ECMT 2130 – Financial
Econometrics
Lecture 3
The CAPM Model
These slides were partially based on Sinan Deng’s slides. I’m very grateful for her kind contribution!
Instructor: Felipe Pelaio
Email: felipe.queirozpelaio@sydney.edu.au
Overview
Mean-Variance Analysis
Efficient Frontier
Capital Allocation Line
Capital Market Line
Security Market Line
Capital Asset Pricing Model
Empirical CAPM Applications
• A portfolio is a combination of assets. Portfolio weights (w) add up to 1
(100%) and can be zero, positive, or negative.
• A common assumption in portfolio theory is that investors like return and
dislike risk. They seek the highest returns for a given risk or the same returns
for a lower risk.
Portfolio Mean and Variance
• The expected return of a portfolio with 2 stocks is:
() = = 11 + 22
• The variance of a portfolio with 2 stocks is:

2 = 1
21
2 + 2
22
2 + 2121212
: portfolio weight on stock i.
: expected return of stock i.
: standard deviation of stock i.
12: correlation coefficient for stocks 1 and 2.
Mean-Variance Analysis
Example
Suppose Woolworths shares have an expected return of 1.5% and a standard
deviation of 8%. Coca-Cola shares have an expected return of 2% and a standard
deviation of 10%. Their correlation coefficient is +0.4.
Some relevant questions:
• What is the expected return and standard deviation of a portfolio composed of
100% Woolworths shares? What about 100% Coca-Cola?
• Can we get a better deal? In other words, can we form a portfolio that will
give us either a higher return for the same level of risk, or a lower level of
risk for the same return? In other words, are there any pros in diversification
in this case?
Mean Variance Analysis
Example
How do portfolio returns change with different weights?
1 0 1.50% 8.00%
0.75 0.25 1.63% 7.37%
0.5 0.5 1.75% 7.55%
0.25 0.75 1.88% 8.50%
0 1 2.00% 10.00%
Mean Variance Analysis
Example
Draw the plot using our table:
Efficient Frontier
• If we compute expected returns and standard deviations for any portfolio of
securities, we get the curve F in the graph.
• The curve F illustrates the trade-off between mean and standard deviation of
different portfolios.
• The top half of the curve is called the efficient frontier. It is the combination
(curve) of portfolios that results in the highest return for a given risk. The best
portfolios lie on this curve.
Efficient Frontier
• The efficient frontier is better than the individual stocks A, B and C.
• A portfolio is able to achieve expected returns and risk levels that are not
possible with just the individual stocks. This is one of the many advantages of
diversification.
• An investor who likes return and dislikes risk would only hold a portfolio of
stocks, (almost) never just an individual stock.
Efficient Frontier
Option 1: You can save
some of your money in the
bank and earn some safe
income.
For example, if you have
$100, you can invest $50
in stocks and deposit the
other $50 into the bank.
Option 2: You can borrow
money from the bank and
invest more money in the
stock market.
For example, if you have
$100, you can borrow $50
from the bank and invest all
$150 in the stock market.
• The efficient frontier only considers shares. It ignores a very important factor:
banks (risk-free assets to be more precise).
• When a bank exists, you have more options of securities to compose your
portfolio. What can you do with your portfolio when a bank exists?
Efficient Diversification
• Investors might want to add safer assets to their portfolios. Alternatively,
investors might want to borrow money and invest in the stock market.
• The risk-free rate () is the return on an investment that is risk-less, such as
government debt securities. U.S. Treasury bills can be considered as proxies
for the risk-free rate.
• We plot the risk-free rate on the vertical axis, which gives an expected
return of and 0 risk (variance).
Efficient Diversification
• The blue point is an arbitrary portfolio on the efficient frontier. This
portfolio is made up of only risky assets.
• We can treat this blue point as a new “asset” and construct a new portfolio
with the risk-free asset.
• How does the new portfolio look like when we plot it on the graph?
Efficient Diversification
• The blue line shows all possible combinations between the risky portfolio on
the efficient frontier and the risk-free portfolio. This line is called Capital
Allocation Line.
• Point (intercept) is the portfolio with 100% invested in the risk-free asset
and the light-blue point is 100% in risky assets on the efficient frontier. Point
in the middle of these two circles is 50-50.
Capital Allocation Line (CAL)
• How many capital allocation lines can we have?
• There is an infinite number of capital allocation lines, but which one is the
best?
Capital Allocation Line (CAL)
• The tangent line is the capital market line. It represents portfolios that
optimally combine risk and return.
• It is a theoretical concept that represents all the portfolios that optimally
combine the risk-free rate of return and the market portfolio of risky assets.
The tangent point is called the market portfolio. With a risk-free asset, the
market portfolio is the best portfolio on the efficient frontier.
Capital Market Line (CML)
• But there are many portfolios on the capital market line, which is the best?
• It depends on investors’ preferences. Risk-averse investors prefer less risk, being
happy with sacrificing some return (green dot).
• Riskier investors prefer are happy to accept higher risk in exchange for higher
expected returns (blue dot).
• Points beyond the line mean that you borrow money at the risk-free rate and use the
extra money to buy portfolio M. You have therefore negative weights on risk-free
rates and more than 100% on the portfolio M.
Capital Market Line (CML)
Example
Sinan holds 50% of her portfolio in risk-free assets ( = 0.04) and 50% in
portfolio M (() = 0.12; = 0.16).
What would the portfolio expected return and standard deviation be?
() = + = 0.5 × 0.04 + 0.5 × 0.12 = 0.08

