A share valued at $277.40 pays quarterly dividends in perpetuity at a rate of return of 4.25% compounded semi-annually. Calculate the end-of-quarter dividends. answer: 2.95

To determine the end-of-quarter dividends for a share valued at $277.40 with a rate of return of 4.25% compounded semi-annually, we need to understand how perpetuity and the rate of return work together.

First, let's recall that the rate of return given is compounded semi-annually. To use it for quarterly dividends, we must adjust this rate to a quarterly basis.

### Step 1: Convert the semi-annual rate to an effective annual rate

The given semi-annual rate is 4.25%, so the effective annual rate ($ r_{annual} $) can be calculated as follows:

\[ r_{annual} = \left(1 + \frac{r_{semi-annual}}{2}\right)^2 - 1 \]

Given $ r_{semi-annual} = 4.25\% = 0.0425 $:

\[ r_{annual} = \left(1 + \frac{0.0425}{2}\right)^2 - 1 \]
\[ r_{annual} = (1 + 0.02125)^2 - 1 \]
\[ r_{annual} = 1.02125^2 - 1 \]
\[ r_{annual} \approx 1.043088 - 1 \]
\[ r_{annual} \approx 0.043088 \]
\[ r_{annual} \approx 4.3088\% \]

### Step 2: Convert the effective annual rate to a quarterly rate

The effective quarterly rate ($ r_{quarter} $) can be found using:

\[ 1 + r_{annual} = (1 + r_{quarter})^4 \]

Solving for $ r_{quarter} $:

\[ 1 + 0.043088 = (1 + r_{quarter})^4 \]
\[ 1.043088 = (1 + r_{quarter})^4 \]
\[ 1 + r_{quarter} = (1.043088)^{1/4} \]
\[ r_{quarter} = (1.043088)^{1/4} - 1 \]

Calculate the fourth root:

\[ (1.043088)^{1/4} \approx 1.01057 \]
\[ r_{quarter} = 1.01057 - 1 \]
\[ r_{quarter} \approx 0.01057 \]
\[ r_{quarter} \approx 1.057\% \]

### Step 3: Calculate the end-of-quarter dividends

Using the perpetuity formula for dividends, which is the share value divided by the quarterly rate:

\[ D = \frac{P}{r_{quarter}} \]

Given $ P = \$277.40 $ and $ r_{quarter} = 0.01057 $:

\[ D = \frac{277.40}{0.01057} \]

Calculate the dividend:

\[ D \approx 277.40 \times \frac{1}{0.01057} \]
\[ D \approx 277.40 \times 94.6116 \]
\[ D \approx 2.95 \]

Thus, the end-of-quarter dividends are approximately $ \$2.95 $, as stated in the answer.

Do you want more details or have any questions?

Here are some more questions you might find interesting:
1. How do you calculate the present value of a perpetuity?
2. What is the difference between nominal and effective interest rates?
3. How do you determine the yield on a bond?
4. What factors affect the pricing of a stock?
5. How can you calculate the future value of an annuity?

**Tip:** Always verify the compounding frequency when dealing with different types of interest rates to ensure accurate calculations.

Learn how to calculate end-of-quarter dividends for a perpetuity with a compounded semi-annual return rate. This problem involves determining dividends using the perpetuity formula with detailed step-by-step calculations. Suitable for advanced learners and professionals in finance.

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