Slide 1: Lecture 13
Welcome to Lecture 13: The Capital Asset Pricing Model, or what is more commonly referred
to as the CAPM.
Slide 2: Topics
In this lecture, we will be covering these five topics:
One, we will start by looking at the definition of CAPM and the formula that describes it.
Two, we will discuss the interpretation of the model and how it can be used to determine
expected asset return and expected portfolio return.
Three, we will show a couple of numerical examples that demonstrate how the CAPM can be
applied.
Four, we will demonstrate how to estimate an asset`s beta by using the market model.
Last, as always, we end the lecture with a question to practice your new-found skills.
Slide 3: CAPM defined
What is the Capital Asset Pricing Model? CAPM is a theoretical financial model that says that
the expected return on any asset “i" depends on three factors: risk-free rate (represented
by Rf), the beta of the asset i (represented by βi), and the expected return on the market
portfolio that contains all assets in the market (represented by E(RM)).
Note that the difference between the expected market return and the risk-free rate is called
the market risk premium, which measures the excess return of the market portfolio over
and above the risk-free rate.
Slide 4: Determinants of expected return
So, what do these symbols mean, really?
Rf, the risk-free rate, is the pure-time value of money. That is, it is the return that one
could earn if one were to forgo all risks, including default risk, and purchase the safest of
assets.
The expected market risk premium, E(RM) – Rf, represents the reward one would expect for
bearing any risk when investing in any asset.
The beta of the asset, βi, represents the level of systematic risk on the asset. It is the risk
that an average investor in the market will face if he or she purchases the asset i.
From the formula E(Ri) = Rf + βi (E(RM) – Rf), we can see that, the higher the risk-free rate,
asset beta, and/or market risk premium, the higher will be the expected return on the
asset.
Slide 5: CAPM applications
Seeing that the CAPM is a theoretical model, we might say that the value of the model
comes from how it can help us understand reality. And so, how can CAPM be applied to this
end?
CAPM can be used to project the return on any asset or portfolio, as long as we are able to
obtain the three relevant and important factors: risk-free rate, expected market return, and
the beta on the asset or portfolio.
Slide 6: Numerical Example #1 – CAPM with one asset
Let‘s go through a numerical example to see how we can apply the capital asset pricing
model with one asset.
We are given the market data: T-bill rate of 5% and S&P 500 return of 15% and we are also
given asset-specific information (i.e., data specific to the asset we are looking at): beta = 2.
As usual, the first thing we do is write down the information that we have been given:
Rf = 0.05
That is, the T-bill rate is usually used to represent the risk-free rate, because T-bills are
issued by governments, and government-issued assets rarely, if ever, have any default risk.
The next bit of information we write down is:
E(RM) = 0.15
That is, the S&P 500 return is used to represent the expected market return, because the
S&P 500 index has a sufficiently large number of companies to represent the market.
Next, we have asset-specific information relating to the beta of the asset i:
i = 2
The market risk premium is calculated as:
E(RM) – Rf = 0.15 – 0.05 = 0.1
Using the CAPM formula,
E(Ri) = Rf + Beta[E(RM) – Rf]
= 0.05 + 2(0.1)
= 0.05 + 0.2
= 0.25 = 25%.
That is, the CAPM predicts that, with the current market conditions yielding 15% expected
market return and 5% risk-free rate, an asset with a beta of 2 will have a projected return
of 25%.
That is how we can use the CAPM to find the expected return on an asset, given the three
input factors of risk-free rate, expected market return, and asset beta.
Slide 7: Numerical Example #2 – CAPM with a portfolio of assets
In the previous numerical example, we worked on a simple one-asset model. Now, let us
move to work on something a bit more complicated. Let us look at a numerical example
wherein we use the CAPM to calculate the expected return on a portfolio containing multiple
assets.
In this example, we are given market data consisting of the risk-free rate of 0.05 and
market return of 0.15:
Rf = 0.05
E(RM) = 0.15
We are also given the portfolio data. In this portfolio, we have 5 stocks: Stock A, Stock B,
Stock C, Stock D, and Stock E. (If it helps, think of these as abbreviations for stocks of
Agrium Inc., Bombardier, CIBC, Dell, and Enbridge.)
The betas on Stocks A, B, C, D, and E are, respectively,
0.5
0.95
1.52
2.1
2.32
The dollar amounts invested in Stocks A, B, C, D, and E are, respectively,
$100,000
$200,000
$400,000
$200,000
$100,000
Given all this information, what is the expected return on this portfolio?
Slide 8: Numerical Example #2 – cont.
To calculate the expected portfolio return, we follow a three-step process:
Step 1: We calculate the weights on each of the five assets in our portfolio by using the
dollar amount invested in each asset and the total amount invested in the portfolio (What
we will call the ‘total portfolio value’).
