6.6 Nominal and effective annual interest rate
6.6 Nominal and Effective Annual Interest Rate
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Introduction to nominal and effective annual interest rates.
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Study Design Dot Points:
Explicit rule of arithmetic/geometric sequence evaluation.
Learning Intention: Convert nominal interest rates to compounding period interest rates.
Success Criteria:
Differentiate between nominal and effective annual interest rates.
Calculate the effective annual interest rate.
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Worked Example 2:
Formula to compute effective annual interest rate.
Calculation of 'real' annual interest rate considering compounding periods.
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Differences Between Nominal and Effective Rates:
Nominal Interest Rate: Advertised by financial institutions, expressed as a percentage per annum, does not account for compounding periods.
Effective Annual Interest Rate (EAR): Compares annual nominal interest between different loans or investments, taking into account different compounding periods.
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The effective annual interest rate is a realistic measure of interest rates.
Effective Annual Interest Rate Formula:
( r_{eff} = (1 + i/n)^n - 1 )
Where:
( r_{eff} ): effective annual interest rate (%)
( i ): nominal annual interest rate (as decimal)
( n ): number of compounding periods per year.
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Example Calculation:
For a $100 loan at 10% p.a. compounded quarterly over 2 years:
Calculate effective rate: ( r_{eff} = (1 + 0.10/4)^{4 imes 2} - 1 = 0.1038 ) or 10.38%.
Amount calculations for each period until the total due.
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Worked Example 1:
Example of calculating effective annual interest rate for Jason’s loan at 12% p.a. for 4 years.
Method 1:
Values: ( n = 4, i = 0.12 ).
Apply effective annual rate formula: ( r_{eff} = (1 + 0.12/4)^4 - 1 = 0.1255 ) or 12.55%.
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Worked Example 1 Continued:
Method 2: Using CASIO calculator for effective interest rate.
Input formula on Main screen to obtain 12.55% correct to 2 decimal places.
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Your Turn:
Emma’s car loan calculation at 9% p.a. with monthly repayments.
Using formula: ( r_{eff} = (1 + 0.09/12)^{12} - 1 = 0.09380689... ) or 9.38%.