11.9Summary and Conclusions

This chapter analyzed the rules for computing the marketvalues of the future cash flows of risky investment proj-ects. Academics often recommend and practitioners imple-ment two equivalent discounted cash flow methods—therisk-adjusted discount rate method and the certainty equiv-alent method—to value future cash flows. As a simplepractical approach, we also recommend a particularimplementation—the risk-free scenario method—of thecertainty equivalent method.

The major theme of this chapter is that practical ratherthan theoretical considerations dictate which valuation ap-proach to use and how to implement it. The risk-adjusteddiscount rate method, which obtains the discount rate (thatis, the cost of capital) from commonly used theories of riskand return, such as the CAPM and APT, is impracticalwhen the betas of comparison firms are hard to estimate.Also, a variety of nuances require adjustments to the betaestimates. These adjustments can make this seeminglysimple valuation method extremely complicated. In caseswhere comparison firms do not exist and scenarios are re-quired to estimate risk, practical considerations dictate thatthe certainty equivalent method is the better valuation

method to use. Once the cash flow’s certainty equivalent isobtained, there are no further nuances and complications towatch out for. Hence, whenever observable forward pricesor internal estimation procedures lead to certainty equiva-lents, the certainty equivalent is the preferred valuationmethod.

Despite a thorough treatment of real asset valuation inthe last two chapters, our coverage of this important topicremains incomplete; a number of additional issues thathave a major impact on the capital allocation decision re-main. Chapter 12 studies the impact of growth options andother strategic options. It also explores an alternative valu-ation approach that is quite popular in a number of practi-cal settings: the ratio comparison approach. Chapters13–15 analyze financing and dividend policies and theirimpact on corporate tax liabilities in deciding betweenprojects. The effect of capital structure and dividend policyon incentives for choosing positive NPVprojects, as wellas bankruptcy costs, are studied in the latter half of Part IV.Managerial incentives and information asymmetries aredealt with in Part Vof the text.

²⁸Chapter

12 further examines how to identify present values from futures and forward prices.

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Key Concepts

Result	11.1:	Whenever a tracking portfolio for the	compute the true present value of the
		future cash flows of a project generates tracking error with zero systematic (or factor) risk and zero expected value, the market value of the tracking portfolio isthe present value of the project’s future cash flows.	project. Since the profitability index exceeds 1 for positive NPVprojects and is below 1 for negative NPVprojects, this error in beta computation does not affect project selection in the absence of project selection constraints.
Result Result	11.2:11.3:	To find the present value of next period’scash flow using the risk-adjusted discountrate method: (1) compute the expected ˜ future cash flow next period, E(C); (2)compute the beta of the return of theproject, ; (3) compute the expectedreturn of the project by substituting the beta calculated in step 2 into the tangencyportfolio risk-expected return equation;(4)divide the expected future cash flow instep 1 by one plus the expected returnfrom step 3. In algebraic terms E(˜ C) PV 1r (R r) fTf Increasing the firm’s debt (raising Dandreducing E) increases the (beta and standard deviation) risk per dollar ofequity investment. It will increase linearlyin the D/Eratio if the debt is risk free.	Result 11.6:To obtain a certainty equivalent, subtract the product of the cash flow beta and the tangency portfolio risk premium from the expected cash flow; that is ˜C˜ CE(C)E() b(R r) Tf where cov(˜ C, R) T b 2 T Result 11.7:(The certainty equivalent present value formula.) PV, the present value of next period’s cash flow, can be found by (1)computingE(˜)the expected future C cash flow and the beta of the future cash flow, (2) subtracting the product of this beta and the risk premium of the tangency portfolio from the expected future cash flow, and (3) dividing by (1the risk- free return); that is
Result	11.4:	The cost of equity rr(D/E) (r r) EAAD	˜) b(R r) E(C PVTf 1r f

Result 11.8:(Estimating the certainty equivalent with a

increases as the firm’s leverage ratio D/E

risk-free scenario.)If it is possible to

increases. It will increase linearly in the

estimate the expected future cash flow of

ratio D/Eif the debt is default free and if

an investment or project under a scenario

r,the expected return of the firm’s

A

where all securities are expected to

assets, does not change as the leverage

appreciate at the risk-free return, then the

ratio increases.

present value of the cash flow is computedResult 11.5:The betas of the actual returns of projects

by discounting the expected cash flow for

equal the project’s profitability index

the risk-free scenario at the risk-free rate.

