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The Capital Asset Pricing Model
©2000 P. LeBel
I. Evaluating Individual Stock and Market Risk
Whether to buy or sell a stock is more complicated than looking at the implied and actual
stock market price. Apart from inflation and taxes, one critical variable is the given equity
rate of return. Usually, this refers to the overall sector or industry, rate of return against which
one can evaluate a particular stock, or group of sector stocks. Stocks as a whole embody two
types of variability: one is the level of risk associated with a particular stock and the other is the
level of risk associated with the market as a whole. The former is known as diversifiable risk
since one can make a decision to buy or sell an individual stock. While one also can make a
decision whether to buy or sell any stock portfolio as a whole, it is useful in the first instance to
compare the individual diversifiable rate of return from the system wide rate of return. Given
one's risk preferences, one then has a consistent basis as to whether to hold any stocks or
other assets for a given time period.
A. Portfolio Selection
Selection of a particular portfolio begins with a reference portfolio. This can be an index of stocks
such as the Dow-Jones industrial average, the S&P 500 portfolio of stocks, the Russell 2000,
the Russell 5000, or any number of other stock market indices. Each of these portfolios reflects
a basket of assets whose value is traced over time. While the overall correlation among these
indices is expected to be positive, because each index has stocks reflecting different characteristics,
there will be a different level of volatility, and a different corresponding rate of return for each.
With this caveat, one can then proceed to the selection of an individual portfolio. We review here
some better known portfolio models, each of which enable us to consider what equity rate of return
might be considered in the preceding formulas. Most of them build on Harry M. Markowitz'
Portfolio Selection (New Haven: Yale University Press, 1959)
B. The Basic CAPM, or Capital Asset Pricing Model
One of the more popular models used in stock portfolio construction is the capital asset pricing
model, or CAPM for short. This was first developed by William F. Sharpe, in "Capital Asset Prices:
A Theory of Market Equilibrium Under Conditions of Risk," Journal of Finance, 19 (September 1964),
pp. 425-42. Sharpe's approach uses a linear regression model of individual stocks to generate
deviations from the excess returns of a portfolio of assets. A portfolio's excess rate of return is
calculated in reference to a risk-free rate of return. The risk-free rate is usually a benchmark asset
such as the interest rate on U.S. Treasury securities. The idea of equilibrium is that when one
regresses the excess rate of return of individual stocks against the excess return of a market portfolio,
the resulting characteristic line should go through the origin to reflect the fact that investors can
diversify stock portfolios in such a way that they can eliminate any level of unsystematic risk.
It is this orthogonal regression expectation that forms the basis of the efficient market hypothesis,
i.e., that all stocks embody all relevant information about present and future conditions such that
economic agents have valued individual stocks according to their underlying level of systematic risk.
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Figure 1
The Basic Capital Asset Pricing Model
Excess Return on Stock
0.08
0.06
f(x) = 0.004717391304348 x − 0.052717391304348
0.04 R² = 0.816001524944981
0.02
0.00
4 0 9 8 6 4 3 3 1 1 0 1 2 3 4 4 5 5 6 7 0 1 3 5
1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1
-0.02 . . . . . . . . . . . . . . . . . . . . . . . .
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
- - - - - - - - - -
-0.04
-0.06
-0.08
Excess Return on Market Portfolio
Time Series Sample for Estimation of the Capital Asset Pricing Model
Portfolio: Stock: Characteristic Line:
Quarter Excess Returns Excess Returns Estimated Individual Returns
15 -0.14 -0.06 -0.06 (Van Horne, 1989, p.108)
4 -0.10 -0.05 -0.04 SUMMARY OUTPUT
14 -0.09 -0.06 -0.04
9 -0.08 -0.02 -0.04 Regression Statistics
3 -0.06 -0.04 -0.03 Multiple R 0.9208
16 -0.04 -0.02 -0.02 R Square 0.8479
6 -0.03 0.00 -0.01 Adj. R Square 0.8410
23 -0.03 -0.01 -0.01 Standard Error 0.0147
8 -0.01 -0.01 0.00 Observations 24
13 -0.01 0.01 0.00 F 122.687782395
10 0.00 0.04 0.00
22 0.01 -0.01 0.01 Coefficients t Stat
5 0.02 0.02 0.01 Intercept 0.0011 0.3681
21 0.03 0.01 0.01 X Variable 1 0.4561 11.0765
12 0.04 -0.01 0.02
24 0.04 0.02 0.02
1 0.05 0.04 0.02
20 0.05 0.03 0.02
18 0.06 0.02 0.03
7 0.07 0.02 0.03
2 0.10 0.05 0.05
19 0.11 0.04 0.05
11 0.13 0.07 0.06
17 0.15 0.07 0.07
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Estimation Procedure:
1 Compile time series data on excess returns on the market portfolio and on excess returns
of an individual stock. Excess returns for an individual stock refer to the rate of return relative
to the risk-free rate of return. The risk-free rate of return generally is based on a widely held
asset such as the rate on government securities. Excess returns for a market portfolio are
calculated in the same way as for an individual stock. The Excess Return for the market portfolio
can be computed using a basket of representative stocks such as the S&P 500, or the Dow Jones
Industrial Average.
