Portfolio Theory and the Capital Asset Pricing Model CAPM

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Topic 8

• Portfolio Theory and the Capital Asset Pricing
  Model (CAPM)

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Outline

• Learning Objectives
• Excel Features Statistical Functions
• Individual Securities
• The Return and Risk for Portfolios
• Systematic versus Unsystematic Risk
• Capital Asset Pricing Model (CAPM)

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1 Learning Objectives

• LO 8.1 Explain and calculate covariance and
  correlation in an Excel model.
• LO 8.2 Distinguish systematic and unsystematic
  risk.
• LO 8.3 Explain the idea of diversification and
  its importance for portfolio theory.
• LO 8.4 Explain the security market line and the
  capital asset pricing model (CAPM).
• LO 8.5 Discuss the assumptions and flaws in that
  model.
• LO 8.6 Calculate the expected return on a stock
  using the capital asset pricing model (CAPM)

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2 Excel Features Statistical Functions

• SLOPE(known_y's,known_x's)
• NORMDIST(x,mean,standard_dev,cumulative)
• Normal distribution for the specified mean and
  standard deviation
• TRUE CDF
• FALSE PDF
• NORMSDIST(z)
• Standard normal distribution
• Histogram

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Excel Features Histogram

• A histograms plots the number of data points
  within certain ranges (bins).
• In our example we will plot a histogram for the
  annualized returns on the SP 500 index.

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Excel Features Histogram

• 1) Your data should be arranged in a column.

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Excel Features Histogram

• 2) Click on Data Analysis under the Tools
  drop-down menu to open the Data Analysis window.
  Then select Histogram and click on OK.
• (NOTE if you do not find Data Analysis under
  your Tools menu, go to Add-Ins under the Tools
  menu and install Data Analysis.)

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Excel Features Histogram

• 3) The Histogram window will appear.

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Excel Features Histogram

• 4) In the Histogram window, input the range of
  cells for data in the Input Range box, then
  click on OK.

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Excel Features Histogram

• 5) A new worksheet will appear with the results.

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Excel Features Histogram

• 6) You can then plot the frequencies using the
  chart wizard.

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3 Individual Securities

• The characteristics of individual securities that
  are of interest are the
• Expected Return
• Variance and Standard Deviation
• Covariance or ______ (to another security or
  index)

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Expected Return, Variance, and Covariance

• Consider the following two risky asset world.
  There is a 1/3 chance of each state of the
  economy, and the only assets are a stock fund and
  a bond fund.

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Expected Return
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Expected Return
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Variance
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Variance
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Standard Deviation
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Covariance
Deviation compares return in each state to the
expected return.
Weighting takes the product of the deviations
multiplied by the ______ of that state.
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4 The Return and Risk for Portfolios
Note that stocks have a higher/lower expected
return than bonds and higher/lower risk. Let us
turn now to the risk-return tradeoff of a
portfolio that is 50 invested in bonds and 50
invested in stocks.
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Portfolios
The rate of return on the portfolio is a ______
of the returns on the stocks and bonds in the
portfolio
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Portfolios
The expected rate of return on the portfolio is a
weighted average of the expected returns on the
securities in the portfolio.
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Portfolios
The variance of the rate of return on the two
risky assets portfolio is
where ?BS is the ______ between the returns on
the stock and bond funds.
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Portfolios
Observe the decrease in risk that diversification
offers. An equally weighted portfolio (50 in
stocks and 50 in bonds) has more/less risk than
either stocks or bonds held in isolation.
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4a The Efficient Set
100 stocks
100 bonds
We can consider other portfolio weights besides
50 in stocks and 50 in bonds
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The Efficient Set
100 stocks
100 bonds
Note that some portfolios are better than
others. They have higher returns for the same
level of risk or less.
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Portfolios with Various Correlations

• Relationship depends on correlation coefficient
• -1.0 lt r lt 1.0
• If r 1.0, no risk reduction is possible
• If r 1.0, complete risk reduction is possible

100 stocks
return
? -1.0
? 1.0
? 0.2
100 bonds
?
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The Efficient Set for Many Securities
return
Individual Assets
?P

• Consider a world with many risky assets we can
  still identify the opportunity set of risk-return
  combinations of various portfolios.

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The Efficient Set for Many Securities
return
efficient frontier
minimum variance portfolio
Individual Assets
?P

• The section of the opportunity set ______ the
  minimum variance portfolio is the efficient
  frontier.

