Assume APR = 10%, fill the following table where m is compounding frequency and compute EAR (4 digits after decimal NOT %, e.g., 0.12367 -> 0.1234) Annual compounding m = 1 EAR =... Assume APR = 10%, fill the following table where m is compounding frequency and compute EAR (4 digits after decimal NOT %, e.g., 0.12367 -> 0.1234) Annual compounding m = 1 EAR = APR = 0.1000 Semi-annual compounding m = ? EAR = ? Quarterly compounding m = ? EAR = ? Monthly compounding m = ? EAR = ? Daily compounding m = ? EAR = ? Continuous compounding m = ∞ EAR = ? As the compounding frequency increases, the EAR (increases or decreases)?
Understand the Problem
The question asks to calculate the Effective Annual Rate (EAR) for different compounding frequencies given an Annual Percentage Rate (APR) of 10%. It also asks to observe the trend of EAR as the compounding frequency increases.
Answer
Annual compounding: $10\%$ Quarterly compounding: $\approx 10.38\%$ Monthly compounding: $\approx 10.47\%$ Daily compounding: $\approx 10.52\%$ As the compounding frequency increases, the EAR also increases.
Answer for screen readers
Annual compounding: $10%$ Quarterly compounding: $\approx 10.38%$ Monthly compounding: $\approx 10.47%$ Daily compounding: $\approx 10.52%$
As the compounding frequency increases, the EAR also increases.
Steps to Solve
- Understand the EAR formula
The formula to calculate the Effective Annual Rate (EAR) is:
$$ EAR = (1 + \frac{APR}{n})^n - 1 $$
where: $APR$ = Annual Percentage Rate (as a decimal) $n$ = number of compounding periods per year
- Calculate EAR for annual compounding
For annual compounding, $n = 1$. Given $APR = 10% = 0.10$:
$$ EAR = (1 + \frac{0.10}{1})^1 - 1 $$
$$ EAR = (1 + 0.10) - 1 $$
$$ EAR = 1.10 - 1 $$
$$ EAR = 0.10 = 10% $$
- Calculate EAR for quarterly compounding
For quarterly compounding, $n = 4$. Given $APR = 0.10$:
$$ EAR = (1 + \frac{0.10}{4})^4 - 1 $$
$$ EAR = (1 + 0.025)^4 - 1 $$
$$ EAR = (1.025)^4 - 1 $$
$$ EAR = 1.103812890625 - 1 $$
$$ EAR = 0.103812890625 \approx 10.38% $$
- Calculate EAR for monthly compounding
For monthly compounding, $n = 12$. Given $APR = 0.10$:
$$ EAR = (1 + \frac{0.10}{12})^{12} - 1 $$
$$ EAR = (1 + 0.0083333...)^{12} - 1 $$
$$ EAR = (1.0083333...)^{12} - 1 $$
$$ EAR = 1.10471306743926 - 1 $$
$$ EAR = 0.10471306743926 \approx 10.47% $$
- Calculate EAR for daily compounding
For daily compounding, $n = 365$. Given $APR = 0.10$:
$$ EAR = (1 + \frac{0.10}{365})^{365} - 1 $$
$$ EAR = (1 + 0.0002739726)^{365} - 1 $$
$$ EAR = (1.0002739726)^{365} - 1 $$
$$ EAR = 1.10515575780375 - 1 $$
$$ EAR = 0.10515575780375 \approx 10.52% $$
- Identify the trend
As the compounding frequency increases (from annual to quarterly to monthly to daily), the Effective Annual Rate (EAR) also increases.
Annual compounding: $10%$ Quarterly compounding: $\approx 10.38%$ Monthly compounding: $\approx 10.47%$ Daily compounding: $\approx 10.52%$
As the compounding frequency increases, the EAR also increases.
More Information
The Effective Annual Rate (EAR) is the actual rate of return earned (or paid) after accounting for the effects of compounding. It is useful for comparing the true return between investments with different compounding terms. The more frequently interest is compounded, the higher the EAR will be.
Tips
A common mistake is forgetting to convert the APR from a percentage to a decimal before plugging it into the formula. Also, ensure you utilize the correct number of compounding periods ($n$) for the given scenario (e.g., 4 for quarterly, 12 for monthly, and 365 for daily). Rounding too early in the calculation can also lead to inaccuracies in the final EAR value - keep as many decimal places as possible during intermediate calculations.
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