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The risk and return model that has been in use the longest and is still the standard
in most real world analyses is the capital asset pricing model (CAPM). While it has come
in for its fair share of criticism over the years, it provides a useful starting point for our
discussion of risk and return models.

1. Assumptions
While diversification has its attractions in terms
Riskless Asset: A riskless asset is
of reducing the exposure of investors to firm specific
one, where the actual return is
risk, most investors limit their diversification to
always equal to the expected return.
holding relatively few assets. Even large mutual funds
are reluctant to hold more than a few hundred stocks, and many of them hold as few as 10
to 20 stocks. There are two reasons for this reluctance. The first is that the marginal
benefits of diversification become smaller as the portfolio gets more diversified - the
twenty-first asset added will generally provide a much smaller reduction in firm specific
risk than the fifth asset added, and may not cover the marginal costs of diversification,
which include transactions and monitoring costs. The second is that many investors (and
funds) believe that they can find under valued assets and thus choose not to hold those
assets that they believe to be correctly or over valued.
The capital asset pricing model assumes that there are no transactions costs, all
assets are traded and that investments are infinitely divisible (i.e., you can buy any
fraction of a unit of the asset). It also assumes that there is no private information and that
investors therefore cannot find under or over valued assets in the market place. By
making these assumptions, it eliminates the factors that cause investors to stop
diversifying. With these assumptions in place, the logical end limit of diversification is to
hold every traded risky asset (stocks, bonds and real assets included) in your portfolio, in
proportion to their market value8. This portfolio of every traded risky asset in the market
place is called the market portfolio.

2. Implications for Investors
If every investor in the market holds the same market portfolio, how exactly do
investors reflect their risk aversion in their investments? In the capital asset pricing
model, investors adjust for their risk preferences in their allocation decisions, where they

8 If investments are not held in proportion to their market value, investors are still losing some
diversification benefits. Since there is no gain from over weighting some sectors and under weighting
others in a market place where the odds are random of finding under valued and over valued assets,
investors will not do so.

decide how much to invest in an asset with guaranteed returns “ a riskless asset - and how
much in risky assets (market portfolio). Investors who are risk averse might choose to put
much or even all of their wealth in the riskless asset. Investors who want to take more
risk will invest the bulk or even all of their wealth in the market portfolio. Those
investors who invest all their wealth in the market portfolio and are still desirous of
taking on more risk, would do so by borrowing at the riskless rate and investing in the
same market portfolio as everyone else.
These results are predicated on two additional assumptions. First, there exists a
riskless asset. Second, investors can lend and borrow at this riskless rate to arrive at their
optimal allocations. There are variations of the CAPM that allow these assumptions to be
relaxed and still arrive at conclusions that are consistent with the general model.

˜: 3.5. Efficient Risk Taking
In the capital asset pricing model, the most efficient way to take a lot of risk is to
a. Buy a well-balanced portfolio of the riskiest stocks in the market
b. Buy risky stocks that are also undervalued
c. Borrow money and buy a well diversified portfolio

3. Measuring the Market Risk of an Individual Asset
The risk of any asset to an investor is the risk added on by that asset to the
investor™s overall portfolio. In the CAPM world, where all investors hold the market
portfolio, the risk of an individual asset to an investor will be the risk that this asset adds
on to the market portfolio. Intuitively, assets that move more with the market portfolio
will tend to be riskier than assets that move less, since the movements that are unrelated
to the market portfolio will not affect the overall value of the portfolio when an asset is
added on to the portfolio. Statistically, this added risk is measured by the covariance of
the asset with the market portfolio.
The covariance is a non-standardized measure of market risk; knowing that the
covariance of Disney with the Market Portfolio is 55% does not provide a clue as to
whether Disney is riskier or safer than the average asset. We therefore standardize the
risk measure by dividing the covariance of each asset with the market portfolio by the
variance of the market portfolio. This yields the beta of the asset:

Covariance of asset i with Market Portfolio
Beta of an asset i =
Variance of the Market Portfolio
Since the covariance of the market portfolio with itself is its variance, the beta of the
market portfolio, and by extension, the average asset in it, is one. Assets that are riskier
than average (using this measure of risk) will have betas that exceed one and assets that
are safer than average will have betas that are lower than one . The riskless asset will
have a beta of zero.

