You have 100 shares of stock A and 1 bond B. The total value of the two holdings is $5,993.3. Bond B has a coupon rate of 12.30 percent, par value of $1000, YTM of 8.58 percent, 17.0 years until maturity, and semi-annual coupons with the next coupon expected in 6 months. Stock A is expected to pay a dividend of $8.32 every year forever and has an expected return of X percent per year. The stock's next dividend is expected in 1 year. What is X?

To determine $ X $, the expected return of Stock A, let's break down the problem and solve it step-by-step.

### Step 1: Find the Present Value of Bond B
Since we know the total value of Stock A and Bond B combined, we first need to determine the current value of Bond B.

#### Given information about Bond B:
- **Coupon rate** = 12.30% (annual), so the semi-annual coupon payment is:
  \[
  \text{Coupon payment} = \frac{1000 \times 0.123}{2} = 61.5
  \]
- **Yield to Maturity (YTM)** = 8.58% (annual), or 4.29% per semi-annual period.
- **Par value** = $1,000
- **Time to maturity** = 17 years, or $ 17 \times 2 = 34 $ semi-annual periods.

The present value (PV) of Bond B, given that it pays semi-annual coupons, can be calculated as the sum of the present values of the annuity (coupon payments) and the par value (face value) at maturity.

\[
\text{PV of coupons} = 61.5 \times \frac{1 - (1 + 0.0429)^{-34}}{0.0429}
\]
\[
\text{PV of face value} = 1000 \times (1 + 0.0429)^{-34}
\]

Calculating these values:

1. **PV of coupons**:
   \[
   61.5 \times \frac{1 - (1 + 0.0429)^{-34}}{0.0429} \approx 1032.16
   \]

2. **PV of face value**:
   \[
   1000 \times (1 + 0.0429)^{-34} \approx 291.85
   \]

Adding these gives the present value of Bond B:
\[
\text{PV of Bond B} = 1032.16 + 291.85 = 1324.01
\]

### Step 2: Find the Value of Stock A
The total value of the two holdings is given as $5,993.3, so the value of Stock A alone is:
\[
\text{Value of Stock A} = 5993.3 - 1324.01 = 4669.29
\]

### Step 3: Calculate $ X $, the Expected Return on Stock A
Stock A is a perpetuity since it pays a dividend of $8.32 forever. The value of a perpetuity is given by:
\[
\text{Value of Stock A} = \frac{\text{Dividend}}{X}
\]
Solving for $ X $:
\[
X = \frac{\text{Dividend}}{\text{Value of Stock A}}
\]
Substitute the values:
\[
X = \frac{8.32}{\frac{4669.29}{100}} \approx 8.32 \text{% per year}
\]

### Answer:
The expected return $ X $ on Stock A is approximately **8.32% per year**.

---

Would you like details on any of the calculations or any questions about the process?

### Additional Questions:
1. How would the expected return change if the dividend increased?
2. What impact would a decrease in Bond B’s YTM have on Stock A’s expected return?
3. How would semi-annual dividends affect Stock A’s expected return calculation?
4. What is the effect of a shorter time to maturity for Bond B on the overall portfolio value?
5. How could changes in interest rates influence the valuation of both Bond B and Stock A?

#### Tip:
In perpetuity calculations, always confirm the dividend frequency aligns with the expected return frequency (e.g., both annually).

Explore how to calculate the expected return on Stock A using perpetuity valuation, bond valuation, and discounting cash flows. Given a bond with a 12.30% coupon rate, par value of $1000, 8.58% YTM, and 17 years to maturity, along with a stock paying $8.32 in dividends annually, we calculate Stock A's expected return. Suitable for undergraduate finance students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation