You have 100 shares of stock A and 1 bond B. The total value of the two holdings is $4,359.67. Bond B has a coupon rate of 14.40 percent, par value of $1000, YTM of 18.26 percent, 21.0 years until maturity, and semi-annual coupons with the next coupon expected in 6 months. Stock A is expected to pay a dividend of $6.19 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 find the expected return $ X $ of Stock A, we need to determine its current price using the given information and then solve for $ X $.

First, let's calculate the price of Bond B using its given characteristics.

**Bond B:**

- Coupon rate ($ C $): 14.40% of $ \$1000 $ par value = $ 0.144 \times 1000 = \$144 $
- Number of coupon payments per year: 2
- Semi-annual coupon payment: $ \frac{144}{2} = \$72 $
- Yield to Maturity (YTM) ($ r $): 18.26% per year, or 9.13% semi-annually (since it is a semi-annual coupon bond)
- Number of semi-annual periods ($ n $): $ 21 $ years $\times 2 = 42 $ periods

The price of Bond B ($ P_B $) is calculated using the present value of annuities (for the coupons) and the present value of a lump sum (for the par value):

\[ P_B = \sum_{i=1}^{42} \frac{72}{(1 + 0.0913)^i} + \frac{1000}{(1 + 0.0913)^{42}} \]

To simplify, we use the formula for the present value of an annuity and the present value of a lump sum:

\[ P_B = 72 \times \left( \frac{1 - (1 + 0.0913)^{-42}}{0.0913} \right) + \frac{1000}{(1 + 0.0913)^{42}} \]

Calculating this step by step:

1. Present value of coupons:
\[ 72 \times \left( \frac{1 - (1 + 0.0913)^{-42}}{0.0913} \right) \]

2. Present value of the par value:
\[ \frac{1000}{(1 + 0.0913)^{42}} \]

Using a calculator or financial software to compute these values:

1. Present value of coupons:
\[ 72 \times 10.196 = 734.112 \]

2. Present value of the par value:
\[ \frac{1000}{21.714} \approx 46.055 \]

Therefore, the price of Bond B:
\[ P_B = 734.112 + 46.055 = 780.167 \]

Now, we know the value of Bond B is \$780.167. The total value of the holdings is \$4,359.67. Hence, the value of Stock A can be found by subtracting the value of Bond B from the total value:

\[ \text{Value of Stock A} = 4359.67 - 780.167 = 3579.503 \]

This is the total value of 100 shares of Stock A. Therefore, the price per share of Stock A ($ P_A $) is:

\[ P_A = \frac{3579.503}{100} = 35.79503 \]

Stock A is expected to pay a dividend of \$6.19 every year forever (a perpetuity). The price of a perpetuity is given by:

\[ P_A = \frac{D}{r} \]

where $ D $ is the dividend and $ r $ is the expected return. Rearranging to solve for $ r $:

\[ r = \frac{D}{P_A} \]

Substituting the known values:

\[ r = \frac{6.19}{35.79503} \approx 0.1729 \]

Converting to percentage:

\[ X = 0.1729 \times 100 = 17.29\% \]

So, the expected return $ X $ of Stock A is approximately **17.29%** per year.

Would you like more details or have any questions? 

Here are five related questions you might find interesting:
1. How do you calculate the present value of an annuity?
2. What factors influence the yield to maturity of a bond?
3. How is the price of a perpetuity derived?
4. What is the significance of a bond's coupon rate compared to its YTM?
5. How does the dividend discount model help in valuing stocks?

**Tip:** Understanding the concept of present value is crucial in finance as it allows you to determine the value of future cash flows in today's terms.

You have 100 shares of stock A and 1 bond B. The total value of the two holdings is $4,359.67. Bond B has a coupon rate of 14.40 percent, par value of $1000, YTM of 18.26 percent, 21.0 years until maturity, and semi-annual coupons with the next coupon expected in 6 months. Stock A is expected to pay a dividend of $6.19 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?

This problem involves calculating the expected return for Stock A using bond valuation techniques and the perpetuity formula. The solution includes bond pricing with semi-annual coupons, present value of future cash flows, and stock valuation using a perpetuity. Suitable for university-level finance or economics 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