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.

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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.

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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:

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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)

where,

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.

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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

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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

component.

9 Ross, Stephen A., 1976, The Arbitrage Theory Of Capital Asset Pricing, Journal of Economic Theory,

v13(3), 341-360.

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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

where

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 .....

where,

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.

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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]

where

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)

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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: