11.5Pitfalls in Using the Comparison Method

As the discussion below indicates, there are a number of pitfalls to watch for when

implementing the comparison approach.

Project Betas Are Not the Same as Firm Betas

Most firms use their own cost of capital as a discount rate for evaluating specific

investment projects. This could be appropriate, for example, if the analyst is estimat-

ing the value of one new Sears outlet: Each Sears store is largely a clone of the oth-

ers and its risk probably closely matches the overall risk of the Sears corporation,

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which is basically a collection of these cloned stores. Hence, the cost of capital for

Sears as a whole is probably a good discount rate for evaluating the profitability of

opening up a Sears store in a new location.

In most cases, however, using the firm’s cost of capital as the discount rate for a

new project is inappropriate. For example, new projects may have higher beta risk than

the firm’s mature projects. This would tend to occur if the project has more R&D invest-

ment in its early years, has more operating leverage, the ratio of fixed to variable costs,

or more options associated with it than the firm as a whole. Alternatively, a project may

be less risky than the firm’s existing projects or it may support a higher percentage of

debt than the firm as a whole.¹⁶This would argue for using a lower cost of capital for

the project than that experienced by the firm. Most importantly, however, and especially

in newer industries, a firm’s market value is determined both by its existing projects and

by the firm’s perceived ability to develop new profitable projects, even projects that are

not yet “on the drawing board.” There is no reason to believe that the discount rate for

a firm’s perceived ability should be the discount rate for any of its projects.

Growth Opportunities Are Usually the Source of High Betas

The previous subsection observed that when a firm’s value is largely generated by its

perceived ability to develop new profitable projects—more commonly referred to as

growth opportunitiesor growth options, one must be cautious about using it as a

comparison firm for a project, even one of its own. Consider Wal-Mart, for example.

The value of this firm’s assets can be regarded as the value of the existing Wal-Mart

outlets in addition to the value of any outlets that Wal-Mart may open in the future.

The option to open new stores is known as a growth option. Because growth options

tend to be most valuable in good times and have implicit leverage (as Chapter 8 noted

for call options in general), which tends to increase beta, they contain a great deal of

systematic risk. Hence, individual projects can differ in their risk from the firm as a

whole because they lack the growth options that are embedded in the firm’s stock price.

Sears, however, has been contracting in recent years, so the growth option is prob-

ably small, indicating that Sears’s cost of capital may be appropriate for valuing a Sears

store.¹⁷On the other hand, Wal-Mart, which has been opening new stores at a phe-

nomenal rate, may have most of its value generated by growth options. An individual

Wal-Mart store thus has assets that look quite different from the assets of the Wal-Mart

Corporation, which consist of the relatively low-risk existing stores and high-risk

growth options to open new stores. One would be exaggerating the systematic risk of

an individual Wal-Mart store by using the risk of Wal-Mart’s stock as the appropriate

comparison.

There is no good rule of thumb for adjusting the risk of comparison firms for

growth options. Usually, but not always, growing franchises like Starbucks, promising

biotech and internet firms such as Amgen or Yahoo!—which have high price-earnings

ratios—or firms with high ratios of market value of equity to book value of equity

have valuable growth options. However, the systematic risk of these growth options,

which is the risk that is relevant for discounting, depends on how strongly the growth

is tied to the health of the economy. The stronger the tie to the health of the economy,

the riskier the growth option.

16

See Chapter 13.

¹⁷One

must be cautious with this example because Sears also derives substantial profits from in-home

remodeling and maintenance, such as carpet cleaning and painting. Also, Sears is currently experiencing

a modest turnaround and may soon experience the same comparison problems as Wal-Mart.

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With a company like Amgen, whose success depends largely on successful clini-

cal trials and approval by the Food and Drug Administration, the growth options are

tied less directly to the health of the economy. For this reason, the beta of Amgen’s

growth options may be low. On the other hand, the potential advertising and subscrip-

tion revenues of Yahoo!, or the willingness of individuals to regularly pay more than

$4 for their morning coffee and pastry at Starbucks, are probably tied to the health of

the economy—implying relatively high betas for the growth options of Yahoo! and

Starbucks. Reaching conclusions with casual observations like this is deceiving, how-

ever. Despite the observations above, Amgen’s beta is quite high.