The portfolio weight on asset i is equal to the dollar amount invested in asset i, divided by
the total portfolio value, which means that, since we do not have it yet, we must first
calculate the total portfolio value:
Total portfolio value
= $ invested A + $ invested in B + $ invested in C + $ invested in D + $ invested in E
= $100,000 + $200,000 + $400,000 + $200,000 + $100,000
= $1,000,000
The weight on each asset is then calculated as:
Weight on Asset A = $100,000/$1,000,000 = 0.1
Weight on Asset B = $200,000/$1,000,000 = 0.2
Weight on Asset C = $400,000/$1,000,000 = 0.4
Weight on Asset D = $200,000/$1,000,000 = 0.2
Weight on Asset E = $100,000/$1,000,000 = 0.1
Note that we must always check that the portfolio weights add up to a sum of 1:
0.1 + 0.2 + 0.4 + 0.2 + 0.1 = 1.
Slide 9: Numerical Example #2 – cont.
Step 2: We calculate the beta of the portfolio.
The portfolio beta is calculated as the weighted sum of the betas of the assets in the
portfolio, wherein the weights are the portfolio weights we have just calculated in Step 1.
The formula for calculating the beta of our portfolio looks like this:
Portfolio beta = βP = wA(βA) + wB(βB) + wC(βC) + wD(βD) + wE(βE)
Writing out again the information we have on the weights and betas of the assets,
wA = 0.1 βA = 0.5
wB = 0.2 βB = 0.95
wC = 0.4 βC = 1.52
wD = 0.2 βD = 2.1
wE = 0.1 βE = 2.32
We plug these numbers into the portfolio beta formula, and we have
Portfolio beta = 0.1(0.5) + 0.2(0.95) + 0.4(1.52) + 0.2(2.1) + 0.1(2.32) = 1.5
Therefore, the beta of this portfolio is 1.5. We normally assume that the beta on the
market portfolio, M, is equal to 1. As such, we can say that our portfolio is a bit riskier than
the market portfolio, and therefore our portfolio should have a higher expected return than
the expected market return of 15%.
Slide 10: Numerical Example #2 – cont.
Step 3: We calculate the expected return on the portfolio using the CAPM formula:
E(RP) = Rf + βi (E(RM) – Rf)
Plugging the numbers of Rf = 0.05, E(RM) = 0.15, and βP of 1.5 into the CAPM formula, we
get
E(RP) = 0.05 + 1.5(0.15 – 0.05) = 0.05 + 1.5(0.1) = 0.05 + 0.15 = 0.2 = 20%.
The projected return on this portfolio is 20% which, as we have predicted, is higher than the
projected market return of 15%.
Slide 11: Estimation of Beta
You might ask: How did we get the asset betas in our previous example in the first place?
That is a very good question! Most frequently, they are estimated by the market model:
Ri = a + bRM
Where
Ri = returns on asset i
RM = returns on proxy of market portfolio, such as a market index
Market model is a simple regression model. It allows us to run a regression model to obtain
the beta, where the Ri is the y-variable (dependent variable), and RM is the x-variable
(independent variable). We regress Ri against RM, and the coefficient b, estimated in the
model, is our estimated beta.
How is this achieved practically? We collect historical time series data on the asset’s
returns, Ri, and the returns on a sufficiently large representation market index, such as the
S&P 500 composite index, which we use to represent market returns, R M.
Most of the time, researchers use 5 or 10 years of monthly data to estimate beta. They
then run an Ordinary Least Squares (OLS) regression to obtain the coefficients a and b, and
the coefficient b is then treated as the beta (β) of the asset.
Note that in this lecture you are assumed to be already familiar with regression analysis.
Slide 12: Data required for Market Model
What are the data required to run the market model?
Let‘s say that we wish to estimate the beta for the common stock of Telus. What data do
we need in order to use the market model to estimate Telus‘s beta?
We will need two sets of time-series historical data. First, we will need historical data on
Telus‘s common stock. We can find this information from financial websites such as
finance.yahoo.com. Let us find the monthly stock price for Telus on finance.yahoo.com, and
let’s do this for the 10-year period between December 2000 and December 2010. This
should give us 121 months of stock price data on Telus.
Second, we will need to find the return on a market portfolio. Frequently, we use a market
index such as the S&P 500 Composite Index to approximate or represent the market
portfolio. We can do this because this index contains quite a lot of the representative
companies traded in the stock market. As before, we can find historical data on the S&P
500 composite index from websites such as finance.yahoo.com. Let us do this now. Let’s
obtain the monthly S&P 500 index value for the 10-year period between December 2000
and December 2010. This should give us 121 data points of S&P 500 index values.
