Effective annual rate: Difference between revisions

From ACT Wiki
Jump to navigationJump to search
imported>Doug Williamson
(Add link.)
imported>Doug Williamson
(Add examples.)
Line 45: Line 45:
==Conversion formulae==
==Conversion formulae==


====Nominal annual rate to periodic rate====


r = R / n
====Converting periodic interest rate or yield to Effective annual rate====
 
EAR = (1 + r)<sup>n</sup> - 1




''Where:''
''Where:''
EAR = effective annual rate or yield


r = periodic interest rate or yield
r = periodic interest rate or yield


R = nominal annual rate
n = number of times the interest calculation period fits into a calendar year of 365 days (or 366 days in a leap year)
 
 
 
<span style="color:#4B0082">'''Example 1: EAR from periodic rate of 1% per week'''</span>
 
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)<sup>n</sup> - 1
 
EAR = 1.01<sup>52</sup> - 1
 
EAR = '''67.8%'''.
 
 
 
 
<span style="color:#4B0082">'''Example 2: EAR from periodic rate of 1% per month'''</span>
 
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).


n = number of times the period fits into a conventional year (for example, 360 or 365 days)
1 + r = 1 + 0.01 = 1.01


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


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


EAR = (1 + r)<sup>n</sup> - 1
EAR = (1 + r)<sup>n</sup> - 1
EAR = 1.01<sup>12</sup> - 1
EAR = '''12.68%'''.
====Converting nominal annual rate to periodic rate====
r = R / n




''Where:''
''Where:''


EAR = effective annual rate or yield
r = periodic interest rate or yield


r = periodic interest rate or yield, as before
R = nominal annual rate


n = number of times the interest calculation period fits into a calendar year of 365 days (or 366 days in a leap year)
n = number of times the period fits into a conventional year (for example, 360 or 365 days)




==Calculating EAR from overnight quotes==
==Calculating EAR from overnight nominal annual quotes==


<span style="color:#4B0082">'''Example 1: EAR from overnight quote'''</span>
<span style="color:#4B0082">'''Example 3: EAR from overnight quote'''</span>


GBP overnight interest is conventionally quoted on a simple interest basis for a 365-day fixed year.
GBP overnight interest is conventionally quoted on a simple interest basis for a 365-day fixed year.
Line 107: Line 163:




<span style="color:#4B0082">'''Example 2: EAR from 360-day overnight quote'''</span>
<span style="color:#4B0082">'''Example 4: EAR from 360-day overnight quote'''</span>


USD short term interest is conventionally quoted on a simple interest basis for a 360-day year.
USD short term interest is conventionally quoted on a simple interest basis for a 360-day year.
Line 139: Line 195:




<span style="color:#4B0082">'''Example 3: EAR in a leap year'''</span>
<span style="color:#4B0082">'''Example 5: EAR in a leap year'''</span>


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.
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.

Revision as of 16:09, 4 March 2022

(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

Converting 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

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%.



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%.


Converting 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)


Calculating EAR from overnight nominal annual quotes

Example 3: 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 4: 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 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%.


See also

Retrieved from ‘https://wiki.treasurers.org/w/index.php?title=Effective_annual_rate&oldid=17316’