When You Cannot Use a Single Risk Adjusted Discount Rate for Long Lived Assets

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Sometimes you will encounter problems where risk does change as time passes, and the use of a single risk-adjusted discount rate will then get you into trouble. For example, later in the book we will look at how options are valued. Because an option's risk is continually changing, the certainty-equivalent method needs to be used.

Here is a disguised, simplified, and somewhat exaggerated version of an actual project proposal that one of the authors was asked to analyze. The scientists at Veg-etron have come up with an electric mop, and the firm is ready to go ahead with pilot production and test marketing. The preliminary phase will take one year and cost $125,000. Management feels that there is only a 50 percent chance that pilot production and market tests will be successful. If they are, then Vegetron will build a $1 million plant that would generate an expected annual cash flow in perpetuity of $250,000 a year after taxes. If they are not successful, the project will have to be dropped.

The expected cash flows (in thousands of dollars) are

Ci = 50% chance of -1,000 and 50% chance of 0 = .5( -1,000) + .5(0) = - 500 Ct for t = 2, 3, • • • = 50% chance of 250 and 50% chance of 0 = .5(250) + .5(0) = 125

Management has little experience with consumer products and considers this a project of extremely high risk.25 Therefore management discounts the cash flows at 25 percent, rather than at Vegetron's normal 10 percent standard:

500 " 125

This seems to show that the project is not worthwhile.

Management's analysis is open to criticism if the first year's experiment resolves a high proportion of the risk. If the test phase is a failure, then there's no risk at all—the project is certain to be worthless. If it is a success, there could well be only normal risk from then on. That means there is a 50 percent chance that in one year Vegetron will

25We will assume that they mean high market risk and that the difference between 25 and 10 percent is not a fudge factor introduced to offset optimistic cash-flow forecasts.

CHAPTER 9 Capital Budgeting and Risk have the opportunity to invest in a project of normal risk, for which the normal discount rate of 10 percent would be appropriate. Thus the firm has a 50 percent chance to invest $1 million in a project with a net present value of $1.5 million:

Success -

1000

Pilot production and market tests

Pilot production and market tests

Failure -

Thus we could view the project as offering an expected payoff of .5(1,500)+.5(0) = 750, or $750,000, at t = 1 on a $125,000 investment at t = 0. Of course, the certainty equivalent of the payoff is less than $750,000, but the difference would have to be very large to justify rejecting the project. For example, if the certainty equivalent is half the forecasted cash flow and the risk-free rate is 7 percent, the project is worth $225,500:

This is not bad for a $125,000 investment—and quite a change from the negative-NPV that management got by discounting all future cash flows at 25 percent.

In Chapter 8 we set out some basic principles for valuing risky assets. In this chapter we have shown you how to apply these principles to practical situations.

The problem is easiest when you believe that the project has the same market risk as the company's existing assets. In this case, the required return equals the required return on a portfolio of all the company's existing securities. This is called the company cost of capital.

Common sense tells us that the required return on any asset depends on its risk. In this chapter we have defined risk as beta and used the capital asset pricing model to calculate expected returns.

The most common way to estimate the beta of a stock is to figure out how the stock price has responded to market changes in the past. Of course, this will give you only an estimate of the stock's true beta. You may get a more reliable figure if you calculate an industry beta for a group of similar companies.

Suppose that you now have an estimate of the stock's beta. Can you plug that into the capital asset pricing model to find the company's cost of capital? No, the stock beta may reflect both business and financial risk. Whenever a company borrows money, it increases the beta (and the expected return) of its stock. Remember, the company cost of capital is the expected return on a portfolio of all the firm's securities, not just the common stock. You can calculate it by estimating the expected return on each of the securities and then taking a weighted average of these separate returns. Or you can calculate the beta of the portfolio of securities and then plug this asset beta into the capital asset pricing model.

