Estimating the Cost‐of‐Equity Capital Using Empirical Asset Pricing Models

• Published date: 01 March 2019
• DOI: http://doi.org/10.1111/irfi.12179
• Date: 01 March 2019
• Author: Chris Kirby

Estimating the Cost-of-Equity

Capital Using Empirical Asset

Pricing Models*

CHRIS KIRBY

Department of Finance, University of North Carolina Charlotte at Charlotte, NC

ABSTRACT

Empirical asset pricing models seek to capture characteristic-based patterns

in the cross-section of average stock returns. I propose a new approach for

constructing these models, and investigate its performance with respect to

estimating the cost-of-equity capital. Using a model that accounts for the

cross-sectional relation between five characteristics and average stock returns,

I obtain cost-of-equity estimates that outperform those produced by the

Fama-French five-factor model in out-of-sample tests. Because the proposed

approach builds directly on standard cross-sectional regression techniques, it

provides complete flexibility in choosing the firm characteristics used to for-

mulate the cost-of-equity estimates.

JEL Code: G12

Accepted: 18 December 2017

I. INTRODUCTION

The cost-of-equity capital (i.e., the expected rate of return on a firm’s stock)

plays a central role in capital budgeting, firm valuation, portfolio selection, and

a host of other applications. Estimating this quantity is therefore of widespread

interest to academics and practitioners alike. The most commonly used

methods for constructing cost-of-equity estimates are finding the internal rate

of return that equates a firm’s stock price to the present value of its estimated

expected future cash flows (the internal-cost-of-capital method), and fitting lin-

ear asset pricing models to historical stock returns.

1

My analysis focuses on the

latter of these two methods. In particular, I investigate the cost-of-equity esti-

mates produced by linear asset pricing models that are explicitly designed to

capture characteristic-based patterns in average individual stock returns.

* I thank an anonymous referee for helpful comments, and Adriana Cordis for her contributions to

early versions of the paper.

1 Studies that use the former approach include Claus and Thomas (2001), Gebhardt

et al. (2001), Easton (2004), and Hou et al. (2012), while those that use the latter approach

include Elton et al. (1994), Ferson and Locke (1998), Pastor and Stambaugh (1999), Cummins

and Phillips (2005), and Barth et al. (2013).

© 2018 International Review of Finance Ltd. 2018

International Review of Finance, 19:1, 2019: pp. 105–154

DOI: 10.1111/irfi.12179

The three-factor model of Fama and French (1993), which builds on the

findings of Fama and French (1992), is the most famous example of such a

specification.

2

It is motivated by cross-sectional regression evidence that two

easily-measured firm characteristics, the market value of equity (ME) and the

book-to-market (B/M) equity ratio, dominate the market beta with respect to

explaining the cross-section of average individual stock returns. To capture this

feature of the historical data, Fama and French (1993) add the returns on two

characteristic-based hedge portfolios to the right-hand side of the market model

regression. By restricting the regression intercept to be zero for every stock, they

obtain an empirical asset pricing model that implies a linear multi-beta repre-

sentation for expected excess stock returns.

Unlike Fama and French (1993), who use cross-sectional regression evidence

to motivate a separate and distinct time-series specification, I focus on the time-

series implications of the cross-sectional regressions themselves. My strategy is

inspired by a connection between cross-sectional regression estimators and the

type of hedge portfolios that are employed in the three-factor model. Building

on the ideas developed by Fama (1976), I demonstrate that each element of the

estimated vector of slope coefficients for a cross-sectional regression specifica-

tion is proportional to the return generated by a characteristic-based hedge

portfolio. This insight lies at the core of my approach for constructing empirical

asset pricing models.

