We should warn you that being a financial expert has its occupational hazards. One is being cornered at cocktail parties by people who are eager to explain their system for making creamy profits by investing in common stocks. One of the few good things about a financial crisis is that these bores tend to disappear, at least temporarily.
We may exaggerate the perils of the trade. The point is that there is no easy way to ensure superior investment performance. Therefore, when in this chapter we use present value calculations to value common stocks, we are not providing a key to investment success nor attempting to turn you into a cocktail-party bore. We are just explaining the fundamental reasons some stocks are more valuable than others.
Why should you care about fundamentals? If you want to know what a stock is worth, why can’t you just look up the price on the Internet?
There are at least three reasons you should care. First, changes in the price of a public company’s shares reveal how well the company is doing financially, at least in the eyes of investors, and usually determine a large fraction of top management’s compensation. If your company is going to use its stock price to assess performance, you had better understand what determines the price.
Second, many companies are not public. A private company may want to understand what its stock is worth or what it would be worth if it were traded.
Third, a firm that acts in it shareholders’ interest should accept capital investments that increase the value of their stake in the firm. In order to do this, the financial manager needs to understand what determines the value of the firm’s shares.
We begin with a look at how stocks are traded. Then we explain the basic principles of share valuation and the use of discounted-cash-flow (DCF) models to estimate expected rates of return. Later in the chapter, we show how these DCF models can be used to value entire businesses rather than individual shares.
We will also explain the fundamental difference between growth and income stocks. A growth stock doesn’t just grow; its future investments are also expected to earn rates of return that are higher than the cost of capital. It’s the combination of growth and superior returns that generates high price–earnings ratios for growth stocks.
Still another warning: Everybody knows that common stocks are risky and that some are more risky than others. Therefore, investors will not commit their hard-earned cash to stocks unless the expected rates of return are commensurate with the risks. But we say next to nothing in this chapter about the linkages between risk and expected return. A more careful treatment of risk starts in Chapter 7.
4-1How Common Stocks Are Traded
Boeing has 596 million shares outstanding. Shareholders include large pension funds and insurance companies that each own millions of shares, as well as individuals who own a handful. If you owned one Boeing share, you would own .0000002% of the company and have a claim on the same tiny fraction of its profits. Of course, the more shares you own, the larger your “share” of the company.
If Boeing wishes to raise new capital, it can do so either by borrowing or by selling new shares to investors. Sales of shares to raise new capital are said to occur in the primary market. But most trades in Boeing take place on the stock exchange, where investors buy and sell existing Boeing shares. Stock exchanges are really markets for secondhand shares, but they prefer to describe themselves as secondary markets, which sounds more important.
The two principal U.S. stock exchanges are the New York Stock Exchange and Nasdaq. Both compete vigorously for business and just as vigorously tout the advantages of their trading systems. In addition to the NYSE and Nasdaq, there are electronic communication networks (ECNs) that connect traders with each other. Large U.S. companies may also arrange for their shares to be traded on foreign exchanges, such as the London exchange or the Euronext exchange in Paris. At the same time, many foreign companies are listed on the U.S. exchanges. For example, the NYSE trades shares in Sony, Royal Dutch Shell, Canadian Pacific, Tata Motors, Deutsche Bank, Telefonica Brasil, China Eastern Airlines, and more than 500 other companies.
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Major world stock exchanges
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Suppose that Ms. Jones, a long-time Boeing shareholder, no longer wishes to hold her shares. She can sell them via the stock exchange to Mr. Brown, who wants to increase his stake in the firm. The transaction merely transfers partial ownership of the firm from one investor to another. No new shares are created, and Boeing will neither care nor know that the trade has taken place.
Ms. Jones and Mr. Brown do not trade the Boeing shares themselves. Instead, their orders must go through a brokerage firm. Ms. Jones, who is anxious to sell, might give her broker a market order to sell stock at the best available price. On the other hand, Mr. Brown might state a price limit at which he is willing to buy Boeing stock. If his limit order cannot be executed immediately, it is recorded in the exchange’s limit order book until it can be executed.
Table 4.1 shows a portion of the limit order book for Boeing from the BATS Exchange, one of the largest electronic markets. The bid prices on the left are the prices (and number of shares) at which investors are currently willing to buy. The ask prices on the right are those at which investors are prepared to sell. The prices are arranged from best to worst, so the highest bids and lowest asks are at the top of the list. The broker might electronically enter Ms. Jones’s market order to sell 100 shares on the BATS Exchange, where it would be matched with the best offer to buy, which at that moment was $263.76 a share. A market order to buy would be matched with the best ask price, $264.07. The bid-ask spread at that moment was, therefore, 31 cents per share.
|
Bid |
Ask |
||
|
Price |
Shares |
Price |
Shares |
|
263.76 |
1,100 |
264.07 |
200 |
|
263.73 |
100 |
264.12 |
1 |
|
263.67 |
100 |
264.13 |
100 |
|
263.61 |
100 |
264.18 |
200 |
TABLE 4.1 A portion of the limit order book for Boeing on the BATS BZX Exchange, November 20, 2017, at 09:51:03
When they transact on the NYSE or one of the electronic markets, Brown and Jones are participating in a huge auction market in which the exchange matches up the orders of thousands of investors. Most major exchanges around the world, including the Tokyo, Shanghai, London stock exchanges and the Deutsche Börse, are also auction markets, but the auctioneer in these cases is a computer.1 This means that there is no stock exchange floor to show on the evening news and no one needs to ring a bell to start trading.
Nasdaq is not an auction market. All trades on Nasdaq take place between the investor and one of a group of professional dealers who are prepared to buy and sell stock. Dealer markets are common for other financial instruments. For example, most bonds are traded in dealer markets.
Trading Results for Boeing
You can track trades in Boeing or other public corporations on the Internet. For example, if you go to finance.yahoo.com and enter the ticker symbol BA under “Quote Lookup,” you will see results like the table below.2 We will focus here on some of the more important entries.
Boeing’s closing price on November 20, 2017, was $264.63, up $2.37, or .90% from the previous day’s close. Boeing had 595.58 million shares outstanding, so its market cap (shorthand for market capitalization) was 595.58 × $264.63 = $157.61 billion.
|
Boeing Common Stock (NYSE) |
|||
|
264.63 +2.37 (+0.90%) Nov 20 4:00PM EST |
|||
|
Previous close |
262.26 |
Market cap |
157.61B |
|
Open |
263.00 |
Beta |
1.31 |
|
Day’s range |
262.76-265.62 |
P/E ratio (TTM) |
23.2 |
|
52-week range |
146.52-267.62 |
EPS (TTM) |
11.41 |
|
Volume |
2,404,762 |
Forward dividend |
5.68 |
|
Average volume |
3,187,640 |
Dividend yield |
2.17% |
Source: finance.yahoo.com.
Boeing’s earnings per share (EPS) over the previous 12 months were $11.41 (“TTM” stands for “trailing 12 months”). The ratio of stock price to EPS (the P/E ratio) was 23.2. Notice that this P/E ratio uses past EPS. P/E ratios using forecasted EPS are generally more useful. Security analysts forecasted an increase in Boeing’s EPS to 11.68 per share for 2018, which gives a forward P/E of 22.7.3
Boeing paid a cash dividend of $5.68 per share per year, so its dividend yield (the ratio of dividend to price) was 2.17%.
Boeing’s beta of 1.31 measures the market risk of Boeing’s stock. We explain betas in Chapter 7.
Boeing stock was a wonderful investment in 2017: Its price was up 81% in the 12 months previous to the quotes summarized here. Buying stocks is a risky occupation, however. Take GE as an example. GE used to be one of the most powerful and admired U.S. companies. But GE stock fell by 40% over the 12 months ending on November 16, 2017—a period in which the S&P 500 market index gained 18.5%. The stock fell by 23% in the last month of that period when its new CEO cut GE’s dividend in half and announced a plan for drastic restructuring. There weren’t many GE stockholders piping up at cocktail parties over the 2017 holiday season; they either kept quiet or were not invited.
Most of the trading on the NYSE and Nasdaq is in ordinary common stocks, but other securities are traded also, including preferred shares, which we cover in Chapter 14, and warrants, which we cover in Chapter 21. Investors can also choose from hundreds of exchange-traded funds (ETFs), which are portfolios of stocks that can be bought or sold in a single trade. With a few exceptions ETFs are not actively managed. Many simply aim to track a well-known market index such as the Dow Jones Industrial Average or the S&P 500. Others track specific industries or commodities. (We discuss ETFs more fully in Chapter 14.) You can also buy shares in closed-end mutual funds4 that invest in portfolios of securities. These include country funds, for example, the Mexico and India funds, that invest in portfolios of stocks in specific countries. Unlike ETFs, most closed-end funds are actively managed and seek to “beat the market.”
