Risk and Return – Part 3
For 9.220, Term 1, 2002/03
02_Lecture14.ppt
Student Version
Outline
1.
2.
3.
4.
Introduction
The Markowitz Efficient Frontier
The Capital Market Line (CML)
The Capital Asset Pricing Model
(CAPM)
5. Summary and Conclusions
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.
The Opportunity Set and
The Efficient Set
Portfolio Expected Return
Expected Return and Standard Deviation for Portfolios of Two Assets Plotted for Different
Portfolio Weights
100%
Stock 1
30%
25%
20%
15%
The portfolios in this
area are all dominated.
100%
Stock 2
10%
5%
0%
0%
5%
10%
15%
Portfolio Standard Deviation
20%
25%
The Opportunity Set when considering
all risky securities
E[R]
Individual Assets

Consider all the risky assets in the world; we
can still identify the Opportunity Set of riskreturn combinations of various portfolios.
The Efficient Set when considering all
risky securities
The section of the
frontier above the
minimum variance E[R]
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.
minimum
variance
portfolio
Individual Assets

Optimal Risky Portfolio with a RiskFree Asset
 In addition to risky
assets, consider a
world that also has E[R]
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.

The riskfree asset:
riskless lending and borrowing
 Consider combinations of
the risk-free asset with a E[R]
portfolio, A, on the
Efficient Frontier.
 With a risk-free asset
available, taking a long f
position (positive
portfolio weight in f)
Rf
gives us risk-free lending
combined with A.
 Taking a short f position
(negative portfolio
weight in f) gives us riskfree borrowing combined
with A.
Portfolio A
P
The riskfree asset:
riskless lending and borrowing
 Which combination of
f and portfolios on
the Efficient Frontier
are best?
E[R]
Rf
P
What is the optimal strategy
for every investor?
M: The Market Portfolio
CML
The combination of f and
E[R]
portfolios on the Efficient
Frontier that are best are…
M
All investors choose a point
along the line…
CML stands for
the
Capital Market
Line
Rf
In a world with
homogeneous expectations,
M is the same for all
investors.
P
A new separation theorem
CML
This separation
E[R]
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
Rf
total portfolio.
All investors should
have the same risky
component, M!
M
P
Given Separation, what does an
investor choose?
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.
CML
E[R]
M
Rf
P
The Capital Market Line
(CML) Equation
The CML equation can
be written as follows:
 
E REPi
 ERM   R f
 R f   EPi 
M

Where
 EPi = efficient portfolio i (a portfolio on the CML composed of
the risk-free asset, f, and M)
Note: the CML is our first
 E[ ] is the expectation operator
formal relationship between
 R = return
risk and expected return.
 σ = standard deviation of return Unfortunately it is limited in
its use as it only works for
 f denotes the risk-free asset
perfectly efficient portfolios:
 M denotes the market portfolio
composed of f and M.



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 j’s 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 ρjM, the correlation
between the returns of asset j and the returns of M.
 Asset j’s total risk is defined by σj
 The part of σj that can be diversified away is (1-ρjM)● σj
 The part of σj that remains is ρjM● σj
Non-diversifiable risk and the
relation to expected return.
We can extend the CML to a single asset by substituting in the
asset’s non-diversifiable risk for σEPi:
 
CML : E REPi  R f   EPi
 E RM   R f
 
M




sub in ρ jM  σ j for σ EPi
 
SML : E R j  R f 
Let  j 
ρ jM  σ j
σM
 E RM   R f 
ρ jM  σ j
M
 
SML : E R j  R f  β j  E RM   R f 
SML stands for Security
Market Line. It relates
expected return to β and
is the fundamental
relationship specified by
the CAPM.
The Security’s Beta
 The important measure of the risk of a security in a large
portfolio is thus the beta ()of the security.
 Beta measures the non-diversifiable risk of a security –
i.e., the risk related to movements in the market
portfolio.
iM   i Cov( Ri , RM ) iM   i   M
i 


2
2
M
M
M
Security Returns
Estimating  with regression
Return on
market
Know your betas!




The possible range for β is -∞ to +∞
The value of βM is…
The value of βf is…
For a portfolio, if you know the individual
securities’ β’s, then the portfolio β is…
n
 p   xi   i  x11  x2  2  ...xn  n
i 1
where the xi values are the security weights.
Estimates of  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
Examples
 What would be your portfolio beta, βp, if you had
weights in the first four stocks of 0.2, 0.15, 0.25, and
0.4 respectively.
 What would be E[Rp]? Calculate it two ways.
 Suppose σM=0.3 and this portfolio had ρpM=0.4, what
is the value of σp?
 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 β 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?
Summary and Conclusions
 The CAPM is a theory that provides a relation between
expected return and an asset’s 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.