Amgen’s Beta and the Discount Rate forIts Projects: The Perils of

NegativeCashFlows

Amgen Corporation is a biotechnology company. As of spring 2001, it was selling three drugs

and had a market capitalization of about $70 billion. Each of Amgen’s projects requires sub-

stantial R&D that is not expected to generate profits for many years. Atypical project tends

to generate substantial negative cash flows in its initial years after inception and significant

positive cash flows in the far-distant future as the drug developed from the project’s R&D

efforts is sold. Although the positive cash flows depend on the success of early research and

clinical trials, assume for the moment that these cash flows are certain. In this case, the future

cash flows of any one of Amgen’s projects can be tracked by a short position in short-term

debt, which has a beta close to zero, and a long position in long-term default-free debt, such

as government-backed zero-coupon bonds with maturities from 10 to 30 years. Such long-

term bonds have positive but modest betas, as discussed earlier in this chapter. It is useful

to think of the negative cash flows that arise early in the life of the project as leverage gen-

erated by short-term debt and the positive cash flows, even though they are assumed to be

certain, as risky assets with a modest amount of beta risk. As a leveraged position in posi-

tive beta long-term debt, the combination of these long and short positions results in a beta

for the project’s value, and ultimately for the firm’s value, that is quite high.

As Amgen has matured in the last decade, its R&D efforts have started to bear fruit.

Amgen now generates $3 billion in revenue from its three drug products. The profit margin

on these products is very high because the heavy expense phase for these drugs is now past.

Amgen also has a host of drugs “in the pipeline.” These projects have passed through much

of their large negative cash flow stage and are now much closer to the prospect of generat-

ing significant profits. The positive cash flows of these “old” and “middle-aged” projects

will arrive soon or have already arrived. This implies that the values of their positive cash

flows,even if riskless, have lower betas than if they were expected in the distant future.

Moreover, these successful projects are leveraged less than a project just starting out (that

is, they do not have a series of many early years with substantial negative cash flows). This

also contributes to a much lower beta. As a result of mixing these more mature projects with

Amgen’s usual portfolio of start-up projects, there has been a steady drop in Amgen’s beta

over time. Amgen’s year 2001 beta was about 1.3, down from about 2 in the last decade.

Although Amgen’s beta, and hence the firm’s overall cost of capital, dropped consider-

ably during this time period, the risk of the projects it has been implementing has not

changed appreciably. Hence, the discount rate used to value Amgen projects should not have

changed. Indeed, the discount rates used to value Amgen projects may have nothing to do

with Amgen’s beta, which is tied to Amgen’s rate of return (a return made riskier because

of the leverage discussed above). Recall that the typical Amgen product has negative cash

flows in its early years, and positive cash flows far down the road. Although this creates a

high beta for the return on the value of projects in their early years, and consequently for

the return of Amgen stock, that value, like the equity in a leveraged firm, is the difference

between the value of the long-term positive cash flows and the implicit cost of financing

those positive cash flows (in the form of the present value of the costs represented by the

negative near-term cash flows). That difference is relatively small, but bears all the risk

from the high value long-term positive cash flows.

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Clearly, one cannot track both the expected negative cash flows (occurring early on) and

the expected positive cash flows (occurring later on) of a start-up Amgen project with the

same tracking portfolio. Atracking portfolio with a fixed expected rate of return cannot gen-

erate both a positive expected value at one horizon and a negative expected value at another

horizon. If separate tracking portfolios are used for the near-term and distant cash flows,

one with long positions in securities to track the positive cash flows, and the other with

short positions in securities to track the negative cash flows, it is likely that both tracking

portfolios would have modest and perhaps negligible betas. The negative early cash flows

of a typical Amgen project are largely predetermined. If they vary, it is because there is a

need for more or less R&D and testing of the product. Thus, the typical negative cash flow

is unlikely to have any systematic risk and has a tracking portfolio consisting of a short

positionin a short-term risk-free asset. The long-term cash flows that are forecasted to be

positive have positive betas, both because of their horizon (long-term bonds have betas of

around 0.2) and because the demand for drugs and drug reimbursement rates from HMOs

and the government are likely to be modestly tied to the health of the economy. However,

the betas of the typical long-term cash flow are probably small, say 0.5, as most of its risk

is due to the success of clinical trials and medical needs, not the economy. Such a tracking

portfolio would have positiveweights of 0.5 on both the market portfolio and a short-term

risk-free asset.