4-2How Common Stocks Are Valued
Finding the value of the stock of Boeing or GE may sound like a simple problem. Public companies publish quarterly and annual balance sheets, which list the value of the company’s assets and liabilities. For example, at the end of September 2017, the book value of all GE’s assets—plant and machinery, inventories of materials, cash in the bank, and so on—was $378 billion. GE’s liabilities—money that it owes the banks, taxes that are due to be paid, and the like—amounted to $298.5 billion. The difference between the value of the assets and the liabilities was just over $79.5 billion. This was the book value of GE’s equity.5
Book value is a reassuringly definite number. Each year KPMG, one of America’s largest accounting firms, gives its opinion that GE’s financial statements present fairly in all material respects the company’s financial position, in conformity with U.S. generally accepted accounting principles (commonly called GAAP). However, the book value of GE’s assets measures only their original (or “historical”) cost less an allowance for depreciation. This may not be a good guide to what those assets are worth today.
One can go on and on about the deficiencies of book value as a measure of market value. Book values are historical costs that do not incorporate inflation. (Countries with high or volatile inflation often require inflation-adjusted book values, however.) Book values usually exclude intangible assets such as trademarks and patents. Also accountants simply add up the book values of individual assets, and thus do not capture going-concern value. Going-concern value is created when a collection of assets is organized into a healthy operating business.
Book values can nevertheless be a useful benchmark. Suppose, for example, that the aggregate value of all of Holstein Oil’s shares is $900 million. Its book value of equity is $450 million. A financial analyst might say, “Holstein sells for two times book value. It has doubled shareholders’ cumulative past investment in the company.” She might also say, “Holstein’s market value added is $900 – 450 = $450 million. (There is more on market value added in Chapter 28.)
Book values may also be useful clues about liquidation value. Liquidation value is what investors get when a failed company is shut down and its assets are sold off. Book values of “hard” assets like land, buildings, vehicles, and machinery can indicate possible liquidation values.
Intangible “soft” assets can be important even in liquidation, however. Eastman Kodak provides a good example. Kodak, which was one of the Nifty Fifty growth stocks of the 1960s, suffered a long decline and finally filed for bankruptcy in January 2012. What was one of its most valuable assets in bankruptcy? Its portfolio of 79,000 patents, which was subsequently sold for $525 million.
Valuation by Comparables
When financial analysts need to value a business, they often start by identifying a sample of similar firms as potential comparables. They then examine how much investors in the comparable companies are prepared to pay per dollar of earnings or book assets. They see what the business would be worth if it traded at the comparables’ price–earnings or price-to-bookvalue ratios. This valuation approach is called valuation by comparables.
Table 4.2 tries out this valuation method for three companies and industries. Let’s start with Union Pacific (UNP). In November 2017, UNP’s stock was trading around $117. Security analysts were forecasting earnings per share (EPS) for 2018 at $6.55, giving a “forward” price–earnings ratio of P/E = 17.85.6 UNP’s market–book ratio (price divided by book value per share) was P/B = 4.73.
TABLE 4.2 Stock price, price–earnings (P/E), and market-book (P/B) in November 2017 for three companies and potential comparables
Source: finance.yahoo.com.
P/Es and P/Bs for several of UNP’s competitors are reported on the right-hand side of the table. Notice that UNP’s P/E is close to the P/Es of these comparables. If you didn’t know UNP’s stock price, you could get an estimate by multiplying UNP’s forecasted EPS of $6.55 by 17.49, the average P/E for the comparables. The resulting estimate of $114.56 would be almost spot on. On the other hand, UNP’s P/B is higher than all the comparables’ P/Bs except for that of Canadian Pacific. If you had used the average price-to-book ratio for the comparables to value UNP, you would have come up with an underestimate of UNP’s actual share price.
Look now at Johnson & Johnson (J&J) and its four comparables in Table 4.2. In this case, the P/B ratio for J&J is higher than for the comparables (5.2 versus an average of 4.1). The average P/E for J&J is also higher (17.8 versus 15.5). An estimate of the value of J&J based on the comparables’ P/Es and P/Bs would be too low. Comparing the estimate to J&J’s actual stock price could still be worthwhile, however, if it leads you to ask why J&J was more attractive to investors.
The ratios for Devon Energy in Table 4.2 illustrate the potential difficulties with valuation by comparables. There is a huge variation in the P/E ratios of the comparables. Anadarko and Marathon had negative P/E ratios; their stock prices were of course positive, but their operations had been battered by a sudden fall in oil prices, and their forecasted earnings for the next year were negative. The average P/E of –.58 for the comparables is meaningless.7 The comparables’ average P/B is more informative.
The P/E ratios for the oil companies in Table 4.2 illustrate what can go wrong with the ratios in hard times when firms makes losses. P/E ratios are also almost useless as a guide to the value of new start-ups, most of which do not have any earnings to compare.
Such difficulties do not invalidate the use of comparables to value businesses. Maybe Table 4.2 doesn’t show the companies most closely similar to Devon. A financial manager or analyst would need to dig deeper to understand Devon’s industry and its competitors. Also, the method might work better with different ratios.8
Of course, investors did not need valuation by comparables to value Devon Energy or the other companies in Table 4.2. They are all public companies with actively traded shares. But you may find valuation by comparables useful when you don’t have a stock price. For example, in August 2017 the mining giant, BHP Billiton, announced plans to sell its U.S. shale business. Preliminary estimates put the business’s value at $8 to $10 billion. It’s a safe bet that BHP and its advisers were burning the midnight oil and doing their best to identify the best comparables and check what the assets would be worth if they traded at the comparables’ P/E and P/B ratios.
But BHP would need to be cautious. As Table 4.2 shows, these ratios can vary widely even within the same industry. To understand why this is so, we need to look more carefully at what determines a stock’s market value. We start by connecting stock prices to the cash flows that stockholders receive from the company in the form of cash dividends. This will lead us to a discounted-cash-flow (DCF) model of stock prices.
Stock Prices and Dividends
Not all companies pay dividends. Rapidly growing companies typically reinvest earnings instead of paying out cash. But most mature, profitable companies do pay regular cash dividends.
Think back to Chapter 3, where we explained how bonds are valued. The market value of a bond equals the discounted present value (PV) of the cash flows (interest and principal payments) that the bond will pay out over its lifetime. Let’s import and apply this idea to common stocks. The future cash flows to the owner of a share of common stock are the future dividends per share that the company will pay out. Thus, the logic of discounted cash flow suggests
PV (share of stock) = PV (expected future dividends per share)
At first glance, this statement may seem surprising. Investors hope for capital gains as well as dividends. That is, they hope to sell stocks for more than they paid for them. Why doesn’t the PV of a stock depend on capital gains? As we now explain, there is no inconsistency.
Today’s Price If you own a share of common stock, your cash payoff comes in two forms: (1) cash dividends and (2) capital gains or losses. Suppose that the current price of a share is P0, that the expected price at the end of a year is P1, and that the expected dividend per share is DIV1. The rate of return that investors expect from this share over the next year is defined as the expected dividend per share DIV1 plus the expected price appreciation per share P1 – P0, all divided by the price at the start of the year P0:
Suppose Fledgling Electronics stock is selling for $100 a share (P0 = 100). Investors expect a $5 cash dividend over the next year (DIV1 = 5). They also expect the stock to sell for $110 a year, hence (P1 = 110). Then the expected return to the stockholders is 15%:
On the other hand, if you are given investors’ forecasts of dividend and price and the expected return offered by other equally risky stocks, you can predict today’s price:
For Fledgling Electronics, DIV1 = 5 and P1 = 110. If r, the expected return for Fledgling is 15%, then today’s price should be $100:
What exactly is the discount rate, r, in this calculation? It’s called the market capitalization rate or cost of equity capital, which are just alternative names for the opportunity cost of capital, defined as the expected return on other securities with the same risks as Fledgling shares.
Many stocks will be safer than Fledgling and many riskier. But among the thousands of traded stocks, there will be a group with essentially the same risks. Call this group Fledgling’s risk class. Then all stocks in this risk class have to be priced to offer the same expected rate of return.
Let’s suppose that the other securities in Fledgling’s risk class all offer the same 15% expected return. Then $100 per share has to be the right price for Fledgling stock. In fact, it is the only possible price. What if Fledgling’s price were above P0 = $100? In this case, the expected return would be less than 15%. Investors would shift their capital to the other securities and, in the process, would force down the price of Fledgling stock. If P0 were less than $100, the process would reverse. Investors would rush to buy, forcing the price up to $100. Therefore, at each point in time, all securities in an equivalent risk class are priced to offer the same expected return. This is a condition for equilibrium in well-functioning capital markets. It is also common sense.
Next Year’s Price? We have managed to explain today’s stock price P0 in terms of the dividend DIV1 and the expected price next year P1. Future stock prices are not easy things to forecast directly. But think about what determines next year’s price. If our price formula holds now, it ought to hold then as well:
That is, a year from now, investors will be looking out at dividends in year 2 and price at the end of year 2. Thus, we can forecast P1 by forecasting DIV2 and P2, and we can express P0 in terms of DIV1, DIV2, and P2:
Take Fledgling Electronics. A plausible explanation for why investors expect its stock price to rise by the end of the first year is that they expect higher dividends and still more capital gains in the second. For example, suppose that they are looking today for dividends of $5.50 in year 2 and a subsequent price of $121. That implies a price at the end of year 1 of
Today’s price can then be computed either from our original formula
or from our expanded formula
We have succeeded in relating today’s price to the forecasted dividends for two years (DIV1 and DIV2) plus the forecasted price at the end of the second year (P2). You will not be surprised to learn that we could go on to replace P2 by (DIV3 + P3)/(1 + r) and relate today’s price to the forecasted dividends for three years (DIV1, DIV2, and DIV3) plus the forecasted price at the end of the third year (P3). In fact, we can look as far out into the future as we like, removing Ps as we go. Let us call this final period H. This gives us a general stock price formula:
The expression 
indicates the sum of the discounted dividends from year 1 to year H.
