Math Problem Statement

You have just received a $370 dividend from your common stock. The dividends are expected to grow at a rate of 11% annually for the next four years, and at 4.2% thereafter. You anticipate earning a 5.6% return on this stock. Calculate dividend yield and capital gains yield.

Solution

To calculate the dividend yield and capital gains yield, we must first understand the formulas:

  1. Dividend Yield is the ratio of the dividend received to the stock price: Dividend Yield=D0P0\text{Dividend Yield} = \frac{D_0}{P_0} where D0D_0 is the dividend just received, and P0P_0 is the current stock price.

  2. Capital Gains Yield is the expected percentage increase in the stock price, often represented by the dividend growth rate for the stock: Capital Gains Yield=g\text{Capital Gains Yield} = g where gg is the dividend growth rate after the period of higher growth (after year 4 in this case).

Step-by-Step Breakdown:

  • We know:
    • Initial dividend, D0=370D_0 = 370
    • Dividend growth for the first 4 years = 11%
    • Dividend growth rate after 4 years = 4.2%
    • Expected return on stock, r=5.6%r = 5.6\%

Finding the Stock Price (P0P_0):

The current stock price (P0P_0) is determined by the Dividend Discount Model (DDM), which factors in both the high-growth period and the stable growth period.

The price is given by the present value of future dividends: P0=t=14Dt(1+r)t+P4(1+r)4P_0 = \sum_{t=1}^{4} \frac{D_t}{(1 + r)^t} + \frac{P_4}{(1 + r)^4} where P4P_4 is the stock price at the end of the fourth year and can be calculated using: P4=D5rg2P_4 = \frac{D_5}{r - g_2} where D5=D4×(1+g2)D_5 = D_4 \times (1 + g_2) is the dividend in the fifth year and g2=4.2%g_2 = 4.2\% is the long-term growth rate.

Let's calculate the stock price first, and then we will calculate the yields.### Results:

  1. Current Stock Price (P0P_0): $35,297.58
  2. Dividend Yield: 1.05% (or 0.0105)
  3. Capital Gains Yield: 4.2% (or 0.042)

The dividend yield is based on the initial dividend relative to the current stock price, while the capital gains yield corresponds to the long-term growth rate of 4.2%.

Would you like further details on the calculation or any additional questions?

Here are 5 related questions to expand the topic:

  1. How does the dividend growth rate impact the stock price?
  2. What factors can influence the dividend growth rate in a company?
  3. How would an increase in the required rate of return affect the dividend yield?
  4. What is the relationship between dividend yield and overall stock returns?
  5. How do different stock valuation models compare, such as the Gordon Growth Model vs. the Dividend Discount Model?

Tip: When estimating stock returns, it's essential to distinguish between dividend income and potential capital appreciation to get a holistic view of expected returns.

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Math Problem Analysis

Mathematical Concepts

Finance
Stock Valuation
Dividend Growth
Capital Gains

Formulas

Dividend Yield = D_0 / P_0
Capital Gains Yield = g
P_0 = Σ(D_t / (1 + r)^t) + (P_4 / (1 + r)^4)
P_4 = D_5 / (r - g_2)

Theorems

Dividend Discount Model (DDM)
Gordon Growth Model

Suitable Grade Level

Undergraduate (Finance, Economics, or Business Studies)