Effective annual rates

In depth

From any one rate to the others

With $m$ compounding periods a year, the periodic rate is $r = \text{APR} \div m$, where APR is the nominal annual rate, and the EAR follows by compounding it $m$ times. F00 wrote this formula with $r$ for the quoted annual rate; this course writes APR for that and keeps $r$ for the rate per period. Going backwards, take the $m$th root of $1 + \text{EAR}$ to find the periodic rate, then multiply by $m$:

What nominal rate compounded monthly gives an EAR of exactly 8%? The monthly rate is $1.08^{1/12} - 1 \approx 0.643403\%$, so the APR is $12 \times 0.643403\% \approx 7.7208\%$. It is not $8\% \div 12$: a monthly rate of 0.6667% compounds to about 8.30% a year, more than intended.

Over several years, either route gives the same future value: $\text{FV} = \text{PV}(1 + \text{APR}/m)^{mt} = \text{PV}(1 + \text{EAR})^{t}$, where $t$ is the time in years. For example, $24,000 for 18 months at 7.2% compounded monthly is $24{,}000 \times 1.006^{18} \approx 26{,}728.52$ dollars, and $24{,}000 \times 1.074424^{1.5}$ gives the same $26,728.52.

Equivalent rates at another frequency

Two rates are equivalent rates when they give the same growth over a year, and so have the same EAR. To turn a rate quoted with $m_1$ periods a year into one with $m_2$, set the two growth factors equal:

Lantern’s case gives an example. A loan like its note would cost 8% a year compounded quarterly in March 2027: a quarterly rate of 2% and an EAR of $1.02^4 - 1 \approx 8.2432\%$. The equivalent monthly rate is $1.02^{1/3} - 1 \approx 0.662271\%$, a nominal 7.9473% a year compounded monthly. On the same monthly basis, then, the market rate for such a loan is about 0.75 of a percentage point above the 7.2% Coastline offers on the brewer.

The same idea gives a rate for a period of any length. A period of $d$ days is $d/365$ of a year, so the rate for it that matches an EAR is $(1 + \text{EAR})^{d/365} - 1$; turning a rate for $d$ days into an EAR raises $1 + r$ to the power $365/d$. The applied example uses this to price an early-payment discount.

Continuous compounding in formulas

Why does the limit involve $e$? The number $e$ is defined as the value that $(1 + 1/m)^m$ approaches as $m$ grows: 2 when $m = 1$, 2.593742 when $m = 10$, 2.704814 at 100, 2.716924 at 1,000 and 2.718146 at 10,000. A nominal rate APR compounded $m$ times a year behaves the same way, its growth factor over a year approaching $e^{\text{APR}}$. Over $t$ years:

The last form converts an effective rate into its continuously compounded equivalent: an EAR of 6% is a continuously compounded rate of $\ln 1.06 \approx 5.8269\%$. Continuously compounded rates have a convenient property: rates for successive periods add, because $e^{a} \times e^{b} = e^{a + b}$.

The annual percentage yield on US deposits

In the United States, deposit rates are disclosed in a standard way under the Truth in Savings Act, implemented for banks and savings institutions by Regulation DD (12 CFR part 1030); credit unions follow the NCUA’s parallel rule (12 CFR part 707). The regulation defines the annual percentage yield as a percentage rate reflecting the total interest paid on an account, based on the interest rate and the frequency of compounding for a 365-day period (Appendix A lets a bank use 366 days in a leap year). It defines the interest rate as the annual rate paid on an account, not reflecting compounding. In this course’s words, a deposit’s interest rate is its nominal annual rate and its APY is an effective annual rate.

Appendix A to the regulation gives the general formula, in which Interest is the total interest earned on the Principal over the term and Days in term is the actual number of days:

For an account with no fixed term, the regulation treats the term as 365 days, so the APY is a year’s interest as a percentage of the principal. Two of the appendix’s examples show the formula at work. An account paying $61.68 on $1,000 over a 365-day year has an APY of 6.17%, which is what 6% compounded monthly pays: $1{,}000 \times (1.005^{12} - 1) \approx 61.68$. A six-month certificate of deposit paying $30.37 on $1,000 over 182 days has an APY of $100 \times (1.03037^{365/182} - 1) \approx 6.18\%$. The formula compounds the six-month return over a whole year, so the APY of a short deposit can be compared directly with that of a longer one.

The same formula handles terms longer than a year by taking a root instead of a power. Suppose a fictional two-year certificate of deposit, a term of 730 days, pays $832 of interest on $10,000 at maturity. Its APY is $100 \times (1.0832^{365/730} - 1) = 100 \times (\sqrt{1.0832} - 1) \approx 4.08\%$, not $8.32\% \div 2 = 4.16\%$. Halving the two-year return would ignore the interest earned in the second year on the first year’s interest, and so overstate the yearly rate.

Comparing offers quoted differently

A reliable procedure for any comparison of rates:

1. For each offer, find the rate per period and the number of periods a year, or note that the rate is continuously compounded.
2. Convert each to an effective annual rate.
3. Compare: the lowest EAR is the cheapest loan; the highest is the best deposit.
4. Check what the rate leaves out: fees, the timing of payments, penalties for early repayment or withdrawal, and whether the rate is fixed or can change.

Some offers state no rate at all. The supplier’s plan for Lantern’s brewer charges “0% interest” but asks for 24 payments of $1,050, $24 \times 1{,}050 = 25{,}200$ dollars in all, against a cash price of $24,000. Its true rate is hidden in the payments, and finding it needs the annuity formulas of module 3. An early-payment discount on an invoice, the applied example below, is a simpler case of the same idea: a price difference that is really an interest rate.