Table 4.3 continues the Fledgling Electronics example for various time horizons, assuming that the dividends are expected to increase at a steady 10% compound rate. The expected price Pt increases at the same rate each year. Each line in the table represents an application of our general formula for a different value of H.Figure 4.1 is a graph of the table. Each column shows the present value of the dividends up to the time horizon and the present value of the price at the horizon. As the horizon recedes, the dividend stream accounts for an increasing proportion of present value, but the total present value of dividends plus terminal price always equals $100.
TABLE 4.3 Applying the stock valuation formula to Fledgling Electronics
Assumptions:
1. Dividends increase at 10% per year, compounded.
2. Discount rate (cost of equity or market capitalization rate) is 15%.
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Try It! Figure 4.1: Value and the investor’s horizon
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How far out could we look? In principle, the horizon period H could be infinitely distant. Common stocks do not expire of old age. Barring such corporate hazards as bankruptcy or acquisition, they are immortal. As H approaches infinity, the present value of the terminal price ought to approach zero, as it does in the final column of Figure 4.1. We can, therefore, forget about the terminal price entirely and express today’s price as the present value of a perpetual stream of cash dividends. This is usually written as
where ∞ indicates infinity. This formula is the DCF or dividend discount model of stock prices. It’s another present value formula.9 We discount the cash flows—in this case, the dividend stream—by the return that can be earned in the capital market on securities of equivalent risk. Some find the DCF formula implausible because it seems to ignore capital gains. But we know that the formula was derived from the assumption that price in any period is determined by expected dividends and capital gains over the next period.
Notice that it is not correct to say that the value of a share is equal to the sum of the discounted stream of earnings per share. Earnings are generally larger than dividends because part of those earnings is reinvested in new plant, equipment, and working capital. Discounting earnings would recognize the rewards of that investment (higher future earnings and dividends) but not the sacrifice (a lower dividend today). The correct formulation states that share value is equal to the discounted stream of dividends per share. Share price is connected to future earnings per share, but by a different formula, which we cover later in this chapter.
FIGURE 4.1 As your horizon recedes, the present value of the future price (shaded area) declines but the present value of the stream of dividends (unshaded area) increases. The total present value (future price and dividends) remains the same.
Although mature companies generally pay cash dividends, thousands of companies do not. For example, Amazon has never paid a dividend, yet it is a successful company with a market capitalization in January 2018 of $620 billion. Why would a successful company decide not to pay cash dividends? There are at least two reasons. First, a growing company may maximize value by investing all its earnings rather than paying out a dividend. The shareholders are better off with this policy, provided that the investments offer an expected rate of return higher than shareholders could get by investing on their own. In other words, shareholder value is maximized if the firm invests in projects that can earn more than the opportunity cost of capital. If such projects are plentiful, shareholders will be prepared to forgo immediate dividends. They will be happy to wait and receive deferred dividends.10
The dividend discount model is still logically correct for growth companies, but difficult to use when cash dividends are far in the future. In this case, most analysts switch to valuation by comparables or to earnings-based formulas, which we cover in Section 4-4.
Second, a company may pay out cash not as dividends but by repurchasing shares from stockholders. We cover the choice between dividends and repurchases in Chapter 16, where we also explain why repurchases do not invalidate the dividend discount model.11
Nevertheless, the dividend discount model can be difficult to deploy if repurchases are irregular and unpredictable. In these cases, it can be better to start by calculating the present value of the total free cash flow available for dividends and repurchases. Discounting free cash flow gives the present value of the company as a whole. Dividing by the current number of shares outstanding gives present value per share. We cover this valuation method in Section 4-5.
The next section considers simplified versions of the dividend discount model.
4-3Estimating the Cost of Equity Capital
In Chapter 2, we encountered some simplified versions of the basic present value formula. Let us see whether they offer any insights into stock values. Suppose, for example, that we forecast a constant growth rate for a company’s dividends. This does not preclude year-to-year deviations from the forecast: It means only that expected dividends grow at a constant rate. Such an investment would be just another example of the growing perpetuity that we valued in Chapter 2. To find its present value we must divide the first year’s cash payment by the difference between the discount rate and the growth rate:
Remember that we can use this formula only when g, the anticipated growth rate, is less than r, the discount rate. As g approaches r, the stock price becomes infinite. Obviously, r must be greater than g if growth really is perpetual.
Our growing perpetuity formula explains P0 in terms of next year’s expected dividend DIV1, the projected growth trend g, and the expected rate of return on other securities of comparable risk r. Alternatively, the formula can be turned around to obtain an estimate of r from DIV1, P0, and g:
The expected return equals the dividend yield (DIV1/P0) plus the expected rate of growth in dividends (g).
These two formulas are much easier to work with than the general statement that “price equals the present value of expected future dividends.”12 Here is a practical example.
Using the DCF Model to Set Water, Gas, and Electricity Prices
In the United States, the prices charged by local water, electric, and gas utilities are regulated by state commissions. The regulators try to keep consumer prices down but are supposed to allow the utilities to earn a fair rate of return. But what is fair? It is usually interpreted as r, the market capitalization rate for the firm’s common stock. In other words, the fair rate of return on equity for a public utility ought to be the cost of equity—that is, the rate offered by securities that have the same risk as the utility’s common stock.13
Small variations in estimates of this return can have large effects on the prices charged to the customers and on the firm’s profits. So both the firms’ managers and regulators work hard to estimate the cost of equity. They’ve noticed that most utilities are mature, stable companies that pay regular dividends. Such companies should be tailor-made for application of the constant-growth DCF formula.
Suppose you wished to estimate the cost of equity for Aqua America, a local water distribution company. Aqua’s stock (ticker symbol WTR) was selling for $33.62 per share at the end of September 2017. Dividend payments for the next year were expected to be $1.18 a share. Thus, it was a simple matter to calculate the first half of the DCF formula:
The hard part is estimating g, the expected rate of dividend growth. One option is to consult the views of security analysts who study the prospects for each company. Analysts are rarely prepared to stick their necks out by forecasting dividends to kingdom come, but they often forecast growth rates over the next five years, and these estimates may provide an indication of the expected long-run growth path. In the case of Aqua, analysts in 2017 were forecasting an annual growth of 6.6%.14 This, together with the dividend yield, gave an estimate of the cost of equity capital:
An alternative approach to estimating long-run growth starts with the payout ratio, the ratio of dividends to earnings per share (EPS). For Aqua, this ratio has averaged about 60%. In other words, each year the company was plowing back into the business about 40% of earnings per share:
Also, Aqua’s ratio of earnings per share to book equity per share has averaged about 12.6%. This is its return on equity, or ROE:
If Aqua earns 12.6% on book equity and reinvests 40% of earnings, then book equity will increase by .40 × .126 = .05, or 5%. Earnings and dividends per share will also increase by 5%:
Dividend growth rate = g = plowback ratio × ROE = .40 × .126 = .05
That gives a second estimate of the market capitalization rate:
Although these estimates of Aqua’s cost of equity seem reasonable, there are obvious dangers in analyzing any single firm’s stock with the constant-growth DCF formula. First, the underlying assumption of regular future growth is at best an approximation. Second, even if it is an acceptable approximation, errors inevitably creep into the estimate of g.
Remember, Aqua’s cost of equity is not its personal property. In well-functioning capital markets, investors capitalize the dividends of all securities in Aqua’s risk class at exactly the same rate. But any estimate of r for a single common stock is “noisy” and subject to error. Good practice does not put too much weight on single-company estimates of the cost of equity. It collects samples of similar companies, estimates r for each, and takes an average. The average gives a more reliable benchmark for decision making.
The next-to-last column of Table 4.4 gives DCF cost-of-equity estimates for Aqua and seven other water companies. These are all stable, mature companies for which the constant-growth DCF formula ought to work. Notice the variation in the cost-of-equity estimates. Some of the variation may reflect differences in the risk, but some is just noise. The average estimate is 9.9%.
TABLE 4.4 Cost-of-equity estimates for water companies in September 2017. The long-term growth rate is based on security analysts’ forecasts. In the multistage DCF model, growth after five years is assumed to adjust gradually to the estimated long-term growth rate of gross domestic product (GDP).
aDividend and analysts’ long-term growth-rate forecasts at the end of September 2017.
bSum of dividend yield and long-term growth rate. This column contains some small rounding differences.
cLong-term growth rate of GDP was forecasted at 4.1% by Blue-Chip Economic Indicators.
Source: The Brattle Group, Inc.
Estimates of this kind are only as good as the long-term forecasts on which they are based. For example, several studies have observed that security analysts are subject to behavioral biases and their forecasts tend to be overoptimistic. If so, such DCF estimates of the cost of equity should be regarded as upper estimates of the true figure.