2 =
2
2 +
2
2 + 2
Note: Risk-free assets have 0 variance ( = 0 and = 0).

2 =
2
2 = 0.520.162
= 0.5 × 0.16 = 0.08
Capital Market Line (CML)
• The combination between the risk-free asset and portfolio M can be
better than the minimum variance portfolio.
• For the same return, CML gives a lower standard deviation.
• For the same risk, CML gives a higher return.
Capital Market Line (CML)
• We can measure the portfolio's risk-return tradeoff by dividing the portfolio's
risk premium by the volatility.
• The risk premium is the return on an asset minus the risk-free rate.
• The Sharpe Ratio is a measure of a portfolio's risk-return trade-off equal to
the portfolio's risk premium divided by its volatility.
ℎ =
() −

(): expected return of portfolio.
: risk-free rate.
: standard deviation of portfolio.
Sharpe Ratio
• When we rearrange the Capital Market Line equation, we obtain:
() = +


−
(): expected return of portfolio.
: risk-free rate.
: standard deviation of portfolio.
: standard deviation of market.
(): expected return of market.
• We can generalise the equation above to any individual stock/security
that bears risk (a portfolio P of one share i, for example).
Capital Asset Pricing Model (CAPM)
• The CAPM is fundamental to modern asset pricing.
• It can serve to calculate expected returns of assets that have not yet been traded, e.g.,
expected returns on infrastructure projects.
• It can help understand if a security if fairly priced or not.
• It provides an intuitive framework for understanding the risk-return
relationship.
• The Capital Asset Pricing Model can also be written as:
() = + −
(): expected return of stock i.
: risk-free rate.
(): expected market return.
: beta for stock i.
=
[(), ()]