In conclusion, given the unsystematic nature of the risk associated with the cash flows

from most of Amgen’s projects, if one were to use a single discount rate for the cash flows

of its projects, it would have to be relatively low, even though Amgen’s beta as a firm is

relatively high, as is the beta of the valueof a typical Amgen project.

Multiperiod Risk-Adjusted Discount Rates

Virtually all projects have cash flows over multiple periods. Traditionally, multiperiod

valuation problems have been viewed as simple extensions of the one-period analysis.

The Approach Used by Practitioners.To value the multiperiod cash flow stream

from a project, practitioners typically use the following approach:

1.Estimate the equity beta from a comparison firm using historical data,

typically of weekly or monthly frequency. Usually, the comparison firm is the

firm doing the project.

2.Compute the expected return using the risk-expected return formula of choice

(CAPM or APT) with parameters estimated from historical data.

3.Adjust for leverage and taxes¹⁸to obtain a cost of capital.

4.Use the cost of capital as a single discount rate for each period in the way

that we used the risk-free rate (assuming a flat term structure) in Chapter 10

to discount multiperiod cash flows.

Example 11.4 illustrates this approach.

Example 11.4:Applying a One-Period Cost of Capital to MultiyearCash Flows

Example 11.2 identified three comparison firms for Marriott’s restaurant division and from

these estimated 10.55 percent per year as the cost of capital for Marriott.Assume that

Marriott’s restaurant division is expected to produce $5 million in cash flows at the end of

this year and that this number will grow by 5 percent per year forever.At what price is

Marriott willing to sell its restaurant division?

¹⁸See

Chapter 13.

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Answer:The present value, using a risk-adjusted discount rate of 10.55 percent per

year,is

$5 million(1.05)$5 million(1.05)²

$5 million

PV. . .

1.10551.1055²³

1.1055

Recognizing this as a growing perpetuity (see equation 9.10), the restaurant division’s value

would be

$5 million

PV$90.09 million

.1055 .05

Pitfalls in Using the PractitionerApproach.Using a single cost of capital to dis-

count each of the expected future cash flows—the approach taken in Example 11.4—

is popular with analysts because it is simpler than using a different discount rate for

each individual cash flow. However, in many cases a single cost of capital tends to mis-

value cash flows.

This can be true even when cash flows are riskless. Chapter 10 emphasized that

analysts should discount the cash flows in different years at different discount rates.

For example, if the rate on a default-free zero-coupon bond maturing in the year 2005

is 7 percent, then a certain cash flow occurring in 2005 must be discounted at a 7 per-

cent rate. Similarly, if the rate on a default-free zero-coupon bond maturing in 2010 is

9 percent, then cash flows occurring in 2010 also must be discounted at 9 percent.

Hence, if analysts used a single discount rate of 8 percent, the near-term cash flows

would be undervalued while the cash flows in the more distant future would be over-

valued.¹⁹Misvaluation will be extreme if some future cash flows are negative and oth-

ers are positive, as the Amgen case aptly illustrated.

The same lesson applies to risky projects. In seeking a comparison firm for the

cash flows of its restaurant division, Marriott can use the 10.55 percent expected rate

of return of comparison stocks only if the cash flows of its restaurant division, at each

horizon, have expected values and market (or factor) risk identical to those of the track-

ing portfolio. For example, if the expected cash flows of Church’s Chicken and Wendy’s

have faster growth rates than the cash flows of Marriott’s restaurant division, then they

represent an inappropriate comparison and should not be used in the tracking portfolio.