To see why cross-sectional regression estimators are closely related to hedge

portfolios, let r

i,t

and x

i,t

denote the stock return of firm ifor period tand some

characteristic of firm ithat is predetermined with respect to period t. The Fama

and MacBeth (1973) methodology for assessing whether the characteristic helps

to explain the cross-section of expected stock returns is simple. Fit a cross-

sectional regression of r

i,t

on x

i,t

for a range of different time periods, and use

the average values of the resulting coefficient estimates to draw inferences. It is

well known that the ordinary least squares (OLS) estimator of the slope coeffi-

cient for a regression of r

i,t

on x

i,t

is simply a weighted sum of the stock returns

used to fit the regression. If one has Nstocks in total (e.g., the available number

of NYSE, AMEX, and NASDAQ firms for period t), then the weight for firm iis

proportional to xi,t−^μx,t, where ^μx,t=1=NðÞ

PN

i=1xi,t. It follows, therefore, that

the estimated slope coefficient is the payoff on a zero-cost position in the

Nstocks.

With a little algebra, this zero-cost payoff can be expressed as a scaled version

of the return on a characteristic-based hedge portfolio, that is, the return

obtained by taking a long position in one unit-cost portfolio and a short posi-

tion in another unit-cost portfolio, both of which have nonnegative

characteristic-dependent weights. Consequently, one can show that fitting a

cross-sectional regression of r

i,t

on x

i,t

decomposes the stock return for each firm

into a sum of three components, all of which have a simple interpretation. The

2 Other examples include the four-factor model of Carhart (1997), the four-factor model of

Hou et al. (2014), and the five-factor model of Fama and French (2015).

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International Review of Finance

first component is the return on an equally weighted portfolio of the Nstocks,

the second is proportional to the return on a characteristic-based hedge portfo-

lio, and the third is a firm-specific residual.

This approach readily generalizes to the case of multiple regressors. Suppose,

for example, that the regressors consist of log ME and the log B/M ratio, as in

Fama and French (1992). It is easy to verify that the estimated slope for log ME

is proportional to the return on a characteristic-based hedge portfolio. To con-

struct the long and short ends of the portfolio, one fits a cross-sectional regres-

sion of log ME on the log B/M ratio, gathers all the firms that have positive

residuals into one group (stocks purchased) and all the firms that have negative

residuals into another group (stocks shorted), and weights the returns of the

firms in each group in proportion to their absolute residuals. Similarly, the esti-

mated slope for the log B/M ratio is proportional to the return on a

characteristic-based hedge portfolio whose long and short ends are constructed

via the reverse regression (log B/M ratio on log ME). Thus the stock return for

each firm can be expressed as a linear function of the return on an equally

weighted portfolio, the returns on two characteristic-based hedge portfolios,

and a firm-specific residual. By averaging each of these components for a given

firm over time, I obtain a regression-based decomposition of the firm’s average

stock return.

The decompositions of average stock returns obtained by averaging cross-

sectional regressions are exact, that is, they hold by construction for ev ery

firm. However, they can be transformed into empirical asset pricing models by

adopting suitable assumptions. Under the proposed assumptions, specifying

log ME and the log B/M ratio as regressors produces a model that mirrors the

basic structur e of the Fama and Fren ch (1993) three- factor model. Tha t is, it

decomposes the expected excess return for each stock into a linear combina-

tion of the expected excess returns on the market portfolio and two

characteristic-based hedge portfolios. The key distinction between the two

models is how risk is measured. My approach implies that the loadings on the

risk factors are directly linked to the characteristic values. In effect, I obt ain a

modelinwhichtheaveragecharacteristicvaluesserveasobservableproxies

for systematic risk.

The empirical tests focus on a model that uses five characteristic-based hedge

portfolios as factors. My choice of characteristics is motivated by prior research.

I consider log ME and the log B/M ratio (Fama and French 1992, 1993), and the

ratio of gross profits to total assets (GP/TA). Novy-Marx (2013) shows that the

cross-sectional explanatory power of the GP/TA ratio rivals that of the log B/M

ratio. I also consider two other characteristics from the recent literature: the rate

of growth in total assets (TAG), and the ratio of cash equivalents to total assets

(CH/TA). The former is of interest because the literature reports that capital

investment is inversely related to subsequent stock returns (see, e.g., Titman

et al. 2004; Cordis and Kirby 2015), and TAG provides a comprehensive picture

of a firm’s investment and disinvestment activities (Cooper et al. 2008). The

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Empirical Asset Pricing Models