The company cost of capital is the correct discount rate for projects that have the same risk as the company's existing business. Many firms, however, use the company cost of capital to discount the forecasted cash flows on all new projects. This

.summary

PART II Risk is a dangerous procedure. In principle, each project should be evaluated at its own opportunity cost of capital; the true cost of capital depends on the use to which the capital is put. If we wish to estimate the cost of capital for a particular project, it is project risk that counts. Of course the company cost of capital is fine as a discount rate for average-risk projects. It is also a useful starting point for estimating discount rates for safer or riskier projects.

These basic principles apply internationally, but of course there are complications. The risk of a stock or real asset may depend on who's investing. For example, a Swiss investor would calculate a lower beta for Merck than an investor in the United States. Conversely, the U.S. investor would calculate a lower beta for a Swiss pharmaceutical company than a Swiss investor. Both investors see lower risk abroad because of the less-than-perfect correlation between the two countries' markets.

If all investors held the world market portfolio, none of this would matter. But there is a strong home-country bias. Perhaps some investors stay at home because they regard foreign investment as risky. We suspect they confuse total risk with market risk. For example, we showed examples of countries with extremely volatile stock markets. Most of these markets were nevertheless low-beta investments for an investor holding the U.S. market. Again, the reason was low correlation between markets.

Then we turned to the problem of assessing project risk. We provided several clues for managers seeking project betas. First, avoid adding fudge factors to discount rates to offset worries about bad project outcomes. Adjust cash-flow forecasts to give due weight to bad outcomes as well as good; then ask whether the chance of bad outcomes adds to the project's market risk. Second, you can often identify the characteristics of a high- or low-beta project even when the project beta cannot be calculated directly. For example, you can try to figure out how much the cash flows are affected by the overall performance of the economy: Cyclical investments are generally high-beta investments. You can also look at the project's operating leverage: Fixed production charges work like fixed debt charges; that is, they increase beta.

There is one more fence to jump. Most projects produce cash flows for several years. Firms generally use the same risk-adjusted rate to discount each of these cash flows. When they do this, they are implicitly assuming that cumulative risk increases at a constant rate as you look further into the future. That assumption is usually reasonable. It is precisely true when the project's future beta will be constant, that is, when risk per period is constant.

But exceptions sometimes prove the rule. Be on the alert for projects where risk clearly does not increase steadily. In these cases, you should break the project into segments within which the same discount rate can be reasonably used. Or you should use the certainty-equivalent version of the DCF model, which allows separate risk adjustments to each period's cash flow.

FURTHER READING

There is a good review article by Rubinstein on the application of the capital asset pricing model to capital investment decisions:

M. E. Rubinstein: "A Mean-Variance Synthesis of Corporate Financial Theory," Journal of Finance, 28:167-182 (March 1973).

There have been a number of studies of the relationship between accounting data and beta. Many of these are reviewed in:

G. Foster: Financial Statement Analysis, 2nd ed., Prentice-Hall, Inc., Englewood Cliffs, N.J., 1986.

CHAPTER 9 Capital Budgeting and Risk

For some ideas on how one might break down the problem of estimating beta, see: W. F. Sharpe: "The Capital Asset Pricing Model: A 'Multi-Beta' Interpretation," in H. Levy and M. Sarnat (eds.), Financial Decision Making under Uncertainty, Academic Press, New York, 1977.

Fama and French present estimates of industry costs of equity capital from both the CAPM and APT

models. The difficulties in obtaining precise estimates are discussed in: E. F. Fama and K. R. French, "Industry Costs of Equity," Journal of Financial Economics, 43:153-193 (February 1997).

The assumptions required for use of risk-adjusted discount rates are discussed in: E. F. Fama: "Risk-Adjusted Discount Rates and Capital Budgeting under Uncertainty," Journal of Financial Economics, 5:3-24 (August 1977).

S. C. Myers and S. M. Turnbull: "Capital Budgeting and the Capital Asset Pricing Model: Good News and Bad News," Journal of Finance, 32:321-332 (May 1977).

1. Suppose a firm uses its company cost of capital to evaluate all projects. Will it underestimate or overestimate the value of high-risk projects?

