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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?

Short Answer

Expert verified
Martell Mining's stock value is approximately \$23.75.

Step by step solution

01

Identify the Given Variables

The problem gives us several key variables that we'll use to calculate the stock price:- Initial dividend, \( D_0 = \$5 \)- Required rate of return, \( r_s = 15\% = 0.15 \)- Decline rate of dividends, \( g = -5\% = -0.05 \).
02

Apply the Gordon Growth Model

The Gordon Growth Model (or Dividend Discount Model for a constant growth rate) is used to determine the intrinsic value of the stock. The formula is: \( P_0 = \frac{D_1}{r_s - g} \). Where:- \( P_0 \) is the price of the stock today.- \( D_1 \) is the expected dividend next year.- \( r_s \) is the required rate of return.- \( g \) is the growth rate (or decline rate in this case) of the dividend.
03

Calculate Next Year's Dividend (\(D_1\))

Next year's dividend, \( D_1 \), can be calculated using the formula: \( D_1 = D_0 (1 + g) \).Substitute the given figures: \[ D_1 = 5 \times (1 - 0.05) = 5 \times 0.95 = 4.75 \]
04

Calculate the Stock Price (\(P_0\))

Substitute \( D_1 \), \( r_s \), and \( g \) into the Gordon Growth Model formula:\[ P_0 = \frac{4.75}{0.15 - (-0.05)} = \frac{4.75}{0.15 + 0.05} = \frac{4.75}{0.20} = 23.75 \]
05

Interpretation of the Result

The calculated \(P_0\) represents the intrinsic value of Martell Mining's stock given the current financial situation and the future earnings expectations. Therefore, the current stock price is expected to be \$23.75.

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Key Concepts

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

Gordon Growth Model
The Gordon Growth Model, sometimes referred to as the Dividend Discount Model for a constant growth rate, is an effective method for determining the intrinsic value of a stock. This model is especially useful when dividends are expected to grow (or decline) at a constant rate indefinitely. It simplifies stock valuation by focusing on future dividends and the required rate of return. The formula of the Gordon Growth Model is given as: \[ P_0 = \frac{D_1}{r_s - g} \]Where:
  • \(P_0\) represents the current price of the stock.
  • \(D_1\) is the expected dividend in the next year.
  • \(r_s\) is the required rate of return.
  • \(g\) is the growth rate of dividends.
This model assumes that the only return on the stock is the dividend, and that it grows at a constant rate forever. Thus, it helps investors find out whether a stock is priced fairly for its potential future earnings.
Dividend Discount Model
At its core, the Dividend Discount Model (DDM) values a stock by assuming the present value of all future dividends. In its simplest form (Gordon Growth Model), it works best with companies that pay consistent dividends. The DDM is useful as it streamlines the valuation process to concentrate on dividend payouts, providing a clear picture of potential returns. The formula, as used for constant growth, is similar:\[ P_0 = \frac{D_1}{r_s - g} \]If dividends do not grow at a constant rate, the formula becomes more complex. However, for companies with steady dividend growth, this model provides a straightforward means to gauge value. Investors favor this model when dividends are a significant portion of earnings and can be continuously forecasted. It falls short when dividend growth isn't stable, or the company reinvests earnings instead of distributing them.
Required Rate of Return
The required rate of return is a key factor in determining the attractiveness of an investment. It represents the minimum return an investor expects to achieve to compensate for risk. In stock valuation, this rate helps weigh the future benefits of holding a stock against its risks. The rate can vary based on market conditions, risk appetite, and interest rates. Calculating a realistic required rate often involves considering:
  • The risk-free rate, like government bond yields.
  • The stock's beta, indicative of its risk compared to the market.
  • Market risk premium, which is the additional return expected from a risky investment over the risk-free rate.
For Martell Mining, with a required rate of return of 15%, investors need at least this percentage to consider it worthwhile, reflecting both the risks and potential returns associated with decreased dividends.
Intrinsic Value of Stock
Intrinsic value is the estimated worth of a stock based on fundamental financial analysis and expected future cash flows, rather than current market price. It aids in discerning whether a stock is over or undervalued. Intrinsic value calculation involves predicting factors like dividends, cash flows, and growth rates. Using models like the Gordon Growth Model, investors estimate a fair stock price:\[ P_0 = \frac{D_1}{r_s - g} \]Once calculated, comparing this intrinsic value against the market price reveals investment opportunities—if the intrinsic value exceeds market price, the stock might be undervalued and vice versa. Martell Mining's stock, with an intrinsic value of $23.75, indicates what's deemed a fair valuation, considering its reduced future dividends due to declining operations.

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Most popular questions from this chapter

Harrison Clothiers' stock currently sells for \(\$ 20\) a share. It just paid a dividend of \(\$ 1.00\) a share (that is, \(D_{0}=\$ 1.00\) ). The dividend is expected to grow at a constant rate of 6 percent a year. What stock price is expected 1 year from now? What is the required rate of return?

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?

