Portfolio Theory and the Capital Asset
Pricing Model
Sinan Deng
The University of Sydney
Page 2The University of Sydney
References and Readings
2022 CFA level 1:
• SS17 Portfolio Management: R48 Portfolio Management: An Overview
• SS17 Portfolio Management: R49 Portfolio Risk and Return: Part I
• SS17 Portfolio Management: R50 Portfolio Risk and Return: Part II
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Overview
Portfolio
Theory and
the Capital
Asset Pricing
Model
Mean
Variance
Analysis
Portfolio Overview
Mean Variance Analysis
Two Asset Portfolio
Markowitz’s
Theory
Minimum-Variance Frontier
Efficient Frontier
Capital
Market
Theory
Capital Allocation Line
Capital Market Line
Capital Asset Pricing Model (CAPM)
Security Market Line
Nonsystematic Risk and Systematic Risk
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Overview
Mean
Variance
Analysis
Portfolio Overview
Mean Variance Analysis
Two Asset Portfolio
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Portfolio Overview
• Portfolio management is about creating a diversified approach to meet
one's investment goals.
• Diversification provides an investor with a way to reduce the risk without
necessarily decreasing their expected rate of return.
• A portfolio is a combination of assets.
• Portfolio weights add up to 1 (100%) and can be zero, positive, or
negative.
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Portfolio Overview
• A common assumption in portfolio theory is that investors: like return and
hate risk.
• Investors want the highest returns for a given risk or the same returns for a
lower risk.
• Dominance principle: Portfolio A is preferred or dominates portfolio B if:() ≥ () & () ≤ ()
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Mean Variance Analysis
• The expected return of a portfolio with 2 stocks is: (!) = ! ="" + ##
• The variance of a portfolio with 2 stocks is !# = "#"# + #### +2"#"# = "#"# + #### + 2"#"#"#$: portfolio weight on stock $: expected return of stock $: standard deviation of stock "#: correlation coefficient for stocks 1 and 2 "# = %&'!"(!("
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Two Asset Portfolio
• Suppose Woolworths has an expected return of 1.5% and a standard
deviation of 8%. Coca-Cola has an expected return of 2% and a
standard deviation of 10%. Their correlation coefficient is 0.4.
1. If you invest all money in Woolworths, what is the expected return and
standard deviation of your portfolio?
2. If you invest all money in Coca-Cola, what is the expected return and
standard deviation of your portfolio?
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Two Asset Portfolio
• How does your portfolio do with different weights? What conclusions can
you draw from this table?
()
1 0 1.5% 8.00%
0.75 0.25 1.625% ?
0.5 0.5 1.75% ?
0.25 0.75 1.875% ?
0 1 2% 10.00%
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Two Asset Portfolio
• How does your portfolio do with different weights? What conclusions can
you draw from this table?
()
1 0 1.5% 8.00%
0.75 0.25 1.625% 7.37%
0.5 0.5 1.75% 7.55%
0.25 0.75 1.875% 8.50%
0 1 2% 10.00%
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Two Asset Portfolio
• Let’s draw the plot:
0
0.5
1
1.5
2
2.5
0 2 4 6 8 10 12
Ex
pe
ct
ed
R
et
ur
n
Standard Deviation
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Overview
Markowitz’s
Theory
Minimum-Variance Frontier
Efficient Frontier
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Minimum-Variance Frontier
• If we compute all possible portfolio expected returns and standard
deviations, we get the curve F in the graph. The curve F represents the
mean-standard deviation trade-off, and we can also call it the minimum-
variance frontier if we have a many-asset portfolio.
• Global minimum-variance portfolio: The portfolio with the minimum
variance among all portfolios of risky assets, which is the left-most point
on the F curve.
Expected
return
Global minimum
variance portfolio
Standard deviation
F Curve
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Efficient Frontier
• The curve that lies above and to the right of the global minimum-variance
portfolio is referred to as the Markowitz efficient frontier.
• Those portfolios that have the greatest expected return with a given level
of risk make up the efficient frontier. Efficient portfolio are well-
diversified or fully-diversified.
Expected
return
Global minimum
variance portfolio
Standard deviation
F Curve
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Efficient Frontier
• The efficient frontier is better than the individual stocks A, B and C. The
portfolio is able to obtain expected returns and risks that are not possible
with just the individual stocks.
• An investor who likes return and dislikes risk would only hold a portfolio
of stocks, never just an individual stock.
Expected
return
Global minimum
variance portfolio
Standard deviation
. A
.C . B
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Efficient Frontier
• Based on this analysis, a rational investor will only choose portfolios on the
efficient frontier – but which portfolio?
• The portfolio chosen will depend on each individual investor’s attitude
toward risk:
• Those who are more risk-averse will choose portfolios close to point G –
these portfolios have less risk and lower returns.
• Those who are less risk-averse will choose portfolios further to the right on
the efficient frontier – these portfolios offer higher expected returns but
carry greater risk.
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Overview
Capital
Market
Theory
Capital Allocation Line
Capital Market Line
Capital Asset Pricing Model (CAPM)
Security Market Line
Nonsystematic Risk and Systematic Risk
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Capital Allocation Line
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 stocks but ignores a very important
factor, which is the bank.
• When a bank exists, you have more options for your portfolio. What can
you do with your portfolio when a bank exists?
