# Annual Percentage Rate (APR)

Annual percentage rate (APR) is the annualized interest rate on a loan or investment which does not account for the effect of compounding. It is the annualized form of the periodic rate which when applied to a loan or investment balance gives the interest expense or income for the period. In most cases it is the interest rate quoted by banks and other financial intermediaries on various products i.e. loans, mortgages, credit cards, deposits, etc. It is also called the nominal annual interest rate or simple interest rate.

Annual percentage rate (APR) is a useful measure when comparing different loans and investments because it standardizes the interest rates with reference to time. It is useful to quote an annual rate instead of quoting a 14-day rate for a 14-day loan or 30-year rate for a 30-year mortgage. Due to its simplicity, annual percentage rate is the most commonly quoted rate even though effective annual interest rate is a better measure when there are more than one compounding periods per year.

Let us say you obtained two loans, one for $150,000 requiring 6% interest rate for six months and another for $200,000 requiring 3.5% interest rate for three months. Annual percentage rate is helpful in this situation because it helps us compare the cost of loans. Annual percentage rate for the first loan is 12% (periodic rate of 6% multiplied by number of relevant periods in a year i.e. 2). Similarly, annual percentage rate for the second loan is 14% (periodic rate of 3.5% multiplied by number of periods in a year of 4). It helps us conclude that the second loan is expensive.

## Formula

Even though annual percentage rate (APR) is simple in concept, its calculation might be tricky. It depends on whether the loan is based on simple interest or discount.

When periodic interest rate is given, we can use the following formula to calculate APR:

Annual percentage rate (interest-based loan)

= Periodic Interest Rate for m Months × 12/m

If the interest amount is deducted from the loan amount at the start of the loan period as in discount loans, the periodic rate is calculated by dividing the finance charge by the amount financed.

Annual percentage rate (discount loan)

= Finance Charge/Amount Financed × 12/Term of Loan in Months

Where the finance charge is the product of principal, interest rate and time factor:

Finance Charge = Principal × Interest Rate × Term of Loan in Months/12

Amount financed equals the difference between principal and total finance charge:

Amount Financed

= Principal − Finance Charge

= Principal – Principal × Periodic Rate × Term of Loan in Months/12

### Converting Effective Interest Rate to Nominal Annual Percentage Rate

If you know the effective annual interest rate, you can find APR as follows:

APR = m × ((1 + EAR)^{(1/m)} – 1)

Where m is the number of compounding periods per year and n is number of years.

## Example

You are a personal finance expert advising three clients:

- Angela, who must choose between two payday loans, each for $3,000 and 14-days: Loan A with financial charge of $100 payable at the end of 14th day and Loan B with finance charge of $90 deducted from the principal balance at the start of the loan.
- Ahsan, who must decide between two credit cards: Card C with 2.5% monthly charge and Card D with 7.1% quarterly charge.
- Antonio, who wants to identify better investment for his $50,000 for 5 years: Investment E paying APR of 10.6% compounded semiannually and Investment F with effective interest rate of 11% compounded monthly.

### Solution

In case of Angela, Loan B is better. This is because annual percentage rate (APR) of Loan B is lower than APR for the Loan A.

APR of Loan A is 86.9% worked out through the following steps:

- calculating periodic interest rate, which equals 3.33% (=$100/$3,000) for 14-day period,
- annualizing the rate by dividing it by the term of the loan (i.e. 14) and multiplying by the number of days in a year (i.e. 3.33%/14×365 = 86.9%).

APR of Loan B is 80.63% calculated as follows:

- finding financial charge for 14 days which is $90,
- finding amount financed, which is $2,910 ($3,000 total amount minus $90 interest because it is paid at the start of the loan),
- finding periodic rate for the 14-days which is 3.093% (=$90/$2,910), and
- annualizing the rate (i.e. 3.093%/14×365=80.63%).

In case of Ahsan, Card D is better because APR for Card C is 30% (=periodic rate of 2.5% × 12/1) and APR for Card D is 28.4% (= periodic rate of 7.1% × 12/3), which is lower.

In case of Antonio, we need to find out APR for Investment F to make a comparison.

Annual percentage rate (APR) – Investment F = 12 × ((1 + 11%)^{(1/12)} – 1) = 10.48%

## Weakness of Annual Percentage Rate

We may quickly conclude that Investment E is better because it has higher annual percentage rate. However, this is exactly where the weakness of APR lies: it ignores the effect of compounding. In such a situation, we need to make a comparison based on effective annual interest rate. Effective annual interest rate (EAR) in case of Investment E is just 10.88% (as shown below) which is lower than the effective interest rate on Investment F i.e. 11%. Antonio should choose Investment F paying 11% effective rate instead of Investment E paying 10.6% annual percentage rate (APR) compounded semiannually.

Effective annual interest rate = (1 + 10.60%/2)^{2} – 1 = 10.88%

by Obaidullah Jan, ACA, CFA and last modified on

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