# Calculating effective annual rates

Effective annual rate (EAR) is a conventional measure which expresses the returns on different instruments on a comparable basis.

The EAR basis of comparison is the *equivalent* rate of interest paid and compounded annually, which would give the same all-in rate of return as the instrument under review.

For this reason, 'EAR' is sometimes expressed as __equivalent__ annual rate.

## Conversion formulae

#### Nominal annual rate to periodic rate

r = R / n

*Where:*

r = periodic interest rate or yield

R = nominal annual rate

n = number of times the period fits into a conventional year (for example, 360 or 365 days)

#### Periodic interest rate or yield to Effective annual rate

EAR = (1 + r)^{n} - 1

*Where:*

EAR = effective annual rate or yield

r = periodic interest rate or yield, as before

n = number of times the period fits into a calendar year

## EAR from semi-annual quote

We can calculate EAR from a semi-annual (half-year) quote.

**Example 1: EAR from semi-annual quote**

GBP semi-annual interest is conventionally quoted on a simple interest basis for half-years, using half-years to calculate interest for each period of six months, rather than an exact daycount.

So GBP semi-annual interest quoted at R = 5.00% means:

(i)

Interest of:

r = R / n

r = 5.00 / 2

r = 2.50% is paid per six months.

(ii)

The *equivalent* effective annual rate is:

EAR = (1 + r)^{n} - 1

EAR = 1.025^{2} - 1

EAR = **5.0625%**.

## EAR from USD overnight quote

We can calculate EAR from a USD overnight quote which has a 360-day year.

**Example 2: EAR from USD overnight quote**

USD overnight interest is conventionally quoted on a simple interest basis for a 360-day year.

So USD overnight interest quoted at R = 5.04% means:

(i)

Interest of:

r = R / n

r = 5.04% / 360

r = 0.014% is paid per day.

(ii)

The *equivalent* effective annual rate is:

EAR = (1 + r)^{n} - 1

EAR = 1.00014^{365} - 1

EAR = **5.2424%**.

## See also

- ACT/365 fixed
- Annual effective rate
- Annual effective yield
- Annual percentage rate
- Capital market
- Certificate in Treasury Fundamentals
- Certificate in Treasury
- Continuously compounded rate of return
- Effective annual rate
- Effective annual yield
- Equivalent Annual Rate
- Nominal annual rate
- Periodic discount rate
- Periodic rate of interest
- Periodic yield
- Rate of return
- Real
- Return
- Semi-annual rate
- SONIA