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1. It specifies the number of common factors that affected the historical data that it
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.
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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 + βGNP (E(RGNP)-Rf) + βi (E(Ri)-Rf) ...+ βδ (E(Rδ)-Rf)
where
βGNP = 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
βi = 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 “diversified” 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.
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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.
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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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