Chapter 8
Portfolio Selection
Learning Objectives
• State three steps involved in building a portfolio.
• Apply the Markowitz efficient portfolio selection
model.
• Describe the effect of risk-free borrowing and
lending on the efficient frontier.
• Discuss the separation theorem and its
importance to modern investment theory.
• Separate total risk into systematic and nonsystematic risk.
Portfolio Selection
• Diversification is key to optimal risk
management
• Analysis required because of the infinite
number of portfolios of risky assets
• How should investors select the best risky
portfolio?
• How could riskless assets be used?
• Investors can invest in both risky and riskless
assets and buy assets on margin or with
borrowed funds
Building a Portfolio
• Step 1: Use the Markowitz portfolio selection
model to identify optimal combinations
• Step 2: Consider borrowing and lending
possibilities
• Step 3: Choose the final portfolio based on
your preferences for return relative to risk
Portfolio Theory
• Optimal diversification takes into account all
available information (as opposed to random
diversification)
• Assumptions in portfolio theory



A single investment period (e.g., one year)
Liquid position (e.g., no transaction costs)
Preferences based only on a portfolio’s expected
return and risk
An Efficient Portfolio
• Smallest portfolio risk for a given level of
expected return
• Largest expected return for a given level of
portfolio risk
• From the set of all possible portfolios

Only locate and analyze the subset known as
the efficient set
•
Lowest risk for given level of return
An Efficient Portfolio
• Fig. 8.1 pg 219
• All other portfolios in attainable set are dominated
by efficient set
• Global minimum variance portfolio

Smallest risk of the efficient set of portfolios
• Efficient set (frontier)


Segment of the minimum variance frontier above
the global minimum variance portfolio
The set of efficient portfolios composed entirely of
risky securities generated by the Markowitz
portfolio model
Efficient Portfolios
x
E(R) A
y
C
Risk = 
B
• Efficient frontier or
Efficient set
(curved line from A
to B)
• Global minimum
variance portfolio
(represented by
point A)
Efficient Portfolios
• The basic Markowitz model is solved by a
complex technique called quadratic
programming
• The expected returns, standard deviations,
and correlation coefficients for the securities
being considered are inputs in the Markowitz
analysis
• The portfolio weights are the only variables
that can be manipulated to solve the portfolio
problem of determining efficient portfolios
1- Selecting an Optimal Portfolio of
Risky Assets
• In finance we assume investors are risk averse (i.e.,
they require additional expected return for assuming
additional risk)
• Indifference curves describe investor preferences for
risk and return (Fig. 8.2 pg 221)
• Indifference curves





Cannot intersect since they represent different levels of
desirability
Are upward sloping for risk-averse investors
Greater slope implies greater risk aversion
Investors have an infinite number of indifference curves
Higher indifference curves are more desirable
Selecting an Optimal Portfolio of
Risky Assets
• The optimal portfolio for a risk-averse
investor occurs at the point of tangency
between the investor’s highest indifference
curve and the efficient set of portfolios (Fig.
8.3 pg 221)
• This portfolio maximizes investor utility
because the indifference curves reflect
investor preferences, while the efficient set
represents portfolio possibilities
Selecting an Optimal Portfolio of
Risky Assets
• Markowitz portfolio selection model



Generates a frontier of efficient portfolios which are
equally good
Does not address the issue of riskless borrowing
or lending (investors are not allowed to use leverage)
Different investors will estimate the efficient frontier
differently (this results from estimating the inputs to
the Markowitz model differently)
•
Element of uncertainty in application (i.e., uncertainty
is inherent in security analysis)
Alternative Methods of Obtaining the
Efficient Frontier
• For a portfolio of n securities:



The full variance-covariance model of Markowitz
requires [n (n + 3)] / 2 estimates
The single-index model requires (3n + 2)
estimates
Example: calculate the required number of
estimates needed by both models for a portfolio
of 250 securities
Selecting Optimal Asset Classes
• Another way to use the Markowitz model is with
asset classes

Allocation of portfolio assets to broad asset
categories (i.e., how much of the portfolio’s assets are to
be invested in stocks, bonds, money market securities,
etc.)
• Asset class rather than individual security decisions
most important for investors

The rationale behind the asset allocation approach is
that different asset classes offer various returns and
levels of risk
•
Correlation coefficients may be quite low
Example: Selecting Optimal Asset
Classes
• (Pg 223) Consider the performance of two Canadian
portfolio managers, A and B, between 1999 and 2003.
• Manager A maintained an equally weighted portfolio with
respect to T-bills, long-term government bonds, and
common stocks.
• Manager B was more conservative and allocated 20% of
funds to each stocks and bonds, with the remaining 60% to
T-bills.
• Assume each manager matched the risk-return
performance on the relevant asset class benchmark index
for the proportion of their portfolio invested in each of the
three asset classes.
Example: Selecting Optimal Asset
Classes
• Over the period, the average annual return and
standard deviation for the asset classes were:
• T-bills: 4.01% & 1.29%
• Government bonds: 5.6% & 7.96%
• Common stocks: 7.68% & 20.42%
• Calculate the annual return earned by managers A
& B and the standard deviations of their portfolios.
Optimal Risky Portfolios
Investor Utility Function
E (R)
Efficient Frontier
*