2. "A stock's beta can be estimated by plotting past prices against the level of the market index and drawing the line of best fit. Beta is the slope of this line." True or false? Explain.

3. Look back to the top-right panel of Figure 9.2. What proportion of Dell's return was explained by market movements? What proportion was unique or diversifiable risk? How does the unique risk show up in the plot? What is the range of possible error in the beta estimate?

4. A company is financed 40 percent by risk-free debt. The interest rate is 10 percent, the expected market return is 18 percent, and the stock's beta is .5. What is the company cost of capital?

5. The total market value of the common stock of the Okefenokee Real Estate Company is $6 million, and the total value of its debt is $4 million. The treasurer estimates that the beta of the stock is currently 1.5 and that the expected risk premium on the market is 9 percent. The Treasury bill rate is 8 percent. Assume for simplicity that Okefenokee debt is risk-free.

a. What is the required return on Okefenokee stock?

b. What is the beta of the company's existing portfolio of assets?

c. Estimate the company cost of capital.

d. Estimate the discount rate for an expansion of the company's present business.

e. Suppose the company wants to diversify into the manufacture of rose-colored spectacles. The beta of unleveraged optical manufacturers is 1.2. Estimate the required return on Okefenokee's new venture.

6. Nero Violins has the following capital structure:

QUIZ

Total Market Value,

Security

Beta

$ millions

Debt

0

100

Preferred stock

.20

40

Common stock

1.20

200

a. What is the firm's asset beta (i.e., the beta of a portfolio of all the firm's securities)?

b. How would the asset beta change if Nero issued an additional $140 million of common stock and used the cash to repurchase all the debt and preferred stock?

c. Assume that the CAPM is correct. What discount rate should Nero set for investments that expand the scale of its operations without changing its asset beta? Assume a risk-free interest rate of 5 percent and a market risk premium of 6 percent.

PART II Risk

7. True or false?

a. Many foreign stock markets are much more volatile than the U.S. market.

b. The betas of most foreign stock markets (calculated relative to the U.S. market) are usually greater than 1.0.

c. Investors concentrate their holdings in their home countries. This means that companies domiciled in different countries may calculate different discount rates for the same project.

8. Which of these companies is likely to have the higher cost of capital?

a. A's sales force is paid a fixed annual rate; B's is paid on a commission basis.

b. C produces machine tools; D produces breakfast cereal.

9. Select the appropriate phrase from within each pair of brackets: "In calculating PV there are two ways to adjust for risk. One is to make a deduction from the expected cash flows. This is known as the [certainty-equivalent; risk-adjusted discount rate] method. It is usually written as PV = [CEQt/(1 + rf)1; CEQt/(1 + rm)t]. The certainty-equivalent cash flow, CEQt, is always [more than; less than] the forecasted risky cash flow. Another way to allow for risk is to discount the expected cash flows at a rate r. If we use the CAPM to calculate r, then r is [rf + prm; Tf + p(rm — rf); rm + p(rm — rf)]. This method is exact only if the ratio of the certainty-equivalent cash flow to the forecasted risky cash flow [is constant; declines at a constant rate; increases at a constant rate]. For the majority of projects, the use of a single discount rate, r, is probably a perfectly acceptable approximation."

10. A project has a forecasted cash flow of $110 in year 1 and $121 in year 2. The interest rate is 5 percent, the estimated risk premium on the market is 10 percent, and the project has a beta of .5. If you use a constant risk-adjusted discount rate, what is a. The PV of the project?

b. The certainty-equivalent cash flow in year 1 and year 2?

c. The ratio of the certainty-equivalent cash flows to the expected cash flows in years 1 and 2?

PRACTICE QUESTIONy

"The cost of capital always depends on the risk of the project being evaluated. Therefore the company cost of capital is useless." Do you agree?

Look again at the companies listed in Table 8.2. Monthly rates of return for most of these companies can be found on the Standard & Poor's Market Insight website (www.mhhe. com/edumarketinsight)—see the "Monthly Adjusted Prices" spreadsheet. This spreadsheet also shows monthly returns for the Standard & Poor's 500 market index. What percentage of the variance of each company's return is explained by the index? Use the Excel function RSQ, which calculates R2.