Dangers Lurk in Constant-Growth Formulas
The simple constant-growth DCF formula is an extremely useful rule of thumb, but no more than that. Naive trust in the formula has led many financial analysts to silly conclusions.
We have stressed the difficulty of estimating r by analysis of one stock only. Try to use a large sample of equivalent-risk securities. Even that may not work, but at least it gives the analyst a fighting chance because the inevitable errors in estimating r for a single security tend to balance out across a broad sample.
Also, resist the temptation to apply the formula to firms having high current rates of growth. Such growth can rarely be sustained indefinitely, but the constant-growth DCF formula assumes it can. This erroneous assumption leads to an overestimate of r.
Example The U.S. Surface Transportation Board (STB) tracks the “revenue adequacy” of U.S. railroads by comparing the railroads’ returns on book equity with estimates of their costs of equity. To estimate the cost of equity, the STB traditionally used the constant-growth formula. It measured g by stock analysts’ forecasts of long-term earnings growth. The formula assumes that earnings and dividends grow at a constant rate forever, but the analysts’ “long-term” forecasts looked out five years at most. As the railroads’ profitability improved, the analysts became more and more optimistic. By 2009, their forecasts for growth averaged 12.5% per year. The average dividend yield was 2.6%, so the constant-growth model estimated the industry-average cost of capital at 2.6 + 12.5 = 15.1%.
So the STB said, in effect, “Wait a minute: Railroad earnings and dividends can’t grow at 12.5% forever. The constant-growth formula no longer works for railroads. We’ve got to find a more accurate method.” The STB now uses a multistage growth model.15 Let us look at an example of such a model.
DCF Models with Two or More Stages of Growth Consider Growth-Tech Inc., a firm with DIV1 = $.50 and P0 = $50. The firm has plowed back 80% of earnings and has had a return on equity (ROE) of 25%. This means that in the past
Dividend growth rate = plowback ratio × ROE = .80 × .25 = .20
The temptation is to assume that the future long-term growth rate g also equals .20. This would imply
But this is silly. No firm can continue growing at 20% per year forever, except possibly under extreme inflationary conditions. Eventually, profitability will fall and the firm will respond by investing less.
In real life, the return on equity will decline gradually over time, but for simplicity, let’s assume it suddenly drops to 16% at year 3 and the firm responds by plowing back only 50% of earnings. Then g drops to .50 × .16 = .08.
Table 4.5 shows what’s going on. Growth-Tech starts year 1 with book equity of $10.00 per share. It earns $2.50, pays out 50 cents as dividends, and plows back $2. Thus, it starts year 2 with book equity of $10 + 2 = $12. After another year at the same ROE and payout, it starts year 3 with equity of $14.40. However, ROE drops to .16, and the firm earns only $2.30. Dividends go up to $1.15 because the payout ratio increases, but the firm has only $1.15 to plow back. Therefore, subsequent growth in earnings and dividends drops to 8%.
TABLE 4.5 Forecasted earnings and dividends for Growth-Tech. Note the changes in year 3: ROE and earnings drop, but payout ratio increases, causing a big jump in dividends. However, subsequent growth in earnings and dividends falls to 8% per year. Note that the increase in equity equals the earnings not paid out as dividends.
Now we can use our general DCF formula:
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Try it! Table 4.5: Valuing Growth-Tech
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Investors in year 3 will view Growth-Tech as offering 8% per year dividend growth. So we can use the constant-growth formula to calculate P3:
We have to use trial and error to find the value of r that makes P0 equal $50. It turns out that the r implicit in these more realistic forecasts is just over .099, quite a difference from our “constant-growth” estimate of .21.
Our present value calculations for Growth-Tech used a two-stage DCF valuation model. In the first stage (years 1 and 2), Growth-Tech is highly profitable (ROE = 25%), and it plows back 80% of earnings. Book equity, earnings, and dividends increase by 20% per year. In the second stage, starting in year 3, profitability and plowback decline, and earnings settle into long-term growth at 8%. Dividends jump up to $1.15 in year 3, and then also grow at 8%.
Growth rates can vary for many reasons. Sometimes, growth is high in the short run not because the firm is unusually profitable, but because it is recovering from an episode of low profitability. Table 4.6 displays projected earnings and dividends for Phoenix Corp., which is gradually regaining financial health after a near meltdown. The company’s equity is growing at a moderate 4%. ROE in year 1 is only 4%, however, so Phoenix has to reinvest all its earnings, leaving no cash for dividends. As profitability increases in years 2 and 3, an increasing dividend can be paid. Finally, starting in year 4, Phoenix settles into steady-state growth, with equity, earnings, and dividends all increasing at 4% per year.
TABLE 4.6 Forecasted earnings and dividends for Phoenix Corp. The company can initiate and increase dividends as profitability (ROE) recovers. Note that the increase in book equity equals the earnings not paid out as dividends.
Assume the cost of equity is 10%. Then Phoenix shares should be worth $9.13 per share:
You could go on to valuation models with three or more stages. For example, the far right column of Table 4.4 presents multistage DCF estimates of the cost of equity for our local water companies. In this case the long-term growth rates reported in the table do not continue forever. After five years, each company’s growth rate gradually adjusts down to an estimated long-term 4.1% growth rate for gross domestic product (GDP). The reduced growth rate cuts the average cost of equity to 6.6%.
We must leave you with two more warnings about DCF formulas for valuing common stocks or estimating the cost of equity. First, it’s almost always worthwhile to lay out a simple spreadsheet, like Table 4.5 or 4.6, to ensure that your dividend projections are consistent with the company’s earnings and required investments. Second, be careful about using DCF valuation formulas to test whether the market is correct in its assessment of a stock’s value. If your estimate of the value is different from the market value, it is probably because you have used poor dividend forecasts. Remember what we said at the beginning of this chapter about simple ways of making money on the stock market: There aren’t any.
4-4The Link between Stock Price and Earnings per Share
Investors separate growth stocks from income stocks. They buy growth stocks primarily for the expectation of capital gains, and they are interested in the future growth of earnings rather than in next year’s dividends. They buy income stocks primarily for the cash dividends. Let us see whether these distinctions make sense.
Imagine first the case of a company that does not grow at all. It does not plow back any earnings and simply produces a constant stream of dividends. Its stock would resemble the perpetual bond described in Chapter 2. Remember that the return on a perpetuity is equal to the yearly cash flow divided by the present value. So the expected return on our share would be equal to the yearly dividend divided by the share price (i.e., the dividend yield). Since all the earnings are paid out as dividends, the expected return is also equal to the earnings per share divided by the share price (i.e., the earnings–price ratio). For example, if the dividend is $10 a share and the stock price is $100, we have16
The price equals
The expected return for growing firms can also equal the earnings–price ratio. The key is whether earnings are reinvested to provide a return equal to the market capitalization rate. For example, suppose our monotonous company suddenly hears of an opportunity to invest $10 a share next year. This would mean no dividend at t = 1. However, the company expects that in each subsequent year the project would earn $1 per share, and therefore the dividend could be increased to $11 a share.
Let us assume that this investment opportunity has about the same risk as the existing business. Then we can discount its cash flow at the 10% rate to find its net present value at year 1:
Thus, the investment opportunity will make no contribution to the company’s value. Its prospective return is equal to the opportunity cost of capital.
What effect will the decision to undertake the project have on the company’s share price? Clearly none. The reduction in value caused by the nil dividend in year 1 is exactly offset by the increase in value caused by the extra dividends in later years. Therefore, once again the market capitalization rate equals the earnings–price ratio:
Table 4.7 repeats our example for different assumptions about the cash flow generated by the new project. Note that the earnings–price ratio, measured in terms of EPS1, next year’s expected earnings, equals the market capitalization rate (r) only when the new project’s NPV = 0. This is an extremely important point—managers can make poor financial decisions because they confuse earnings–price ratios with the market capitalization rate.
TABLE 4.7 Effect on stock price of investing an additional $10 in year 1 at different rates of return. Notice that the earnings–price ratio overestimates r when the project has negative NPV and underestimates it when the project has positive NPV.
aProject costs $10.00 (EPS1). NPV = –10 + C/r, where r = .10.
bNPV is calculated at year 1. To find the impact on P0, discount for one year at r = .10.
In general, we can think of stock price as the capitalized value of average earnings under a no-growth policy, plus PVGO, the net present value of growth opportunities:
The earnings–price ratio, therefore, equals
It will underestimate r if PVGO is positive and overestimate it if PVGO is negative. The latter case is less likely, since firms are rarely forced to take projects with negative net present values.
Calculating the Present Value of Growth Opportunities for Fledgling Electronics
In our last example, both dividends and earnings were expected to grow, but this growth made no net contribution to the stock price. The stock was, in this sense, an “income stock.” Be careful not to equate firm performance with the growth in earnings per share. A company that reinvests earnings at below the market capitalization rate r may increase earnings but will certainly reduce the share value.