Since these data are stock prices and index values, we will need to convert them to stock
and index returns using these two formulas shown here:
Return on Telus stock over time t = (Price at time t – Price at time t-1) / Price at time t-1
Return on market portfolio = (Index value at time t – Index value at time t-1) / Index value
at time t-1
This is the basic formula for calculating any return over a single period. We do this for all
121 months in our raw data sets, and we should obtain 120 returns for Telus stock and 120
returns for the market portfolio (S&P 500 index).
Slide 13: Data Sample
Our data sets should look something like this table.
Month
Ri
Rm
Jan-01
0.010539
0.034637
Feb-01
-0.13006
-0.09229
Mar-01
-0.10226
-0.0642
Apr-01
-0.00079
0.076814
May-01
0.052673
0.00509
Jun-01
0.005975
-0.02504
Jul-01
-0.21232
-0.01074
Aug-01
-0.15174
-0.06411
Sep-01
-0.17444
-0.08172
Oct-01
0.273217
0.018099
Nov-01
0.096195
0.075176
Dec-01
-0.07425
0.007574
Jan-10
-0.04093
-0.03697
Feb-10
0.051141
0.028514
Mar-10
0.156108
0.058796
Apr-10
-0.01423
0.014759
May-10
-0.02181
-0.08198
Jun-10
0.062166
-0.05388
Jul-10
0.038427
0.068778
Aug-10
0.04583
-0.04745
Sep-10
0.090093
0.087551
Oct-10
-0.00449
0.036856
Nov-10
0.032606
-0.00229
Dec-10
0.012388
0.0653
and
For each month, there are two data points: one for the return on Telus shares, and one for
the return on the S&P 500 Index.
Note that I have only included the first 12 and last 12 data points for each data set here.
Slide 14: Estimating Telus Beta using Market Model
So, we regress the data series for Return on Telus stock against the data series for Market
Return, and what we get is:
RTelus = 0.0129 + (1.11 x RM)
Where
RTelus = monthly return on Telus
RM = monthly return on S&P 500 Index
The coefficient beside the Market Return is what we want: our estimated beta for Telus.
Therefore, the beta on Telus stock is estimated to be 1.11 with our 10-year monthly data
sets.
Slide 15: Using estimated Beta
So, now that we have the estimated beta for Telus, we can use it, as in our Numerical
Example #1 from before, to calculate a projected return on the Telus stock using the CAPM
formula: E(RTelus) = Rf + βTelus (E(RM) – Rf). That is, provided that we know the expected
return on the market portfolio and the risk-free rate.
Let’s say that we use a similar method to the one from before to collect data on the last
twelve months of returns on the S&P 500 index (proxying as the market portfolio) and the 1
month t-bill rate (proxying as the risk-free rate). Using these data sets, we calculate the
average return on the S&P 500 index to be 11% and the average return on the t-bill to be
8%. We then take a giant leap of faith and assume that the expected return on the market
over the next 12 months will be the same as our average S&P return of 11%. Now we are
driving with gas!!
So, we now have
E(RM) = 0.11
Rf = 0.08
Estimated Beta of Telus = 1.11
We plug these numbers into the CAPM formula, and we get
E(RTelus) = Rf + βTelus (E(RM) – Rf)
 Expected return on Telus stock = 0.08 + 1.11(0.11 – 0.08) = 0.1133 = 11.33%.
That is, given a beta of 1.11, Telus stock is expected to return 11.33% over the next twelve
months.
Slide 16: Practice Question
Since practice is more fun than skimming textbook, or for the love of all that is sacred in
knowledge, listening to audio lectures, here is a practice question for you to try your newlylearned skills. Press the pause button to pause this lecture, and take down the information
shown on this slide, try to solve the problem, and then move on to the next slide to see the
solution provided.
Given the following information, calculate the expected return on this portfolio:
S&P 500 expected return = 13%
3-month T-bill rate = 6%
There are three assets, Stock X, Stock Y, and Stock Z, in this portfolio.
Dollars invested in Stock X, Y, and Z are, respectively, $5000, $6000, and $7000.
The betas of Stock X, Y, and Z are, respectively, 1.1, 2.1 and 2.6.
Solve for the expected return on this portfolio.
Slide 17: Solution to Practice Question
The solution to the practice question:
Step 1: Calculate portfolio weights:
Total portfolio value = $5000 + $6000 + $7000 = $18000
Weight on X = wx = 5/18
Weight on Y = wy = 6/18
Weight on Z = wz = 7/18
Step 2: Calculate portfolio beta:
βP = (5/18)(1.1) + (6/18)(2.1) + (7/18)(2.6) = 2.016666667
Step 3: Calculate expected portfolio return:
E(RP) = Rf + βP(E(RM) – Rf)
= 0.06 + (2.016666667)(0.13 – 0.06)
= 0.201166667
= 20.12% (approximately)
Happy practicing!
Slide 18: End of Lecture 13
Here ends Lecture 13 on the Capital Asset Pricing Model.