Converting APY to APR

Financial institutions love to advertise impressive-sounding annual percentage yields, or APYs, on savings accounts and certificates of deposit. APY accounts for the effects of compounding within a year, so it reflects the actual percentage increase in your balance over twelve months. However, if you are comparing two products with different compounding frequencies or want to understand the underlying nominal rate used to compute monthly interest, you need to convert that APY back into an APR. This calculator performs that conversion by reversing the compounding formula. Simply enter the APY and the number of compounding periods per year—monthly compounding uses 12, quarterly uses 4, and so on—and the tool outputs the equivalent APR.

The math behind the conversion stems from the relationship between the nominal rate and the effective annual yield. When a nominal rate $r$ is compounded $n$ times per year, the APY is given by

$\mathrm{APY}={\left(1+\frac{r}{n}\right)}^n-1$

To solve for the nominal rate, rearrange the formula:

$r=n\left({1+\mathrm{APY}}^{\frac{1}{n}},-,1\right)$

Here, $\mathrm{APY}$ and $r$ are expressed as decimals. The calculator converts the APY input from a percentage to a decimal, applies this equation, and then multiplies the nominal rate by 100 to report the APR in familiar percentage terms. This APR is the rate you would see if the bank did not account for compounding in its promotional materials.

Why might you want to perform this conversion? Suppose one bank offers a savings account with a 5.12% APY compounded monthly, while another advertises a 5% APR with simple interest. At first glance, the APY looks better, but to compare apples to apples you must translate the APY into APR. Using the formula above with $\mathrm{APY}=0.0512$ and $n=12$ yields $r\approx 0.05$, revealing that both accounts effectively offer the same nominal rate. Understanding this relationship prevents confusion and helps you choose the option that truly pays more, especially when different institutions use APY and APR interchangeably in marketing.

Consider the following table showing how APY and APR compare across common compounding frequencies. It assumes a 5% APR and calculates the corresponding APY for each frequency. The inverse process, which this calculator performs, takes the APY and frequency and returns the APR:

Compounds per Year	APR (%)	APY (%)
1 (annual)	5.00	5.00
4 (quarterly)	5.00	5.09
12 (monthly)	5.00	5.12
365 (daily)	5.00	5.13

Although the differences may appear small, they can meaningfully affect long-term savings. For example, a certificate of deposit with a 5% APR compounded daily will earn slightly more over a year than one compounded annually. Conversely, if you receive an APY quote for a loan, converting it to APR reveals the nominal rate used to compute periodic interest charges. This can be crucial for mortgages, auto loans, and credit cards where regulations require APR disclosure.

To ensure accuracy, the calculator assumes that compounding periods are evenly spaced throughout the year and that the APY is accurate. In reality, banks may have quirks like 360-day years or promotional periods. Furthermore, APY does not account for fees or minimum balance requirements that could reduce your actual earnings. Nonetheless, converting between APY and APR provides a solid starting point for evaluating financial products and making informed decisions.

Try experimenting with different inputs. Enter a high APY and observe how increasing the number of compounding periods narrows the gap between APR and APY. Alternatively, keep the APY fixed and adjust the frequency to see how the nominal rate changes. These exercises build intuition about the mechanics of compounding and help you spot marketing tricks that rely on compounding to inflate apparent returns.

Ultimately, knowing how to convert APY to APR empowers you as a consumer. It demystifies financial jargon and allows you to focus on meaningful comparisons. While the difference between 5.00% and 5.12% may seem negligible, over large balances or long timeframes the extra yield compounds significantly. Use this calculator whenever you encounter an APY and want to understand the underlying nominal rate driving the advertised return.

Worked Example

Suppose a credit union advertises an APY of 4.5% compounded quarterly. To find the APR, plug 4.5 for APY and 4 for the number of periods into the formula. First convert the APY to decimal form: 0.045. The expression ${1+0.045}^{\frac{1}{4}}-1$ evaluates to 0.01106. Multiplying by 4 yields an APR of roughly 4.42%. This is the nominal rate the institution uses to compute each quarterly interest credit.

Comparison of APY and APR

The table below demonstrates how different APY values translate into APR when compounded monthly. Use it as a quick reference for common rates.

APY (%)	Equivalent APR (%)
2.00	1.98
5.00	4.88
8.00	7.72

Limitations and Assumptions

This calculator presumes interest compounds at evenly spaced intervals and that the APY accurately reflects all fees. Products with teaser rates, tiered yields, or daily balances may not conform perfectly to the formula. Still, the conversion offers a solid starting point for evaluating offers.

For deeper savings analysis, explore related tools such as the Compound Interest Calculator or the 401(k) Growth Calculator. Combining these calculators clarifies how nominal rates, compounding frequency, and contribution schedules interact over time.