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Chapter 9: Problem 2

Thomas Brothers is expected to pay a \(\$ 0.50\) per share dividend at the end of the year (that is, \(D_{1}=\$ 0.50\) ). The dividend is expected to grow at a constant rate of 7 percent a year. The required rate of return on the stock, \(r_{v}\) is 15 percent. What is the stock's value per share?

Short Answer

Expert verified
The stock's value per share is $6.25.

Step by step solution

01

Identify Known Values

First, identify all the given values in the problem statement. We have:- The expected dividend at the end of the year, \(D_1 = \$0.50\).- The constant growth rate of the dividend is 7%, which is \(g = 0.07\).- The required rate of return on the stock is 15%, so \(r_v = 0.15\).
02

Apply Dividend Discount Model

The stock's value per share can be calculated using the Gordon Growth Model (a type of Dividend Discount Model for stocks with dividends that grow at a constant rate). The formula is:\[ P_0 = \frac{D_1}{r_v - g} \]where \(P_0\) is the present value per share, \(D_1\) is the expected dividend, \(r_v\) is the required rate of return, and \(g\) is the growth rate.
03

Substitute Values into Formula

Substitute the values into the formula: \[ P_0 = \frac{0.50}{0.15 - 0.07} \] This will give you the current value of the stock based on the information given.
04

Calculate the Stock Value

Calculate the expression:\[ P_0 = \frac{0.50}{0.08} \] Perform the division to find:\[ P_0 = 6.25 \] Thus, the stock is valued at \$6.25 per share.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gordon Growth Model
Are you curious about how to value a stock with dividends? The Gordon Growth Model is a popular method used for this. You might encounter this model under the umbrella of the Dividend Discount Model, especially when dealing with dividends that grow at a consistent rate over time. The essential formula of the Gordon Growth Model is: \[ P_0 = \frac{D_1}{r_v - g} \] where:
  • \( P_0 \) is the current stock value per share.
  • \( D_1 \) is the expected dividend payment in the next period.
  • \( r_v \) represents the required rate of return.
  • \( g \) stands for the dividend growth rate.
This model is particularly useful because it estimates future dividends to determine a stock's value. By assuming dividends grow at a steady rate, the Gordon Growth Model simplifies stock evaluation for many investors.
dividend growth rate
The dividend growth rate is an essential part of understanding the future potential of a stock. It represents how much you can expect the dividend to increase annually. This is incredibly important because consistent growth implies increasing cash flows from the stock, which should ultimately enhance the stock's value. So, how is it applied? Assume a company is pretty stable and can reliably increase its dividends by a certain percentage every year. This growth rate is denoted by \( g \). In our example, we have \( g = 0.07 \), meaning the dividends are expected to grow by 7% each year. This constant growth shows financial health and optimism from a company's performance.Picking stocks with a good dividend growth rate can be a smart move if you're looking for long-term investments. Remember, growth in dividends often translates to growth in your investment returns!
required rate of return
The required rate of return is a critical component for evaluating investment opportunities. It determines the minimum return an investor would accept to invest in a stock instead of a risk-free asset. In the Dividend Discount Model and Gordon Growth Model, it's denoted by \( r_v \). This rate takes into account the risk associated with the stock compared to a risk-free alternative, such as government bonds. For instance, in our exercise, we have \( r_v = 0.15 \), or 15%. This indicates that, for the given risk, investors expect a return of at least 15% per year.By linking the required rate of return with the dividend growth model, investors can decide if a stock's current price is attractive compared to its calculated value:
  • If the calculated value, using the model, is higher than the current market price, the stock may be underpriced.
  • Conversely, if it is lower, the stock might be overpriced.
Understanding your required rate of return helps ensure your investment goals align with the level of risk you're prepared to accept.

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Most popular questions from this chapter

Hart Enterprises recently paid a dividend, \(\mathrm{D}_{0},\) of \(\$ 1.25 .\) It. It expects to have nonconstant growth of 20 percent for 2 years followed by a constant rate of 5 percent thereafter. The firm's required return is 10 percent. a. How far away is the terminal, or horizon, date? b. What is the firm's horizon, or terminal, value? c. What is the firm's intrinsic value today, \(\hat{P}_{0}\) ?

