# Effective annual rate

(EAR).

1.

A quoting convention under which interest at the quoted effective annual rate is calculated and added to the principal annually.

EAR is the most usual conventional quotation basis for instruments with maturities of greater than one year.

2.

A conventional measure which usefully 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 - or borrowing cost - as the instrument under review.

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

## Comparing effective annual rates

For depositing, a greater effective annual rate (EAR) means a better (higher) rate of return.

For borrowing, a lower EAR means a lower (better, cheaper) cost of borrowing.

If the opportunities being compared were identical in all other ways, the better EAR would generally be the choice.

In practice, however, other characteristics will usually be relevant, in addition to the EAR.

Examples include flexibility and risk.

If flexibility or risk were different, these characteristics would need to be weighed against the EAR, to make a final decision.

Treasury policy would also be relevant to investment or borrowing decisions in practice.

For example, higher risk investments are likely to be prohibited.

## 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 interest calculation period fits into a calendar year of 365 days (or 366 days in a leap year)

## Calculating EAR from overnight quotes

Example 1: EAR from overnight quote

GBP overnight interest is conventionally quoted on a simple interest basis for a 365-day fixed year.

So GBP overnight interest quoted at R = 5.11% means:

(i)

Interest of:

r = R / n

r = 5.11% / 365

r = 0.014% (= 0.00014) is paid per day.

(ii)

The equivalent effective annual rate is calculated from (1 + r).

1 + r = 1 + 0.00014 = 1.00014

EAR = (1 + r)n - 1

EAR = 1.00014365 - 1

EAR = 5.2424%.

Example 2: EAR from 360-day overnight quote

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

So USD overnight interest quoted at R = 5.11% means:

(i)

Interest of:

r = R / n

r = 5.11% / 360

r = 0.01419444% (= 0.0001419444) is paid per day.

(ii)

The equivalent effective annual rate is calculated from (1 + r).

1 + r = 1 + 0.0001419444 = 1.0001419444

EAR = (1 + r)n - 1

EAR = 1.0001419444365 - 1

EAR = 5.3171%.

Example 3: EAR in a leap year

The strict calculation of the effective annual rate is based on the prevailing calendar year, which is 365 days in a normal year, and 366 days in a leap year.

For the same periodic rate of interest (r), the effective annual rate is greater in a leap year.

For example, where (r) = 0.00014 overnight (as in Example 1).

The number of times (n) that the one-day period fits into the calendar year in a leap year = 366.

EAR = (1 + r)n - 1

EAR = 1.00014366 - 1

EAR = 5.2572%.