11A.2Geometric versus Arithmetic Means and the Compounding-Based Bias

The previous subsection demonstrated that, on average, estimated present values tend to be

higher than true present values. To correct for this bias, one should increase the estimate of the

cost of capital. In a multiperiod setting, estimated present values tend to be too low rather than

too high. This occurs because, as Section 11.5 noted, the financial manager generally compounds

an imprecise cost of capital to discount a long-horizon cash flow in a multiperiod setting. The

size of this overestimate depends on the number of times the estimated discount rate is com-

pounded. The bias also depends on the size of the estimation error.

The type of bias described in this subsection does not arise if the discount rate is not com-

pounded. There is no compounding of the discount rate whenever the cash flow has the same

horizon as the returns used to estimate beta. Hence, in Example 11A.1, a financial manager using

six-month returns as the basis for estimating the cost of capital for Marriott’s restaurant division

needs only to adjust the cost of capital upward for the denominator-based bias described in the

last subsection. The opposite bias, from compounding the cost of capital, does not exist in this

case.

However (see Example 11A.1), if Marriott’s financial manager uses monthly returns to esti-

mate the monthly cost of capital as .8393 percent per month (1.1055^1/12

1), a present value

would be obtained by first compounding the monthly rate six times (that is, 1.008393⁶) and then

discounting the six-month expected cash flow E(˜)with the formula

C

E˜˜)

(C)E(C

PV

1.008393⁶1.1055^1/2

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420Part IIIValuing Real Assets

In this case, if the compounding-based bias just balances or is larger than the opposite, denom-

inator-based bias, there may be no need to adjust the cost of capital at all or, possibly, the cost

of capital may require a downward adjustment to correct for the stronger bias generated by com-

pounding.

To understand the rough magnitude of the compounding-based bias, assume that expected

returns do not change with the horizon of the cash flow; for example, if the expected one-year

return on an investment is 10 percent per year, its expected 10-year return is 1.1¹⁰ 1. Although

this seems to suggest that it is sufficient to collect data that estimates annual mean returns and

to adjust the mean for the appropriate horizon with the standard compound interest formula, it

would be the wrong thing to do. Why is this the case?

One statistical estimate of the annual mean return is the arithmetic sample mean, which

looks at annual returns and then averages them. Exhibit 11A.1 illustrates the arithmetic mean

returns for the S&P500 Index, a portfolio of stocks, over four decades: the 1960s, 1970s, 1980s,

and 1990s. These are averages of the 10 yearly returns in each decade. Below these average

annual returns are the compounded 10-year returns computed from the averages. Note that the

actual 10-year return, directly below the 10-year compounded return, is always less than the

compounded return computed from the arithmetic averages.

Ageneral property of the arithmetic sample mean is that, when compounded, it always over-

estimates the true return. Hence, while the arithmetic sample mean is a good estimate of the

one-period mean return, errors in the estimate are greatly magnified when the analyst compounds

the sample mean to produce an estimate of the long-horizon mean. Alittle algebra illustrates

this bias with greater generality. Suppose that an asset has returns in two consecutive years

denoted as ˜and r˜. One dollar invested over two years in the asset would earn

r

12

˜r˜r˜ r˜r˜r˜r˜ r˜

r

(1r˜)(1r˜)1¹²¹21^{12 1}2

1²22 22

22

˜r˜r˜ r˜

r

1¹2 12

22

˜r˜

r

Taking the one-period sample mean, ¹2

,and compounding it over two periods indicates

2

that one dollar earns

2

˜r˜

r

1¹2

2

2

˜ r˜

r

which exceeds the actual amount earned over the two periods by 12.This overesti-

2

mate of the long-run return occurs because any positive deviations of the sample mean from the

true mean are exacerbated more than negative deviations as a result of compounding. Thus, it

may be important to reduce the estimated cost of capital to eliminate the bias.

The cost of capital adjustment that achieves unbiased present values depends on the frequency

with which means are estimated, the length of time over which data are available, and the hori-

zon of the cash flow. If the horizon of the cash flow is fairly long, it may be better to estimate

the per period mean as the geometric mean (discussed earlier in this chapter), instead of the

arithmetic mean. For short-horizon returns, the arithmetic mean may provide a better estimate.

Blume (1974) argued that there is an appropriate weighting of the arithmetic mean and the geo-

metric mean that provides the correct estimate of the long-run mean return on a comparison

investment. The following considerations determine the weights:

•	If pairs of consecutive returns are negatively correlated, more weight should be placedon the geometric mean.
•	If a great number of small-return horizons are averaged to obtain the arithmetic mean(for example, a day, a week), more weight should be placed on the geometric mean.
•	If the horizon of the cash flow is long term, more weight should be placed on thegeometric mean.

