Risk and Return

1
Risk and Return Part 3

• For 9.220, Term 1, 2002/03
• 02_Lecture14.ppt
• Student Version

2
Outline

1. Introduction
2. The Markowitz Efficient Frontier
3. The Capital Market Line (CML)
4. The Capital Asset Pricing Model (CAPM)
5. Summary and Conclusions

3
Introduction

• We have seen that holding portfolios of more than
  one asset gives the potential for
  diversification.
• We will now look at what might be an optimal
  strategy for portfolio construction being well
  diversified.
• We extend the results from this into a model of
  Risk and Return called the Capital Asset Pricing
  Model (CAPM) that theoretically holds for
  individual securities and for portfolios.

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The Opportunity Set and The Efficient Set
100 Stock 1
The portfolios in this area are all dominated.
100 Stock 2
5
The Opportunity Set when considering all risky
securities
ER
Individual Assets
?

• Consider all the risky assets in the world we
  can still identify the Opportunity Set of
  risk-return combinations of various portfolios.

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The Efficient Set when considering all risky
securities

• The section of the frontier above the minimum
  variance portfolio is the efficient set. It is
  named the Markowitz Efficient Frontier after
  researcher Harry Markowitz (Nobel Prize in
  Economics, 1990) who first discussed it in 1959.

ER
efficient frontier
minimum variance portfolio
Individual Assets
?
7
Optimal Risky Portfolio with a Risk-Free Asset

• In addition to risky assets, consider a world
  that also has risk-free securities like T-bills.
• We can now consider portfolios that are
  combinations of the risk-free security, denoted
  with the subscript f and risky portfolios along
  the Efficient Frontier.

ER
?
8
The riskfree asset riskless lending and
borrowing

• Consider combinations of the risk-free asset with
  a portfolio, A, on the Efficient Frontier.
• With a risk-free asset available, taking a long f
  position (positive portfolio weight in f) gives
  us risk-free lending combined with A.
• Taking a short f position (negative portfolio
  weight in f) gives us risk-free borrowing
  combined with A.

ER
Portfolio A
Rf
?P
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The riskfree asset riskless lending and
borrowing

• Which combination of f and portfolios on the
  Efficient Frontier are best?

ER
Rf
?P
What is the optimal strategy for every investor?
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M The Market Portfolio
CML

• The combination of f and portfolios on the
  Efficient Frontier that are best are
• All investors choose a point along the line
• In a world with homogeneous expectations, M is
  the same for all investors.

ER
M
CML stands for the Capital Market Line
Rf
?P
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A new separation theorem
CML
ER

• This separation theorem states that the market
  portfolio, M, is the same for all investors. They
  can separate their level of risk aversion from
  their choice of the risky component of their
  total portfolio.
• All investors should have the same risky
  component, M!

M
Rf
?P
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Given Separation, what does an investor choose?
CML

• While all investors will choose M for the risky
  part of their portfoio, the point on the CML
  chosen depends on their level of risk aversion.

ER
M
Rf
?P
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The Capital Market Line (CML) Equation

• The CML equation can
• be written as follows
• Where
• EPi efficient portfolio i (a portfolio on the
  CML composed of the risk-free asset, f, and M)
• E is the expectation operator
• R return
• s standard deviation of return
• f denotes the risk-free asset
• M denotes the market portfolio

Note the CML is our first formal relationship
between risk and expected return. Unfortunately
it is limited in its use as it only works for
perfectly efficient portfolios composed of f and
M.
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The Capital Asset Pricing Model (CAPM)

• If investors hold the market portfolio, M, then
  the risk of any asset, j, that is important is
  not its total risk, but the risk that it
  contributes to M.
• We can divide asset js risk into two components
  the risk that can be diversified away, and the
  risk that remains even after maximum
  diversification.
• The division is found by examining ?jM, the
  correlation between the returns of asset j and
  the returns of M.
• Asset js total risk is defined by sj
• The part of sj that can be diversified away is
  (1-?jM)? sj
• The part of sj that remains is ?jM? sj

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Non-diversifiable risk and the relation to
expected return.

• We can extend the CML to a single asset by
  substituting in the assets non-diversifiable
  risk for sEPi

SML stands for Security Market Line. It relates
expected return to ß and is the fundamental
relationship specified by the CAPM.
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The Securitys Beta

• The important measure of the risk of a security
  in a large portfolio is thus the beta (b)of the
  security.
• Beta measures the non-diversifiable risk of a
  security i.e., the risk related to movements in
  the market portfolio.

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Estimating b with regression
Security Returns
Return on market
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Know your betas!

• The possible range for ß is -8 to 8
• The value of ßM is
• The value of ßf is
• For a portfolio, if you know the individual
  securities ßs, then the portfolio ß is
• where the xi values are the security weights.

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Estimates of b for Selected Stocks
Stock Beta
C-MAC Industries 1.85
Nortel Networks 1.61
Bank of Nova Scotia 0.83
Bombardier 0.71
Investors Group. 1.22
Maple Leaf Foods 0.83
Roger Communications 1.26
Canadian Utilities 0.50
TransCanada Pipeline 0.24
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Examples

• What would be your portfolio beta, ßp, if you had
  weights in the first four stocks of 0.2, 0.15,
  0.25, and 0.4 respectively.
• What would be ERp? Calculate it two ways.
• Suppose sM0.3 and this portfolio had ?pM0.4,
  what is the value of sp?
• Is this the best portfolio for obtaining this
  expected return?
• What would be the total risk of a portfolio
  composed of f and M that gives you the same ß as
  the above portfolio?
• How high an expected return could you achieve
  while exposing yourself to the same amount of
  total risk as the above portfolio composed of the
  four stocks. What is the best way to achieve it?

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Summary and Conclusions

• The CAPM is a theory that provides a relation
  between expected return and an assets risk.
• It is based on investors being well-diversified
  and choosing non-dominated portfolios that
  consist of combinations of f and M.
• While the CAPM is useful for considering the
  risk/return tradeoff, and it is still used by
  many practitioners, it is but one of many
  theories relating return to risk (and other
  factors) so it should not be regarded as a
  universal truth.