times the appropriate beta needed to

Key Terms

book return on equity389	geometric mean398
capital structure382	Gordon Growth Model388
cash flow beta404	gross return400
certainty equivalent372	growth opportunities (growth options)392
certainty equivalent method372	leverage ratio382
comparison approach378	operating leverage392
conditional expected cash flows409	plowback ratio389
cost of debt383	plowback ratio formula389
cost of equity383	risk-adjusted discount rate method371
expected cash flow371	risk-free scenario method408

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416Part IIIValuing Real Assets

Exercises

11.1.Aproject has an expected cash flow of $1 million

one year from now. The standard deviation of this

cash flow is $250,000. If the expected return of the

market portfolio is 10 percent, the risk-free rate is

5 percent, the standard deviation of the market

return is 5 percent, and the correlation between

this future cash flow and the return on the market

is .5, what is the present value of the cash flow?

Assume the CAPM holds. Hint:Use the certainty

equivalent method.

Exercises 11.2–11.6 make use of the following information.

Assume that Marriott’s restaurant division has the

following joint distribution with the market return:

Year 1

Restaurant

Market MarketCash Flow

ScenarioProbabilityReturn (%)Forecast

a.Compute the percentage increase in the value

of equity if the firm is financed with $50

million in debt.

b.Compute the leverage ratio of this firm in 2002.11.9.Explain intuitively why the certainty equivalent of

a cash flow with a negative beta exceeds the cash

flow’s expected value.

Exercises 11.10–11.14 make use of the following data.

In 1989, General Motors (GM) was evaluating the

acquisition of Hughes Aircraft Corporation.

Recognizing that the appropriate discount rate for

the projected cash flows of Hughes was different

than its own cost of capital, GM assumed that

Hughes had approximately the same risk as

Lockheed or Northrop, which had low-risk

defense contracts and products that were similar to

Hughes. Specifically, assume the following inputs:

				Comparison		D/E
Bad	.25	15	$40 million		E

Good	.50	5	$50 million
				GM	1.20	.40
Great	.25	25	$60 million
				Lockheed	0.90	.90

Northrop

0.85

.70

		D
	Assume also that the CAPM holds.	Target for Hughes’s acquisition1
		E
11.2.	Compute the expected year 1 restaurant cash flow
		Hughes’s expected cash flow next year
	for Marriott.
		$300 million
11.3.	Find the covariance of the cash flow with the
		Growth rate of Hughes’s cash flows
	market return and its cash flow beta.
		5 percent per year
11.4.	Assuming that historical data suggests that the
		Appropriate discount rate on debt (riskless
	market risk premium is 8.4 percent per year and
		rate)8 percent
	the market standard deviation is 40 percent per
		Expected return of the tangency portfolio
	year, find the certainty equivalent of the year 1
		14 percent
	cash flow. What are the advantages and
	disadvantages of using such historical data for
	market inputs as opposed to inputs from a set of
		11.10.Analyze the Hughes acquisition (which took
	scenarios, like those given in the table above
		place) by first computing the betas of the
	exercise 11.2?
		comparison firms, Lockheed and Northrop, as if
11.5.	Discount your answer in exercise 11.4 at a risk-	they were all equity financed. Assume no taxes.
	free rate of 4 percent per year to obtain the present
		11.11.Compute the beta of the assets of the Hughes
	value.
		acquisition, assuming no taxes, by taking the
11.6.	Explain why the answer to exercise 11.5 differs	average of the asset betas of Lockheed and
	from the answer in Example 11.2.	Northrop.
11.7.	Start with the risk-adjusted discount rate formula.	11.12.Compute the cost of capital for the Hughes
	Derive the certainty equivalent formula by	acquisition, assuming no taxes.
	rearranging terms and noting that b PV.
		11.13.Compute the value of Hughes with the cost of
11.8.	In Section 11.3’s illustration, asset values	capital estimated in exercise 11.12.
	increased 10 percent from 2001 to 2002, from
	$100 million to $110 million.

