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worked on.

2. It measures the beta of each investment relative to each of the common factors, and

provides an estimate of the actual risk premium earned by each factor.

The factor analysis does not, however, identify the factors in economic terms.

In summary, in the arbitrage pricing model the market or non-diversifiable risk in

an investment is measured relative to multiple unspecified macro economic factors, with

the sensitivity of the investment relative to each factor being measured by a factor beta.

The number of factors, the factor betas and factor risk premiums can all be estimated

using a factor analysis.

C. Multi-factor Models for risk and return

The arbitrage pricing model's failure to identify specifically the factors in the

model may be a strength from a statistical standpoint, but it is a clear weakness from an

intuitive standpoint. The solution seems simple: Replace the unidentified statistical

factors with specified economic factors, and the resultant model should be intuitive while

still retaining much of the strength of the arbitrage pricing model. That is precisely what

multi-factor models do.

Deriving a Multi-Factor Model

Multi-factor models generally are not based on extensive economic rationale but

are determined by the data. Once the number of factors has been identified in the

arbitrage pricing model, the behavior of the factors over time can be extracted from the

data. These factor time series can then be compared to the time series of macroeconomic

Unanticipated Inflation: This is

the difference between actual

inflation and expected inflation.

30

variables to see if any of the variables are correlated, over time, with the identified

factors.

For instance, a study from the 1980s suggested that the following macroeconomic

variables were highly correlated with the factors that come out of factor analysis:

industrial production, changes in the premium paid on corporate bonds over the riskless

rate, shifts in the term structure, unanticipated inflation, and changes in the real rate of

return.10 These variables can then be correlated with returns to come up with a model of

expected returns, with firm-specific betas calculated relative to each variable. The

equation for expected returns will take the following form:

E(R) = Rf + ОІGNP (E(RGNP)-Rf) + ОІi (E(Ri)-Rf) ...+ ОІОґ (E(RОґ)-Rf)

where

ОІGNP = Beta relative to changes in industrial production

E(RGNP) = Expected return on a portfolio with a beta of one on the industrial

production factor, and zero on all other factors

ОІi = Beta relative to changes in inflation

E(Ri) = Expected return on a portfolio with a beta of one on the inflation factor,

and zero on all other factors

The costs of going from the arbitrage pricing model to a macroeconomic multi-

factor model can be traced directly to the errors that can be made in identifying the

factors. The economic factors in the model can change over time, as will the risk

premium associated with each one. For instance, oil price changes were a significant

economic factor driving expected returns in the 1970s but are not as significant in other

time periods. Using the wrong factor(s) or missing a significant factor in a multi-factor

model can lead to inferior estimates of cost of equity.

In summary, multi factor models, like the arbitrage pricing model, assume that market

risk can be captured best using multiple macro economic factors and estimating betas

relative to each. Unlike the arbitrage pricing model, multi factor models do attempt to

identify the macro economic factors that drive market risk.

10 Chen, N., R. Roll and S.A. Ross, 1986, Economic Forces and the Stock Market, Journal of Business,

1986, v59, 383-404.

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D. Proxy Models

All of the models described so far begin by thinking about market risk in

economic terms and then developing models that might best explain this market risk. All

of them, however, extract their risk parameters by looking at

Book-to-Market Ratio: This

historical data. There is a final class of risk and return

is the ratio of the book value

models that start with past returns on individual stocks, and

of equity to the market value

then work backwards by trying to explain differences in of equity.

returns across long time periods using firm characteristics.

In other words, these models try to find common characteristics shared by firms that have

historically earned higher returns and identify these characteristics as proxies for market

risk.

Fama and French, in a highly influential study of the capital asset pricing model

in the early 1990s, note that actual returns over long time periods have been highly

correlated with price/book value ratios and market capitalization.11 In particular, they

note that firms with small market capitalization and low price to book ratios earned

higher returns between 1963 and 1990. They suggest that these measures and similar ones

developed from the data be used as proxies for risk and that the regression coefficients be

used to estimate expected returns for investments. They report the following regression

for monthly returns on stocks on the NYSE, using data from 1963 to 1990:

Rt = 1.77% - 0.11 ln (MV) + 0.35 ln (BV/MV)

where

MV = Market Value of Equity

BV/MV = Book Value of Equity / Market Value of Equity

The values for market value of equity and book-price ratios for individual firms, when

plugged into this regression, should yield expected monthly returns. For instance, a firm

with a market value of $ 100 million and a book to market ratio of 0.5 would have an

expected monthly return of 1.02%.

Rt = 1.77% - 0.11 ln (100) + 0.35 ln (0.5) = 1.02%

11 Fama, E.F. and K.R. French, 1992, The Cross-Section of Expected Returns, Journal of Finance, v47,

427-466.

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As data on individual firms has becomes richer and more easily accessible in recent

years, these proxy models have expanded to include additional variables. In particular,

researchers have found that price momentum (the rate of increase in the stock price over

recent months) also seems to help explain returns; stocks with high price momentum tend

to have higher returns in following periods.

In summary, proxy models measure market risk using firm characteristics as

proxies for market risk, rather than the macro economic variables used by conventional

multi-factor models12. The firm characteristics are identified by looking at differences in

returns across investments over very long time periods and correlating with identifiable

characteristics of these investments.

A Comparative Analysis of Risk and Return Models

All the risk and return models developed in this chapter have common

ingredients. They all assume that only market-wide risk is rewarded, and they derive the

expected return as a function of measures of this risk. Figure 3.7 presents a comparison of

the different models:

12 Adding to the confusion, researchers in recent years have taken to describing proxy models also as multi

factor models.