Long-Term Risk-Free Rate orShort-Term Risk-Free Rate?The expected rate of

return of the firms comparable to Marriott’s restaurant division—McDonald’s,

Wendy’s, and Church’s Chicken—are based on risk-expected return models like the

CAPM and APT. Aportfolio of these comparison firms is assumed to track the Mar-

riott restaurant cash flows. In turn, the cost of capital for a given comparison firm is a

statement that the firm is tracked by a weighted average of the risk-free asset and the

market portfolio (or factor portfolios). For these risk-expected return formulas, there is

no theoretical reason to select a short-term risk-free rate over a long-term risk-free

rate, or vice versa.Moreover, the decision about which horizon to use for the risk-free

return in these formulas is not at all tied to the horizon of the cash flow one is trying

to value.

¹⁹With

a riskless cash flow stream, it is possible that all of the overvaluations and undervaluations

would cancel out if the single 8 percent discount rate were used. However, this would only be true if

(1)the project’s cash flows were identical to those of the tracking portfolio and (2) the internal rate of

return of the tracking portfolio was 8 percent.

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In general, the use of the long-term versus the short-term risk-free rate in the

CAPM or APTrisk-expected return relation depends on practical considerations, not

on the horizon of the cash flow. As an illustration, consider the valuation of a long-

horizon certain cash flow. It is very clear that the yield of a default-free zero-coupon

bond of matched horizon provides the appropriate rate for discounting the certain

cash flow. As Chapter 10 emphasized, this discounting is equivalent to finding a

tracking portfolio, comprised entirely of a default-free zero-coupon bond, that per-

fectly tracks the certain cash flow. For practical reasons,valuation with perfect

tracking is generally more accurate than valuation using the market portfolio or fac-

tor portfolios.

Thus, for a long horizon, the beta of the certain cash flow, measured over long

horizons, is zero, and the risk-free rate is the long-term riskless rate. However, if one

believes the CAPM is correct, it is also appropriate to track a long-horizon certain cash

flow with short-horizonriskless bonds and the market portfolio.

Since, over short intervals of time, the values of both the certain cash flow and the

market portfolio tend to decrease when expected inflation increases, and vice versa, the

certain cash flow is likely to have a positive beta when measured against the short-term

return of the market portfolio. Indeed, as this chapter noted earlier, a typical default-

free long-term zero coupon bond has a beta, measured over short horizons, of about .2.

With a market risk premium of 8 percent per year, this generates a reasonable forecast

of the typical 1.5 to 2 percent higher yield of default-free long-term bonds over short-

term bonds. Thus, the beta of .2 obtained from regressions using short-horizon returns

suggests that the long-horizon certain cash flow of the long-term bond, and by exten-

sion the long-term certain cash flow of any real asset, is tracked by a portfolio that is

20 percent invested in the market portfolio and 80 percent invested in a rollover posi-

tion in short-term riskless bonds.

Note that the beta of the cash flow of the long-term bond above depends on the

maturity of the debt used in the tracking portfolio. The bond cash flow’s short-horizon

CAPM beta of .2 identifies the best mix of the market portfolio and a short-term risk-

free asset that, with rollovers in the position, tracks the long-horizon certain cash flow.

However, the weight of .2 on the market portfolio and .8 on the short-term risk-free

asset changes to 0 and 1, respectively, when the market portfolio is combined with a

long-term risk-free asset.

While technically correct, the short-horizon CAPM-based method of valuing a risk-

less long-horizon cash flow has substantial tracking error. Because of the difference in

tracking error with the two approaches, it is preferable to use a long-horizon riskless

bond as the sole instrument in the tracking portfolio for a long-term riskless cash flow

and to avoid the CAPM-based approach altogether.

The same considerations apply when evaluating a risky long-horizon cash flow.

Whether it is better to include a long-term risk-free bond or a short-term risk-free bond

in the tracking portfolio depends on which bond generates a better tracking portfolio.

The better tracking portfolio is the one that generates the tracking error with a present

value that is closest to zero. The outcome of this horse race will be determined partly

by the amount of tracking error and partly by how closely the risk-expected return rela-

tion described by the tracking portfolio approximates reality.