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Chapter 11

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417

11.14.	Compute the value of Hughes if GM’s cost of	b.What is the present value of an expected $1
	capital is used as a discount rate instead of the cost	million HSCC cash flow one year from now,
	of capital computed from the comparison firms.	assuming that Dell is the appropriate
11.15.	In a two-factor APTmodel, Dell Computer has a	comparison? Assume no taxes and no need for
	factor beta of 1.15 on the first factor portfolio,	leverage adjustments.
	which is highly correlated with the change in	c.What is the cash flow beta and the certainty
	GDP, and a factor beta of .3 on the second factor	equivalent for the HSCC project?
	portfolio, which is highly correlated with interest	11.16.Risk-free rates at horizons of one year, two years,
	rate changes. If the risk-free rate is 5 percent per	and three years are 6.00 percent per year, 6.25
	year, the first factor portfolio has a risk-premium	percent per year, and 6.75 percent per year,
	of 2 percent per year and the second has a risk	respectively. The manager of the space shuttle at
	premium of .5 percent per year,	Rockwell International forecasts respective cash
	a.Compute the cost of capital for the HSCC	flows of $200 million, $250 million, and $300
	project that uses Dell as the appropriate	million for these three years under the risk-free
	comparison firm. Assume no taxes and no need	scenario. Value each of these cash flows
	for leverage adjustments.	separately.

References and Additional Readings

Brennan, Michael. “The Term Structure of Discount

Rates.” Financial Management26 (Spring 1997),

pp.81–90.

Copeland, Tom; Tim Koller; and Jack Murrin. Valuation:

Measuring and Managing the Value of Companies.

New York: John Wiley, 1994.

Cornell, Bradford. Corporate Valuation: Tools for

Effective Appraisal and Decision Making.Burr

Ridge, IL: Business One Irwin, 1993.

———. “Risk, Duration, and Capital Budgeting: New

Evidence on Some Old Questions.” Journal of

Business72, no. 2 (April 1999), pp. 183–200.

Cornell, Bradford, and Simon Cheng. “Using the DCF

Approach to Analyze Cross-Sectional Variation in

Expected Returns.” Working paper, University of

California, Los Angeles, 1995.

Damodoran, Aswath. Investment Valuation.New York:

John Wiley, 1996.

Elton, Edwin; Martin Gruber; and Jiangping Mei. “Cost

of Capital Using Arbitrage Pricing Theory: ACase

Study of Nine New York Utilities.” Financial

Markets, Institutions, and Instruments3, no. 3

(1994), pp. 46–73.

Gordon, Myron. The Investment Financing and Valuation

of the Corporation.Burr Ridge, IL: Richard D.

Irwin, 1962.

Harris, Robert. “Using Analysts’Growth Forecasts to

Estimate Shareholder Required Rate of Return.”

Financial Management15, no. 1 (1986), pp. 58–67.

Rappaport, Alfred. Creating Shareholder Value: The New

Standard for Business Performance.New York: Free

Press, 1986.

Ross, Stephen A. “Mutual Fund Separation in Financial

Theory—The Separating Distributions.” Journal of

Economic Theory17, no. 2 (1978), pp. 254–86.Ruback, Richard. “Marriott Corporation: The Cost of

Capital.” Harvard Case Study 289-047. In Case

Problems in Finance,William Fruhan et al., eds.

Burr Ridge, IL: Richard D. Irwin, 1992.

Rubinstein, Mark. “AMean-Variance Synthesis of

Corporate Financial Theory.” Journal of Finance28,

no. 1 (1973), pp. 167–181.

Shapiro, Alan. “Creating Shareholder Value.” Working

paper, University of Southern California, 1995.

Grinblatt850Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw850Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

418Part IIIValuing Real Assets

APPENDIX11A

STATISTICALISSUESINESTIMATINGTHECOSTOFCAPITALFORTHE

RISK-ADJUSTEDDISCOUNTRATEMETHOD

The implementation of the risk-adjusted discount rate method uses an estimate of the cost of

capital. Error in the cost of capital estimate can arise from several sources, including:

•	Having the wrong comparison firm (or portfolio) for computing beta.
••	Using historical data to estimate beta, which does not estimate beta perfectly. Adjusting for leverage with estimated leverage ratios instead of true leverage ratios.1
•	Knowing that inherent flaws are in the model of how to adjust equity risk for leverage(for reasons discussed in Chapter 12 and Part IVof the text).
•	Having an improper model of how risk relates to return.

The mere fact that errors exist in the cost of capital estimate means that the process of esti-

mation itself leads the firm to reject good projects and to accept bad projects. In the presence

of such cost of capital estimation error, it would be desirable to have a valuation procedure that

leads to an unbiased estimate of the present value, implying that the expected NPVof an esti-

mated positive NPVproject is still positive and that the expected NPVof a negative NPVproj-

ect is negative. However, an estimation procedure that yields unbiased estimates of the cost of

capital is a procedure that generates biased present values.