Now let us turn to that well-known growth stock, Fledgling Electronics. You may remember that Fledgling’s market capitalization rate, r, is 15%. The company is expected to pay a dividend of $5 in the first year, and thereafter, the dividend is predicted to increase indefinitely by 10% a year. We can use the simplified constant-growth formula to work out Fledgling’s price:
Suppose that Fledgling has earnings per share of EPS1 = $8.33. Its payout ratio is then
In other words, the company is plowing back 1 – .6, or 40% of earnings. Suppose also that Fledgling’s ratio of earnings to book equity is ROE = .25. This explains the growth rate of 10%:
Growth rate = g = plowback ratio × ROE = .4 × .25 = .10
The capitalized value of Fledgling’s earnings per share if it had a no-growth policy would be
But we know that the value of Fledgling stock is $100. The difference of $44.44 must be the amount that investors are paying for growth opportunities. Let’s see if we can explain that figure.
Each year Fledgling plows back 40% of its earnings into new assets. In the first year, Fledgling invests $3.33 at a permanent 25% return on equity. Thus, the cash generated by this investment is .25 × 3.33 = $.83 per year starting at t = 2. The net present value of the investment as of t = 1 is
Everything is the same in year 2 except that Fledgling will invest $3.67, 10% more than in year 1 (remember g = .10). Therefore, at t = 2, an investment is made with a net present value of
Thus, the payoff to the owners of Fledgling Electronics stock can be represented as the sum of (1) a level stream of earnings, which could be paid out as cash dividends if the firm did not grow, and (2) a set of tickets, one for each future year, representing the opportunity to make investments having positive NPVs. We know that the first component of the value of the share is
The first ticket is worth $2.22 in t = 1; the second is worth $2.22 × 1.10 = $2.44 in t = 2; the third is worth $2.44 × 1.10 = $2.69 in t = 3. These are the forecasted cash values of the tickets. We know how to value a stream of future cash values that grows at 10% per year: Use the constant-growth DCF formula, replacing the forecasted dividends with forecasted ticket values:
Now everything checks:
Why is Fledgling Electronics a growth stock? Not because it is expanding at 10% per year. It is a growth stock because the net present value of its future investments accounts for a significant fraction (about 44%) of the stock’s price.
Today’s stock price reflects investor expectations about the earning power of the firm’s current and future assets. For example, take Alphabet, the parent company of Google. Alphabet has never paid a dividend. It plows back all its earnings into its business. In early 2018, its stock sold for $1,130 per share at a forward P/E of about 27. EPS forecasted for 2018 were $41.54.
Suppose that Alphabet did not grow and that future EPS were expected to stay constant at $41.54. In this case, Alphabet could pay a constant dividend of $41.54 per share. If the cost of equity is, say, 8%, market value would be PV = 41.54/.08 = $519.25 per share, about $611 less than the actual stock price of $1,130. So it appears that investors were valuing Alphabet’s future investment opportunities at $611 per share, about half of the stock price. Alphabet is a growth stock because that large fraction of its market value comes from the expected NPV of its future investments.
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Valuing Alphabet
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4-5Valuing a Business by Discounted Cash Flow
Investors buy or sell shares of common stock. Companies often buy or sell entire businesses or major stakes in businesses. For example, we have noted BHP Billiton’s plans to sell its U.S. shale business. Both BHP and potential bidders were doing their best to value that business by discounted cash flow.
DCF models work just as well for entire businesses as for shares of common stock. It doesn’t matter whether you forecast dividends per share or the total free cash flow of a business. Value today always equals future cash flow discounted at the opportunity cost of capital.
Valuing the Concatenator Business
Rumor has it that Establishment Industries is interested in buying your company’s concatenator manufacturing operation. Your company is willing to sell if it can get the full value of this rapidly growing business. The problem is to figure out what its true present value is.
Table 4.8 gives a forecast of free cash flow (FCF) for the concatenator business. Free cash flow is the amount of cash that a firm can pay out to investors after paying for all investments necessary for growth. As we will see, free cash flow can be negative for rapidly growing businesses.
TABLE 4.8 Forecasts of free cash flow in $ millions for the concatenator division. Free cash flow is zero for periods 1 to 3 because investment absorbs all of net income. Free cash flow turns positive when growth slows down after period 3. Inputs required for the table’s calculations are in bold type.
Notes:
1. Starting asset value is $10 million. Assets grow at 12% to start, then at 9%, and finally at 6% in perpetuity. Profitability is assumed constant at 12%.
2. Free cash flow equals earnings minus net investment. Net investment equals total capital outlays minus depreciation. We assume that investment for replacement of existing assets is covered by depreciation and that net investment is devoted to growth. Earnings are also net of depreciation.
Table 4.8 is similar to Table 4.5, which forecasted earnings and dividends per share for Growth-Tech, based on assumptions about Growth-Tech’s equity per share, return on equity, and the growth of its business. For the concatenator business, we also have assumptions about assets, profitability—in this case, after-tax operating earnings relative to assets—and growth. Growth starts out at a rapid 12% per year, then falls in two steps to a moderate 6% rate for the long run. The growth rate determines the net additional investment required to expand assets, and the profitability rate determines the earnings thrown off by the business.
Free cash flow, the fourth line in Table 4.8, is equal to the firm’s earnings less any new investment expenditures. Free cash flow is zero in years 1 to 3, even though the parent company is investing over $4 million during this period.
Are the early zeros for free cash flow a bad sign? No: Free cash flow is zero because the business is growing rapidly, not because it is unprofitable. Rapid growth is good news, not bad, because the business is earning 12%, 2 percentage points over the 10% cost of capital. If the business could grow at 20%, Establishment Industries and its stockholders would be happier still, although growth at 20% would mean still higher investment and negative free cash flow.
Valuation Format
The value of a business is usually computed as the discounted value of free cash flows out to a valuation horizon (H), plus the forecasted value of the business at the horizon, also discounted back to present value. That is,
Of course, the concatenator business will continue after the horizon, but it’s not practical to forecast free cash flow year by year to infinity. PVH stands in for free cash flow in periods H + 1, H + 2, and so on.
Valuation horizons are often chosen arbitrarily. Sometimes the boss tells everybody to use 10 years because that’s a round number. We will try year 6, because growth of the concatenator business seems to settle down to a long-run trend after year 7.
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Try It! Table 4.8: Valuing the concatenator division
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Estimating Horizon Value
There are two common approaches to estimating horizon value. One uses valuation by comparables, based on P/E, market-to-book, or other ratios. The other uses DCF. We will start with valuation by comparables.
Horizon Value Based on P/E Ratios Suppose you can observe stock prices for good comparables, that is, for mature manufacturing companies whose scale, risk, and growth prospects today roughly match those projected for the concatenator business in year 6.17 Suppose further that these companies tend to sell at price–earnings ratios of about 11. Then you could reasonably guess that the price–earnings ratio of a mature concatenator operation will likewise be 11. That implies:
The present value of the business up to the horizon is $.9 million.Therefore
PV (business) = .9 + 13.5 = $14.4 million
Horizon Value Based on Market–Book Ratios Suppose also that the market–book ratios of the sample of mature manufacturing companies tend to cluster around 1.5. If the concatenator business market–book ratio is 1.5 in year 6,
It’s easy to poke holes in these last two calculations. Book value, for example, is often a poor measure of the true value of a company’s assets. It can fall far behind actual asset values when there is rapid inflation, and it often entirely misses important intangible assets, such as your patents for concatenator design. Earnings may also be biased by inflation and a long list of arbitrary accounting choices. Finally, you never know when you have found a sample of truly similar companies to use as comparables.
But remember, the purpose of discounted cash flow is to estimate market value—to estimate what investors would pay for a stock or business. When you can observe what they actually pay for similar companies, that’s valuable evidence. Try to figure out a way to use it. One way to use it is through valuation by comparables, based on price–earnings or market–book ratios. Valuation rules of thumb, artfully employed, sometimes beat a complex discounted-cash-flow calculation hands down.
Horizon Value Based on DCF Now let us try the constant-growth DCF formula. This requires free cash flow for year 7, which we have at $1.09 million from Table 4.8; a long-run growth rate, which appears to be 6%; and a discount rate, which some high-priced consultant has told us is 10%. Therefore,
The PV of the near-term free cash flows is $.9 million. Thus the present value of the concatenator division is
Now, are we done? Well, the mechanics of this calculation are perfect. But doesn’t it make you just a little nervous to find that 94% of the value of the business rests on the horizon value? Moreover, a little checking shows that horizon value can change dramatically in response to small changes in the assumed long-term growth rate.
Suppose the growth rate is 7% instead of 6%. That means that asset value has to grow by an extra 1% per year, requiring extra investment of $.18 million in period 7, which reduces FCF7 to $.91 million. Horizon value increases to PVH = $30.3 million in year 6 and to $17.1 million discounted to year zero. The PV of the concatenator business increases from $16 .3 million to $.9 + 17.1 = $18.0 million.
Warning 1: When you use the constant-growth DCF formula to calculate horizon value, always remember that faster growth requires increased investment, which reduces free cash flow. Slower growth requires less investment, which increases free cash flow.
So 7% instead of 6% growth increases PV by $18.0 – 16.3 = $1.7 million. Why? We did not ignore warning 1: We accounted for the increased investment required for faster growth. Therefore the additional investment in periods 7 and beyond must have generated additional positive NPV. In other words, we must have assumed expanded growth opportunities and added more PVGO to the value of the business.