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4b Optimal Portfolio with a Risk-Free Asset
return
100 stocks
rf
100 bonds
?

• In addition to stocks and bonds, consider a world
  that also has risk-free securities like ______.

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Riskless Borrowing and Lending
100 stocks
Capital Market Line
return
Balanced fund
100 bonds
rf
?

• Now investors can allocate their money across the
  T-bills and a balanced mutual fund.

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Riskless Borrowing and Lending
CML
return
efficient frontier
rf
?P

• With a risk-free asset available and the
  efficient frontier identified, we choose the
  capital allocation line with the ______ slope.

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Expected vs. Unexpected Returns

• Realized returns are generally not equal to
  expected returns.
• There is the expected component and the
  unexpected component.
• At any point in time, the unexpected return can
  be either positive or negative.
• Over time, the average of the unexpected
  component is ______.

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4c Announcements and News

• Announcements and news contain both an expected
  component and a surprise component.
• It is the surprise component that affects a
  stocks price and, therefore, its return.
• This is very obvious when we watch how stock
  prices move when an unexpected announcement is
  made or earnings are different than anticipated

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5 Systematic versus Unsystematic Risk

• Systematic Risk
• Risk factors that affect a large number of assets
• Also known as non-diversifiable risk or market
  risk
• Includes such things as changes in ______,
  ______, ______, etc.

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Systematic versus Unsystematic Risk

• Unsystematic Risk
• Risk factors that affect a limited number of
  assets
• Also known as unique risk and asset-specific risk
• Includes such things as ______, ______, etc.

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Returns

• Total Return expected return unexpected
  return
• Unexpected return systematic portion
  unsystematic portion
• Therefore, total return can be expressed as
  follows
• Total Return ______ ______ ______

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5a Diversification and Portfolio Risk

• Diversification can substantially reduce the
  variability of returns with/without an equivalent
  reduction in expected returns.
• This reduction in risk arises because worse than
  expected returns from one asset are offset by
  better than expected returns from another.
• However, there is a minimum level of risk that
  cannot be diversified away, and that is the
  ______ portion.

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Portfolio Risk and Number of Stocks
In a large portfolio the variance terms are
effectively diversified away, but the ______
terms are not.
?
Diversifiable Risk Nonsystematic Risk Firm
Specific Risk Unique Risk
Portfolio risk
Nondiversifiable risk Systematic Risk Market
Risk
n
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Diversifiable Risk

• The risk that can be eliminated by combining
  assets into a portfolio
• Often considered the same as unsystematic,
  unique, or asset-specific risk
• If we hold only one asset, or assets in the same
  industry, then we are exposing ourselves to risk
  that we could/could not diversify away.

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Total Risk

• Total risk
• systematic risk unsystematic risk
• The standard deviation of returns is a measure of
  total risk.
• For well-diversified portfolios, unsystematic
  risk is very ______.
• Consequently, the total risk for a diversified
  portfolio is essentially equivalent to the ______
  risk.

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5b Market Equilibrium
return
CML
efficient frontier
M
rf
?P

• With the capital allocation line identified, all
  investors choose a point along the linesome
  combination of the ______ and the ______, M. In
  a world with homogeneous expectations, M is the
  same for all investors.

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Market Equilibrium
CML
return
100 stocks
Balanced fund
rf
100 bonds
?

• Just where the investor chooses along the Capital
  Market Line depends on his risk tolerance. The
  central point is that all investors have the same
  CML.

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6 Capital Asset Pricing Model (CAPM)

• The Capital Asset Pricing Model uses beta to
  capture the systematic risk of an asset and then
  provides an equation that prices an assets with
  respect to the level of its beta.

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6a Beta (b)

• Researchers have shown that the best measure of
  the risk of a security in a large portfolio is
  the beta (b)of the security.
• Beta measures the responsiveness of a security to
  movements in the market portfolio (i.e.,
  systematic risk).

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Estimating b with Regression
Security Returns
Return on market
Ri a i biRm ei
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The Formula for Beta
Clearly, your estimate of beta will depend upon
your choice of a ______ for the market portfolio.
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6b Capital Asset Pricing Model Equation

• Expected Return on the Market

• Expected return on an individual security

Market Risk Premium
This applies to individual securities held within
well-diversified portfolios!
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Expected Return on a Security

• This formula is called the Capital Asset Pricing
  Model (CAPM)

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Relationship Between Risk Return
Expected return
Security Market Line
b
1.0
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Relationship Between Risk Return
SML
Expected return
b
1.5