In contrast with a riskless cash flow, the decision about using a long-term or a

short-term risk-free rate in valuing a risky cash flow that cannot be perfectly tracked

is less clear-cut. In this case, which of the two imperfect tracking portfolios to use—

market portfolio and short-term risk-free investment or market portfolio and long-term

risk-free investment (with a different mix of the two in each pairing)—is an empirical

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397

issue best left to the analyst’s judgment. Given our ambivalence on this topic, we are

fortunate to be able to present a compromise position: the two-factor APTallows the

tracking of a future cash flow with a short-term risk-free investment, a long-term risk-

free investment, and a proxy for the market portfolio—if these are indeed the appro-

priate factors. If including each one of these three financial assets improves the track-

ing ability of the overall tracking portfolio, then such a tracking portfolio would be

superior to one that leaves out either of the two risk-free investments.

Cash Flow Horizon and Beta Risk.The last subsection illustrated that the horizon

of the returns used in the tracking portfolio may affect the computed beta for a cash

flow of a given horizon. This subsection considers whether beta risk depends on the

horizon of the cash flow. In practice, financial analysts who implement the risk-adjusted

discount rate method almost always use the same beta for every cash flow in a cash

flow stream. Below, we argue that in many cases this practical shortcut leads to major

valuation errors.

The prices of comparison stocks represent the present values of the cash flow

streams. Unless the cash flow pattern of the comparison stream matches well with the

stream of cash flows being valued, the beta risk of comparison firms may not provide

an appropriate discount rate for the project.²⁰

Moreover, as we saw with Amgen, when

the cash flows of a project at some horizons are expected to be negative and those at

other horizons are expected be positive, comparison firm betas, even in the case of a

good match of the cash flow streams, tend to be too high for both the positive and the

negative cash flows.

Such cautions about using comparison firms apply doubly to individual cash flows.

Asingle cash flow, 10 years out, is unlikely to have the same beta risk as a firm in the

same line of business. The risk of the comparison firm depends on the risk of each of

its future cash flows. Hence, the beta of the comparison firm is, in essence, a blend of

the betas of each of the cash flows in its cash flow stream. These cash flows occur at

many different horizons.

There are no hard-and-fast rules for how betas vary with the cash flow horizon.

For some projects, the initial cash flows are relatively safe, but the cash flows

in the future depend much more on market returns. In this case, other things equal,

it is best to use a lower discount rate for the shorter-horizon cash flows. In other

cases, long-horizon cash flows tend to have less systematic risk than similar cash

flows of short horizons because many cash flows are highly correlated with the con-

temporaneous returns of traded securities and the health of the economy at the time

the cash flow is produced, but are not significantly correlated with cumulative past

returns.

For example, the cash flows of a brokerage firm such as Merrill Lynch are primar-

ily determined by same-year and prior-year transaction volume, which is highly corre-

lated with the market return. This would imply that the returns of the market over the

next 8 years probably have little impact on Merrill Lynch’s brokerage cash flows 10

years from now. As a consequence, Merrill Lynch’s year 10 cash flow should be dis-

counted back to year 8 at a risky rate of interest, and from year 8 to date 0 discounted

at a much lower rate of interest, perhaps even the risk-free rate.

²⁰This

remains an issue whether we believe that an appropriate tracking portfolio (that is, a tracking

portfolio with zero-PVtracking error) is composed of a long-horizon riskless bond and the market

portfolio (or factor portfolios) or whether it is composed of a short-horizon riskless bond rolled over

each period and the market portfolio (or factor portfolios).

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To understand this, consider the case where the risk-free rate over all horizons was

8 percent per year and the risky rate was 15 percent per year. The per year discount

rate for the year 10 cash flow should be the geometric mean of eight years of 8 per-

cent returns and two years of 15 percent returns or

[(1.08)⁸2^1/10 1

(1.15)]

More generally, the geometric meanof Treturns, ˜, r˜, . . . , r˜is

r

12T

[(1r

(1

. . .

(1)]^1/T 1

˜)r˜)r˜

12T

This is the short-horizon rate of return which, when compounded, gives the return over

the long horizon.

Competitive considerations also suggest that long-horizon cash flows contain rel-

atively little systematic (or factor) risk. When market returns are high, business is

often good. However, good times often encourage entry into the market by competi-

tors who can erode profits in subsequent years. Hence, if the economy is doing well

in year 5 and General Motors is selling a record number of Chevy Suburban sport

utility vehicles in that year, Ford, DaimlerChrysler, Toyota, and Mitsubishi also might

decide to expand their sport utility line to include an eight-seat, 8,000-pound vehicle

like the Suburban. By the time year 10 rolls around, the competitors’products will

be eroding GM’s profits in this line of vehicles. This means that although the year 5

market return is positively correlated with the year 5 cash flow for the Suburban proj-

ect, it may actually be negatively correlated with the year 10 cash flow. Hence, the

year 10 cash flow may not be highly sensitive to the cumulative 10-year return of

the market portfolio.

