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Chapter 9: Problem 29

Growth Opportunities The annual earnings of Avalanche Skis, Inc., will be \(\$ 7\) per share in perpetuity if the firm makes no new investments. Under such a situation, the firm would pay out all of its earnings as dividends. Assume the first dividend will be received exactly one year from now. Alternatively, assume that three years from now, and in every subsequent year in perpetuity, the company can invest 30 percent of its earnings in new projects. Each project will earn 20 percent at year-end in perpetuity. The firm's discount rate is 11 percent. 1\. What is the price per share of Avalanche Skis, Inc., stock today without the company making the new investment? 2\. If Avalanche announces that the new investment will be made, what will the per-share stock price be today?

Short Answer

Expert verified
The price per share of Avalanche Skis, Inc., stock today without the company making the new investment is approximately $63.64. If Avalanche announces that the new investment will be made, the per-share stock price today will be approximately $62.38.

Step by step solution

01

Scenario 1: Without making the new investment

In this case, the company will pay all of its earnings as dividends, with each share receiving $7 per year in perpetuity, starting one year from now. Since the dividends are constant, we can directly use the perpetuity formula. PV = D / r = $7 / 0.11 PV = $\( \frac{7}{0.11} \approx 63.64 \) So, the price per share today without making the new investment is approximately $63.64.
02

Scenario 2: With the new investment

If the firm decides to make the new investment, the earnings will have a different distribution starting from the fourth year. 1. For the first three years, the earnings will remain the same, as the company will not yet invest in new projects. Therefore, the total dividends earned in the first three years, discounted back to today are: \( D_1 \div (1 + r) + D_2 \div (1 + r)^2 + D_3 \div (1 + r)^3 = 7 \div (1 + 0.11) + 7 \div (1 + 0.11)^2 + 7 \div (1 + 0.11)^3 \) 2. From the fourth year onward, the company will invest 30% of its earnings in new projects that generate a 20% return at the end of the year. Thus, the earnings at the end of the fourth year will be: Future earnings = Initial earnings + (Investment return * Investment percentage * Initial earnings) In this case: Future earnings = \(7 + (0.2 * 0.3 * 7) = \)7 * (1 + 0.2 * 0.3) = \(7 * 1.06\) Since these earnings will be paid in perpetuity, starting from the fourth year, we can use the perpetuity formula to find the present value of these dividends at the end of the third year. PV = D / r = ($7 * 1.06) / 0.11 PV = $\( \frac{7 * 1.06}{0.11} \approx 67.32 \) Now, we need to discount this value back to today: PV = $67.32 / (1 + 0.11)^3 The stock price today will be the sum of the present value of the first three years of dividends and the present value of the fourth year onward earnings: Stock price = \(D_1 \div (1 + r) + D_2 \div (1 + r)^2 + D_3 \div (1 + r)^3 + PV\) Stock price = $\( 7 \div (1 + 0.11) + 7 \div (1 + 0.11)^2 + 7 \div (1 + 0.11)^3 + \frac{67.32}{(1 + 0.11)^3} \approx 62.38 \) If Avalanche announces the new investment, the per-share stock price today will be approximately $62.38.

