11.4Implementing the Risk-Adjusted Discount Rate Formula with

Comparison Firms

Suppose that the CAPM is correct and that HSCC (from Example 11.1) has operations

similar to those of Dell Computer, a stock traded on Nasdaq. HSCC is a wholly owned

subsidiary of Novel, Inc., so it is not traded, but Dell is. In this case, it might be pos-

sible to use Dell’s beta—computed by regressing Dell’s past stock returns on the returns

of a market proxy—to determine beta and, thus, the expected return for HSCC. How-

ever, one needs to be cautious about drawing this connection because the way in which

a firm is financed can affect its equity beta, as indicated in Section 11.3. It is impor-

tant not only for Dell Computer and HSCC to have similar lines of business but also

for both to have similar leverage ratios.

The CAPM, the Comparison Method, and Adjusting for Leverage

If an acquisition (or project) and its comparison firm(s) are financed differently, it may

be possible to adjust the comparison firm’s beta for the difference in leverage ratios.

However, making this type of adjustment can be tricky, especially when one takes cor-

porate taxes into account. We will discuss beta adjustments in the absence of taxes

below, and will examine how these adjustments are affected by taxes in Chapter 13.

(In the absence of taxes, the weighted average cost of capitaland the unlevered cost

of capital,terms commonly used in the financial services industry, are identical to the

expected return on assets, r.)

A

An Illustration of the Necessary Leverage Adjustment without Taxes.Example

11.2 illustrates the simpler task of how to adjust for leverage differences in the absence

of taxes.

Example 11.2:Using the Comparison Approach to Obtain Beta and r

This example is based on a Harvard case about the Marriott Corporation,¹⁰

although the

data here are fictitious.In this case, Marriott has identified three comparison firms for its

¹⁰Ruback

(1992).

Grinblatt782Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw782Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

Chapter 11

Investing in Risky Projects

385

restaurant division.For these comparison firms, the table below provides CAPM-based

equity beta estimates ( ), book values of debt (D), and market values of equity (E).

E

	DE ($billions)($billions)
E

Church’s Chicken	.75	.004	.096
McDonald’s	1.00	2.300	7.700
Wendy’s	1.08	.210	.790

Assume that the risk-free rate is 4 percent per year, the risk premium on the market port-

folio is 8.4 percent per year, the CAPM holds, the debt of the comparison firms is risk free,

and all three firms provide equally good comparisons for Marriott’s restaurant division.Esti-

mate the cost of capital for Marriott’s restaurant division.

E

Answer:Using equation (11.2a), ,we first find the three firms’asset betas:

^ADE^E

A

	.096
Church’s Chicken	.72 .75
	.100

	7.7
McDonald’s	.77 1.00
	10

	.79
Wendy’s	.85 1.08
	1.0

	.72.77.85
Marriott (average of above)	.78
	3

Applying the CAPM risk-expected return equation, using the .78 estimate of Marriott’s restau-

rant asset beta gives the restaurant cost of capital, 10.55 percent per year:

.1055 .04 .78(.084)

In Example 11.2, , the equity beta, determines the cost of equity capital. Recall from

the last section that if is positive, this beta and the associated expected return on

the equity increase as the leverage of the firm increases. The cash flows that are dis-

counted by the risk-adjusted discount rate method are the cash flows from the project’s

assets, which do not have debt interest payments subtracted from them. Hence, it is

inappropriate to discount these cash flows at a rate used for discounting the cash flows

that belong to the leveraged equity of comparison firms.

By multiplying the betas by E/(DE), Example 11.2 identifies the beta of the

portfolio of debt and equity of the comparison firms (with 0). Since assets equal

D

debt plus equity, this portfolio beta is indeed the asset beta that generates an appropri-

ate discount rate for the cash flows of the assets.

