Question
1 Assignment \( 02- \) Questions Question 1 Determine the equivalent continuous compounding rate (to two decimal places) of an interest rate of \( 17,5 \% \) per year, compounded quarterly.
Ask by Vargas Love.
Mar 21,2025 19:10
UpStudy AI Solution
Tutor-Verified Answer
Answer
The equivalent continuous compounding rate is approximately **17.13% per year**.
Solution
**Step 1: Express the quarterly compounded effective annual growth**
An annual nominal interest rate of \(17.5\%\) compounded quarterly gives a quarterly rate of
\[
\frac{0.175}{4} = 0.04375.
\]
The amount after 1 year is
\[
\left(1 + 0.04375\right)^4.
\]
**Step 2: Equate to the continuously compounded growth**
For continuous compounding at rate \(r_c\), the growth over 1 year is given by
\[
e^{r_c}.
\]
Setting the two expressions equal, we have
\[
e^{r_c} = \left(1 + 0.04375\right)^4.
\]
**Step 3: Solve for \(r_c\)**
Taking the natural logarithm on both sides yields:
\[
r_c = 4 \ln\left(1 + 0.04375\right).
\]
Thus,
\[
r_c = 4 \ln\left(1.04375\right).
\]
**Step 4: Compute the numerical value**
Calculate \(\ln(1.04375)\).
Using an approximation,
\[
\ln(1.04375) \approx 0.04282.
\]
Then,
\[
r_c \approx 4 \times 0.04282 = 0.17128.
\]
**Step 5: Express as a percentage**
The equivalent continuous compounding rate is approximately
\[
0.17128 \times 100\% \approx 17.13\%\text{ per year}.
\]
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Simplify this solution The Deep Dive
To find the equivalent continuous compounding rate from a nominal rate that is compounded quarterly, you can use the formula:
\[ r_c = n \cdot \ln\left(1 + \frac{r}{n}\right) \]
where \( r \) is the nominal annual interest rate (in decimal form), and \( n \) is the number of compounding periods per year. For an interest rate of \( 17.5\% \) per annum compounded quarterly, \( r = 0.175 \) and \( n = 4 \). Plugging in those values:
\[ r_c = 4 \cdot \ln\left(1 + \frac{0.175}{4}\right) \]
This works out to approximately \( r_c \approx 0.1783 \), or \( 17.83\% \) when expressed as a percentage.
Thus, the equivalent continuous compounding rate is approximately \( 17.83\% \) per year.
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