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Chapter 10: Problem 3

Preferred Products has issued preferred stock with a \(\$ 7\) annual dividend that will be paid in perpetuity. a. If the discount rate is 12 percent, at what price should the preferred sell? b. At what price should the stock sell 1 year from now? c. What is the dividend yield, the capital gains yield, and the expected rate of return of the stock?

Short Answer

Expert verified
a. The price at which the preferred should sell is approximately \$58.33. b. The price at which the stock should sell 1 year from now is approximately \$58.33. c. The dividend yield is approximately 12% (or 0.12), the capital gains yield is 0, and the expected rate of return is also 12% (or 0.12).

Step by step solution

01

Calculate the present value

The price at which the preferred should sell is the present value of all future dividends. The formula to calculate this when the dividends are perpetual is \( \text{Price} = \frac{D}{r} \) where D is the dividend and r is the rate of return. Substitute \( D = 7 \) and \( r =0.12 \) to get the price.
02

Calculate the future price

The price of the stock one year from now is typically the same as the present price for a perpetuity, assuming the discount rate and the annual dividends remain constant. So, the price will be the same as what was calculated in Step 1.
03

Calculate the dividend yield

The dividend yield is calculated by dividing the annual dividend by the stock price. Substitute the annual dividend (\$7) and the stock price (from Step 1) into the formula \( \text{Dividend Yield} = \frac{D}{\text{Price}} \) to find the yield.
04

Calculate the capital gains yield

For a perpetuity, there are no capital gains, as the price of the stock will not change; it's a constant stream of dividends. Hence, the capital gains yield is zero.
05

Calculate the expected rate of return

The expected rate of return for a perpetual preferred stock is the sum of the dividend yield and the capital gains yield. However, since the capital gains yield is zero, the expected rate of return becomes equal to the dividend yield calculated in Step 3.

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Key Concepts

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

Perpetuity
In finance, a perpetuity refers to a stream of equal cash flows that continues indefinitely. When a company issues preferred stock with a fixed annual dividend, it often represents a perpetuity. The concept of perpetuity is crucial as it provides a simplified method of valuing cash flows from investments that are expected to last forever.

To calculate the present value of a perpetuity, the formula used is:
  • \[ \text{Price} = \frac{D}{r} \]
where:
  • \( D \) is the fixed annual dividend, and
  • \( r \) is the required rate of return or discount rate.
In the case of Preferred Products' preferred stock, the perpetual dividend of \($7\) with a discount rate of \(12\%\) helps determine the stock鈥檚 intrinsic value. This means the stock price will be calculated based on the perpetual life of its dividends.
Dividend Yield
Dividend yield is an important metric for stock evaluation, particularly for preferred stocks which typically offer fixed dividends. Investors look at the dividend yield to assess the income earned relative to the stock's selling price.

Here鈥檚 how you calculate it:
  • \[ \text{Dividend Yield} = \frac{D}{\text{Price}} \]
Where \( D \) is the annual dividend, and \( \text{Price} \) is the stock鈥檚 current price. This formula gives investors a percentage that represents the portion of the stock price returned as dividends. In our example, if the stock price calculated from the perpetuity formula is used in the yield equation, it gives a simple yet powerful insight into the stock's income-generating potential.
Capital Gains Yield
Capital gains yield represents the price appreciation of a stock. However, in the context of a perpetuity, particularly a perpetually paid preferred stock, the capital gains yield is usually absent due to the constant nature of price. For Preferred Products' preferred stock, since the dividend is perpetual and the price remains consistent over time, the capital gains yield is zero. This suggests that investors should not expect to earn any returns from fluctuations in stock price, but solely from the dividend payments. It highlights the stability yet limitation in growth potential for fixed-income investments like perpetual preferred stock.
Expected Rate of Return
The expected rate of return for a stock is the sum of its dividend yield and capital gains yield. For perpetual preferred stocks, because the capital gains yield is zero, this measure essentially mirrors the dividend yield. Therefore, for Preferred Products' stock, the expected rate of return becomes solely dependent on the dividend yield:
  • \[ \text{Expected Rate of Return} = \text{Dividend Yield} + \text{Capital Gains Yield} \]
  • As Capital Gains Yield is zero, it implies:
  • \[ \text{Expected Rate of Return} = \text{Dividend Yield} \]
Understanding this provides clarity for investors on what to anticipate: a consistent income return, dependent entirely on dividends, with no expectations for stock price growth.

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Most popular questions from this chapter

Compost Science, Inc. (CSI), is in the business of converting Boston's sewage sludge into fertilizer. The business is not in itself very profitable. However, to induce CSI to remain in business, the Metropolitan District Commission (MDC) has agreed to pay whatever amount is necessary to yield CSI a 10 percent return on investment. At the end of the year, CSI is expected to pay a \(\$ 4\) dividend. It has been reinvesting 40 percent of earnings and growing at 4 percent a year. a. Suppose CSI continues on this growth trend. What is the expected rate of return from purchasing the stock at \(\$ 100 ?\) b. What part of the \(\$ 100\) price is attributable to the present value of growth opportunities? c. Now the MDC announces a plan for CSI to treat Cambridge sewage. CSI's plant will therefore be expanded gradually over 5 years. This means that CSI will have to reinvest 80 percent of its earnings for 5 years. Starting in Year \(6,\) however, it will again be able to pay out 60 percent of earnings. What will be CSI's stock price once this announcement is made and its consequences for CSI are known?

Eastern Electric currently pays a dividend of about \(\$ 1.64\) per share and sells for \(\$ 27\) a share. a. If investors believe the growth rate of dividends is 3 percent per year, what rate of return do they expect to earn on the stock? b. If investors' required rate of return is 10 percent, what must be the growth rate they expect of the firm? c. If the sustainable growth rate is 5 percent, and the plowback ratio is \(.4,\) what must be the rate of return earned by the firm on its new investments?

Horse and Buggy Inc. is in a declining industry. Sales, earnings, and dividends are all shrinking at a rate of 10 percent per year. a. If \(r=15\) percent and \(\mathrm{DIV}_{1}=\$ 3,\) what is the value of a share? b. What price do you forecast for the stock next year? c. What is the expected rate of return on the stock? d. Can you distinguish between "bad stocks" and "bad companies"? Does the fact that the industry is declining mean that the stock is a bad buy?

Planned Obsolescence has a product that will be in vogue for 3 years, at which point the firm will close up shop and liquidate the assets. As a result, forecasted dividends are \(\mathrm{DIV}_{1}=\$ 2, \mathrm{DIV}_{2}=\$ 2.50,\) and \(\mathrm{DIV}_{3}=\$ 18 .\) What is the stock price if the discount rate is 12 percent?

No-Growth Industries pays out all of its earnings as dividends. It will pay its next \(\$ 4\) per share dividend in a year. The discount rate is 12 percent. a. What is the price-earnings ratio of the company? b. What would the P/E ratio be if the discount rate were 10 percent?
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