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Chapter 11

Investing in Risky Projects

421

EXHIBIT11A.1The Bias in Compounded Average Returns

Ending
Digit	S&P500 Returns (%)
of Year
in Decade1960s	1970s1980s	1990s

0	0.47%	4.01%	32.42%	3.17%
1	26.89	14.31	4.91	30.55
2	8.73	18.98	21.41	7.67
3	22.80	14.66	22.51	9.99
4	16.48	26.47	6.27	1.31
5	12.45	37.20	32.16	37.43
6	10.06	23.84	18.47	23.07
7	23.98	7.18	5.23	33.36
8	11.06	6.56	16.81	28.58
9	8.50	18.44	31.49	21.04

Average annual return (%)

8.68

7.50

31.64

18.98

Average return compounded for 10
years (%)	129.96	106.16	431.55	568.66

Actual 10-year return (%)

112.07

76.68

403.53

532.41

Source: Ibbotson Associates, Bonds, Bills, Stocks, and Inflation: 2000 Yearbook.

These insights apply even when the short-horizon expected return is obtained with an equi-

librium model that relates expected return to beta risk. For example, the three considerations

concerning the appropriate weighting of arithmetic and geometric means still apply in estimat-

ing the expected return of the tangency portfolio. Moreover, because of error in estimating beta

along with the expected return of the tangency portfolio, the short-horizon cost of capital of a

stock is estimated with error. As a result, adjustments need to be made to the cost of capital to

correct for the compounding-based bias in present values.

The compounding-based bias is avoided only if one estimates returns by averaging (or esti-

mating a risk-return model like the CAPM), over periods of the same length as the cash flow

horizon. For example, if the horizon of the cash flow is five years, an estimate of the five-year

expected return of an investment, obtained by averaging the five-year returns during 1980–85,

1981–86, 1982–1987, and so on is not subject to the compounding-based bias. Hence, an unbi-

ased present value estimate of a cash flow five years out in this case requires an upward adjust-

ment to the cost-of-capital estimate to compensate for the denominator-based bias, but it would

not require an adjustment for the bias induced by compounding.

Key Terms

arithmetic sample mean420

References and Additional Readings

Blume, Marshall. “Unbiased Estimators of Long-RunIbbotson Associates. Bonds, Bills, Stocks, and Inflation:

Expected Rates of Return.” Journal of the American2000 Yearbook.Chicago: Ibbotson Associates, 2000.

Statistical Association69 (Sept. 1974), pp. 634–38.

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Strategy, Second Edition

CHAPTER

Allocating Capital and

12

Corporate Strategy

Learning Objectives

After reading this chapter, you should be able to:

1.Identify the sources of positive net present value.

2.Implement the real options approach to value projects, the options inherent in

mines and vacant land, and the options to wait or to expand a project. Know the

effect of these options on a firm’s choice to diversify and select different

production techniques from its competitors.

3.Use the ratio comparison approach to value real assets, and in the case of

price/earnings ratios, how to adjust the ratio to make appropriate comparisons

between firms with different leverage ratios.

4.Compare the virtues and pitfalls of the competitive analysis approach to evaluate

real investments and know how to apply it.

“My father and I started a cosmetic cream factory in the late 1940s. At the time, no

company could supply us with plastic caps of adequate quality for cream jars, so we

had to start a plastic business. Plastic caps alone were not sufficient to run the

plastic-molding plant, so we added combs, toothbrushes, and soap boxes. This plastic

business also led us to manufacture electric fan blades and telephone cases, which in

turn led us to manufacture electrical and electronic products and telecommunication

equipment. The plastics business also took us into oil refining which needed a

tanker-shipping company. The oil-refining company alone was paying an insurance

premium amounting to more than half the total revenue of the then largest insurance

company in Korea. Thus an insurance company was started. This natural step-by-

step evolution through related businesses resulted in the Lucky-Goldstar group as we

see it today. For the future, we will base our growth primarily on chemicals, energy,

and electronics. Our chemical business will continue to expand toward fine

chemicals and genetic engineering while the electronics business will grow in the

direction of semiconductor manufacturing, fiber optic telecommunications, and

eventually satellite telecommunications.”

Koo Cha-kyung, CEO, LG Group (Chaebol).

422

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Strategy, Second Edition

Chapter 12

Allocating Capital and Corporate Strategy

423

Chapters 10 and 11 examined the traditional discounted cash flow (DCF) method for

valuing real assets. DCFis useful in many cases, but it does not get to the heart of

how intimately linked capital allocation is to long-term corporate strategy. This chapter’s

opening vignette, a quotation from the CEO of the Lucky Star Chaebol conglomerate in

Korea, illustrates the way in which past investment projects generate future as well as

current opportunities—an important consideration that DCFrarely takes into account.