4. Getting Expected Returns
The fact that every investor holds some Beta: The beta of any investment
combination of the riskless asset and the market in the CAPM is a standardized
measure of the risk that it adds to the
portfolio leads to the next conclusion, which is that the
market portfolio.
expected return on an asset is linearly related to the
beta of the asset. In particular, the expected return on an asset can be written as a function
of the risk-free rate and the beta of that asset;
Expected Return on asset i
= Rf + βi [E(Rm) - Rf]
= Risk-free rate + Beta of asset i * (Risk premium on market portfolio)
E(Ri) = Expected Return on asset i
Rf = Risk-free Rate
E(Rm) = Expected Return on market portfolio
βi = Beta of asset i
To use the capital asset pricing model, we need three inputs. While we will look at the
estimation process in far more detail in the next chapter, each of these inputs is estimated
as follows:
The riskless asset is defined to be an asset where the investor knows the expected

return with certainty for the time horizon of the analysis. Consequently, the riskless
rate used will vary depending upon whether the time period for the expected return is
one year, five years or ten years.

The risk premium is the premium demanded by investors for investing in the market

portfolio, which includes all risky assets in the market, instead of investing in a
riskless asset. Thus, it does not relate to any individual risky asset but to risky assets
as a class.
The beta, which we defined to be the covariance of the asset divided by the market

portfolio, is the only firm-specific input in this equation. In other words, the only
reason two investments have different expected returns in the capital asset pricing
model is because they have different betas.
In summary, in the capital asset pricing model all of the market risk is captured in one
beta, measured relative to a market portfolio, which at least in theory should include all
traded assets in the market place held in proportion to their market value.

˜: 3.6. What do negative betas mean?
In the capital asset pricing model, there are assets that can have betas that are less than
zero. When this occurs, which of the following statements describes your investment?
a. This investment will have an expected return less than the riskless rate
b. This investment insures your “diversified portfolio” against some type of market risk
c. Holding this asset makes sense only if you are well diversified
d. All of the above

In Practice: Index Funds and Market Portfolios
Many critics of the capital asset pricing model seize on its conclusion that all
investors in the market will hold the market portfolio, which includes all assets in
proportion to their market value, as evidence that it is an unrealistic model. But is it? It is
true that not all assets in the world are traded and that there are transactions costs. It is
also true that investors sometimes trade on inside information and often hold
undiversified portfolios. However, we can create portfolios that closely resemble the
market portfolio using index funds. An index fund replicates an index by buying all of the
stocks in the index, in the same proportions that they form of the index. The earliest and
still the largest one is the Vanguard 500 Index fund, which replicates the S&P 500 index.
Today, we have access to index funds that replicate smaller companies in the United
States, European stocks, Latin American markets and Asian equities as well as bond and

commodity markets An investor can create a portfolio composed of a mix of index funds
“ the weights on each fund should be based upon market values of the underlying market
- which resembles the market portfolio; the only asset class that is usually difficult to
replicate is real estate.

B. The Arbitrage Pricing Model
The restrictive assumptions in the capital asset pricing model and its dependence
upon the market portfolio have for long been viewed with skepticism by both academics
and practitioners. In the late seventies, an alternative and more general model for
measuring risk called the arbitrage pricing model was developed.9

1. Assumptions
Arbitrage: An investment that
The arbitrage pricing model is built on the
requires no investment, involves no
simple premise that two investments with the same risk but still delivers a sure profit.
exposure to risk should be priced to earn the same
expected returns. An alternate way of saying this is that if two portfolios have the same
exposure to risk but offer different expected returns, investors can buy the portfolio that
has the higher expected returns and sell the one with lower expected returns, until the
expected returns converge.
Like the capital asset pricing model, the arbitrage pricing model begins by
breaking risk down into two components. The first is firm specific and covers
information that affects primarily the firm. The second is the market risk that affects all
investment; this would include unanticipated changes in a number of economic variables,
including gross national product, inflation, and interest rates. Incorporating this into the
return model above
R = E(R) + m + µ
where m is the market-wide component of unanticipated risk and µ is the firm-specific

