Chapter 13
The Capital Asset Pricing Model
How to Measure Risk
Security Prices
100000
Stock
Bond
Stock_Mu
Bond_Mu
Value (Log)
10000
1000
100
10
0
5
10
15
20
25
30
35
40
Years
Main Observations:
1. The stock price is more volatile than the bond price.
2. The expected (or average) return on stock is higher than the expected (or
average) return on the bond. Expected returns can be measured by the
slopes of the fitted straight lines.
Risk/Return Analysis
Consider two stocks, Anheuser-Busch and Microsoft, with following expected
returns, standard deviations, and variances measured in annual units.
Expected
return
Standard
deviation of
the return
Variance
of the return
Anheuser- 0.225
Busch
0.08292
0.006875
Microsoft
0.22776
0.051875
0.375
Correlation coefficient between the two stocks = -0.364
Though on average you earn less on Anheuser-Busch stock, this stock also gives
you a much lower risk given by a variance of only 0.006875. The Microsoft stock
earns 15% more on the average, but it also has more risk given by a variance of the
return equal to 0.051875. So, which stock do you invest in? Typically, in the
financial markets, you find it optimal to invest in a portfolio of stocks, and not
single stocks. This is due to the benefits of diversification that allow you to
eliminate some risk. In the above example, constructing a portfolio would be
helpful since correlation between the two stock returns is –0.364, a negative
number. Generally, diversification is helpful, whenever correlation between any
two stocks is less than one (correlation coefficient must lie between –1 and + 1, by
definition). The two questions that are important are:
1. How does one determine the optimal portfolio, i.e., what proportion of the
portfolio to invest in which security?
2. What are the measures of risk at the portfolio level and at the individual
security level?
As shown by the following example, the answer to the second question is
dependent on the answer to the first question. One has to first determine the
optimal portfolio, by minimizing portfolio risk (or portfolio variance) for some
level of expected return on the portfolio. This can be accomplished using a
quadratic programming technique. Once the optimal portfolio has been found
(more on this later), than portfolio risk and individual security risk can be
appropriately defined. Consider the following example in which the optimal
portfolio is assumed to be equal investments in Anheuser-Busch and Microsoft
stocks (this assumption is made for expositional purpose – a two stock portfolio is
generally not optimal).
Example 1.
Probability
Anheuser-Busch Microsoft
Optimal Portfolio
(50%-50% allocation)
0.15
0.45
0.30
0.30
0.25
0.25
0.25
0.25
0.30
0.30
0.10
0.20
0.00
0.60
0.50
0.40
Expected
Return
0.225
0.375
0.30
Standard
deviation
0.08292
0.22776
0.10607
Variance
0.006875
0.051875
0.01125
Covariance
with Optimal
Portfolio
0
0.0225
Definition of Risk – Portfolio level
Portfolio risk is defined as the variance of the return on your optimal portfolio.
You only care about how your portfolio performs, and not how each and every
individual security performs in the portfolio. In the above example, the portfolio
risk equals the variance of the portfolio return, or 0.01125.
Note that the variance of the portfolio return is not a linear weighted average of the
variances of the individual stock returns. In other words,
0.5 * 0.006875 + 0.5 * 0.051875 = 0.029375  0.01125
In fact, the variance of the portfolio return (0.01125) is much less than the linear
weighted average of the variances of the individual stock returns (0.029375). This
is due to the diversification effect. Some of the risk inherent in the individual
stock returns has been diversified away. The risk that has been diversified away is
called the unsystematic risk. The risk that cannot be diversified away is called
systematic risk. In efficient markets, investors holding an optimal portfolio only
worry about systematic risks, and ignore unsystematic risks.
Definition of Risk – Individual securities
The appropriate measure of risk for an individual security is its systematic risk.
The systematic risk of an individual security is the contribution of that stock’s risk
to the portfolio risk. To obtain a precise measure of systematic risk we note the
following. The variance of the portfolio return is equal to the linear weighted
average of the covariances of the individual stock returns with the portfolio return.
Covariance of Microsoft stock return with the portfolio return is computed as
follows:
0.25 *(0.00 - 0.375)*(0.15 - 0.30)
+ 0.25 *(0.60 - 0.375)*(0.45 - 0.30)
+ 0.25 *(0.50 - 0.375)*(0.30 - 0.30)
+ 0.25 *(0.40 - 0.375)*(0.30 - 0.30)
= 0.0225
Similarly, covariance of Anheuser-Busch stock return with the portfolio return is
computed as follows:
0.25 *(0.30 - 0.225)*(0.15 - 0.30)
+ 0.25 *(0.30 - 0.225)*(0.45 - 0.30)
+ 0.25 *(0.10 - 0.225)*(0.30 - 0.30)
+ 0.25 *(0.20 - 0.225)*(0.30 - 0.30)
=0
Note that the variance of the portfolio return is equal to the linear weighted
average of the covariances of the individual stock returns with the portfolio return.