Suppose Chance Chemical Company's management conducted a study and concluded that if it expands its consumer products division (which is less risky than its primary business, industrial chemicals), its beta would decline from 1.2 to \(0.9 .\) However, consumer products have a somewhat lower profit margin, and this would cause its constant growth rate in earnings and dividends to fall from 6 percent to 4 percent. The following also apply: \(\mathrm{r}_{\mathrm{M}}=9 \% ; \mathrm{r}_{\mathrm{RF}}=6 \% ;\) and \(\mathrm{D}_{0}=\$ 2.00\). a. Should management expand the consumer products division? b. Assume all the facts as given above except the change in the beta coefficient. How low would the beta have to fall to cause the expansion to be a good one? (Hint: Set \(\hat{\mathrm{P}}_{0}\) under the new policy equal to \(\hat{\mathrm{P}}_{0}\) under the old one, and find the new beta that will produce this equality.

Robert Balik and Carol Kiefer are senior vice presidents of the Mutual of Chicago Insurance Company. They are co-directors of the company's pension fund management division, with Balik having responsibility for fixed income securities (primarily bonds) and Kiefer being responsible for equity investments. A major new client, the California League of Cities, has requested that Mutual of Chicago present an investment seminar to the mayors of the represented cities, and Balik and Kiefer, who will make the actual presentation, have asked you to help them. To illustrate the common stock valuation process, Balik and Kiefer have asked you to analyze the Bon Temps Company, an employment agency that supplies word-processor operators and computer programmers to businesses with temporarily heavy workloads. You are to answer the following questions: a. Describe briefly the legal rights and privileges of common stockholders. b. (1) Write out a formula that can be used to value any stock, regardless of its dividend pattern. (2) What is a constant growth stock? How are constant growth stocks valued? (3) What are the implications if a company forecasts a constant \(g\) that exceeds its \(r_{s} ?\) Will many stocks have expected \(g > r_{s}\) in the short run (that is, for the next few years)? In the long run (that is, forever)? c. Assume that Bon Temps has a beta coefficient of \(1.2,\) that the risk-free rate (the yield on T-bonds) is 7 percent, and that the required rate of return on the market is 12 percent. What is Bon Temps' required rate of return? d. Assume that Bon Temps is a constant growth company whose last dividend (D), which was paid yesterday) was \(\$ 2.00\) and whose dividend is expected to grow indefinitely at a 6 percent rate. (1) What is the firm's expected dividend stream over the next 3 years? (2) What is its current stock price? (3) What is the stock's expected value 1 year from now? (4) What are the expected dividend yield, capital gains yield, and total return during the first year? e. Now assume that the stock is currently selling at \(\$ 30.29\). What is its expected rate of return? f. What would the stock price be if its dividends were expected to have zero growth? g. Now assume that Bon Temps is expected to experience nonconstant growth of 30 percent for the next 3 years, then to return to its long-run constant growth rate of 6 percent. What is the stock's value under these conditions? What are its expected dividend and capital gains yields in Year 1? Year 4? h. Suppose Bon Temps is expected to experience zero growth during the first 3 years and then to resume its steady-state growth of 6 percent in the fourth year. What would its value be then? What would its expected dividend and capital gains yields be in Year \(1 ?\) In Year \(4 ?\) i. Finally, assume that Bon Temps' earnings and dividends are expected to decline at a constant rate of 6 percent per year, that is, \(g=-6 \%\). Why would anyone be willing to buy such a stock, and at what price should it sell? What would be its dividend and capital gains yields in each year? j. Suppose Bon Temps embarked on an aggressive expansion that requires additional capital. Management decided to finance the expansion by borrowing \(\$ 40\) million and by halting dividend payments to increase retained earnings. Its WACC is now 10 percent, and the projected free cash flows for the next 3 years are \(-\$ 5\) million, \(\$ 10\) million, and \(\$ 20\) million. After Year \(3,\) free cash flow is projected to grow at a constant 6 percent. What is Bon Temps' total value? If it has 10 million shares of stock and \(\$ 40\) million of total debt, what is the price per share? k. For Bon Temps' stock to be in equilibrium, what relationship must exist between its estimated intrinsic value and its current stock price and between its expected and required rates of return? Are the equilibrium intrinsic value and expected rate of return the values that management estimates or values as estimated by some other entity? Explain. l. If equilibrium does not exist, how will it be established? m. Suppose Bon Temps decided to issue preferred stock that would pay an annual dividend of \(\$ 5,\) and the issue price was \(\$ 50\) per share. What would the expected return be on this stock? Would the expected rate of return be the same if the preferred was a perpetual issue or if it had a 20 -year maturity?

Microtech Corporation is expanding rapidly and currently needs to retain all of its earnings, hence it does not pay dividends. However, investors expect Microtech to begin paying dividends, beginning with a dividend of \(\$ 1.00\) coming 3 years from today. The dividend should grow rapidly-at a rate of 50 percent per year-during Years 4 and \(5,\) but after Year 5 growth should be a constant 8 percent per year. If the required return on Microtech is 15 percent, what is the value of the stock today?

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