Empirical Failures of the CAPM and APT

The tracking portfolio metaphor applies even if one questions the validity of the CAPM

and APT. If, after reading the empirical evidence in Chapters 5 and 6, one believes that

market-to-book ratios and firm size are better determinants of a stock’s futureexpected

rate of return than market betas or factor betas, then the expected return of each com-

parison firm is identical to that of a portfolio of firms with similar firm size and market-

to-book ratios. In the case of Marriott, this means that the average historical returns of

firm size and market-to-book matched portfolios would be good estimates for the

expected returns of McDonald’s, Church’s Chicken, and Wendy’s.

In early 2001, for example, the McDonald’s Corporation had a market-to-book ratio

of about 4 and a market capitalization of about $35 billion. If the average historical

return of a portfolio of firms with this same market-to-book ratio and market capital-

ization is 15 percent per year, then the appropriate required return for projects with the

same risk as McDonald’s is 15 percent per year. Because this is only an estimate of

the project’s cost of capital, we may want to do the same computation for Church’s

Chicken and Wendy’s, averaging their required rates of return with McDonald’s 15 per-

cent per year estimate to obtain a more statistically precise estimate of the cost of cap-

ital for Marriott’s restaurant division.²¹

However, if one believes that the empirical evidence from the past is the result of

a psychological fad, a statistical accident, poor research, or some other anomaly that is

²¹For

simplicity, we have ignored the usual adjustments for leverage that need to be made.

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unlikely to repeat in the future, then one should be more cautious in discarding CAPM-

or APT-based expected return estimates.

What If No Comparable Line of Business Exists?

Using comparisons to identify beta risk is fine if portfolios of traded assets exist. How-

ever, for some real assets there is no suitable comparison line of business in which to

search for the components of a tracking portfolio. In October 1996, for example, NASA

announced a $7 billion deal with Lockheed Martin and Rockwell International, effec-

tively privatizing the operation of the space shuttle. To value such a deal either from

the perspective of NASAor the two companies, one would need to estimate beta risk,

yet no comparison firm to the NASAspace shuttle existed. In instances like this, beta

risk can be computed by estimating the project’s cash flows in each of many scenar-

ios. Each scenario is associated with a particular realization of the return of the tan-

gency portfolio.

Using Scenarios to Estimate Betas forthe Risk-Adjusted Discount Rate Method.

Example 11.5 illustrates one way to estimate betas with scenarios. As we shall shortly

see, there are some pitfalls to this approach.

Example 11.5:Estimating Betas with Scenarios

The Adonis Travel Agency wishes to estimate the present value of next year’s cash flow

from the purchase of 10 new airline reservation computers, at a cost of $10,000 per com-

puter.The new computers, which are faster than the current ones at the agency, are

expected to increase the number of reservations that each agent can handle.For sim-

plicity, assume that the additional cash flows associated with the increase in booking

capacity are all received one year from now.The size of the increase is tied to the state

of the economy.Over the next year, three possible economic scenarios, described in the

following table, are considered:

		Incremental
	Market	Cash Flow
Outcome	ProbabilityReturn (%)	in One Year	Return on Computers

	3			$150,000 $100,000
Recovery		25%	$150,000
				50%
	4			$100,000

	3			$35,000 $100,000
Recession		1	35,000	65%
	16			$100,000

	1			$5,000 $100,000
Depression		15	5,000	95%
	16			$100,000

What is the present value of the additional cash flow one year from now if the risk-free return

over the next year is 8.625 percent and the CAPM determines the expected returns of traded

securities?

