Math Problem Statement

Suppose we are interested in the price of a stock in five years, P5. We just paid a dividend of $2.20 and the growth rate is 4%. The required return is 12%. What is the price of the stock in five years?

Multiple Choice

$17.40

$24.00

$32.70

$34.80

$38.20

Solution

We can use the Gordon Growth Model (Dividend Discount Model) to determine the price of the stock in five years, P5P_5. This model assumes that dividends grow at a constant rate indefinitely and that the price of the stock is based on the present value of all future dividends.

The formula for the stock price based on the Gordon Growth Model is:

Pt=Dt+1rgP_t = \frac{D_{t+1}}{r - g}

Where:

  • PtP_t is the price of the stock at time tt,
  • Dt+1D_{t+1} is the dividend at time t+1t+1,
  • rr is the required return,
  • gg is the growth rate of the dividends.

Step-by-Step Solution:

  1. Current Dividend (D0) = $2.20
  2. Growth Rate (g) = 4% = 0.04
  3. Required Return (r) = 12% = 0.12

The dividend in year 6 (because we are interested in P5P_5, which depends on the dividend in year 6) can be found by growing the current dividend:

D6=D0×(1+g)6=2.20×(1+0.04)6=2.20×1.265319=2.783702D_6 = D_0 \times (1 + g)^6 = 2.20 \times (1 + 0.04)^6 = 2.20 \times 1.265319 = 2.783702

Now, applying the Gordon Growth Model formula to find P5P_5:

P5=D6rg=2.7837020.120.04=2.7837020.08=34.80P_5 = \frac{D_6}{r - g} = \frac{2.783702}{0.12 - 0.04} = \frac{2.783702}{0.08} = 34.80

Final Answer:

The price of the stock in five years is $34.80.

Would you like more details or have any other questions?

Here are 5 related questions for practice:

  1. How would the price change if the growth rate were 5% instead of 4%?
  2. What would the stock price be if the required return were 10%?
  3. How would the price change if the dividend payout were $3.00 instead of $2.20?
  4. What is the present value of the stock today using the same parameters?
  5. How does a higher growth rate affect the stock price in the Gordon Growth Model?

Tip: When solving finance problems involving growth models, always double-check that you are calculating dividends at the correct future time!

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Math Problem Analysis

Mathematical Concepts

Finance
Gordon Growth Model
Dividend Discount Model

Formulas

P_t = D_{t+1} / (r - g)
D_t = D_0 * (1 + g)^t

Theorems

Gordon Growth Model

Suitable Grade Level

College Level / Finance Courses