imported>Doug Williamson |
imported>Administrator |
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| (EAR).
| | Statements of fact, which are capable of proof or disproof. |
| __NOTOC__
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| 1.
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| A quoting convention under which interest at the quoted effective annual rate is calculated and added to the principal annually.
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| EAR is the most usual conventional quotation basis for instruments with maturities of greater than one year.
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| 2.
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| A conventional measure which usefully expresses the returns on different instruments on a comparable basis.
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| 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.
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| For this reason, 'EAR' is sometimes expressed as <u>equivalent</u> annual rate.
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| ==Comparing effective annual rates==
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| For depositing, a greater effective annual rate (EAR) means a better (higher) rate of return.
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| For borrowing, a lower EAR means a lower (better, cheaper) cost of borrowing.
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| If the opportunities being compared were identical in all other ways, the better EAR would generally be the choice.
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| In practice, however, other characteristics will usually be relevant, in addition to the EAR.
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| Examples include flexibility and risk.
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| If flexibility or risk were different, these characteristics would need to be weighed against the EAR, to make a final decision.
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| Treasury policy would also be relevant to investment or borrowing decisions in practice.
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| For example, higher risk investments are likely to be prohibited.
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| <span style="color:#4B0082">'''''CONVERSION from other rates to Effective annual rate'''''</span>
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| <span style="color:#4B0082">'''''(i) Converting periodic interest rate or yield (r) to Effective annual rate (EAR)'''''</span>
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| ''EAR = (1 + r)<sup>n</sup> - 1''
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| ''Where:''
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| EAR = effective annual rate or yield
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| r = periodic interest rate or yield
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| n = number of times the interest calculation period fits into a calendar year of 365 days (or 366 days in a leap year)
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| <span style="color:#4B0082">'''Example 1: EAR from periodic rate of 1% per week'''</span>
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| Interest is payable on a borrowing at a rate of 1% per week, compounded once per week.
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| What is the effective annual rate?
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| Assume exactly 52 weeks in a year.
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| r = 1% (= 0.01) is paid per week.
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| The ''equivalent'' effective annual rate is calculated from (1 + r).
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| 1 + r
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| = 1 + 0.01 = 1.01
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| n = 52, the number of times interest is compounded per year
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| EAR = (1 + r)<sup>n</sup> - 1
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| EAR = 1.01<sup>52</sup> - 1
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| EAR = '''67.8%'''.
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| Out of this total, the amount relating to interest on the original principal - simple interest - is 52 weeks x 1% per week = 52%.
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| The rest of the total of 67.8% is the additional amount due to compounding - interest on interest.
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| <span style="color:#4B0082">'''Example 2: EAR from periodic rate of 1% per month'''</span>
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| Interest is payable on a borrowing at a rate of 1% per month, compounded once per month.
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| What is the effective annual rate?
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| r = 1% (= 0.01) is paid per month.
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| The ''equivalent'' effective annual rate is calculated from (1 + r).
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| 1 + r
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| = 1 + 0.01 = 1.01
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| n = 12, the number of times interest is compounded per year
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| EAR = (1 + r)<sup>n</sup> - 1
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| EAR = 1.01<sup>12</sup> - 1
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| EAR = '''12.68%'''.
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| Out of this total, the amount relating to interest on the original principal - simple interest - is 12 months x 1% per month = 12%.
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| The rest of the total of 12.68% is the additional amount due to compounding - interest on interest.
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| <span style="color:#4B0082">'''''(ii) Converting nominal annual rate (R) to periodic rate (r)'''''</span>
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| ''r = R / n''
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| ''Where:''
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| r = periodic interest rate or yield
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| R = nominal annual rate
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| n = number of times the period fits into a conventional year (for example, 360 or 365 days)
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| ---
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| Examples 3 and 4 illustrate the conversion from an interest rate quoted on a nominal annual basis, to an EAR.
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| <span style="color:#4B0082">'''Example 3: EAR from overnight quote (R)'''</span>
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| GBP overnight interest is conventionally quoted on a simple interest basis for a 365-day fixed year.
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| Let's assume for this example that the overnight interest rate quoted for GBP is 5.11%.
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| GBP overnight interest quoted at R = 5.11% means:
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| (i)
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| Interest of:
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| r = R / n
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| r = 5.11% / 365
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| r = 0.014% (= 0.00014) is paid per day.
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| (ii)
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| The ''equivalent'' effective annual rate is calculated from (1 + r).
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| 1 + r = 1 + 0.00014 = 1.00014
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| EAR = (1 + r)<sup>n</sup> - 1
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| EAR = 1.00014<sup>365</sup> - 1
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| EAR = '''5.2424%'''.
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| <span style="color:#4B0082">'''Example 4: EAR from 360-day overnight quote'''</span>
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| USD short term interest is conventionally quoted on a simple interest basis for a 360-day year.
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| 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).
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| USD overnight interest quoted at R = 5.11% means:
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| (i)
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| Interest of:
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| r = R / n
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| r = 5.11% / 360
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| r = 0.01419444% (= 0.0001419444) is paid per day.
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| (ii)
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| The ''equivalent'' effective annual rate is calculated from (1 + r).
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| 1 + r = 1 + 0.0001419444 = 1.0001419444
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| EAR = (1 + r)<sup>n</sup> - 1
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| EAR = 1.0001419444<sup>365</sup> - 1
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| EAR = '''5.3171%'''.
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| (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.)
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| <span style="color:#4B0082">'''Example 5: EAR in a leap year'''</span>
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| 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.
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| For the same periodic rate of interest (r), the effective annual rate is greater in a leap year.
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| For example, where (r) = 0.00014 overnight (as in Example 1).
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| The number of times (n) that the one-day period fits into the calendar year in a leap year = 366.
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| EAR = (1 + r)<sup>n</sup> - 1
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| EAR = 1.00014<sup>366</sup> - 1
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| EAR = '''5.2572%'''.
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| == See also == | | == See also == |
| * [[ACT/365 fixed]] | | * [[Normative statement]] |
| * [[Annual effective rate]] (AER)
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| * [[Annual effective yield]]
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| * [[Annual percentage rate]] (APR)
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| * [[Basis]]
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| * [[Benchmark]]
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| * [[Calculating effective annual rates]]
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| * [[Capital market]]
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| * [[Certificate in Treasury Fundamentals]]
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| * [[Certificate in Treasury]]
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| * [[Compound]]
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| * [[Compound interest]]
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| * [[Continuously compounded rate of return]]
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| * [[Effective annual yield]]
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| * [[Headline ]]
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| * [[Leap year]]
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| * [[LIBOR]]
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| * [[Nominal annual rate]]
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| * [[Periodic discount rate]]
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| * [[Periodic rate of interest]]
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| * [[Periodic yield]]
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| * [[Rate of return]]
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| * [[Real]]
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| * [[Return]]
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| * [[Risk]]
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| * [[Semi-annual rate]]
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| * [[Simple interest]]
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| [[Category:Long_term_funding]]
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| [[Category:Cash_management]]
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