This chapter presents a variety of advanced valuation techniques that remedy some

of the deficiencies inherent in traditional DCF.The main emphasis will be on tech-

niques that emphasize the role that the adoption of projects plays in the overall long-

term strategy of a corporation. Among these advanced techniques are

•	Areal options approach, which refers to the application of the derivatives valuation methodology introduced in Chapters 7 and 8 to value real assets.
•	Aratio comparison approach, which values an investment at approximatelythe same ratio of value to a salient economic variable as an existing comparable investment for which the same ratio is observable.
•	Acompetitive analysis approach, which attributes positive net present valueto any project of a firm that can identify its competitive advantages and anegative NPVto any project where competitors have the advantages.

These valuation approaches are not, in any way, inconsistent with the traditional dis-

counted cash flow approach for valuing real investments. The real options approach

and the ratio comparison approach, however, both recognize that financial market infor-

mation can be useful for determining the expected cash flows of a project as well as

the appropriate discount rate and can thus provide more accurate estimates of value.

For example, the futures price for copper, used to value a copper mine with the real

options approach, provides information about the expected future profits of a copper

mine and can be used to compute the certainty equivalent of a copper mine’s cash flows.

Or the ratio of Compaq’s stock price to current earnings—used with the ratio com-

parison approach—may provide information about how the market assesses the future

potential of the personal computer business.

Some of these advanced techniques, particularly the real options approach, also

make an explicit attempt to value new opportunities typically missed in the direct cash

flow forecasts of traditional valuation methods. These opportunities, which arise as a

result of undertaking the project, are often referred to as strategic options.

The value of strategic options, and the real options approach used in their valua-

tion, is attracting considerable attention. The valuation technique has been discussed in

considerable detail in the financial press (see, for example, the BusinessWeekarticle,

“Exploiting Uncertainty: The ‘Real Options’Revolution in Decision-Making”) and has

been picked up by many of the leading management consulting firms. There are also

Web sites specifically devoted to the latest real options methodology (see, for exam-

ple, www.real-options.com).

In a few industries and in some strategic arenas, these advanced valuation tech-

niques have had a major impact on the way projects are evaluated. The real options

approach, for example, has taken the energy industry by storm, and the ratio compar-

ison approach has been common in the real estate industry for a while. In valuing poten-

tial acquisitions, the competitive analysis approach now plays a major role. For exam-

ple, the 1996 acquisition of Duracell by Gillette was based in part on the competitive

analysis approach.¹

¹This acquisition is discussed in Chapter 20.

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424	Part IIIValuing Real Assets

12.1	Sources of Positive Net Present Value

It is worthwhile to step back and ask where positive net present value (NPV) projects

come from. To a large extent, firms are in positions to generate positive net present

value projects because of situations arising from prior investments. For example, in the

early and mid 1990s, Microsoft was in the best position to profit from an investment

in a new operating system (Windows 95) and to compete with Netscape to become the

dominant Internet browser because it already had a large customer base owing to its

DOS and Windows 3.1 programs.

In valuing potential investment projects, most firms do not adequately recognize

that the adoption of an investment project generates future investment opportunities that

often are quite valuable. Firms typically evaluate projects by discounting only the cash

flows directly tied to the project under consideration, thereby underestimating the proj-

ect’s total value. However, the most successful corporations, like Microsoft, grow and

prosper as a result of new opportunities serendipitously arising from the company’s past

investment decisions.

Result 12.1

New opportunities for a firm often arise as a result of information and relationships devel-oped in its past investment projects. Therefore, firms should evaluate investment projectson the basis of their potential to generate valuable information and to develop importantrelationships, as well as on the basis of the direct cash flows they generate.

Unocal’s Yadana Project

Consider Unocal’s current investments in southeast Asia. Unocal, the California-based oil

company, has been doing business with both the Burmese and Thai governments for many

years. From the contacts developed in its various business ventures, the company learned

that Thailand’s rapidly growing economy had brought about a severe shortage of electrical

power in Bangkok, Thailand’s capital. Unocal also was aware of a substantial natural gas

field (the Yadana natural gas field) in Burma (Myanmar), which borders Thailand. Poten-

tially, the natural gas in Burma could be used to power an electrical generator in Bangkok,

but transporting the gas from Burma to Thailand would require a pipeline costing more than

$1 billion. Such a project would not only require a firm with the substantial financial

resources and expertise needed to build a pipeline, but also with the experience and con-

tacts required to deal with the officials of two governments.