As an alternative to the process of unlevering the betas described in Example 11.2,

one can compute the CAPM-based expected returns of the equity of the comparison

firms. Use the equation

Grinblatt784Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw784Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

386Part IIIValuing Real Assets

DE

r r r

^ADE^DDE^E

(obtained from the expected values of both sides of (equation 11.3a)) to unlever

the equity expected returns and obtain the expected asset return of the comparable

firms.

Weighting the Betas of Comparison Firms.Note that Example 11.2 averaged the

betas of the three comparison firms to estimate the beta of Marriott’s restaurant assets.

This averaging is appropriate if each comparison firm provides an equally valid esti-

mate of the Marriott restaurant asset beta. However, if some firms provide better com-

parisons than others, their betas should be weighted more than those of the less closely

matched firms. For example, if Church’s Chicken was a less appropriate match than

McDonald’s or Wendy’s, we might give the .72 beta of Church’s Chicken a lower

weight than the betas of McDonald’s or Wendy’s.

Obtaining a Cost of Capital from the Arbitrage Pricing Theory (APT)

Earlier, we learned that when the tangency portfolio is the market portfolio, the cost of

capital—that is, the discount rate in the denominator of the present value formula, equa-

tion (11.1)—is obtained from the Capital Asset Pricing Model. An alternative to the

CAPM is the Arbitrage Pricing Theory (APT), developed in Chapter 6, which is the

correct theory to use when a combination of factor portfolios, instead of the market

portfolio, is the tangency portfolio.

The MultifactorAPTVersion of the Risk-Adjusted Discount Rate Formula.

When computing costs of capital using the expected returns generated by the APT, the

present value of the project’s future cash flow is

E˜
(C)
PV	(11.4)
1r . . .
f1122KK

The project’s net present value is computed by subtracting the project’s initial cost from

this present value.

The discount rates provided by the APTgenerally differ from those of the CAPM.

Thus, they can generate different capital allocation decisions. The Marriott Corpora-

tion, for example, has an APT-based cost of capital of 9.3 percent, resulting in a denom-

inator of 1.093 for equation (11.4) and hence equation (11.1) as well.¹¹However, Mar-

riott’s CAPM-based cost of capital is 10 percent, resulting in a denominator of 1.1 for

equation (11.1). If the APTis correct and the CAPM is incorrect, Marriott would be

missing out on some good projects by using the higher discount rate from the CAPM.

As the following case study illustrates, such differences in the cost of capital between

the two models are not uncommon.

Arbitrage Pricing Theory versus Capital Asset Pricing Model

Alcar’s APT!, a consulting firm, provides its clients with the costs of equity capital and costs

of capital for a variety of firms, using both the CAPM and the APT. The Alcar version of

the APTis based on a five-factor model, where the five prespecified factors are changes in:

¹¹Based on third-quarter 1995 data for the Marriott Corporation.

Grinblatt786Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw786Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

Chapter 11

Investing in Risky Projects

387

EXHIBIT11.6Cost of Equity Capital

CAPMArbitrage Pricing Theory (APT)

		Equity	Equity	Premiums from Sensitivity to Five Factors
	Equity	Expected	Expected
Firm	Beta	Returns (%)	Returns (%)	SINFLINFINTPREMGDP

Coca Cola	1.03	11.42	12.61	0.83	1.25	1.39	0.95	1.22
Con Edison NY	0.60	9.57	10.41	0.52	0.76	0.86	0.56	0.74
CSX Corp.	1.24	12.34	11.89	0.57	1.13	1.24	0.82	1.15
Fed Nat Mtg Assn	1.52	13.56	11.79	0.39	1.16	1.31	0.69	1.26
Microsoft Corp.	1.05	11.53	8.95	0.03	0.54	0.61	0.22	0.64
Northrop	0.98	11.19	8.54	0.07	0.45	0.49	0.17	0.52

EXHIBIT11.7CAPM and APTCosts of Capital with Leverage Ratios (D/E) forSix Firms

	Debt to	CAPM Cost	APTCost	Difference between APT
	Equity Ratio	of Capital	of Capital	and CAPM Cost of
Firm	(%)	(%)	(%)	Capital (%)

Coca-Cola	6.36	11.02	12.14	1.12
Consolidated Edison	46.90	7.93	8.51	0.58
CSX Corp.	51.31	9.85	9.55	0.30
Federal National Mortgage Association	778.74	5.65	5.45	0.20

Microsoft Corp.

