for a 200K loan will be:

200K * 5% (30*12 + 1) / 24 = 150416.7 ~ 150K

However, using the loan calculator online, they gave the result of

186K total interest
apr186

I guess the formula that I derived might only work for short term loan calculation as a good approximation. The people at the bank might have done something more sophisticated than what I assumed. After I quickly did a Google search, loan formula, a page gave me a really complex equation, I guess it will be a homework for me tomorrow to interpret and reverse engineer how that works.

READING LIST:

(1) Finance Formulas: Loan Payment

(2) Derivation of Mortgage Loan Formula

(3) Wikipedia: Fixed Rate Mortgate

(4) Loan or Investment Formulas

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