Pick at least five of the companies identified in Practice Question 2. The "Monthly Adjusted Prices" spreadsheets should contain about four years of monthly rates of return for the companies' stocks and for the Standard & Poor's 500 index.

a. Split the rates of return into two consecutive two-year periods. Calculate betas for each period using the Excel SLOPE function. How stable was each company's beta?

b. Suppose you had used these betas to estimate expected rates of return from the CAPM. Would your estimates have changed significantly from period to period?

c. You may find it interesting to repeat your analysis using weekly returns from the "Weekly Adjusted Prices" spreadsheets. This will give more than 100 weekly rates of return for each two-year period.

The following table shows estimates of the risk of two well-known British stocks during the five years ending July 2001:

Standard

Standard Error

Deviation

R 2

Beta

of Beta

British Petroleum (BP)

25

.25

.90

.17

British Airways

38

.25

1.37

.22

Brealey-Meyers: Principles of Corporate Finance, Seventh Edition

II. Risk

9. Capital Budgeting and Risk

© The McGraw-H Companies, 2003

CHAPTER 9 Capital Budgeting and Risk a. What proportion of each stock's risk was market risk, and what proportion was unique risk?

b. What is the variance of BP? What is the unique variance?

c. What is the confidence level on British Airways beta?

d. If the CAPM is correct, what is the expected return on British Airways? Assume a risk-free interest rate of 5 percent and an expected market return of 12 percent.

e. Suppose that next year the market provides a zero return. What return would you expect from British Airways?

5. Identify a sample of food companies on the Standard & Poor's Market Insight website

(www.mhhe.com/edumarketinsight). For example, you could try Campbell Soup (CPB),

General Mills (GIS), Kellogg (K), Kraft Foods (KFT), and Sara Lee (SLE).

a. Estimate beta and R2 for each company from the returns given on the "Monthly Adjusted Prices" spreadsheet. The Excel functions are SLOPE and RSQ.

b. Calculate an industry beta. Here is the best procedure: First calculate the monthly returns on an equally weighted portfolio of the stocks in your sample. Then calculate the industry beta using these portfolio returns. How does the R2 of this portfolio compare to the average R2 for the individual stocks?

c. Use the CAPM to calculate an average cost of equity (requity) for the food industry. Use current interest rates—take a look at footnote 8 in this chapter— and a reasonable estimate of the market risk premium.

6. Look again at the companies you chose for Practice Question 5.

a. Calculate the market-value debt ratio (D/V) for each company. Note that V = D + E, where equity value E is the product of price per share and number of shares outstanding. E is also called "market capitalization"—see the "Monthly Valuation Data" spreadsheet. To keep things simple, look only at long-term debt as reported on the most recent quarterly or annual balance sheet for each company.

b. Calculate the beta for each company's assets (Passets), using the betas estimated in Practice Question 5(a). Assume that pdebt = .15.

c. Calculate the company cost of capital for each company. Use the debt beta of .15 to estimate the cost of debt.

d. Calculate an industry cost of capital using your answer to question 5(c). Hint: What is the average debt ratio for your sample of food companies?

e. How would you use this food industry cost of capital in practice? Would you recommend that an individual food company, Campbell Soup, say, should use this industry rate to value its capital investment projects? Explain.

7. You are given the following information for Lorelei Motorwerke. Note: €300,000 means

300,000 euros.

Long-term debt outstanding: €300,000

Current yield to maturity (rdebt): 8%

Number of shares of common stock: 10,000

Book value per share: €25

Expected rate of return on stock (requity): 15%

a. Calculate Lorelei's company cost of capital. Ignore taxes.

b. How would requity and the cost of capital change if Lorelei's stock price fell to €25 due to declining profits? Business risk is unchanged.

