Part 1 Value
31. 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?
32. 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?
33. Valuing free cash flow Construct a new version of Table 4.7, 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 Concatco, 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 Concatco’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?)
3 4 . DCF valuation and PVGO Delhi Big-Box Stores is solidly profitable and growing rapidly. At the same time it faces a relatively high opportunity cost of capital, which the CFO estimates at 14%. Construct a table in the same format as Table 4.7.
Assume a starting asset value of R10 billion, initial ROE of 20%, and an initial growth rate of assets and earnings of 15% per year. But after year 5, the forecasted growth rate of assets falls to 10%.
a. Calculate the value of the company and its PVGO.
b. Why do you think Delhi’s CFO is forecasting a decline in ROE to the 14% cost of capital. Is such a decline normal in competitive markets?
c. How does the company value and PVGO change if ROE is 14% in all future years?
● ● ● ● ●
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 ● ● ● ● ●
Bok Sports
Ten years ago, in Johannesburg, Joost van Hees founded a small mail-order company selling high-quality sports equipment. Since those early days Bok Sports has grown steadily and been consistently profitable. The company has issued 2 million shares, all of which are owned by Joost van Hees and his five children.
For some months Joost 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? Joost’s first instinct is to look at the firm’s balance sheet, which shows that the book value of the equity is R263.4 million, or R131.7 per share. A share price of R131.7 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 Bok’s larger rival, Wenner Corporation.
Joost 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.
Before speaking to her, Joost 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. Joost 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 Wenner announced plans to form a mail- order division. Joost is worried that beyond the next six or so years it might become difficult to find worthwhile investment opportunities.
Joost realizes that Jenny will need to know much more about the prospects for the business before she can put a final figure on Bok’s value, 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 Bok 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 Bok’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, and changed its name to NYSE Euronext.
2Other good sources of trading data are moneycentral.msn.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 “Key Statistics” or “Analyst Estimates”
under “More on GE.”
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.
5Be 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 makes any average meaningless. Often it’s safer to use median P/Es rather than averages.
6Or maybe the table would work better with different financial ratios. For example, analysts may use the ratio of earnings before interest and taxes (EBIT) to enterprise value, defined as the sum of outstanding debt and the market capitalization of equity. This ratio is less sensitive to differences in debt financing policy. In Chapter 19 we discuss valuation when financing comes from a mix of debt and equity. We discuss other financial ratios in Chapter 28.
7Notice 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.
8The 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.
9Notice 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.
10These 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.
11This 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.
12In 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.
13Notice that we use next year’s EPS for E/P and P/E ratios. Thus we are using forward, not trailing, P/E.
14If long-run growth is 7% instead of 6%, an extra 1% of asset value will have to be plowed back into the concatenator business, reducing free cash flow from $1.09 million to $.97 million. The PV of cash flows from dates 1 to 6 stays at $.9 million.
15In other words, we can calculate horizon value as if earnings will not grow after the horizon date, because growth will add no value. But 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 Constant-Growth Valuation Model,” Journal of Applied Corporate Finance 20 (Spring 2008), pp. 66–78.
A
Part 1 Value
Net Present Value and Other Investment Criteria
company’s shareholders prefer to be rich rather than poor. Therefore, they want the firm to invest in every project that is worth more than it costs. The difference between a project’s value and its cost is its net present value (NPV). Companies can best help their shareholders by investing in all projects with a positive NPV and rejecting those with a negative NPV.
We start this chapter with a review of the net present value rule. We then turn to some other measures that companies may look at when making investment decisions. The first two of these measures, the project’s payback period and its book rate of return, are little better than rules of thumb, easy to calculate and easy to communicate. Although there is a place for rules of thumb in this world, an engineer needs something more accurate when designing a 100-story building, and a financial manager needs more than a rule of thumb when making a substantial capital investment decision.
Instead of calculating a project’s NPV, companies often compare the expected rate of return from investing in the project with the return that shareholders could earn on equivalent-risk investments in the capital market. The company accepts those projects that provide a higher return than shareholders could earn for themselves. If used correctly, this rate of return rule should always identify projects that increase firm value. However, we shall see that the rule sets several traps for the unwary.
We conclude the chapter by showing how to cope with situations when the firm has only limited capital. This raises two problems. One is computational. In simple cases we just choose those projects that give the highest NPV per dollar invested, but more elaborate techniques are sometimes needed to sort through the possible alternatives. The other problem is to decide whether capital rationing really exists and whether it invalidates the net present value rule. Guess what?
NPV, properly interpreted, wins out in the end.
5-1 A Review of the Basics
Vegetron’s chief financial officer (CFO) is wondering how to analyze a proposed $1 million investment in a new venture called project X. He asks what you think.
Your response should be as follows: “First, forecast the cash flows generated by project X over its economic life. Second, determine the appropriate opportunity cost of capital (r). This should reflect both the time value of money and the risk involved in project X. Third, use this opportunity cost of capital to discount the project’s future cash flows. The sum of the discounted cash flows is called
present value (PV). Fourth, calculate net present value (NPV) by subtracting the $1 million investment from PV. If we call the cash flows C0, C1, and so on, then
We should invest in project X if its NPV is greater than zero.”
