# 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 from other rates to Effective annual rate

(i) Converting periodic interest rate or yield (r) to Effective annual rate (EAR)

EAR = (1 + r)n - 1

Where:

EAR = effective annual rate or yield

r = periodic interest rate or yield

n = number of times the interest calculation period fits into a calendar year of 365 days (or 366 days in a leap year)

Example 1: EAR from periodic rate of 1% per week

Interest is payable on a borrowing at a rate of 1% per week, compounded once per week.

What is the effective annual rate?

Assume exactly 52 weeks in a year.

r = 1% (= 0.01) is paid per week.

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

1 + r

= 1 + 0.01 = 1.01

n = 52, the number of times interest is compounded per year

EAR = (1 + r)n - 1

EAR = 1.0152 - 1

EAR = 67.8%.

Out of this total, the amount relating to interest on the original principal - simple interest - is 52 weeks x 1% per week = 52%.

The rest of the total of 67.8% is the additional amount due to compounding - interest on interest.

Example 2: EAR from periodic rate of 1% per month

Interest is payable on a borrowing at a rate of 1% per month, compounded once per month.

What is the effective annual rate?

r = 1% (= 0.01) is paid per month.

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

1 + r

= 1 + 0.01 = 1.01

n = 12, the number of times interest is compounded per year

EAR = (1 + r)n - 1

EAR = 1.0112 - 1

EAR = 12.68%.

Out of this total, the amount relating to interest on the original principal - simple interest - is 12 months x 1% per month = 12%.

The rest of the total of 12.68% is the additional amount due to compounding - interest on interest.

(ii) Converting nominal annual rate (R) to periodic rate (r)

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)

---

Examples 3 and 4 illustrate the conversion from an interest rate quoted on a nominal annual basis, to an EAR.

Example 3: EAR from overnight quote (R)

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

Let's assume for this example that the overnight interest rate quoted for GBP is 5.11%.

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 4: EAR from 360-day overnight quote

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

Let's assume for this example the overnight interest rate quoted for USD is 5.11% (the same headline interest rate as in Example 3, but for USD in this case).

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%.

(This is greater than the EAR calculated for GBP in Example 3, because short term USD uses a 360-day conventional year, compared with 365 days for GBP.)

Example 5: 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%.