8. Look again at Table 9.1. This time we will concentrate on Burlington Northern.

a. Calculate Burlington's cost of equity from the CAPM using its own beta estimate and the industry beta estimate. How different are your answers? Assume a risk-free rate of 3.5 percent and a market risk premium of 8 percent.

b. Can you be confident that Burlington's true beta is not the industry average?

c. Under what circumstances might you advise Burlington to calculate its cost of equity based on its own beta estimate?

Brealey-Meyers: Principles of Corporate Finance, Seventh Edition

II. Risk

9. Capital Budgeting and Risk

© The McGraw-H Companies, 2003

PART II Risk

d. Burlington's cost of debt was 6 percent and its debt-to-value ratio, D/V, was .40. What was Burlington's company cost of capital? Use the industry average beta.

Amalgamated Products has three operating divisions:

Division

Percentage of Firm Value

Food

50

Electronics

30

Chemicals

20

To estimate the cost of capital for each division, Amalgamated has identified the following three principal competitors:

Estimated Equity Beta

Debt/(Debt + Equity)

United Foods

.8

.3

General Electronics

1.6

.2

Associated Chemicals

1.2

.4

Assume these betas are accurate estimates and that the CAPM is correct.

a. Assuming that the debt of these firms is risk-free, estimate the asset beta for each of Amalgamated's divisions.

b. Amalgamated's ratio of debt to debt plus equity is .4. If your estimates of divisional betas are right, what is Amalgamated's equity beta?

c. Assume that the risk-free interest rate is 7 percent and that the expected return on the market index is 15 percent. Estimate the cost of capital for each of Amalgamated's divisions.

d. How much would your estimates of each division's cost of capital change if you assumed that debt has a beta of .2?

Look at Table 9.2. What would the four countries' betas be if the correlation coefficient for each was 0.5? Do the calculation and explain.

"Investors' home country bias is diminishing rapidly. Sooner or later most investors will hold the world market portfolio, or a close approximation to it." Suppose that statement is correct. What are the implications for evaluating foreign capital investment projects?

Consider the beta estimates for the country indexes shown in Table 9.2. Could this information be helpful to a U.S. company considering capital investment projects in these countries? Would a German company find this information useful? Explain.

Mom and Pop Groceries has just dispatched a year's supply of groceries to the government of the Central Antarctic Republic. Payment of $250,000 will be made one year hence after the shipment arrives by snow train. Unfortunately there is a good chance of a coup d'état, in which case the new government will not pay. Mom and Pop's controller therefore decides to discount the payment at 40 percent, rather than at the company's 12 percent cost of capital.

a. What's wrong with using a 40 percent rate to offset political risk?

b. How much is the $250,000 payment really worth if the odds of a coup d'état are 25 percent?

An oil company is drilling a series of new wells on the perimeter of a producing oil field. About 20 percent of the new wells will be dry holes. Even if a new well strikes oil, there is still uncertainty about the amount of oil produced: 40 percent of new wells which strike oil produce only 1,000 barrels a day; 60 percent produce 5,000 barrels per day. a. Forecast the annual cash revenues from a new perimeter well. Use a future oil price of $15 per barrel.

Brealey-Meyers: Principles of Corporate Finance, Seventh Edition

II. Risk

9. Capital Budgeting and Risk

© The McGraw-H Companies, 2003

CHAPTER 9 Capital Budgeting and Risk b. A geologist proposes to discount the cash flows of the new wells at 30 percent to offset the risk of dry holes. The oil company's normal cost of capital is 10 percent. Does this proposal make sense? Briefly explain why or why not.

15. Look back at project A in Section 9.6. Now assume that a. Expected cash flow is $150 per year for five years.

b. The risk-free rate of interest is 5 percent.

c. The market risk premium is 6 percent.

d. The estimated beta is 1.2.

Recalculate the certainty-equivalent cash flows, and show that the ratio of these certainty-equivalent cash flows to the risky cash flows declines by a constant proportion each year.