Notice in Table 4.8 that the return on assets (ROA) is forecasted at 12% forever, 2 percentage points higher than the assumed discount rate of 10%. Thus, every dollar invested in period 7 and beyond generates positive NPV and adds to horizon value and the PV of the business.
But is it realistic to assume that any business can keep on growing and making positive-NPV investments forever? Sooner or later you and your competitors will be on an equal footing. You may still be earning a superior return on past investments, but you will find that introductions of new products or attempts to expand profits from existing products trigger vigorous resistance from competitors who are just as smart and efficient as you are. When that time comes, the NPV of subsequent investments will average out to zero. After all, PVGO is positive only when investments can be expected to earn more than the cost of capital.
Warning 2: Always check to see whether horizon value includes post-horizon PVGO. You can check on warning 2 by changing the assumed long-term growth rate. If a higher growth rate increases horizon value—after you have taken care to respect warning 1—then you are assuming post-horizon PVGO. Is it realistic to assume that the firm can earn more than the cost of capital in perpetuity? If not, adjust your forecasts accordingly.
There is an easy way to calculate horizon value if post-horizon PVGO is zero. Recall that PV equals the capitalized value of next period’s earnings plus PVGO:
If PVGO = 0 at the horizon period H, then,
In other words, when the competition catches up and the firm can only earn its cost of equity on new investment, the price–earnings ratio will equal 1/r, because PVGO disappears.
This latest formula for PVH is still DCF. We are valuing the business as if assets and earnings will not grow after the horizon date.18 (The business probably will grow, but the growth can be ignored, because it will add no net value if PVGO goes to zero.) With no growth, there is no net investment,19 and all of earnings ends up as free cash flow.
Therefore, we can calculate the horizon value at period 6 as the present value of a level stream of earnings starting in period 7 and continuing indefinitely. The resulting value for the concatenator business is:
A Value Range for the Concatenator Business We now have four estimates of what Establishment Industries ought to pay for the concatenator business. The estimates reflect four different methods of estimating horizon value. There is no best method, although we like the last method, which forces managers to remember that sooner or later competition catches up.
Our calculated values for the concatenator business range from $13.2 to $16.3 million, a difference of about $3 million. The width of the range may be disquieting, but it is not unusual. Discounted-cash-flow formulas only estimate market value, and the estimates change as forecasts and assumptions change. Managers cannot know market value for sure until an actual transaction takes place.
Free Cash Flow, Dividends, and Repurchases
We assumed that the concatenator business was a division of your company, not a freestanding corporation. But suppose it was a separate corporation with 1 million shares outstanding. How would we calculate price per share? Simple: Calculate the PV of the business and divide by 1 million. If we decide that the business is worth $16.3 million, the price per share is $16.30.
If the concatenator business were a public Concatenator Corp., with no other assets and operations, it could pay out its free cash flow as dividends. Dividends per share would be the free cash flow shown in Table 4.8 divided by 1 million shares: zero in periods 1 to 3, then $.42 per share in period 4, $.46 per share in period 5, etc.
We mentioned stock repurchases as an alternative to cash dividends. If repurchases are important, it’s often simpler to value total free cash flow than dividends per share. Suppose Concatenator Corp. decides not to pay cash dividends. Instead, it will pay out all free cash flow by repurchasing shares. The market capitalization of the company should not change because shareholders as a group will still receive all free cash flow.
Perhaps the following intuition will help. Suppose you own all of the 1 million Concatenator shares. Do you care whether you get free cash flow as dividends or by selling shares back to the firm? Your cash flows in each future period will always equal the free cash flows shown in Table 4.8. Your DCF valuation of the company will, therefore, depend on the free cash flows, not on how they are distributed.
Chapter 16 covers the choice between cash dividends and repurchases (including tax issues and other complications). But you can see why it’s attractive to value a company as a whole by forecasting and discounting free cash flow. You don’t have to ask how free cash flow will be paid out. You don’t have to forecast repurchases.
SUMMARY
In this chapter, we have used our newfound knowledge of present values to examine the market price of common stocks. The value of a stock is equal to the stream of cash payments discounted at the rate of return that investors expect to receive on other securities with equivalent risks.
Common stocks do not have a fixed maturity; their cash payments consist of an indefinite stream of dividends. Therefore, the present value of a share of common stock is
However, we did not just assume that investors purchase common stocks solely for dividends. In fact, we began with the assumption that investors have relatively short horizons and invest for both dividends and capital gains. Our fundamental valuation formula is, therefore,
This is a condition of market equilibrium. If it did not hold, the share would be overpriced or underpriced, and investors would rush to sell or buy it. The flood of sellers or buyers would force the price to adjust so that the fundamental valuation formula holds.
We also made use of the formula for a growing perpetuity presented in Chapter 2. If dividends are expected to grow forever at a constant rate of g, then
It is often helpful to twist this formula around and use it to estimate the market capitalization rate r, given P0 and estimates of DIV1 and g:
Remember, however, that this formula rests on a very strict assumption: constant dividend growth in perpetuity. This may be an acceptable assumption for mature, low-risk firms, but for many firms, near-term growth is unsustainably high. In that case, you may wish to use a two-stage DCF formula, where near-term dividends are forecasted and valued, and the constant-growth DCF formula is used to forecast the value of the shares at the start of the long run. The near-term dividends and the future share value are then discounted to present value.
The general DCF formula can be transformed into a statement about earnings and growth opportunities:
The ratio EPS1/r is the present value of the earnings per share that the firm would generate under a no-growth policy. PVGO is the net present value of the investments that the firm will make in order to grow. A growth stock is one for which PVGO is large relative to the present value of EPS if the firm did not grow. Most growth stocks are stocks of rapidly expanding firms, but expansion alone does not create a high PVGO. What matters is the profitability of the new investments.
The same formulas that we used to value common shares can also be used to value entire businesses. In that case, we discount not dividends per share but the entire free cash flow generated by the business. Usually, a two-stage DCF model is deployed. Free cash flows are forecasted out to a horizon and discounted to present value. Then a horizon value is forecasted, discounted, and added to the value of the free cash flows. The sum is the value of the business.
Valuing a business is simple in principle but not so easy in practice. Forecasting reasonable horizon values is particularly difficult. The usual assumption is moderate long-run growth after the horizon, which allows use of the growing-perpetuity DCF formula at the horizon. Horizon values can also be calculated by the valuation-by-comparables method, for example by assuming normal price–earnings or market-to-book ratios at the horizon date.
The dividend discount models derived in this chapter work best for mature firms that pay regular cash dividends. The models also work when companies pay out cash by share repurchases as well as dividends. That said, it is also true that the dividend discount model is difficult to use if the company pays no dividends at all or if the split of payout between cash dividends and repurchases is unpredictable. In the latter case, it is easier to get price per share by forecasting and valuing the company’s total free cash flow and then dividing by the current number of shares outstanding.
PROBLEM SETS
Select problems are available in McGraw-Hill’s Connect. Answers to questions with an “*” are found in the Appendix.
1. Stock markets True or false?
a. The bid price is always greater than the ask price.
b. An investor who wants to sell his stock immediately should enter a limit order.
c. The sale of shares by a large investor usually takes place in the primary market.
d. Electronic Communications Network refers to the automated ticker tape on the New York Stock Exchange.
2. Stock quotes
a. “I would like to sell 1000 shares of Walmart at best.”
b. “I would like to buy 500 shares of Hattersley at $50 or better.”
Which of these is a limit order and which is a market order? If the price of Walmart is $50 and the price of Hattersley is $60, which, if any, of these orders will be executed?
3. Stock quotes* Here is a small part of the order book for Mesquite Foods:
|
Bid |
Ask |
||
|
Price |
Size |
Price |
Size |
|
103 |
100 |
103.5 |
200 |
|
102.5 |
200 |
103.8 |
200 |
|
101 |
400 |
104 |
300 |
|
99.8 |
300 |
104.5 |
400 |
a. Georgina Sloberg submits a market order to sell 100 shares. What price will she receive?
b. Norman Pilbarra submits a market order to buy 400 shares. What is the maximum price that he will pay?
c. Carlos Ramirez submits a limit bid order at 105. Will it execute immediately?
4. Stock quotes Go to finance.yahoo.com and get trading quotes for IBM.
a. What is the latest IBM stock price and market cap?
b. What is IBM’s dividend payment and dividend yield?
c. What is IBM’s trailing P/E ratio?
d. Calculate IBM’s forward P/E ratio using the EPS forecasted by analysts for the next year.
e. What is IBM’s price–book (P/B) ratio?
5. Valuation by comparables Look up P/E and P/B ratios for Entergy (ticker symbol ETR), using Yahoo! Finance or another Internet source. Calculate the same ratios for the following potential comparables: American Electric Power (AEP), CenterPoint Energy (CNP), and Southern Company (SO). Set out the ratios in the same format as Table 4.2. Are the ratios for these electric companies tightly grouped or scattered? If you didn’t know Entergy’s stock price, would the comparables give a good estimate?
