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:


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]


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

r = (1 + p)^52 - 1


The effective annual rate is 286.11 %

ref: http://en.wikipedia.org/wiki/Effective_interest_rate

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:


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}]


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
  • 2
    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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.