Answer:The market portfolio’s expected return is

331

17.625%(25%)( 1%)( 15%)

41616

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The variance of the market return is

331

.17625)²²²

.017236(.25 ( .01 .17625)( .15 .17625)

41616

The expected return of the incremental cash flow is

331

19.375%(50%)( 65%)( 95%)

41616

The covariance of the cash flow return with the return of the market portfolio is

33

(.5 .19375)(.25 .17625)( .65 .19375) ( .01 .17625)

.0697

416

1

( .95 .19375)( .15 .17625)

16

This yields a return beta ( ) of

.0697

4.045

.017236

With a cost of $100,000 and a return of 19.375 percent, the expected cash flow, E(˜),is

C

$119,375 $100,000(1 .19375)

Thus, by the risk-adjusted discount rate formula

$119,375

PV$82,311

1.086254.045(.17625 .08625)

In Example 11.5, adoption of the new computer system is a negative NPVproj-

ect because the $100,000 cost exceeds the benefit of $82,311. These calculations were

based on beta estimates that generate the correct project adoption/rejection decision in

simple cases. However, these betas are not really correct. As discussed below, the true

beta in Example 11.5 is actually lower than the beta of the return of the project.

Why Betas of Returns Are Not the Correct Betas forthe Project.The beta

computed in Example 11.5 is not the true beta because the project returns used to

compute this beta is computed have a base price of $100,000 instead of a base price

equal to the project’s present value, which is lower than $100,000. The securities

market line relation between beta and expected return holds for the returns of finan-

cial assets, which are zero-NPVinvestments and, by extension, all real asset invest-

ments that are zero NPV.

Suppose that instead of computing beta with scenarios for project returns, we com-

puted scenarios for a portfolio of financial securities that perfectly track the cash flow

of the Adonis Travel Agency project. Example 11.5 resorted to the scenario computa-

tion of beta because a comparison firm, needed to generate a tracking portfolio, did not

exist. However, it is instructive to examine what the beta computation would be if a

comparison-based tracking portfolio did exist.

The Adonis Travel Agency project has a negative NPV,but the hypothetical

comparison-based tracking portfolio has a zero NPV.To indicate that the project has a

negative NPV,the cost of the tracking portfolio (that is, the project’s PV) has to be

smallerthan $100,000. Hence, in each of the three scenarios of Example 11.5, the track-

ing portfolio’s gross return, which is the return plus 100 percent, would be larger than

the gross return of the project.

Grinblatt814Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw814Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

Chapter 11

Investing in Risky Projects

401

EXHIBIT11.8

Gross Returns of the Project and Its Tracking Portfolio forExample 11.5

			Tracking Portfolio
Outcome	Probability	Project Return 100%	Return 100%

	3	$150,000	$150,000
Recovery	4	150% $100,000	PV
Recession	3 16	$35,000 35% $100,000	$35,000 PV
Depression	1 16	$5,000 5% $100,000	$5,000 PV

Exhibit 11.8, along with the observation in the Adonis example that C$100,000,

0

demonstrates that the gross return of the project in each of the three scenarios is

PV

(the gross return of the tracking portfolio)

C

0

Recall from Chapter 10 that PV/( C) is the profitability index. Since, in this exam-

0

ple, the profitability index is less than one, the gross returns of the hypothetical track-

ing portfolio are of larger scale, and thus have a larger beta, than the project’s gross

returns.²²

More generally, we have the following result:

Result 11.5

The betas of the actual returns of projects equal the project’s profitability index times theappropriate beta needed to compute the true present value of the project. Since the prof-itability index exceeds 1 for positive NPVprojects and is below 1 for negative NPVproj-ects, this error in beta computation does not affect project selection in the absence of proj-ect selection constraints.

One can take only momentary comfort in the fact that the erroneous beta and, con-

sequently, the erroneous present value computed from project returns do not alter the

decision about whether a real investment should be accepted or rejected. The discussion

below points out that incorrect capital allocation decisions will be made with mutually

exclusive projects if one does not make corrections for return betas of negative NPV

projects that are too low and those of positive NPVprojects that are too high.

Properties of the Correct Beta and Correct Present Value.How one computes a

correct beta and present value is not immediately obvious. For example, it is tempting

to think that one could do this for Example 11.5 by using the calculated PVof $82,311

as the base number for computing returns and their betas. However, since this PVis still

higher than the true PVof the project, it too is inappropriate as a base number. Also,

the profitability index that, at least in theory, could be used as a divisor to generate the

correct beta and thus the correct PVrequires that the analyst first know the correct PV.