16. A project has the following forecasted cash flows:

Cash Flows, $ Thousands

C0

C1 C2

C3

-100

+40 +60

+50

The estimated project beta is 1.5. The market return rm is 16 percent, and the risk-free rate rf is 7 percent.

a. Estimate the opportunity cost of capital and the project's PV (using the same rate to discount each cash flow).

b. What are the certainty-equivalent cash flows in each year?

c. What is the ratio of the certainty-equivalent cash flow to the expected cash flow in each year?

d. Explain why this ratio declines.

The McGregor Whisky Company is proposing to market diet scotch. The product will first be test-marketed for two years in southern California at an initial cost of $500,000. This test launch is not expected to produce any profits but should reveal consumer preferences. There is a 60 percent chance that demand will be satisfactory. In this case McGregor will spend $5 million to launch the scotch nationwide and will receive an expected annual profit of $700,000 in perpetuity. If demand is not satisfactory, diet scotch will be withdrawn.

Once consumer preferences are known, the product will be subject to an average degree of risk, and, therefore, McGregor requires a return of 12 percent on its investment. However, the initial test-market phase is viewed as much riskier, and McGregor demands a return of 40 percent on this initial expenditure. What is the NPV of the diet scotch project?

1. Suppose you are valuing a future stream of high-risk (high-beta) cash outflows. High risk means a high discount rate. But the higher the discount rate, the less the present value. This seems to say that the higher the risk of cash outflows, the less you should worry about them! Can that be right? Should the sign of the cash flow affect the appropriate discount rate? Explain.

2. U.S. pharmaceutical companies have an average beta of about .8. These companies have very little debt financing, so the asset beta is also about .8. Yet a European investor would calculate a beta of much less than .8 relative to returns on European stock markets. (How do you explain this?) Now consider some possible implications.

a. Should German pharmaceutical companies move their R&D and production facilities to the United States?

b. Suppose the German company uses the CAPM to calculate a cost of capital of

9 percent for investments in the United States and 12 percent at home. As a result it plans to invest large amounts of its shareholders' money in the United States. But its shareholders have already demonstrated their home country bias. Should the German company respect its shareholders' preferences and also invest mostly at home?

CHALLENGE QUESTIONS

PART II Risk c. The German company can also buy shares of U.S. pharmaceutical companies. Suppose the expected rate of return in these shares is 13 percent, reflecting their beta of about 1.0 with respect to the U.S. market. Should the German company demand a 13 percent rate of return on investments in the United States?

3. An oil company executive is considering investing $10 million in one or both of two wells: Well 1 is expected to produce oil worth $3 million a year for 10 years; well 2 is expected to produce $2 million for 15 years. These are real (inflation-adjusted) cash flows.

The beta for producing wells is .9. The market risk premium is 8 percent, the nominal risk-free interest rate is 6 percent, and expected inflation is 4 percent.

The two wells are intended to develop a previously discovered oil field. Unfortunately there is still a 20 percent chance of a dry hole in each case. A dry hole means zero cash flows and a complete loss of the $10 million investment. Ignore taxes and make further assumptions as necessary.

a. What is the correct real discount rate for cash flows from developed wells?

b. The oil company executive proposes to add 20 percentage points to the real discount rate to offset the risk of a dry hole. Calculate the NPV of each well with this adjusted discount rate.

c. What do you say the NPVs of the two wells are?

d. Is there any single fudge factor that could be added to the discount rate for developed wells that would yield the correct NPV for both wells? Explain.

4. If you have access to "Data Analysis Tools" in Excel, use the "regression" functions to investigate the reliability of the betas estimated in Practice Questions 3 and 5 and the industry cost of capital calculated in question 6.

a. What are the standard errors of the betas from questions 3(a) and 3(c)? Given the standard errors, do you regard the different beta estimates obtained for each company as signficantly different? (Perhaps the differences are just "noise.") What would you propose as the most reliable forecast of beta for each company?

b. How reliable are the beta estimates from question 5(a)?

c. Compare the standard error of the industry beta from question 5(b) to the standard errors for individual-company betas. Given these standard errors, would you change or amend your answer to question 6(e)?

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Responses

  • Alberica
    How much is the $250,000 payment really worth if the odds of a coup d’etat are 25%.?
    8 years ago

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