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Key Concepts

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

Perpetuity Valuation
Perpetuity valuation is a method used to determine the value of an investment that generates a constant stream of cash flows indefinitely. Imagine receiving \(7 each year, every year, forever. That's what we'd call a perpetuity. For Avalanche Skis, Inc., this could mean receiving \)7 in dividends yearly per share without any changes or investments.
To find out how much these infinite cash flows are worth today, we use the formula: \[ PV = \frac{D}{r} \]where:
  • \( PV \) is the present value or price per share in today's terms.
  • \( D \) is the annual dividend per share, which is \(7 in this case.
  • \( r \) is the discount rate or required rate of return, here it's 11% or 0.11.
Using these numbers for Avalanche Skis, Inc.:\[ PV = \frac{7}{0.11} \approx 63.64 \]This means, if the company continues in this manner, each share is worth approximately \)63.64 today. This calculation offers a simple way to assess situations where constant returns are expected forever.
Discounted Cash Flow Analysis
Discounted Cash Flow (DCF) Analysis is a powerful tool used to estimate the value of an investment based on its expected future cash flows. It takes into account the time-value of money, meaning a dollar earned today is worth more than a dollar earned in the future.
For Avalanche Skis, Inc., we applied DCF analysis to determine the value of future cash flows from dividends both before and after new investment scenarios.
In the step-by-step solution, the DCF analysis covers two main parts:
  • For the first three years, dividends of \(7 each are received without investment, discounted back to today.
  • From the fourth year onward, the company invests 30% of earnings in projects with a 20% return. This ultimately increases the future dividend to \)7 multiplied by a growth factor of 1.06, again discounted back to today.
The calculation uses the formula:\[ PV = \frac{D_4}{r} \div (1 + r)^3 \]where \( D_4 \) is the adjusted dividend from year four onward, and \( r \) is the discount rate (0.11).
This shows how each year forward is less than without discounting, highlighting the importance of time value.
Investment Opportunities
Investment opportunities are critical for companies like Avalanche Skis, Inc. They allow the firm to grow and potentially increase shareholder value. In this example, the investment strategy involved reinvesting 30% of earnings into new projects that yield a substantial 20% return.
Here's why this matters:
  • By investing in new projects, the company uses current earnings to fuel future growth.
  • This results in increased future dividends, enhancing the value of each share.
  • It shows investors that the firm is actively working towards growth and not just maintaining status quo.
For Avalanche, this means that rather than just distributing all earnings as dividends today, the strategy shifts to long-term gains, which affect the stock price. While initially, the share price with investments is lower than the perpetuity-only scenario, over time, the increased dividends due to these investments can create more overall value for shareholders.

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Most popular questions from this chapter

Growth Opportunities Lewin Skis, Inc., (today) expects to earn \(\$ 6.25\) per share for each of the future operating periods (beginning at time 1) if the firm makes no new investments and returns the earnings as dividends to the shareholders. However, Clint Williams, president and CEO, has discovered an opportunity to retain and invest 20 percent of the earnings beginning three years from today. This opportunity to invest will continue for each period indefinitely. He expects to earn 11 percent on this new equity investment, the return beginning one year after each investment is made. The firm's equity discount rate is 13 percent throughout. 1\. What is the price per share of Lewin Skis, Inc., stock without making the new investment? 2\. If the new investment is expected to be made, per the preceding information, what would the price of the stock be now? 3\. Suppose the company could increase the investment in the project by whatever amount it chose. What would the retention ratio need to be to make this project attractive?

Stock Values The Starr Co. just paid a dividend of \(\$ 1.90\) per share on its stock. The dividends are expected to grow at a constant rate of 5 percent per year, indefinitely. If investors require a 12 percent return on the stock, what is the current price? What will the price be in three years? In 15 years?

Capital Gains versus Income Consider four different stocks, all of which have a required return of 20 percent and a most recent dividend of \(\$ 4.50\) per share. Stocks \(W, X\), and \(Y\) are expected to maintain constant growth rates in dividends for the foreseeable future of 10 percent, 0 percent, and -5 percent per year, respectively. Stock \(Z\) is a growth stock that will increase its dividend by 30 percent for the next two years and then maintain a constant 8 percent growth rate thereafter. What is the dividend yield for each of these four stocks? What is the expected capital gains yield? Discuss the relationship among the various returns that you find for each of these stocks.

Negative Growth Antiques \(R\) Us is a mature manufacturing firm. The company just paid a \(\$ 12\) dividend, but management expects to reduce the payout by 6 percent per year, indefinitely. If you require an 11 percent return on this stock, what will you pay for a share today?

Dividend Growth Four years ago, Bling Diamond, Inc., paid a dividend of \(\$ 1.20\) per share. Bling paid a dividend of \(\$ 1.93\) per share yesterday. Dividends will grow over the next five years at the same rate they grew over the last four years. Thereafter, dividends will grow at 7 percent per year. What will Bling Diamond's cash dividend be in seven years?
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