While correct in theory, the adjustment implied by Result 11.5 obviously is impractical.

The correct PVhas the following property: If the analyst made a lucky guess and

selected the correct PVnumber, the returns (generated by using that PVas a base num-

ber) would have a beta and an associated discount rate from the CAPM or APTthat

22

We know from Chapter 4 that the covariance of a constant times the return of stock iwith the

return of stock jis the constant times the covariance of the returns of stocks iand j.

Grinblatt816Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw816Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

402Part IIIValuing Real Assets

would generate the original PVas the discounted expected future cash flow. Agen-

eral formula for identifying this PVappears in Section 11.6 on the certainty equiva-

lent method where the formula identifying the PVdoes not use a risk-adjusted dis-

count rate.

Mutually Exclusive Projects.If no comparison firms exist in the same line of busi-

ness, forcing the use of scenarios to compute the betas for the risk-adjusted discount

rate formula, Result 11.5 points out that the project’s return betas will be misesti-

mated by a factor equal to the project’s profitability index, PV/( C). This can lead

0

to the wrong choice among a set of mutually exclusive projects, as Example 11.6

demonstrates.

Example 11.6:Mutually Exclusive Projects_Pitfalls in Applying Risk-Adjusted

Discount Rates with Scenarios

The Adonis Travel Agency, discussed in Example 11.5, has a choice between two new soft-

ware reservation systems for its new computers:One is produced by United Airlines, the

other by American Airlines.Both new reservation systems have positive net present values

and thus profitability indexes above 1.The actual returns of United’s system have a beta of

1 while the actual returns of American’s system have a beta of 1.5.Both have an expected

incremental future cash flow of $40,000 one year from now.The cost of initiating United’s

system is $17,679.56 while that of American’s system is $16,555.43.Assume that the CAPM

holds, that the one-year risk-free rate is 8.625 percent, and that the one-year market risk

premium is 9 percent.John Adonis, the son of the owner and, more importantly, the gradu-

ate of a rather backward M.B.A.program that does not use this text, thinks that United’s

system has a higher net present value.He argues that the risk-adjusted discount rate for-

mula implies that the net present value of United’s system is about

$40,000

$16,327 $17,679.56

1.08625(1).09

The NPVof American’s system is

$40,000

$16,198 $16,555.43

1.08625(1.5).09

which is smaller.Is John Adonis correct in his analysis?

Answer:Suppose that the United reservation computer system has a profitability index

of 2.0 and the American system has a profitability index of 2.1.By Result 11.5, the betas

computed for the United and American systems are too high.The true betas are thus

¹

.5() for the United system and .7143 (1.5/2.1) for the American system.Hence, the

2

true NPVof the United system (assuming the CAPM holds) is about

$40,000

$17,680 $17,679.56

1.08625(.5).09

while that of American’s system is

$40,000

$18,211 $16,555.43

1.08625(.7143).09

Thus, in contrast to Mr.Adonis’s assertion, American’s reservation system has the higher

NPVif the profitability index assumptions are correct.However, note that the profitability

indexes of 2.0 and 2.1 are exactly the correct “guesses”because they are consistent with

what the true profitability indexes turned out to be.Thus, the profitability indexes of 2.0

and 2.1 for United and American respectively are not really assumptions, but rather truths

about the projects’profitabilities.Clearly, John Adonis is wrong and needs to read this

text.

Grinblatt820Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw820Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

Chapter 11

Investing in Risky Projects

403

Chapter 10 emphasized that mutually exclusive projects should be chosen on the

basis of the largest NPV,which is an arithmetic difference between the PVof the proj-

ect’s future cash flows and its initial cost. However, the way in which the PVs of pos-

itive NPVprojects are distorted—through the use of the project returns to compute

betas for the risk-adjusted discounted rate formula—is a rather complicated function

of the true PVand the project’s initial cost. The distortion is certainly not an arithmetic

difference. Hence, it should not be surprising that it is possible to construct examples

such as Example 11.6 where the naive manager may select the wrong project by using

the risk-adjusted discount rate method.