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Chapter 9: Problem 11

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?

Short Answer

Expert verified
The expected stock price 4 years from today is approximately $13.13.

Step by step solution

01

Understand the Problem

The problem requires us to find the stock's expected price 4 years from today. We're given the first dividend \(D_1 = 0.50\), a constant growth rate of 7%, and a required return of 12%.
02

Calculate Future Dividends

First, we calculate the dividends for the next few years using the formula \(D_n = D_1 \times (1+g)^{n-1}\), where \(g\) is the growth rate.- \(D_1 = 0.50\)- \(D_2 = 0.50 \times (1+0.07) = 0.50 \times 1.07 = 0.535\)- \(D_3 = 0.535 \times 1.07 \approx 0.57345\)- \(D_4 = 0.57345 \times 1.07 \approx 0.6135915\)
03

Calculate Stock Price in Year 4

Using the Gordon Growth Model, the stock's expected price can be calculated as \(P_4 = \frac{D_5}{r-g}\), where \(D_5\) is the dividend in one year from year 4, \(r\) is the required return, and \(g\) is the growth rate.Calculate \(D_5\):\[D_5 = D_4 \times (1+g) = 0.6135915 \times 1.07 \approx 0.65654\]Then calculate \(P_4\):\[P_4 = \frac{0.65654}{0.12-0.07} = \frac{0.65654}{0.05} = 13.1308\]
04

Conclusion

The expected stock price 4 years from today is approximately \(\$13.13\). This represents the price considering dividends grow at a constant rate of 7% per year and the required return remains at 12%.

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Key Concepts

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

Dividend Growth
Dividend growth refers to the anticipated continuous increase in the dividends a company pays out to its shareholders. It's a crucial aspect of stock valuation because it reflects the expected growth in a company's earnings and cash flow.
This concept assumes that dividends increase at a constant rate. In our exercise, we consider a growth rate of 7% per year.
  • How it works: The future dividends are calculated starting from a known dividend amount, increasing each year by the growth rate. For example, if the first dividend is $0.50, the next year's dividend is calculated as:
    \(D_2 = 0.50 \times (1+0.07) = 0.535\).
  • Significance: Accurately forecasting dividend growth helps investors predict future income streams. It shows potential increases in value, making the stock more attractive.
By understanding dividend growth, investors can gain insights into the company's potential future performance and make informed investment decisions.
Required Return
Required return is the minimum return investors expect to achieve from an investment to compensate for its risk. It's expressed as a percentage and is crucial for evaluating investments.
In our scenario, the required return is 12%.
  • Determining Factors: Investors' expected returns account for the risk-free rate, risks specific to the company or market, and the opportunity cost of capital.
    An investment's risk profile and market conditions can influence these components.
  • Importance in Valuation: Using required return, investors determine the present value of expected returns, like dividends. By comparing this to the stock's current price, they assess if it's a good investment.
Hence, understanding the required return helps investors evaluate whether a stock is appropriately priced compared to the risk it's assumed to carry.
Stock Valuation
Stock valuation estimates a stock's value, reflecting its perceived fair price based on future cash flows such as dividends. In our exercise, stock valuation involves using the Gordon Growth Model, a popular method for valuing dividend-paying stocks.
The model uses the formula: \[P = \frac{D}{r-g}\] where \(P\) is the stock price, \(D\) is the dividend expected next year, \(r\) is the required return, and \(g\) is the growth rate.
Steps in this process include:
  • Calculate Future Dividends: First, predict dividends over each year by applying the growth rate.
  • Determine Value: Apply the formula with these figures to find the expected price 4 years later, which is approximately $13.13 for this exercise.
Thus, stock valuation provides a systematic approach to calculate what investors should be willing to pay for future returns, influencing their buy/hold/sell decisions.

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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, \(\hat{P}_{0}\) ?

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.

Martell Mining Company's ore reserves are being depleted, so its sales are falling. Also, its pit is getting deeper each year, so its costs are rising. As a result, the company's earnings and dividends are declining at the constant rate of 5 percent per year. If \(\mathrm{D}_{0}=\$ 5\) and \(\mathrm{r}_{\mathrm{s}}=15 \%\), what is the value of Martell Mining's stock?

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: \(\left.D_{1} \text { is now } \$ 2.27 .\right)\)

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 \(\hat{P}_{3} ?\))
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