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Figure 3.7: Competing Models for Risk and Return in Finance

Step 1: Defining Risk

The risk in an investment can be measured by the variance in actual returns around an

expected return

Riskless Investment Low Risk Investment High Risk Investment

E(R) E(R) E(R)

Step 2: Differentiating between Rewarded and Unrewarded Risk

Risk that is specific to investment (Firm Specific) Risk that affects all investments (Market Risk)

Can be diversified away in a diversified portfolio Cannot be diversified away since most assets

1. each investment is a small proportion of portfolio are affected by it.

2. risk averages out across investments in portfolio

The marginal investor is assumed to hold a вЂњdiversifiedвЂќ portfolio. Thus, only market risk will be rewarded

and priced.

Step 3: Measuring Market Risk

The CAPM The APM Multi-Factor Models Proxy Models

If there are no

If there is Since market risk affects In an efficient market,

arbitrage opportunities

1. no private information most or all investments, differences in returns

then the market risk of

2. no transactions cost it must come from across long periods must

any asset must be

the optimal diversified macro economic factors. be due to market risk

captured by betas relative

portfolio includes every Market Risk = Risk differences. Looking for

to factors that affect all

traded asset. Everyone exposures of any asset to variables correlated with

investments.

will hold this market portfolio macro economic factors. returns should then give

Market Risk = Risk

Market Risk = Risk added by us proxies for this risk.

exposures of any asset

any investment to the market Market Risk = Captured

to market factors

portfolio: by the Proxy Variable(s)

Beta of asset relative to Betas of asset relative Betas of assets relative Equation relating

Market portfolio (from to unspecified market to specified macro returns to proxy

a regression) factors (from a factor economic factors (from variables (from a

analysis) a regression) regression)

The capital asset pricing model makes the most assumptions but arrives at the simplest

model, with only one risk factor requiring estimation. The arbitrage pricing model makes

fewer assumptions but arrives at a more complicated model, at least in terms of the

parameters that require estimation. In general, the CAPM has the advantage of being a

simpler model to estimate and to use, but it will under perform the richer multi factor

models when the company is sensitive to economic factors not well represented in the

market index. For instance, oil companies, which derive most of their risk from oil price

movements, tend to have low CAPM betas. Using a multi factor model, where one of the

factors may be capturing oil and other commodity price movements, will yield a better

estimate of risk and higher cost of equity for these firms13.

13 Weston, J.F. and T.E. Copeland, 1992, Managerial Finance, Dryden Press. They used both approaches

to estimate the cost of equity for oil companies in 1989 and came up with 14.4% with the CAPM and

19.1% using the arbitrage pricing model.

34

The biggest intuitive block in using the arbitrage pricing model is its failure to

identify specifically the factors driving expected returns. While this may preserve the

flexibility of the model and reduce statistical problems in testing, it does make it difficult

to understand what the beta coefficients for a firm mean and how they will change as the

firm changes (or restructures).

Does the CAPM work? Is beta a good proxy for risk, and is it correlated with

expected returns? The answers to these questions have been debated widely in the last

two decades. The first tests of the model suggested that betas and returns were positively

related, though other measures of risk (such as variance) continued to explain differences

in actual returns. This discrepancy was attributed to limitations in the testing techniques.

In 1977, Roll, in a seminal critique of the model's tests, suggested that since the market

portfolio (which should include every traded asset of the market) could never be

observed, the CAPM could never be tested, and that all tests of the CAPM were therefore

joint tests of both the model and the market portfolio used in the tests, i.e., all any test of

the CAPM could show was that the model worked (or did not) given the proxy used for

the market portfolio.14 He argued that in any empirical test that claimed to reject the

CAPM, the rejection could be of the proxy used for the market portfolio rather than of the

model itself. Roll noted that there was no way to ever prove that the CAPM worked, and

thus, no empirical basis for using the model.

The study by Fama and French quoted in the last section examined the

relationship between the betas of stocks and annual returns between 1963 and 1990 and

concluded that there was little relationship between the two. They noted that market

capitalization and book-to-market value explained differences in returns across firms

much better than did beta and were better proxies for risk. These results have been

contested on two fronts. First, Amihud, Christensen, and Mendelson, used the same data,

performed different statistical tests, and showed that betas did, in fact, explain returns

during the time period.15 Second, Chan and Lakonishok look at a much longer time series

14 Roll, R., 1977, A Critique of the Asset Pricing Theory's Tests: Part I: On Past and Potential Testability

of Theory, Journal of Financial Economics, v4, 129-176.

15 Amihud, Y., B. Christensen and H. Mendelson, 1992, Further Evidence on the Risk-Return Relationship,

Working Paper, New York University.

35

of returns from 1926 to 1991 and found that the positive relationship between betas and

returns broke down only in the period after 1982.16 They attribute this breakdown to

indexing, which they argue has led the larger, lower-beta stocks in the S & P 500 to

outperform smaller, higher-beta stocks. They also find that betas are a useful guide to risk

in extreme market conditions, with the riskiest firms (the 10% with highest betas)

performing far worse than the market as a whole, in the ten worst months for the market

between 1926 and 1991 (See Figure 3.8).

Figure 3.8: Returns and Betas: Ten Worst Months

between 1926 and 1991

May

May

1940

1932

1988

1987

1932

1937

1933

1932

1980

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