6. Dividend discount model True or false?
a. All stocks in an equivalent-risk class are priced to offer the same expected rate of return.
b. The value of a share equals the PV of future dividends per share.
c. The value of a share equals the PV of earnings per share assuming the firm does not grow, plus the NPV of future growth opportunities.
7. Dividend discount model Respond briefly to the following statement:
“You say stock price equals the present value of future dividends? That’s crazy! All the investors I know are looking for capital gains.”
8. Dividend discount model* Company X is expected to pay an end-of-year dividend of $5 a share. After the dividend, its stock is expected to sell at $110. If the market capitalization rate is 8%, what is the current stock price?
9. Dividend discount model Company Y does not plow back any earnings and is expected to produce a level dividend stream of $5 a share. If the current stock price is $40, what is the market capitalization rate?
10. Constant-growth DCF model* Company Z’s earnings and dividends per share are expected to grow indefinitely by 5% a year. If next year’s dividend is $10 and the market capitalization rate is 8%, what is the current stock price?
11. Constant-growth DCF model Consider three investors:
a. Mr. Single invests for one year.
b. Ms. Double invests for two years.
c. Mrs. Triple invests for three years.
Assume each invests in company Z (see Problem 10). Show that each expects to earn a rate of return of 8% per year.
12. Constant-growth DCF model Pharmecology just paid an annual dividend of $1.35 per share. It’s a mature company, but future EPS and dividends are expected to grow with inflation, which is forecasted at 2.75% per year.
a. What is Pharmecology’s current stock price? The nominal cost of capital is 9.5%.
b. Redo part (a) using forecasted real dividends and a real discount rate.
13. Constant-growth DCF model*
Here are forecasts for next year for two stocks:
|
Stock A |
Stock B |
|
|
Return on equity |
15% |
10% |
|
Earnings per share |
$2.00 |
$1.50 |
|
Dividends per share |
$1.00 |
$1.00 |
a. What are the dividend payout ratios for each firm?
b. What are the expected dividend growth rates for each stock?
c. If investors require a return of 15% on each stock, what are their values?
14. Constant-growth DCF model Look up General Mills (GIS), Kellogg (K), Campbell Soup (CPB), and Seneca Foods (SENEA).
a. What are the current P/E and P/B ratios for these food companies? What are the dividend and dividend yield for each company?
b. What are the growth rates of EPS and dividends for each company over the last five years? What EPS growth rates are forecasted by analysts? Do these growth rates appear to be on a steady trend that could be projected for the long run?
c. Would you be confident in applying the constant-growth DCF model to measure these companies’ costs of equity? Why or why not?
15. Cost of equity capital Under what conditions does r, a stock’s market capitalization rate, equal its earnings–price ratio EPS1/P0?
16. Cost of equity capital Each of the following formulas for determining shareholders’ required rate of return can be right or wrong depending on the circumstances:
a. r = DIV1/P0 + g
b. r = EPS1/P0
For each formula, construct a simple numerical example showing that the formula can give wrong answers and explain why the error occurs. Then construct another simple numerical example for which the formula gives the right answer.
17. Two-stage DCF model Company Z-prime is like Z in Problem 10 in all respects save one: Its growth will stop after year 4. In year 5 and afterward, it will pay out all earnings as dividends. What is Z-prime’s stock price? Assume next year’s EPS is $15.
18. Two-stage DCF model* Consider the following three stocks:
a. Stock A is expected to provide a dividend of $10 a share forever.
b. Stock B is expected to pay a dividend of $5 next year. Thereafter, dividend growth is expected to be 4% a year forever.
c. Stock C is expected to pay a dividend of $5 next year. Thereafter, dividend growth is expected to be 20% a year for five years (i.e., years 2 through 6) and zero thereafter.
If the market capitalization rate for each stock is 10%, which stock is the most valuable? What if the capitalization rate is 7%?
19. Two-stage DCF model Company Q’s current return on equity (ROE) is 14%. It pays out one half of earnings as cash dividends (payout ratio = .5). Current book value per share is $50. Book value per share will grow as Q reinvests earnings. Assume that the ROE and payout ratio stay constant for the next four years. After that, competition forces ROE down to 11.5% and the payout ratio increases to 0.8. The cost of capital is 11.5%.
a. What are Q’s EPS and dividends next year? How will EPS and dividends grow in years 2, 3, 4, 5, and subsequent years?
b. What is Q’s stock worth per share? How does that value depend on the payout ratio and growth rate after year 4?
20. Two-stage DCF model 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% book return on equity. At the end of the year, CSI is expected to pay a $4 dividend. It has been reinvesting 40% of earnings and growing at 4% a year.
a. Suppose CSI continues on this growth trend. What is the expected long-run rate of return from purchasing the stock at $100? What part of the $100 price is attributable to the present value of growth opportunities?
b. Now the MDC announces a plan for CSI to treat Cambridge sewage. CSI’s plant will, therefore, be expanded gradually over five years. This means that CSI will have to reinvest 80% of its earnings for five years. Starting in year 6, however, it will again be able to pay out 60% of earnings. What will be CSI’s stock price once this announcement is made and its consequences for CSI are known?
21. Growth opportunities If company Z (see Problem 10) were to distribute all its earnings, it could maintain a level dividend stream of $15 a share. How much is the market actually paying per share for growth opportunities?
22. Growth opportunities Look up Intel (INTC), Oracle (ORCL), and HP (HPQ) on finance.yahoo.com. Rank the companies’ forward P/E ratios from highest to lowest. What are the possible reasons for the different ratios? Which of these companies appears to have the most valuable growth opportunities?
23. Growth opportunities Alpha Corp’s earnings and dividends are growing at 15% per year. Beta Corp’s earnings and dividends are growing at 8% per year. The companies’ assets, earnings, and dividends per share are now (at date 0) exactly the same. Yet PVGO accounts for a greater fraction of Beta Corp’s stock price. How is this possible? (Hint: There is more than one possible explanation.)
24. Growth opportunities Look again at the financial forecasts for Growth-Tech given in Table 4.5. This time assume you know that the opportunity cost of capital is r = .12 (discard the .099 figure calculated in the text). Assume you do not know Growth-Tech’s stock value. Otherwise follow the assumptions given in the text.
a. Calculate the value of Growth-Tech stock.
b. What part of that value reflects the discounted value of P3, the price forecasted for year 3?
c. What part of P3 reflects the present value of growth opportunities (PVGO) after year 3?
d. Suppose that competition will catch up with Growth-Tech by year 4 so that it can earn only its cost of capital on any investments made in year 4 or subsequently. What is GrowthTech stock worth now under this assumption? (Make additional assumptions if necessary.)
25. Free cash flow What do financial managers mean by “free cash flow”? How is free cash flow calculated? Briefly explain.
26. Horizon value What is meant by the “horizon value” of a business? How can it be estimated?
27. Horizon value Suppose the horizon date is set at a time when the firm will run out of positive-NPV investment opportunities. How would you calculate the horizon value? (Hint: What is the P/EPS ratio when PVGO = 0?)
28. Valuing a business Permian Partners (PP) produces from aging oil fields in west Texas. Production is 1.8 million barrels per year in 2018, but production is declining at 7% per year for the foreseeable future. Costs of production, transportation, and administration add up to $25 per barrel. The average oil price was $65 per barrel in 2018. PP has 7 million shares outstanding. The cost of capital is 9%. All of PP’s net income is distributed as dividends. For simplicity, assume that the company will stay in business forever and that costs per barrel are constant at $25. Also, ignore taxes.
a. What is the ending 2018 value of one PP share? Assume that oil prices are expected to fall to $60 per barrel in 2019, $55 per barrel in 2020, and $50 per barrel in 2021. After 2021, assume a long-term trend of oil-price increases at 5% per year.
b. What is PP’s EPS/P ratio, and why is it not equal to the 9% cost of capital?
29. Valuing a business Construct a new version of Table 4.8, assuming that competition drives down profitability (on existing assets as well as new investment) to 11.5% in year 6, 11% in year 7, 10.5% in year 8, and 8% in year 9 and all later years. What is the value of the concatenator business?
30. Valuing a business Mexican Motors’ market cap is 200 billion pesos. Next year’s free cash flow is 8.5 billion pesos. Security analysts are forecasting that free cash flow will grow by 7.5% per year for the next five years.
a. Assume that the 7.5% growth rate is expected to continue forever. What rate of return are investors expecting?
b. Mexican Motors has generally earned about 12% on book equity (ROE = 12%) and reinvested 50% of earnings. The remaining 50% of earnings has gone to free cash flow. Suppose the company maintains the same ROE and investment rate for the long run. What is the implication for the growth rate of earnings and free cash flow? For the cost of equity? Should you revise your answer to part (a) of this question?
31. Valuing a business* Phoenix Corp. faltered in the recent recession but is recovering. Free cash flow has grown rapidly. Forecasts made in 2019 are as follows:
Phoenix’s recovery will be complete by 2024, and there will be no further growth in net income or free cash flow.
a. Calculate the PV of free cash flow, assuming a cost of equity of 9%.
b. Assume that Phoenix has 12 million shares outstanding. What is the price per share?
c. Confirm that the expected rate of return on Phoenix stock is exactly 9% in each of the years from 2020 to 2024.
