APR calculator

How do you calculate APR?

An APR compares the money you receive with the money you pay back, and expresses the difference as a yearly rate. The steps are the same in both conventions:

1. Find the cash you receive. Start with the loan and take away any fees paid at the start. On the defaults, $20,000 less $600 of fees is $19,400.
2. Find the payments. The loan of $20,000 at 6.5% over 60 months costs $391.32 a month.
3. Solve for the rate per period. Find the rate j at which 60 payments of $391.32 are worth exactly $19,400 today: $19,400 = $391.32 x (1 - (1 + j)^-60) / j. There is no way to isolate j, so it is found by trial; here j = 0.6483% a month.
4. Turn it into a yearly figure. The US method multiplies: 0.6483% x 12 = 7.78%. The UK method compounds: (1.006483)^12 - 1 = 8.0635%, shown as 8.1%.

The finance charge, everything paid minus the cash received, is $23,479.38 - $19,400 = $4,079.38: $3,479.38 of interest and the $600 of fees.

How is APR calculated in the US under Regulation Z?

Regulation Z, the CFPB's rule at 12 CFR 1026, says in section 1026.22(a)(1) that the annual percentage rate is a measure of the cost of credit, expressed as a yearly rate, and that it is determined by the actuarial method, with equations and instructions in Appendix J, or by the United States Rule. This calculator uses the actuarial method.

Appendix J builds the rate on a unit-period, the common period, not over a year, that occurs most often in the loan. For a monthly unit-period there are 12 unit-periods a year; for a week or a multiple of a week, 52 divided by the number of weeks in it, so 26 for payments every two weeks; a semimonth gives 24 and a day 365. The APR is the unit-period rate multiplied by the number of unit-periods in a year. That makes the US APR a nominal rate: no compounding inside the year.

Which money counts is set elsewhere in the rule. Section 1026.4 defines the finance charge as the cost of consumer credit as a dollar amount, and lists points, loan fees, assumption fees and finder's fees among its examples. Section 1026.18(b) works out the amount financed by subtracting any prepaid finance charge from the principal. A disclosed APR is treated as accurate within 1/8 of 1 percentage point of the exact figure for a regular loan, and within 1/4 of a point for an irregular one, under 1026.22(a)(2) and (a)(3).

How is APR calculated in the UK under CONC?

In the UK, the FCA's Consumer Credit sourcebook sets out the calculation in CONC App 1.2.6R. The APR is the rate that equates, on an annual basis, the total present value of drawdowns with the total present value of repayments and payments of charges. Times between dates are expressed in years or fractions of a year, with a year of 365 days (366 in a leap year), 52 weeks or 12 equal months, and an equal month of 30.41666 days. The result is expressed to at least one decimal place, and a figure of 5 or more in the next place rounds it up.

Because each payment is discounted at (1 + APR) raised to the time in years, the UK APR is an annual effective rate. For equal payments every month it is (1 + j)^12 - 1, and for weekly payments (1 + j)^52 - 1. CONC App 1.2 covers agreements that are not secured on land; App 1.2.2R excludes those secured on land, which follow separate rules this calculator does not model.

For euro amounts the page opens on the annual effective method. Annex I of the EU Consumer Credit Directive, 2008/48/EC, sets out the same annual equation for the APRC, with the same 365-day, 52-week or 12-equal-month year and the same rounding to at least one decimal place.

Why is the UK APR higher than the US APR for the same loan?

Both start from the same rate per period. The US figure multiplies it by the periods in a year, the UK figure compounds it, and compounding always gives more once the rate is above zero. The gap widens as the rate rises:

Neither is wrong: they answer slightly different questions, and each is correct under its own rules. The mistake is to compare a US-style figure with a UK-style one. Even a loan with no fees at all shows two numbers: 6.5% charged monthly is a 6.5% APR the US way and 6.70% the UK way.

How do fees, the term and the payment frequency move the APR?

Fees. On the $20,000, 6.5%, 5-year loan, $600 of fees taken at the start lifts the US APR from 6.50% to 7.78%. Adding the same $600 to the loan instead gives 7.74%: you then receive the full $20,000, pay $403.06 a month and a finance charge of $4,183.76.