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Capital Allocation Line
• Investors might want to add something very safe to their choices.
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. A U.S. Treasury Bill can be considered as a
proxy 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).
Expected
return
)
Standard deviation
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Capital Allocation Line
• The blue point is an arbitrary portfolio on the efficient frontier. This
portfolio is made up of 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?
Expected
return
)
Standard deviation
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Capital Allocation Line
• 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 (CAL).
• Point ) is 100% in the risk-free asset and the blue point is 100% in risky
assets on the efficient frontier. Point in the middle is 50-50.
Expected
return
)
Standard deviation
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Capital Allocation Line
• How many capital allocation lines can we have?
• There is an infinite number of capital allocation lines, but which one is the
best?
Expected
return
)
Standard deviation
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Capital Market Line
• The tangent line is the capital market line (CML). 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.
Expected
return M
)
Standard deviation
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Capital Market Line
• But there are many portfolios on the capital market line, which is the best?
• It depends on investors’ preferences. Conservative investors prefer less
risk and less return, such as green dot. Riskier investors prefer more risk
and more return, such as blue dot.
• Points beyond the line means that you borrow money under the risk-free
rate and use the extra money to buy portfolio M. You put negative
weights on risk-free rates and more than 100% on the portfolio M.
Expected
return M
)
Standard deviation
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Capital Market Line
• The combination between risk-free assets and the portfolio M can do
better than the global minimum variance portfolio.
• For the same return, CML gives a lower standard deviation.
• For the same risk, CML gives a higher return.
Expected
return M
)
Standard deviation
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The Capital Asset Pricing Model (CAPM)
• 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 ! = ! − )!
Ø !: expected return of portfolio
Ø ): risk-free rate
Ø !: standard deviation of portfolio
Page 27The University of Sydney
The Capital Asset Pricing Model (CAPM)
• When we rearrange the Capital Market Line equation,
! = ) + !* * − )
Ø !: expected return of portfolio
Ø ): risk-free rate
Ø !: standard deviation of portfolio
Ø *: standard deviation of market
Ø *: expected return of market
• This turns out to lead the Capital Asset Pricing Model (CAPM), which is
fundamental to modern asset pricing.
• CAPM provides an intuitive framework for understanding the risk-return
relationship. It describes the theoretical link between the expected rate of
return for an asset and its risk.
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The Capital Asset Pricing Model (CAPM)
• The Capital Asset Pricing Model can also be written as:(!) = ) + ! * − )
Ø (!): expected return of portfolio
Ø ): risk-free rate
Ø !: beta of portfolio
Ø (*): expected return of market
• Another interpretation of CAPM:
• Risk correction for stock = beta * market risk premium
• Required rate of return = risk-free rate + risk correction for stock
Page 29The University of Sydney
The Capital Asset Pricing Model (CAPM)
• Beta is a measure of a stock's market risk. It measures how sensitive the
stock is to market movements.
• 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 beta >1 tend to amplify the overall movements of the market.
• Stocks with beta <1 tend to dampen the overall movements of the
market.
• Stocks with beta =1 tend to move 1-1 with the market.
Page 30The University of Sydney
The Capital Asset Pricing Model (CAPM)
• Alpha is a measure of the skill or performance of an investment. It is the
difference between the expected return predicted by an asset pricing
model, like the CAPM, and the actual return of the investment.
• Alpha is also called abnormal return. The formula is: = ($) − () + $ (*−) )
Ø ($): expected return of stock
Ø ): risk-free rate
Ø $: beta for stock
Ø (*): expected return of market
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The Capital Asset Pricing Model (CAPM)
• Note ($) here is not the same as the CAPM expected return but our
best estimate using the actual returns of the investment. Thus, alpha is
always calculated after returns have been realized.
• It is important to note Alpha can also 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 while taking less risk.
• Similarly, a negative alpha implies that the investment is doing worse than
the CAPM predicts.
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The Security Market Line
• Security market line (SML): Graphical representation of CAPM.
• Investors are rewarded for taking systematic (beta) risk. If a stock has a
higher beta that means the stock should return more.
• A is underpriced and B is overpriced by the CAPM model. The difference
between the expected return implied by the CAPM and the actual return
of the stock is called alpha.
Expected A
return
)
B

market risk beta
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Application of CAPM
• The risk-free rate is 3% p.a., and the expected return on the market
portfolio is 11%. Shares in company A have a beta of 1.2 and are
currently priced to provide a return of 13%. Which of the following
statements is correct?
• a) A is underpriced and lies below the security market line
• b) A is overpriced and lies above the security market line
• c) A is correctly priced and lies on the security market line
• d) A is underpriced and lies above the security market line
• e) A is overpriced and lies below the security market line
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Nonsystematic Risk and Systematic Risk
Nonsystematic risk (or idiosyncratic, diversifiable, company-specific risk):
• Nonsystematic risk is local or limited to a particular asset or industry.
• The risk that disappears in the portfolio construction process.
Systematic risk (or non-diversifiable, market risk)
• The risk that cannot be diversified away.
• Total variance = systematic variance + nonsystematic variance, or
• Total risk = systematic risk + nonsystematic risk
Since nonsystematic risk can be eliminated through diversification, only
systematic risk is compensated.
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Nonsystematic Risk and Systematic Risk
Risk vs. Number of portfolio Assets