0.00

11.53

8.95

2.58

Northrop

27.50

10.12

8.04

2.08

•	Short-term inflation (SINF).
•	Long-term inflation (LINF).
•	The level of short-term interest rates (INT).
•	The premium for default risk (PREM).
•	The monthly Gross Domestic Product (GDP).

Exhibit 11.6 presents Alcar’s equity expected returns from both the CAPM and APT, as

well as the CAPM beta, for six well-known firms, as of the third quarter of 1995.

For each row, the sum of the numbers in the five right-hand columns in Exhibit 11.6

represent the APTrisk premiums. APTequity expected returns (the costs of equity capital)

are computed as the sum of these risk premiums plus the September 7, 1995 risk-free rate

of 6.98 percent.

With adjustments for risky debt and taxes, these numbers translate into Exhibit 11.7’s

comparative costs of capital for the typical existing project of these firms.¹²

The difference

in the cost of capital computed with the CAPM and APTin Exhibit 11.7 is as large as 2.58

percent per year, as you’ll note in the case of Microsoft (2.58% 11.53% 8.95%). Many

projects that are similar to the existing projects of Microsoft typically have large investments

¹²Adjustments

for taxes are discussed in Chapter 13.

Grinblatt788Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw788Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

388Part IIIValuing Real Assets

in the early years—primarily wages for programmers and expenses for advertising and pro-

motion—and substantial revenues from sales of software which are not likely to occur until

much later, perhaps 5 to 15 years after the initial investment. Therefore, adoption decisions

about any prospective project that resembles Microsoft’s existing collection of projects will

be greatly affected by whether one selects the CAPM or the APTto compute Microsoft’s

discount rate.

Costs of Capital Computed with Alternatives to CAPM and APT: Dividend Discount Models

The Capital Asset Pricing Model and the Arbitrage Pricing Theory are the two most

popular models for determining risk-adjusted discount rates. Although both models are

applied in practice, their applications have been criticized because of the difficulties

associated with estimating their essential inputs.

Impediments to Using the CAPM and APT.Specifically, the CAPM requires

knowledge of not only the covariance (or beta) of the return of an investment with the

return of the market portfolio, but also an estimate of the expected return of the mar-

ket portfolio. The APTrequires multiple factor sensitivities and the corresponding

expected returns on multiple factor portfolios.

The Dividend Discount Model.Anumber of financial analysts estimate re-

quired rates of return using analysts’forecasts of future earnings with a special case

of what Chapter 9 referred to as the dividend discount model.This special case,

where dividends grow at a constant rate, is sometimes known as the Gordon Growth

Modelbecause it was first developed by Gordon (1962). According to this model,

the equity of a firm with a dividend stream growing at a constant rate can be val-

ued as follows:

div
1
S	(11.5a)
0(r g)
E

where

Sthe firm’s current stock price per share

0

divthe expected dividend per share one year from now

1

rthe market required rate of return of the firm’s stock (its cost of equity

E

capital)

gthe expected growth rate of dividends

Equation (11.5a) is an application of the growing perpetuity formula developed in Chap-

ter 9. By rearranging this equation, one sees that the expected rate of return of a stock

can be expressed as the sum of the growth rate and the dividend yield

div
rg1	(11.5b)
ES
0

Using Analyst Forecasts to Estimate the Expected Dividend Growth Rate.To

compute the risk-adjusted discount rate for equity from this equation, only g,the

expected rate of growth of the firm’s dividends, and div/S, the firm’s dividend yield,

10

need to be estimated.¹³Analysts’forecasts of the growth rate of a firm’s earnings

¹³A

historical average of the ratio of dividend per share to prior year stock price per share, sometimes

over a period of five years, can be used if the coming year’s dividend payout is expected to be unusual.

