Effective annual rate: Difference between revisions

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(EAR).  
(EAR).  
 
__NOTOC__
1.  
1.  


A quoting convention under which interest at the quoted rate is calculated and added to the principal annually.  
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.
EAR is the most usual conventional quotation basis for instruments with maturities of greater than one year.
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2.  
2.  


A conventional measure which expresses the returns on different instruments on a comparable basis.  
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 as the instrument under review.
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 <u>equivalent</u> annual rate.
For this reason, 'EAR' is sometimes expressed as <u>equivalent</u> annual rate.




===Conversion formulae===


r = R / n
==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.
 
 
<span style="color:#4B0082">'''''CONVERSION from other rates to Effective annual rate'''''</span>
 
 
<span style="color:#4B0082">'''''(i) Converting periodic interest rate or yield (r) to Effective annual rate (EAR)'''''</span>
 
''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%'''.
 
 
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.
 
 
 
 
<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).
 
1 + r


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


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




EAR = (1 + r)<sup>n</sup> - 1
EAR = (1 + r)<sup>n</sup> - 1
EAR = 1.01<sup>12</sup> - 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.
<span style="color:#4B0082">'''''(ii) Converting nominal annual rate (R) to periodic rate (r)'''''</span>
''r = R / n''




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


EAR = effective annual rate or yield
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)


r = periodic interest rate or yield, as before
---


n = number of times the period fits into a calendar year
Examples 3 and 4 illustrate the conversion from an interest rate quoted on a nominal annual basis, to an EAR.




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


===<span style="color:#4B0082">Example 1: 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 year.
Let's assume for this example that the overnight interest rate quoted for GBP is 5.11%.


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


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


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


So GBP semi-annual interest quoted at R = 5.00% means:
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)  
(i)  
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r = R / n
r = R / n


r = 5.00 / 2
r = 5.11% / 360


r = 2.50% is paid per six months.
r = 0.01419444% (= 0.0001419444) is paid per day.




(ii)  
(ii)  


The ''equivalent'' effective annual rate is:
The ''equivalent'' effective annual rate is calculated from (1 + r).
 
1 + r = 1 + 0.0001419444 = 1.0001419444




EAR = (1 + r)<sup>n</sup> - 1
EAR = (1 + r)<sup>n</sup> - 1


EAR = 1.025<sup>2</sup> - 1  
EAR = 1.0001419444<sup>365</sup> - 1  
 
EAR = '''5.0625%'''.
 
 


===<span style="color:#4B0082">Example 3: EAR from USD overnight quote (360-day year)</span>===
EAR = '''5.3171%'''.


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


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


Interest of:


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


r = 5.04% / 360
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.


r = 0.014% is paid per day.
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).


(ii)  
The number of times (n) that the one-day period fits into the calendar year in a leap year = 366.
 
The ''equivalent'' effective annual rate is:
 


EAR = (1 + r)<sup>n</sup> - 1
EAR = (1 + r)<sup>n</sup> - 1


EAR = 1.00014<sup>365</sup> - 1  
EAR = 1.00014<sup>366</sup> - 1  


EAR = '''5.2424%'''.
EAR = '''5.2572%'''.




== See also ==
== See also ==
* [[ACT/365 fixed]]
* [[ACT/365 fixed]]
* [[Annual effective rate]]
* [[Annual effective rate]] (AER)
* [[Annual effective yield]]
* [[Annual effective yield]]
* [[Annual percentage rate]]
* [[Annual percentage rate]]  (APR)
* [[Basis]]
* [[Benchmark]]
* [[Calculating effective annual rates]]
* [[Capital market]]
* [[Capital market]]
* [[Certificate in Treasury Fundamentals]]
* [[Certificate in Treasury Fundamentals]]
* [[Certificate in Treasury]]
* [[Certificate in Treasury]]
* [[Compound]]
* [[Compound interest]]
* [[Continuously compounded rate of return]]
* [[Continuously compounded rate of return]]
* [[Effective annual yield]]
* [[Effective annual yield]]
* [[Equivalent Annual Rate]]
* [[Headline ]]
* [[LIBOR]]
* [[Leap year]]
* [[Nominal annual rate]]
* [[Nominal annual rate]]
* [[Periodic discount rate]]
* [[Periodic discount rate]]
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* [[Real]]
* [[Real]]
* [[Return]]
* [[Return]]
* [[Risk]]
* [[Semi-annual rate]]
* [[Semi-annual rate]]
* [[Simple interest]]
[[Category:Long_term_funding]]

Latest revision as of 08:03, 5 October 2024

(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%.


See also