You are considering an investment in Keller Corp's stock, which is expected to pay a dividend of \(\$ 2\) a share at the end of the year \(\left(\mathrm{D}_{1}=\$ 2.00\right)\) and has a beta of \(0.9 .\) The risk-free rate is 5.6 percent, and the market risk premium is 6 percent. Keller currently sells for \(\$ 25\) a share, and its dividend is expected to grow at some constant rate g. Assuming the market is in equilibrium, what does the market believe will be the stock price at the end of 3 years? (That is, what is \(\hat{P}_{3} ?\))

A stock is expected to pay a dividend of \(\$ 0.50\) at the end of the year (that is, \(D_{1}=0.50\) ), and it should continue to grow at a constant rate of 7 percent a year. If its required return is 12 percent, what is the stock's expected price 4 years from today?

Equilibrium stock price The risk-free rate of return, \(\mathrm{r}_{\mathrm{RF}^{\prime}}\), is 6 percent; the required rate of return on the market, \(\mathbf{r}_{M^{\prime}}\) is 10 percent; and Upton Company's stock has a beta coefficient of 1.5. a. If the dividend expected during the coming year, \(\mathrm{D}_{1},\) is \(\$ 2.25\) and if \(g=a\) constant 5 percent, at what price should Upton's stock sell? b. \(\quad\) Now, suppose the Federal Reserve Board increases the money supply, causing the risk-free rate to drop to 5 percent and \(r_{\mathrm{M}}\) to fall to 9 percent. What would happen to Upton's price? c. In addition to the change in part b, suppose investors' risk aversion declines, and this, combined with the decline in \(r_{R F},\) causes \(r_{M}\) to fall to 8 percent. Now, what is Upton's price? d. Now suppose Upton has a change in management. The new group institutes policies that increase the expected constant growth rate from 5 to 6 percent. Also, the new management smooths out fluctuations in sales and profits, causing beta to decline from 1.5 to \(1.3 .\) Assume that \(r_{R F}\) and \(r_{M}\) are equal to the values in part \(c .\) After all these changes, what is its new equilibrium price? (Note: \(\left.D_{1} \text { is now } \$ 2.27 .\right)\)

Your broker offers to sell you some shares of Bahnsen \& Co. common stock that paid a dividend of \(\$ 2\) yesterday. Bahnsen's dividend is expected to grow at 5 percent per year for the next 3 years, and, if you buy the stock, you plan to hold it for 3 years and then sell it. The appropriate discount rate is 12 percent. a. Find the expected dividend for each of the next 3 years; that is, calculate \(\mathrm{D}_{1}, \mathrm{D}_{2},\) and \(\mathrm{D}_{3} .\) Note that \(\mathrm{D}_{0}=\$ 2.00\) b. Given that the first dividend payment will occur 1 year from now, find the present value of the dividend stream; that is, calculate the \(\mathrm{PV}\) of \(\mathrm{D}_{1}, \mathrm{D}_{2},\) and \(\mathrm{D}_{3},\) and then sum these PVs. c. You expect the price of the stock 3 years from now to be \(\$ 34.73 ;\) that is, you expect \(\hat{\mathrm{P}}_{3}\) to equal \(\$ 34.73 .\) Discounted at a 12 percent rate, what is the present value of this expected future stock price? In other words, calculate the PV of \(\$ 34.73\) d. If you plan to buy the stock, hold it for 3 years, and then sell it for \(\$ 34.73,\) what is the most you should pay for it today? e. Use Equation \(9-2\) to calculate the present value of this stock. Assume that \(\mathrm{g}=5 \%\) and it is constant. f. Is the value of this stock dependent upon how long you plan to hold it? In other words, if your planned holding period were 2 years or 5 years rather than 3 years, would this affect the value of the stock today, \(\hat{P}_{0}\) ? Explain.
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