2 Sort the data set of portfolio and individual stock excess returns by the portfolio excess returns,
which is to be portrayed along the X-axis.
3 Compute a linear regression on the Excess Return of an individual stock as a function of the
Excess Return of the Market Portfolio.
4 Beta, or the slope of the CAPM regression line, defines the responsiveness of the excess returns
for security j in excess of the risk-free rate relative to those of the market. Beta measures systematic,
or unavoidable, risk. If Beta is equal to 1.00, then excess returns for the stock vary proportionally
with excess returns for the market portfolio. A Beta higher than 1.00 means that excess returns vary
by a greater amount than variation in excess returns for the market portfolio. Stocks with a Beta
greater than one are considered to be aggressive, while those with a Beta less than 1 are
considered to be conservative. In general, higher returns to a portfolio require that the individual
Beta values be in excess of 1.
5 In the case of the sample above, the individual stock has less unavoidable risk than the market overall.
6 Selection of the risk free rate in the CAPM depends on the maturity of the security used in the
calculation of the required rate of return. Ideally, one should match the maturity of the risk free
security with the time horizon of the firm's equity-financed investment, a practice that does not
hold true in most instances. Van Horne (1989) uses an intermediate rate such as the yield on
1 or 2 year Treasury bills (Van Horne, 1989, p. 378).
7 The required rate of return on equity capital may be sensitive to the underlying CAPM relationship.
Biases include whether the true relationship is linear or non-linear, and whether the market basket
of securities is representative of the market as a whole. If the S&P has rates of return that are less
than a broader basket such as the Russell 2000 or the Russell 5000, then use of the S&P as the
market excess return measure will be downward biased, as will the true required rate of return
for equity capital.
Components of the Capital Asset Pricing Model: Alpha, Beta, and Variability:
If the alpha of a stock were greater than zero, agents would recognize expected return greater
than that required for the systematic risk involved. They would buy the stock, which would raise
its price and lower its expected return. This would lower the characteristic line until the stock
provides the same expected return as other stocks with that systematic risk. At this point the
vertical intercept of the characteristic line would be zero. Conversely, if the alphs of a stock is
negative, agents will sell it, causing the market price to decline and the expected return to rise,
until the characteristic line passes through zero once again. (cf. Van Horne, 1989, p. 91)
While market sales and purchases tend to revert the alpha (or intercept) of a given stock
through the origin, beta measures the the unavoidable, or systematic risk of a stock in comparison
to the market as a whole. Stocks with a beta (or slope) greater than one have a higher level of
systematic risk, meaning that movements in excess returns for the individual stock are greater than
for the market portfolio as a whole. If the CAPM characteristic line is accurate, one cannot diversify
the level of systematic risk of the stock.
The third component of the CAPM is variability. This is reflected in the degree of dispersion
Page 3
of data used in estimating a stock's characteristic line. The greater the dispersion of points,
the greater will be the unsystematic risk of the stock. One can reduce unsystematic risk by
changing the mix of stocks in a portfolio. The level of portfolio risk is inversely related to the
number of stocks, as illustrated in the following:
Figure 2
Systematic RiskUnsystematic Risk
Systematic, Unsystematic, and Total Market Risk
1 0.6 1.4320 0.6
2 0.6 1.2923
1.6 3 0.6 1.1760
4 0.6 1.0792
5 0.6 0.9987
6 0.6 0.9317
1.4 7 0.6 0.8760
8 0.6 0.8297
9 0.6 0.7911
1.2 10 0.6 0.7590
11 0.6 0.7323
12 0.6 0.7101
13 0.6 0.6916
1 14 0.6 0.6762
15 0.6 0.6634
16 0.6 0.6527
0.8 17 0.6 0.6439
18 0.6 0.6365
Unsystematic risk
19 0.6 0.6304
0.6 20 0.6 0.6253
21 0.6 0.6210
Total Risk 22 0.6 0.6175
23 0.6 0.6146
0.4 24 0.6 0.6121
Systematic risk
25 0.6 0.6101
26 0.6 0.6084
0.2 27 0.6 0.6070
28 0.6 0.6058
29 0.6 0.6048
0 30 0.6 0.6040
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Unsystematic risk can be reduced by increasing the number of stocks in a portfolio. As this occurs,
the relationship between excess returns on an individual stock will correlate more closely with the
excess returns in the market portfolio. As shown in Figure 2, as the number of stocks increases,
the degree of unsystematic risk converges to the level of systematic risk.
Calculating the Required Rate of Return of a Stock in a Portfolio
If financial markets are efficient and investors can efficiently diversify, unsystematic risk becomes
minor. The major risk is the systematic component. The greater the beta of a stock, the greater the
risk of that stock, and the greater the required rate of return. If unsystematic risk is diversified out,
the required rate of return for a given stock is defined as:
R =i+(R −i)β where: Rm = the expected return for themarket portfolio
j m j Bj = the beta coefficient for stock j
I = the risk-free rate of return
Examples:
Risk free rate of return = 5.00% 5.00% 5.00% 5.00%
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