Unocal was in an excellent position to exploit a profitable investment opportunity

because it had the expertise and connections developed from the company’s prior invest-

ments. In other words, Unocal ultimately was able to initiate a positive NPVpipeline proj-

ect because its prior investments generated advantages that its competitors lacked.

Sources of Competitive Advantage

In general, the ability to generate profits in a competitive market is due to the advan-

tages one firm has over its competitors.²

These competitive advantages arise for a vari-

etyof reasons. First, there may be barriers to entry, or obstacles, that preventcom-

petitionby other firms from eroding profits. Consider Merck, which has patents on a

large number of drugs, giving the company a temporary monopoly³on the production

and sale of these drugs which is enforced by the U.S. government and international

agreements. There also may be economies of scale, which arise when per unit

²For a detailed description of these advantages and a plethora of real-world illustrations, see Shapiro

(1985).

³As of 1996, drug patents were valid only for 17 years, but in reality, imitation drugs that carefully

avoid violating the letter of the law often appear within 5 to 10 years of a patent filing.

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Chapter 12

Allocating Capital and Corporate Strategy

425

production costs decline with the scale of production, thus making producers of large

quantities of a good more efficient than small producers. Economies of scale give firms

with a large market share a sustainable advantage. There also are economies of scope,

which arise when a certain product or service can be supplied more efficiently by a

firm that makes a related product. Economies of scope manifest themselves in the supe-

rior knowledge, marketing system, or production technology that are acquired only by

first producing a related product.

Economies of Scope, Discounted Cash Flow, and Options

Much of the discussion in this chapter focuses on economies of scope because they are

the most important source of competitive advantage for a firm. If some of the positive

cash flows that firms achieve from projects are positive only because of the economies

of scope generated by earlier investments, then these positive cash flows should have

been attributed to the earlier related investments. In evaluating the earlier investments,

however, we noted earlier that most firms unnecessarily limit their analysis to the

investment’s direct cash flows and rarely consider the indirect cash flows that might

subsequently follow.

An implication of this limited analysis is that the discounted cash flow method

leads to bad decisions. As currently implemented, DCFis biased against long-term proj-

ects because it ignores many aspects of long-term investment projects that cannot eas-

ily be quantified.⁴Managers who use the discounted cash flow method tend to focus

on what can easily be quantified, and thus tend to ignore the indirect cash flows.⁵

Take the hypothetical case of General Motors considering whether to engage in a

joint venture with Shougang Steel to produce automobiles in China. It is possible to

arrive at some estimate of the joint venture’s cash flows, but the working relationship

between a U.S. and Chinese firm may have other benefits that go beyond the cash flows

of any particular project. After establishing a working relationship with Shougang, Gen-

eral Motors might, for example, have the opportunity to engage in additional joint proj-

ects. Because these opportunities may subsequently prove to be valuable, they need to

be considered when calculating the cash flows of the original investment project. But

making a precise calculation of the value of subsequent opportunities is a heroic task.

This chapter develops a real options approach, based on the derivatives valuation

methodology, that at least can provide rules of thumb about when subsequent oppor-

tunities are likely to significantly enhance a project’s value, and may even provide

“ballpark quantitative estimates” of the degree of this enhancement.

Option Pricing Theory as a Tool for Quantifying Economies of Scope

Because subsequent opportunities will only be pursued further if they prove to be valu-

able, they are options that the firm possesses. Most investment projects include an

optionlike component. In addition to the option to pursue additional projects, financial

managers also need to recognize that the adoption of almost any project contains other

important options—among them, options to cancel, downsize, or expand the project.

The decision about exercising these types of options at a later date can be viewed as

4

Hayes and Abernathy (1980) present the details of this argument.

⁵It is wrong, however, to assert that the discounted cash flow method is flawed as a result of this

short-term focus because the method itself does not tell us to ignore indirect long-term cash flows that

stem from a project.

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426Part IIIValuing Real Assets

an investment project in the same way that GM’s potential future projects with

Shougang Steel are considered projects.

The analysis of whether to “exercise the option” and enter into these new projects

will depend on underlying economic variables such as the demand for the product, the

level of interest rates, the health of the economy, the success of competitors, the polit-

ical climate, and so forth. These underlying variables affect the values of many traded

securities, including the company’s own stock. Hence, if it is possible to model how

the project and other traded securities are affected by underlying economic variables,

it is possible to use real options techniques to value the indirect cash flows. Even in

cases where the modeling is very crude and one can obtain only a few guiding rules

of thumb about the nature of the strategic option, it is important to consider such options

when valuing an investment project. These options contribute to the value of any invest-

ment project in which management has flexibility in future implementation.