9 Ross, Stephen A., 1976, The Arbitrage Theory Of Capital Asset Pricing, Journal of Economic Theory,
v13(3), 341-360.

2. The Sources of Market-Wide Risk
While both the capital asset pricing model and the arbitrage pricing model make a
distinction between firm-specific and market-wide risk, they part ways when it comes to
measuring the market risk. The CAPM assumes that all of the market risk is captured in
the market portfolio, whereas the arbitrage pricing model allows for multiple sources of
market-wide risk, and measures the sensitivity of investments to each source with what a
factor betas. In general, the market component of unanticipated returns can be
decomposed into economic factors:
R = R + m + µ
= R + (β1 F1 + β2 F2 + .... +βn Fn) + µ
βj = Sensitivity of investment to unanticipated changes in factor j
Fj = Unanticipated changes in factor j

3. The Effects of Diversification
The benefits of diversification have been discussed extensively in our treatment of
the capital asset pricing model. The primary point of that discussion was that
diversification of investments into portfolios eliminate firm-specific risk. The arbitrage
pricing model makes the same point and concludes that the return on a portfolio will not
have a firm-specific component of unanticipated returns. The return on a portfolio can
then be written as the sum of two weighted averages -that of the anticipated returns in the
portfolio and that of the factor betas:
Rp = (w1R1+w2R2+...+wnRn)+ (w1β1,1+w2β1,2+...+wnβ1,n) F1 +
(w1β2,1+w2β2,2+...+wnβ2,n) F2 .....
wj = Portfolio weight on asset j
Rj = Expected return on asset j
βi,j= Beta on factor i for asset j
Note that the firm specific component of returns (µ) in the individual firm equation
disappears in the portfolio as a result of diversification.

4. Expected Returns and Betas
The fact that the beta of a portfolio is the weighted average of the betas of the
assets in the portfolio, in conjunction with the absence of arbitrage, leads to the
conclusion that expected returns should be linearly related to betas. To see why, assume
that there is only one factor and that there are three portfolios. Portfolio A has a beta of
2.0, and an expected return on 20%; portfolio B has a beta of 1.0 and an expected return
of 12%; and portfolio C has a beta of 1.5, and an expected return on 14%. Note that the
investor can put half of his wealth in portfolio A and half in portfolio B and end up with a
portfolio with a beta of 1.5 and an expected return of 16%. Consequently no investor will
choose to hold portfolio C until the prices of assets in that portfolio drop and the expected
return increases to 16%. Alternatively, an investor can buy the combination of portfolio
A and B, with an expected return of 16%, and sell portfolio C with an expected return of
15%, and pure profit of 1% without taking any risk and investing any money. To prevent
this “arbitrage” from occurring, the expected returns on every portfolio should be a linear
function of the beta to prevent this f. This argument can be extended to multiple factors,
with the same results. Therefore, the expected return on an asset can be written as
E(R) = Rf + β1 [E(R1)-Rf] + β2 [E(R2)-Rf] ...+ βn [E(Rn)-Rf]
Rf = Expected return on a zero-beta portfolio
E(Rj) = Expected return on a portfolio with a factor beta of 1 for factor j, and zero
for all other factors.
The terms in the brackets can be considered to be risk premiums for each of the factors in
the model.
Note that the capital asset pricing model can be considered to be a special case of
the arbitrage pricing model, where there is only one economic factor driving market-wide
returns and the market portfolio is the factor.
E(R) = Rf + βm (E(Rm)-Rf)

5. The APM in Practice
The arbitrage pricing model requires estimates
Arbitrage: An investment
of each of the factor betas and factor risk premiums in
opportunity with no risk that earns a
addtion to the riskless rate. In practice, these are
return higher than the riskless rate.
usually estimated using historical data on stocks and a
statistical technique called factor analysis. Intuitively, a factor analysis examines the
historical data looking for common patterns that affect broad groups of stocks (rather
than just one sector or a few stocks). It provides two output measures:


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