However, Vegetron’s CFO is unmoved by your sagacity. He asks why NPV is so important.
Your reply: “Let us look at what is best for Vegetron stockholders.” They want you to make their Vegetron shares as valuable as possible.
“Right now Vegetron’s total market value (price per share times the number of shares outstanding) is $10 million. That includes $1 million cash we can invest in project X.” The value of Vegetron’s other assets and opportunities must therefore be $9 million. We have to decide whether it is better to keep the $1 million cash and reject project X or to spend the cash and accept the project. Let us call the value of the new project PV. Then the choice is as follows:
“Clearly project X is worthwhile if its present value, PV, is greater than $1 million, that is, if net present value is positive.”
CFO: “How do I know that the PV of project X will actually show up in Vegetron’s market value?”
Your reply: “Suppose we set up a new, independent firm X, whose only asset is project X.” What would be the market value of firm X?
“Investors would forecast the dividends that firm X would pay and discount those dividends by the expected rate of return of securities having similar risks.” We know that stock prices are equal to the present value of forecasted dividends.
“Since project X is the only asset, the dividend payments we would expect firm X to pay are exactly the cash flows we have forecasted for project X.” Moreover, the rate investors would use to discount
firm X’s dividends is exactly the rate we should use to discount project X’s cash flows.
“I agree that firm X is entirely hypothetical.” But if project X is accepted, investors holding Vegetron stock will really hold a portfolio of project X and the firm’s other assets. We know the other assets are worth $9 million considered as a separate venture. Since asset values add up, we can easily figure out the portfolio value once we calculate the value of project X as a separate venture.
“By calculating the present value of project X, we are replicating the process by which the common stock of firm X would be valued in capital markets.”
CFO: “The one thing I don’t understand is where the discount rate comes from.”
Your reply: “I agree that the discount rate is difficult to measure precisely.” But it is easy to see what we are trying to measure. The discount rate is the opportunity cost of investing in the project rather than in the capital market. In other words, instead of accepting a project, the firm can always return the cash to the shareholders and let them invest it in financial assets.
“You can see the trade-off (Figure 5.1). The opportunity cost of taking the project is the return shareholders could have earned had they invested the funds on their own. When we discount the project’s cash flows by the expected rate of return on financial assets, we are measuring how much investors would be prepared to pay for your project.”
FIGURE 5.1 The firm can either keep and reinvest cash or return it to investors. (Arrows represent possible cash flows or transfers.) If cash is reinvested, the opportunity cost is the expected rate of return that shareholders could have obtained by investing in financial assets.
“But which financial assets?” Vegetron’s CFO queries. “The fact that investors expect only 12% on IBM stock does not mean that we should purchase Fly-by-Night Electronics if it offers 13%.”
Your reply: “The opportunity-cost concept makes sense only if assets of equivalent risk are compared. In general, you should identify financial assets that have the same risk as your project, estimate the expected rate of return on these assets, and use this rate as the opportunity cost.”
Net Present Value’s Competitors
When you advised the CFO to calculate the project’s NPV, you were in good company. These days 75% of firms always, or almost always, calculate net present value when deciding on investment projects. However, as you can see from Figure 5.2, NPV is not the only investment criterion that companies use, and firms often look at more than one measure of a project’s attractiveness.
FIGURE 5.2 Survey evidence on the percentage of CFOs who always, or almost always, use a particular technique for evaluating investment projects.
Source: Reprinted from J. R. Graham and C. R. Harvey, “The Theory and Practice of Finance:
Evidence from the Field,” Journal of Financial Economics 61 (2001), pp. 187–243, © 2001 with permission from Elsevier Science.
About three-quarters of firms calculate the project’s internal rate of return (or IRR); that is roughly the same proportion as use NPV. The IRR rule is a close relative of NPV and, when used properly, it will give the same answer. You therefore need to understand the IRR rule and how to take care when using it.
A large part of this chapter is concerned with explaining the IRR rule, but first we look at two other measures of a project’s attractiveness—the project’s payback and its book rate of return. As we will explain, both measures have obvious defects. Few companies rely on them to make their investment decisions, but they do use them as supplementary measures that may help to distinguish the marginal project from the no-brainer.
Later in the chapter we also come across one further investment measure, the profitability index.
Figure 5.2 shows that it is not often used, but you will find that there are circumstances in which this measure has some special advantages.
Three Points to Remember about NPV
As we look at these alternative criteria, it is worth keeping in mind the following key features of the net present value rule. First, the NPV rule recognizes that a dollar today is worth more than a dollar tomorrow, because the dollar today can be invested to start earning interest immediately. Any investment rule that does not recognize the time value of money cannot be sensible. Second, net present value depends solely on the forecasted cash flows from the project and the opportunity cost of capital. Any investment rule that is affected by the manager’s tastes, the company’s choice of