# How to calculate Annual Percentage Rate

How to calculate Annual Percentage Rate (APR) given the following:

A - Loan borrowed at the beginning (USD 1000),
B - Loan total costs paid at the end (USD 2000),
c - Number of compounding periods per year (52 weeks),
k - Number of periods to pay the loan (60 weeks)

All the formulas found in literature use nominal interest rate but here we do not have it.

By APR I mean: http://en.wikipedia.org/wiki/Annual_percentage_rate

I have constructed a spreadsheet where I can find APR with Excel Solver by changing nominal interest rate. Wouldn't there be more elegant solution for that?

• Your reference gives two definitions for APR. Which do you want? – DJohnM Jan 20 '15 at 17:23
• Is this a real situation you are facing? The rate is too high to reflect anything a lender in the US would charge. – JTP - Apologise to Monica Jan 20 '15 at 17:32
• Are there any payments in addition to the \$2000 at the end of the loan period? – DJohnM Jan 20 '15 at 23:40

The APR for a loan with constant repayments made at regular intervals can be calculated by solving this formula:

http://www.financeformulas.net/Loan_Payment_Formula.html

``````s = 1000;
n = 52;
t = 60/52;
``````

The periodic payments amount to a total cost of \$2,000.

``````pp = 2000.0/60;
``````

This next step solves `pp = (s p)/(1 - (1 + p)^(-n t))` for `p`. (Mathematica used.)

``````p = Last@Reduce[pp == (s p)/(1 - (1 + p)^(-n t)), p, Reals]
``````

0.0263204

Calculating the effective annual interest rate, `r`, from the periodic rate, `p`:

``````r = (1 + p)^52 - 1
``````

2.86112

The effective annual rate is 286.11 %

or, if a nominal rate, `i`, is required:

``````i = p * 52 = 0.0263204 * 52 = 136.9 % nominal rate compounded weekly
``````

Calculation Check

Checking the principal is calculated correctly. This is basically the summation in the page linked in the OP's question:

http://en.wikipedia.org/wiki/Annual_percentage_rate#European_Union

i.e. `s = Σ pp (1 + r)^-(k/n)` for `k = 1 to 60`

The loan formula used earlier is actually inducted from the summation.

So, running the check by back-calculating the loan principal:

``````Sum[pp (1 + r)^-(k/n), {k, 60}]
``````

1000

Yup, checks out.

Note the summation uses the effective annual rate, not the nominal rate.

• Are there any periodic payments, or only the "loan total cost paid at the end"? – DJohnM Jan 20 '15 at 17:55
• Yes, there are 60 periodic payments of `\$2000/60 = \$33.333` – Chris Degnen Jan 20 '15 at 18:09
• 286%. This is why it seemed a hypothetical. Don't we have a maximum rate in the US? – JTP - Apologise to Monica Jan 20 '15 at 18:18
• Lol, yeah, check out pay-day loans! - bbc.co.uk/consumer/24746198 – Chris Degnen Jan 20 '15 at 18:25
• I follow your computations, but the effective APR comes out a bit higher than it seems like it should be to me (particularly when figuring out the rate for using in a basic mortgage type calculator that I modified to work with this structure, which would be much lower - less than half that). I know 'effective interest rate' and actual rate are different, but I wonder if you could explain the meaning behind the terms, particularly when you expand by (1+p)^52 (which is where I am not as sure I understand the meaning of the different 'rates'). – Joe Jan 20 '15 at 20:32