# 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 = 1.0252 - 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 = 1.00014365 - 1

EAR = 5.2424%.