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Loans

How To Calculate APR

The APR is the interest rate that would produce your loan's cash flows if the fee were rolled into the rate. It answers a fairer question than the advertised nominal rate: what single annual rate, applied to the money you actually received, reproduces the payments you actually make?

Quick Answer

Solve for i: P - F = sum of pmt / (1 + i)^k

P
Amount borrowed before fees
F
Upfront fee deducted at closing
pmt
Fixed monthly payment
APR
Annual percentage rate, the monthly IRR times 12

Take the cash you actually receive, which is the amount borrowed minus upfront fees, and find the monthly rate that discounts your payments back to that number. Multiply the monthly rate by 12 to express it annually. A 10,000 loan with a 200 fee and 200 monthly payments for 60 months has an APR above its 7% nominal rate because you paid 200 for the privilege of borrowing 10,000.

What Is APR?

The annual percentage rate expresses the cost of a loan as a single yearly rate that folds in both the interest and the fees charged to set it up. It is the internal rate of return of the loan's cash flows from the borrower's point of view: money in at the start, payments out each month.

Because it includes fees, the APR is always at least as large as the nominal rate, and the gap grows with the size of the fee. Regulators require lenders to disclose it precisely because the nominal rate lets a lender advertise a low rate while recovering the difference in points and fees.

The mechanics come straight from present value. At the moment you sign, you receive the amount borrowed minus any fee taken at closing. Over the life of the loan you pay a fixed amount each month. There is exactly one monthly interest rate that makes the discounted value of those payments equal the cash you actually received, and multiplying that rate by twelve gives the APR. Nothing else enters the calculation.

A worked case makes the fee effect concrete. Borrow 10,000 with a 200 origination fee and pay 200 a month for sixty months. The cash you really get is 9,800, but the payments are unchanged, so the rate that balances the two sides is higher than the loan's nominal rate. The fee is not spread evenly over sixty months; it is a lump taken at time zero, and its whole weight lands on the first period.

This is why the same fee hurts more on a short loan. Spread a 200 fee over two years instead of five and the effective rate rises further, because the smaller number of payments has to carry the whole lump. Two loans can advertise the identical nominal rate and have very different APRs; the APR is the number that exposes the difference.

APR has limits worth knowing. It usually excludes taxes, insurance and many third-party charges, so it is a floor on your true cost rather than the final word. It also assumes you hold the loan to term and make every payment on time; prepaying a loan with a large upfront fee can make the APR a misleading guide to what you actually paid.

The equation has no closed-form solution. You cannot rearrange it to isolate the rate; the rate is found by iteration, which is what any financial calculator or spreadsheet does internally. A quick estimate using the fee divided over the term gets you close enough to sanity-check a lender's figure.

Finally, keep APR and APY apart. APR is a nominal annual borrowing rate with fees; APY is an effective annual saving rate that includes compounding within the year. They answer different questions and are not interchangeable when comparing products.

Formula

P - F = sum_{k=1..n} pmt / (1 + i)^k ; APR = 12 x i

The net cash you receive equals the present value of the payments when discounted at the monthly rate i. There is no closed form; the rate is found by trial or a financial calculator.

SymbolMeaning
PPrincipal borrowed
FUpfront fee
iMonthly IRR
nNumber of payments

How To Calculate APR

  1. 1

    Find the net amount received

    Subtract the upfront fee from the amount borrowed. A 10,000 loan with a 200 origination fee puts 9,800 in your hands.

  2. 2

    List the payment stream

    Write down the fixed monthly payment for every month of the term.

  3. 3

    Search for the monthly rate

    Find the rate i that makes the payments discount back to exactly 9,800. There is no algebra for it, so a calculator iterates until the two sides match.

  4. 4

    Annualise

    Multiply i by 12 to express the result as an APR. If i turns out to be 0.006, the APR is 0.072, or 7.2%.

Examples

Example 1: 10,000 borrowed, 200 fee, 200 a month for 60 months

Amount
10,000
Fee
200
Monthly payment
200
Months
60
StepCalculationResult
Net received10000 - 2009800
Monthly ratei such that 9800 = 200 x annuity factor0.00600
APR0.00600 x 120.072

Result: The APR is 0.072, or 7.2%, higher than the nominal rate because the 200 fee is spread over a smaller net loan.

Example 2: 10,000 with no fee, 200 a month for 60 months

Amount
10,000
Fee
0
Monthly payment
200
Months
60
StepCalculationResult
Net received10000 - 010000
Monthly ratei such that 10000 = 200 x annuity factor0.00551
APR0.00551 x 120.0661

Result: With no fee the APR is 0.0661, equal to the nominal rate, about 6.61%. The fee alone is what widened it in the first example.

Calculator

APR (annual, nominal)

448.98%

Cash actually received
$9,800.00
Monthly rate
37.41%
Total of payments
$12,000.00

Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.

Prefer a full-width tool? Open the APR calculator page.

Common Mistakes

  • Comparing APR with a nominal rate

    They are different quantities. A 7% nominal rate with a big fee can have an 8% APR; compare APR with APR.

  • Forgetting the fee leaves your pocket at time zero

    The fee reduces the cash you receive, not the payment you make. Modelling it as a spread-out charge understates the APR.

  • Assuming a simple formula exists

    The APR equation cannot be solved by rearranging symbols; it is found by iteration. A rough linear guess is fine for a sanity check but not for disclosure.

  • Ignoring that shorter terms raise APR for the same fee

    The same 200 fee spread over 24 months rather than 60 raises the APR more, because the fee is amortised faster.

  • Confusing APR with APY

    APR is a nominal annual rate for borrowing; APY compounds within the year for saving. They are not interchangeable.

  • Leaving out third-party fees

    Some fees are included in the regulated APR and some are not. Read the disclosure to know which fees count.

FAQ

What is the difference between APR and interest rate?

The interest rate is the cost of the money alone. The APR adds the fees charged to arrange the loan, so it reflects the true annual cost. The APR is always at least as high as the nominal rate.

Why is my APR higher than my quoted rate?

Because of fees. Any closing cost, origination fee or points raise the APR above the nominal rate by spreading those charges over the payments.

Is a lower APR always better?

For the same term and loan size, yes, it is the fairest single comparison. But a longer term with a low APR can cost more in total than a shorter term with a slightly higher APR.

Can the APR be calculated exactly?

Not in closed form. The rate that equates the net proceeds with the payment stream is found by iteration, which is what a calculator or spreadsheet does.

Does the APR include taxes and insurance?

Usually not. Taxes, insurance and most third-party charges sit outside the APR in many jurisdictions, so your all-in cost can still be higher.

How is the APR different for a credit card?

Credit card APR is the nominal rate on the revolving balance; because there is no fixed fee per pound borrowed, it is closer to the interest rate than a loan APR is.

References

  1. [1]Consumer Financial Protection Bureau, Truth in Lending Act and Regulation Z — https://www.consumerfinance.gov/rules-policy/regulations/1026/
  2. [2]Investopedia, Annual percentage rate — https://www.investopedia.com/terms/a/apr.asp
  3. [3]Federal Reserve, Consumer credit cost disclosure — https://www.federalreserve.gov/consumerscommunities/consumer-credit.htm