Term. A fee paid once weighs more on a short loan. With the same rate and $600 of fees taken at the start:

Frequency. Paid weekly, the default loan's US APR is 7.79% and its UK APR 8.10%, against 7.78% and 8.06% monthly. The US figure hardly moves, while the UK figure climbs because the rate is compounded 52 times instead of 12. The second chart draws this for your loan.

Worked examples in pounds and dirhams

• Pounds, from the payments. You receive £5,000 and repay 36 monthly payments of £160, £5,760 in all. The rate per month that discounts the payments to £5,000 is 0.7858%. The UK APR is (1.007858)^12 - 1 = 9.848%, shown as 9.8%; the US figure would be 9.43%. Choose GBP and the headline switches to the UK method.
• Dirhams, from the terms. AED 60,000 at 5.25% over 4 years with AED 1,200 of fees taken at the start. The payment on AED 60,000 is AED 1,388.56 a month, while you receive AED 58,800. The APR is 6.28% by the US method and 6.47% by the UK method.

The dirham example shows the arithmetic only; a loan in the UAE follows its own rules, which this calculator does not model. Every currency uses the same steps, and nothing is converted between currencies.

How do I find the APR when I only know the payments?

Choose From the payments. Enter the cash that actually reached you, the payment and the number of payments, and the calculator solves the rate directly, with no need to know how the lender split interest and fees.

With the defaults, $19,400 received and 60 monthly payments of $400 is a finance charge of $4,600 and an APR of 8.71% the US way, 9.07% the UK way (shown as 9.1%). Short, frequent loans can carry far higher rates than their payments suggest. $1,000 received and repaid at $25 a week for 48 weeks costs $200, yet the APR is 40.05% by the US method and 49.02% by the UK method. Paid every two weeks, $2,000 received and 26 payments of $90 is 31.19% US and 36.35% UK. If the payments add up to less than the cash received, the rate is negative and the page says so.

How do you calculate APR in Excel?

Excel's RATE function returns the rate per period of a loan with level payments, found by iteration. Microsoft's documentation gives its syntax as RATE(nper, pmt, pv, [fv], [type], [guess]). Enter the cash you received as pv and the payment as a negative number:

• US method: =RATE(60, -391.32, 19400)*12 returns 7.78%.
• UK method: =(1+RATE(60, -391.32, 19400))^12-1 returns 8.06%, which rounds to 8.1%.

For weekly payments, multiply by 52 or raise to the power 52 instead. Using the loan amount as pv instead of the cash received returns the 6.5% interest rate, not the APR. That is the most common way to get an APR wrong in a spreadsheet.

What this APR calculator leaves out

• Irregular timing. The calculator assumes the first payment comes one full period after the money arrives and the rest follow at equal intervals. Both rules handle odd first periods and uneven gaps, which change the figure slightly.
• The United States Rule. Regulation Z allows it as an alternative to the actuarial method; only the actuarial method is modelled here.
• Which charges count. Each rule decides which fees, insurance and other costs enter the APR. The calculator counts whatever you enter as fees and nothing else.
• Loans secured on land. Mortgages have their own disclosure rules in both countries.
• Variable rates and multiple advances. Credit cards, overdrafts and loans drawn in stages are outside this calculation.
• Other countries. Rules elsewhere differ, and only the US and UK methods are shown.

The APR on a lender's own disclosure is the figure that applies. For the payment, term or rate of a loan without fees, see the loan calculator; for a personal loan's fee and flat-rate options, the personal loan calculator.

Glossary

APR: annual percentage rate, a yearly figure for what credit costs once fees count.Unit-period: in Regulation Z, the most common period between payments, used to state the APR. Amount financed: the credit you actually get the use of, after prepaid finance charges. Finance charge: the dollar cost of credit, interest plus fees. Prepaid finance charge: a fee paid at or before the start, taken from the loan or paid in cash. Nominal rate: a rate per period multiplied by the periods in a year. Effective rate: a rate per period compounded over a year. Present value: what a future payment is worth today at a given rate.

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