Grinblatt790Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw790Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

Chapter 11

Investing in Risky Projects

389

provide one estimate for g.Under the assumption that a firm pays a fixed percentage

of its earnings as dividends, the expected growth rate in dividends equals the forecasted

growth rate in earnings. This growth rate can then be added to the existing dividend

yield to derive the expected return on the firm’s stock.¹⁴

For example, suppose that Value Line, an investment advisory service, forecasts a 12.9

percent rate of growth for IBM’s earnings. Adding this to the firm’s 0.5 percent dividend

yield (as of early 2001) implies an expected rate of return on IBM stock of 13.4 percent.

To obtain a cost of capital for an IBM-like project, it is necessary to adjust this 13.4 per-

cent rate of return for debt in IBM’s capital structure. For example, in the risk-free debt

no taxes case of Section 11.2, one can obtain rby multiplying rby E/(DE).

AE

Using the Plowback Ratio Formula to Estimate the Expected Dividend Growth

Rate.An alternative method for estimating g,the growth rate in dividends, employs

accounting data. This method estimates the growth rate as

	gb ROE	(11.6)
where

bthe plowback ratio, the fraction of earnings retained in the firm

ROEbook return on equity, that is, earnings divided by last year’s

(midyear) book equity¹⁵

The intuition for the plowback ratio formula, given in equation (11.6), is that the

book return on equity (ROE) represents the rate of growth of capital invested in the

firm. When a firm has an ROE of 10 percent, every $1 invested in the firm returns

$1.00 of equity capital and $0.10 of earnings next year, or $1.10. If this $1.10 is entirely

reinvested, it will grow another 10 percent to $1.21 one year later. However, if 75 per-

cent of the earnings are paid out in the form of dividends, implying a plowback ratio

of .25, the capital will only grow at a rate of (1 .75)($.10)—that is, at 25 percent of

the 10 percent growth rate, or 2.5 percent. In this case, at the end of the first year, 75

percent of $0.10 would be paid out in dividends, implying that only $1.025

[$1.10 .75($.10)] is left in the firm for reinvestment. This would grow to

$1.025(1.1), but if 75 percent of the amount over $1.025 (the earnings) is paid out as

a dividend, the amount to be reinvested is just $1.025(1.1) .75($1.025)(.1)$1.025²

.

Thus, paying out a fixed proportion of a company’s earnings as dividends slows the

growth rate of the funds available for reinvestment. Moreover, since earnings and div-

idends are a constant proportion of the reinvestment amount, their growth rates will be

the same as the growth rate of the funds available for investment in the firm.

Assumptions and Pitfalls of the Dividend Discount Model.The plowback ratio for-

mula, equation (11.6), uses the book return on equity in lieu of the return on new

investment, the return that theoretically should be used but which is more difficult to

measure accurately. If old assets and new assets have different returns, ROE in the

plowback ratio formula should be the book return of equity for new asset investment.

If the project is a positive NPVproject, the appropriate book return of equity will

exceed the project’s cost of capital.

¹⁴Note

that using the current dividend yield in equation (11.5b) gives the wrong answer. The formula

requires next year’s expected dividend in the numerator. Multiplying the current dividend per share by

1gand dividing by the current stock price gives the appropriate dividend yield estimate.

¹⁵Alternatively,

it is possible to use forecasts of next year’s earnings divided by this year’s (midyear)

book equity.

Grinblatt792Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw792Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

390Part IIIValuing Real Assets

The implicit assumptions of the dividend discount model’s estimate of the cost of

capital are that:

•	The earnings growth forecasts, whether from analysts or equation (11.6), are unbiased; that is, they do not tend to systematically underestimate or overestimate the earnings growth rate.
•	The earnings growth forecasts are based on the same information that investorsuse to value the firm’s stock.
•	The firm’s earnings and dividends grow at the same constant rate, forever.