CHALLENGE
32. Constant-growth DCF formula The constant-growth DCF formula:
is sometimes written as:
where BVPS is book equity value per share, b is the plowback ratio, and ROE is the ratio of earnings per share to BVPS. Use this equation to show how the price-to-book ratio varies as ROE changes. What is price-to-book when ROE = r?
33. DCF valuation Portfolio managers are frequently paid a proportion of the funds under management. Suppose you manage a $100 million equity portfolio offering a dividend yield (DIV1/P0) of 5%. Dividends and portfolio value are expected to grow at a constant rate. Your annual fee for managing this portfolio is .5% of portfolio value and is calculated at the end of each year. Assuming that you will continue to manage the portfolio from now to eternity, what is the present value of the management contract? How would the contract value change if you invested in stocks with a 4% yield?
34. Valuing a business Construct a new version of Table 4.8, assuming that the concatenator division grows at 20%, 12%, and 6%, instead of 12%, 9%, and 6%. You will get negative early free cash flows.
a. Recalculate the PV of free cash flow. What does your revised PV say about the division’s PVGO?
b. Suppose the division is the public corporation Concatenator Corp, with no other resources. Thus it will have to issue stock to cover the negative free cash flows. Does the need to issue shares change your valuation? Explain. (Hint: Suppose first that Concatenator’s existing stockholders buy all of the newly issued shares. What is the value of the company to these stockholders? Now suppose instead that all the shares are issued to new stockholders, so that existing stockholders don’t have to contribute any cash. Does the value of the company to the existing stockholders change, assuming that the new shares are sold at a fair price?)
FINANCE ON THE WEB
The major stock exchanges have wonderful websites. Start with the NYSE (www.nyse.com) and Nasdaq (www.nasdaq.com). Make sure you know how trading takes place on these exchanges.
MINI-CASE 
Reeby Sports
Ten years ago, in 2010, George Reeby founded a small mail-order company selling high-quality sports equipment. Since those early days, Reeby Sports has grown steadily and been consistently profitable. The company has issued 2 million shares, all of which are owned by George Reeby and his five children.
For some months, George has been wondering whether the time has come to take the company public. This would allow him to cash in on part of his investment and would make it easier for the firm to raise capital should it wish to expand in the future.
But how much are the shares worth? George’s first instinct is to look at the firm’s balance sheet, which shows that the book value of the equity is $26.34 million, or $13.17 per share. A share price of $13.17 would put the stock on a P/E ratio of 6.6. That is quite a bit lower than the 13.1 P/E ratio of Reeby’s larger rival, Molly Sports.
George suspects that book value is not necessarily a good guide to a share’s market value. He thinks of his daughter Jenny, who works in an investment bank. She would undoubtedly know what the shares are worth. He decides to phone her after she finishes work that evening at 9 o’clock or before she starts the next day at 6.00 a.m.
Before phoning, George jots down some basic data on the company’s profitability. After recovering from its early losses, the company has earned a return that is higher than its estimated 10% cost of capital. George is fairly confident that the company could continue to grow fairly steadily for the next six to eight years. In fact, he feels that the company’s growth has been somewhat held back in the last few years by the demands from two of the children for the company to make large dividend payments. Perhaps, if the company went public, it could hold back on dividends and plow more money back into the business.
There are some clouds on the horizon. Competition is increasing and only that morning Molly Sports announced plans to form a mail-order division. George is worried that beyond the next six or so years it might become difficult to find worthwhile investment opportunities.
George realizes that Jenny will need to know much more about the prospects for the business before she can put a final figure on the value of Reeby Sports, but he hopes that the information is sufficient for her to give a preliminary indication of the value of the shares.
QUESTIONS
1. Help Jenny to forecast dividend payments for Reeby Sports and to estimate the value of the stock. You do not need to provide a single figure. For example, you may wish to calculate two figures, one on the assumption that the opportunity for further profitable investment disappears after six years and another assuming it disappears after eight years.
2. How much of your estimate of the value of Reeby’s stock comes from the present value of growth opportunities?
1Trades are still made face to face on the floor of the NYSE, but computerized trading is taking over. In 2006, the NYSE merged with Archipelago, an electronic trading system, and transformed itself into a public corporation. The following year, it merged with Euronext, an electronic trading system in Europe. It is now owned by Intercontinental Exchange Inc., a U.S.-based network of exchanges and clearing houses.
2Other good sources of trading data are moneycentral.msn.com, finance.google.com, or the online edition of The Wall Street Journal at www.wsj.com (look for the “Market” and then “Market Data” tabs).
3Yahoo! Finance provides extensive information and statistics on traded companies, including summaries of analyst forecasts. For example, you can click on “Statistics” or “Analyst Estimates.”
4Closed-end mutual funds issue shares that are traded on stock exchanges. Open-end funds are not traded on exchanges. Investors in open-end funds transact directly with the fund. The fund issues new shares to investors and redeems shares from investors who want to withdraw money from the fund.
5GE’s equity accounts included $3.4 billion for “Misc. stocks options warrants.” The book value of GE’s common stock was $79.5 − 3.4 = $76.1 billion.
6UNP’s EPS for the most recent, or “trailing” 12 months were $5.51, so its trailing P/E was 117/5.51 = 21.15. Trailing P/Es are often quoted, but forward P/Es are more useful if good forecasts are available. Investors learn from the past but are mostly interested in the future.
7The P/E ratios for Devon’s comparables stand as a warning to be extra careful when averaging P/Es. Watch out for companies with earnings close to zero or negative. One company with zero earnings and, therefore, an infinite P/E generates an infinite average P/E. Often, it’s safer to use median P/Es than averages.
8Financial analysts often use ratios of EBIT (earnings before interest and taxes) or EBITDA (earnings before interest, taxes, depreciation, and amortization) to enterprise value (the sum of outstanding debt and the market cap of equity). EBIT or EBITDA ratios are less sensitive than P/E ratios to differences in financing. In Chapter 19, we cover valuation when financing comes from a mix of debt and equity. We discuss other ratios in Chapter 28.
9Notice that this DCF formula uses a single discount rate for all future cash flows. This implicitly assumes that the company is all-equity-financed or that the fractions of debt and equity will stay constant. Chapters 17 through 19 discuss how the cost of equity changes when debt ratios change.
10The deferred payout may come all at once if the company is taken over by another. The selling price per share is equivalent to a bumper dividend.
11Notice that we have derived the dividend discount model using dividends per share. Paying out cash for repurchases rather than cash dividends reduces the number of shares outstanding and increases future earnings and dividends per share. The more shares repurchased, the faster the growth of earnings and dividends per shares. Thus, repurchases benefit shareholders who do not sell as well as those who do sell. We show some examples in Chapter 16.
12These formulas were first developed in 1938 by Williams and were rediscovered by Gordon and Shapiro. See J. B. Williams, The Theory of Investment Value (Cambridge, MA: Harvard University Press, 1938); and M. J. Gordon and E. Shapiro, “Capital Equipment Analysis: The Required Rate of Profit,” Management Science 3 (October 1956), pp. 102–110.
13This is the accepted interpretation of the U.S. Supreme Court’s directive in 1944 that “the returns to the equity owner [of a regulated business] should be commensurate with returns on investments in other enterprises having corresponding risks.” Federal Power Commission v. Hope Natural Gas Company, 302 U.S. 591 at 603.
14In this calculation, we’re assuming that earnings and dividends are forecasted to grow forever at the same rate g. We show how to relax this assumption later in this chapter. The growth rate was based on the average earnings growth forecasted by Value Line and IBES. IBES compiles and averages forecasts made by security analysts. Value Line publishes its own analysts’ forecasts.
15The STB makes two estimates of the cost of equity. One is based on a three-stage DCF model, and the other uses the capital asset pricing model, which we describe in Chapter 8. The STB averages the two estimates.
16Notice that we use next year’s EPS for E/P and P/E ratios. Thus we are using forward, not trailing, P/E.
17We have not asked how the concatenator business is financed. We are implicitly assuming 100% equity and zero debt. Therefore the comparables should also have little or no debt. If they do have debt, EBIT or EBITDA ratios would be better than P/E ratios. See footnote 8 and the examples in Section 19-2.
18But what does “no growth” mean? Suppose that the concatenator business maintains its assets and earnings in real (inflation-adjusted) terms. Then nominal earnings will grow at the inflation rate. This takes us back to the constant-growth formula: Earnings in period H + 1 should be valued by dividing by r – g, where g in this case equals the inflation rate.
We have simplified the concatenator example. In real-life valuations, with big bucks involved, be careful to track growth from inflation as well as growth from investment. For guidance see M. Bradley and G. Jarrell, “Expected Inflation and the ConstantGrowth Valuation Model,” Journal of Applied Corporate Finance 20 (Spring 2008), pp. 66–78.
19The business must invest enough to maintain its assets, even in the no-growth case. We have assumed that a base level of investment equal to depreciation is sufficient to maintain assets. Note that earnings are calculated after depreciation—that is, after paying for the base investment. Depreciation and this investment are not broken out in Table 4.7.