2
• See Chapter 9 (section 9.1) of the Bodie textbook for more details.
Capital Asset Pricing Model (CAPM)
• There are a few ways to interpret the Beta in the CAPM:
• It is a measure of how sensitive the stock is to market fluctuations.
• It can be also viewed as a measure of risk relative to the market portfolio.
• It also measures the proportion of the market portfolio risk that is due to
stock i.
• We measure systematic risk (or market risk) with beta. The risk of a well-
diversified portfolio depends on the market risk of the stocks in the
portfolio.
• Stocks with > 1 tend to amplify the overall movements of the market
(riskier than the market portfolio).
• Stocks with beta < 1 tend to dampen the overall movements of the
market (less risky than the market portfolio).
• Stocks with = 1 tend to move 1-to-1 with the market.
Beta
Another interpretation of CAPM:
• Risk correction for stock = beta * market risk premium
• Required rate of return = risk-free rate + risk correction for stock
() = + −
➢ Beta = 1 implies () = .
➢ Beta = 0 implies () = .
➢ Beta > 1 implies () > .
➢ Beta < 1 implies () < .
• Notice that firm-specific risk is not part of the CAPM model: share prices
already contain all the information publicly available about the firm
(Efficient Market Hypothesis). In well-functioning markets, investors
receive high expected returns only if they’re willing to bear more risk.
Capital Asset Pricing Model (CAPM)
• We graph the relationship between expected returns of a stock and the market
risk beta. This blue line is called the Security Market Line (SML).
• Investors are rewarded for taking systematic (beta) risk. If a stock has a
higher beta that means the stock should have a higher return.
• Share A is underpriced, while share B is overpriced according to the CAPM.
“Faily-priced” assets would consequently “sit” exactly on the SML.
• The difference between the expected return implied by the CAPM and the
actual return of the stock is called alpha.
Securities Market Line (SML)
• Alpha is a measure of the performance of an investment.
• It is the difference between the actual return of the investment and the
expected return predicted by an asset pricing model, like the CAPM.
• Alpha is also called abnormal return. The formula is:
= − + −
• Regression-wise, alpha will be reflected on the intercept of the model.
• Alpha can be positive, negative or zero.
• Then a positive alpha implies that the investment is doing better than the
CAPM predicts. That is, the investment is generating more expected returns
than its risk level suggests.
• Similar idea for a negative alpha.
Alpha
• In practice, we can use/test the CAPM by relating the excess asset returns to
the excess market return, in a simple linear regression.
• Let

be the risk-free rate of return and
be the market return at time t.
• An asset’s excess return at time t is simply −

.
• The CAPM is then written as:
−

= +
−

+
• OR, if = −
and =
−
, it is simply:
= + +
• could express any firm-specific factor we did not account for (because we
didn’t have access to the information at the time of analysis). Hopefully, this
won’t be too large.
Empirical CAPM Applications
• Kenneth French’s data library (link), is an excellent resource for US return
data with proxies for market returns and risk-free rates of return.
• Consider the 5 industry sector asset portfolios in the US: Consumer,
Manufacturing, Hi-Tech, Health and Other, from Jan 2000 till May 2021.
• We consider each sector in the CAPM framework, using daily data. Fit the
single factor CAPM to each excess industry return series.
• The risk-free rate proxy is the 1-month Treasury Bill rate, scaled to be a daily
rate.
Empirical CAPM Applications
• Scatterplot of daily excess returns for Consumer sector vs Market return
plus OLS regression line.
Empirical CAPM Applications
• Each sector is estimated to have a high positive market β that seems
reasonably close to 1.
• Each sector's market α is estimated as positive, except that for the Other
sector; though all estimates are reasonably close to 0.
• All sectors seem to have a fairly similar positive linear relationship with the
market excess return, as posited by the CAPM model.
Empirical CAPM Applications
• The R-squared can be used to report how well and how strongly the CAPM
fits each industry portfolio data set.
• Four sectors have R-squared values above 80%, indicating very strong fit for
the CAPM model.
• The Health sector has an R-squared of 65%, indicating a moderately strong
fit of the CAPM model.
Empirical CAPM Applications
• We can use the 95% confidence intervals (CIs) to assess whether each
industry could be classed as high, medium or low market risk.
• We are 95% confident that the true market beta, in each case is inside the CI.
Thus, we can say we are at least 95% confident that the Consumer, Hi-Tech
and Health sectors exhibit relatively low market risk.
• We are 95% confident that the Other sector exhibits high market risk.
• Finally, the Manufacturing sector has a market beta not significantly different
to 1, indicating it exhibits medium market risk.
Empirical CAPM Applications