0.5 * 0 + 0.5 * 0.0225 = 0.01125
The covariance of the individual stock return with the optimal portfolio return is
the measure of systematic risk of the stock.
Hence, it follows that the risk of the optimal portfolio (measured by the variance of
the return on the optimal portfolio) is equal to the linear weighted average of the
systematic risks of the individual securities (measured by covariances of the
individual security returns with the portfolio return) in the portfolio.
The risk measure Beta of a stock is defined as a ratio of the systematic risk of a
stock to the optimal portfolio’s risk.
Stock Beta = systematic risk of the stock/optimal portfolio’s risk
= covariance of individual stock return with optimal portfolio return/optimal
portfolio variance
Beta of Anheuser-Busch = 0/0.01125 = 0
Beta of Microsoft = 0.0225/0.01125 = 2
Very often stock beta is loosely referred to as the systematic risk of the stock, even
though stock beta is the systematic risk of the stock divided by the portfolio risk.
To generalize these results, note that the optimal portfolio of investors under the
Capital Asset Pricing Model is the market portfolio of all securities. Let us see
why the market portfolio is the optimal portfolio under CAPM.
13.1 The Capital Asset Pricing Model in Brief
See the figure on Capital Market Line in the book chapter
Important points about CAPM:
1. The best combinations of expected return and standard deviation move in the
northwest direction as you go on adding more and more assets.
2. Any risky portfolio combined with the riskless asset gives another portfolio
that lies on the straight line joining the riskless asset and the risky portfolio (in
the expected return and standard deviation space).
3. The tangency portfolio under many assets (or all assets) is the optimal portfolio
for all investors. Every investor holds some combination of the tangency
portfolio and the riskless asset.
4. The line joining the riskless asset and the tangency portfolio is called the
capital market line (CML).
5. More risk averse investors hold less of the tangency portfolio and more of the
risk less asset. Less risk averse investors hold more of the tangency portfolio
and less of the risk less asset.
6. To obtain an expected return higher than that of the tangency portfolio,
investors can invest more than 100% in the tangency portfolio. For example,
lets assume that your initial capital is $100. Now you can borrow $50 from a
bank at the riskless rate. You can then invest $150 ($100 of your own plus the
borrowing of $50) in the tangency portfolio. This will give you a higher
expected return than the tangency portfolio, and also has a higher risk than that
of the tangency portfolio.
7. Since every investor holds the tangency portfolio, this portfolio must be the
market portfolio. In other words, the total market capitalization of Microsoft
stock as a percentage of the total market capitalization of all stocks, must be the
weight of the Microsoft stock in the tangency portfolio. This applies to each
and every stock.
8. The slope of the CML equals [E(rm) – rf]/[m – 0] = [E(rm) – rf]/[m. The slope
of the CML is, thus the risk premium on the market portfolio divided by its
standard deviation:
9. The equation of the CML is given as:
E(r) = rf + Slope of CML * 
E(r) = rf + [[E(rm) – rf ] / m ] 
Problem 1:
Security
Market Value ($)
X
Y
Z
Govt. Risk Free
24
36
24
16
(a) Relative proportions of each asset in the market portfolio
The total value of all assets including the risk free asset = (24 +36 + 24 + 16) =
$100
Asset
Proportion of all assets
X
Y
Z
Gov
24/100 = 0.24
36/100 = 0.36
24/100 = 0.24
16/100 = 0.16
Total value of all risky assets = (24 + 36 + 24 )= $84 [ Here we exclude the
government security which is risk free ]
Risky Asset
X
Y
Z
Proportion of all risky assets
24/84 = 2/7
36/84 = 3/7
24/84 = 2/7
(b) If an investor with $100,000 has $ 30,000 invested in the risk free asset , in
what proportions is the remaining allocated to the risky assets.
After investing $30,000 in the risk free asset the investor is left with $70,000.
This $70,000 will be allocated to the risky assets in the same proportion that each
risky asset makes up the market portfolio of risky assets. i.e. 2/7 in X, 3/7 in Y and
2/7 in Z.
Risky Asset
Amount allocated
X
Y
2/7 (70000) = 20,000
3/7 (70000) = 30,000
Z
2/7 (70000) = 20,000
CAPM says that in equilibrium any investor’s relative holdings of risky assets will
be the same as in the market portfolio
Problem 4:
Treasury bill rate: Rf = 4 %
Expected return on the market portfolio: E [Rm] = 12 %
S.D of return on the market: M = 0.20
Equation of the capital market line:
E(r) = Rf + { [E (Rm) - Rf ] / M }  = 0.04 + [ 0.08 / 0.2 ]  = 0.04 +0.4 
13.2 Determining the Risk Premium on the Market Portfolio
The risk premium on the market portfolio will increase with the risk aversion and
the risk inherent in the market portfolio.