To the extent that these assumptions are valid, the dividend discount model may pro-

vide a better estimate of the expected rate of return on a firm’s stock or project than

either the CAPM or the APTbecause it does not require estimates of beta or estimates

of the expected return of the market portfolio or of factor portfolios. However, these

assumptions, particularly that of a constant growth rate, are stringent and may not apply

to many of the firms or projects that an analyst wants to value.

What If No Pure Comparison Firm Exists?

Many firms are large diversified entities that have many lines of business. In this

instance, the equity returns of potential comparison firms are distorted by other lines

of business and cannot easily be used as comparison firms for projects that represent

only a single line of business. Unfortunately, in many situations, there is no appropri-

ate comparison firm with a single line of business. Afinancial manager in this situa-

tion still may be able to obtain an appropriate comparison by forming portfolios of

firms that generate a “pure” line of business. The mathematics behind the approach

taken in Example 11.3, which illustrates how to create comparison investments in a

pure line of business when none initially exists, is similar in spirit to the formation of

pure factor portfolios in Chapter 6.

Example 11.3:Finding a Comparison Firm from a Portfolio of Firms

Assume that AOL-Time Warner is interested in acquiring the ABC television network from

Disney.It has estimated the expected incremental future cash flows from acquiring ABC

and desires an appropriate beta in order to compute a discount rate to value those cash

flows.However, the two major networks that are most comparable, NBC and CBS, are

owned by General Electric and Viacom—respectively—which have substantial cash flows

from other sources.For these comparison firms, the table below presents hypothetical

equity betas, debt to asset ratios, and the ratios of the market values of the network

assets to all assets:

	D	Network AssetsN
	DE	All AssetsA
E

General Electric	1.1	.1	.25
Viacom	1.3	.4	.50

Estimate the appropriate beta for the ABC acquisition.Assume that the debt of each of the

two comparison firms is risk free.Also assume that the non-network assets of General Elec-

tric and Viacom are substantially similar and thus have the same beta.

Grinblatt794Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw794Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

Chapter 11

Investing in Risky Projects

391

Answer:Using equation (11.2a), [E( DE)] , first find the asset betas of the two

AE

comparison firms.For the two firms, these are, respectively,

A

General Electric.99(.9)(1.1)

Viacom.78(.6)(1.3)

Viewing the comparison firms as portfolios of network and non-network assets, and recog-

nizing that the beta of a portfolio is a portfolio-weighted average of the betas of the portfo-

lio components, implies the following equation

NA N

(network assets’ beta)

(non-network assets’ beta)

^AAA

For the two comparison firms this equation is represented as

General Electric:.99(.25) (.75)

NETWORKNON-NETWORK

Viacom: .78 (.5) (.5)

NETWORKNON-NETWORK

Multiplying both sides of the second equation (Viacom) by 1.5, subtracting it from the first

equation (General Electric), and solving for yields .36, which is used

NETWORKNETWORK

for the ABC acquisition.

The procedure used in Example 11.3 is based on the idea that portfolio betas are

portfolio-weighted averages of the betas of individual securities. If we view firms

with multiple lines of business as portfolios of lines of business, it may be possible

to infer the betas of the individual lines of business by solving systems of linear

equations.

When valuing a potential acquisition, it may be possible to identify an appropri-

ate comparison portfolio using accounting numbers. For example, regression coeffi-

cients from a regression of the historical sales numbers of the acquisition target on

the comparable sales figures of a group of tracking firms generate a portfolio of these

tracking firms that best tracks the historical sales figure of the acquisition target. If

one thought that the critical accounting value to target was an equal weighting of

sales, earnings, assets, and book/market ratio, then regressing this equal weighting of

the historical accounting numbers from the target firm on the historical values from

an equal weighting of the accounting numbers from a group of tracking firms would

also generate an appropriate weighting of these tracking firms. Such a portfolio is the

one that, in a statistical sense, has best tracked the acquisition in the relevant account-

ing dimensions.