E[Rm] – Rf = A 2M
where:
A = risk aversion coefficient of the economy
2M = risk in the market portfolio measure by the variance of its return
Problem 5:
Given
Expected return on the market portfolio: E [Rm] = 0.25
S.D of the return on the market portfolio: M = 0.25
Average degree of risk aversion: A = 3
We need to find the price at which the government can issue a zero-coupon bond
for 1 period with a face value of 100,000.
According to the CAPM:
E[Rm] – Rf = A 2M
Rf = E[Rm] - A 2M = 0.25 – 3 (0.25)2 = 0.0625
The yield on the bond would be 6.25 %.
The bondholder receives 100,000 at maturity. So he would pay (100000/1.0625) =
94,117.65 today.
13.3 Beta and Risk Premiums on Individual Securities
Recall that earlier we defined the beta of a stock as follows:
Stock Beta = systematic risk of the stock/optimal portfolio’s risk
= covariance of individual stock return with optimal portfolio return/optimal
portfolio’s variance
Now we know that the optimal portfolio is the market portfolio under CAPM.
Hence, the beta of the jth stock can be given as:
j = jm /m2
jm = covariance of the jth stock return with market portfolio return
m2 = market portfolio’s variance
According to the CAPM, in equilibrium, the risk premium on any asset is equal to
its beta times the risk premium on the market portfolio. The equation expressing
this relation is
E(rj) – rf = j [ E (rm) – rf ]
The above equation can be rewritten as:
E(rj) = rf + j [ E (rm) – rf ]
This is the security market line (SML) relation. Since this is an equilibrium
relation, it determines what is the expected return on a stock given its beta. The
expected return on a stock is obtained by accounting for the systematic risk of the
stock.
This expected return must be used as the appropriate discount rate to obtain
the present values of any cash flows from the stock. This expected return is
also the cost of equity, or the discount rate used for discounting the dividends
in chapter 9.
k = E(rj) = rf + j [ E (rm) – rf ]
Problem 9:
Expected price of Yablonsky Toy Co. share 1 year from today: 1000 Roubles
Expected return on the market portfolio: E [Rm]= 18 %
Riskless rate: Rf = 10 %
The CAPM security market line relation is given by:
E(rj) – rf = j [ E(rm) – rf ]
(a) j = 3
Expected return on the stock E(rj) = rf + j [ E(rm) – rf ]
= 0.10 + 3 [0.18 - 0.10]
= 0.34
Today’s price will be the present value of 1000 Roubles since this is the price 1year from now.
Today’s price of the stock = 1000/(1 +0.34) = 746.27 Roubles
(b) j = 0.5
Expected return on the stock E(rj) = rf + j [ E(rm) – rf ]
= 0.10 + 0.5 [0.18 –0.10]
= 0.14
Today’s price = 1000/1.14 = 877.19 Roubles
13.4 Using the CAPM in Portfolio Selection
The security market line (SML) relation is given as:
E(rj) = rf + j [ E (rm) – rf ]
If the portfolio manger’s estimate of the expected return on a stock is higher than
the SML expected return (above equation), then this difference is called the alpha.
Portfolio managers search for positive alpha stocks to enhance the return of the
portfolio, given a certain level of risk.
Though SML is generally used for individual stock selection, CML can be used for
portfolio evaluation. Any portfolios that lie above (below) the CML are good
(bad) buys. Evaluating portfolios by using CML allows you to compare two
portfolios with different levels of risk.
Problem 14:
The answer suggested here takes a different approach than that given by the
problem solutions manual by the publisher.
Annualized rate of return earned by Pizzaro Mutual Fund: 12%
Annualized standard deviation: 30%
Average risk free rate: 5 %
Average rate of return on market index: 10%
S.D of the market index return: 20%
The CML return on Pizzaro mutual fund equals:
E(r) = Rf + { [E (Rm) - Rf ] / M } 
= 0.05 + {[0.10 – 0.05]/0.20}* 0.30
= 0.125 = 12.5%
The actual annualized rate of return on Pizzaro equals 12%
Hence, the actual expected return is less than the CML return (or the Pizzaro
portfolio lies below the CML). Hence, Pizzaro is not as good an investment as the
market index. Did you know that more than 75% of mutual funds are beaten year
after year by the popular market index S & P 500!
13.5 Valuation and Regulating Rates of Return:
Discounted Cash Flow Valuation Models
The capitalization rate for the rate (k) used to find the present value of a stock
could be determined using the SML relation.
k = rf +  [ E(rM) – rf ]
Practitioners also often use a CAPM-based method to estimate the cost of equity
capital.