2- Borrowing and Lending Possibilities
• Risk-free assets




Certain-to-be-earned expected return (this is
nominal return and not real return which is
uncertain since inflation is uncertain)
Zero variance
No covariance or correlation with risky assets
(ρ_RF = 0 since the risk-free rate is a constant
which by nature has no correlation with the
changing returns on risky securities)
Usually proxied by a Treasury Bill
•
Amount to be received at maturity is free of default
risk, known with certainty
Borrowing and Lending Possibilities
• Adding a risk-free asset extends and
changes the efficient frontier
• Investors can now invest part of their wealth
in the risk-free asset and the remainder in
any of the risky portfolios in the Markowitz
efficient set
• It allows the Markowitz portfolio theory to be
extended in such a way that the efficient
frontier is completely changed
Risk-Free Lending
L
B
E(R)
T
Z
X
RF
A
Risk
• Riskless assets can be
combined with any
portfolio in the efficient
set AB (comprised only of
risky assets)
 Z implies lending
• Set of portfolios on line
RF to T dominates all
portfolios below it
Impact of Risk-Free Lending
• If wRF placed in a risk-free asset and (1- wRF) in risky
portfolio X:
 Expected portfolio return
E(R p )  w RFRF  (1 - w RF )E(R X )
▪ Risk of the portfolio (correlation and covariance
for the risk-free asset is zero)
 p  (1 - w RF ) X
• Expected return and risk of the portfolio with lending
is a weighted average
Example: Impact of Risk-Free Lending
• (Pg 226) Assume that portfolio X has an expected
return of 15% with a standard deviation of 30%,
and that the risk-free security has an expected
return of 3%.
• If 60% of investable funds is placed in RF and 40%
in portfolio X, calculate the expected return and
standard deviation of the resulting portfolio.
Impact of Risk-Free Lending
• An investor could change positions on the line RF-X
by varying wRF. As more of the investable funds are
placed in the risk-free asset, both the expected
return and the risk of the portfolio decline.
• It is apparent that the segment of the efficient
frontier below X (i.e., A to X) is now dominated by
the line RF-X.
• Therefore, the ability to invest in RF provides
investors with a more efficient set of portfolios from
which to choose which lies along line RF-T.
Borrowing Possibilities
• Investor no longer restricted to own wealth
• One way to accomplish this borrowing is to buy
stocks on margin
• Interest paid on borrowed money


Higher returns sought to cover expense
Assume borrowing at risk-free rate (RF)
• Risk will increase as the amount of borrowing
increases

Financial leverage
Borrowing Possibilities
• Borrowing additional investable funds and investing
them together with the investor’s own wealth allows
investors to seek higher expected returns while
assuming greater risk
• These borrowed funds can be used to leverage the
portfolio position beyond the tangency point T,
which represents 100% of an investor’s wealth in
Risky asset portfolio T
• The straight line RF-T is now extended upward,
and can be designated RF-T-L
The New Efficient Set
• Risk-free investing and borrowing creates a
new set of expected return-risk possibilities
• Addition of risk-free asset results in


A change in the efficient set from an arc to a
straight line tangent to the feasible set without
the riskless asset
Chosen portfolio depends on investor’s riskreturn preferences (i.e., investors can be
anywhere they choose on line RF-T-L,
depending on their risk-return preference)
The New Efficient Set
• In real life, it is unlikely that the typical
investor can borrow at the same rate as that
offered by riskless securities because
borrowing rates generally exceed lending
rates
• The straight line RF-T-L will be transformed
into a line with a “kink” at point T (Fig. 8.6 pg
230)
3- Portfolio Choice
• The more conservative the investor, the more that
is placed in risk-free lending and the less in
borrowing (i.e., closer to the risk-free rate RF)
• The more aggressive the investor, the less that is
placed in risk-free lending and the more in
borrowing (i.e., closer to, or on, point T,
representing full investment in a portfolio of risky
assets)
• Even more aggressive investors could go beyond
point T by using leverage to move up the line RF-TL
The Separation Theorem
• Investors use their preferences (reflected in
an indifference curve) to determine their
optimal portfolio along the new efficient
frontier RF-T-L
• Separation Theorem
 The investment decision (which portfolio
of risky assets to hold) is separate from
the financing decision (how to allocate
investable funds between the risk-free
asset and the risky asset)
Separation Theorem
• All investors


Invest in the same portfolio of risky assets T
Attain any point on the straight line RF-T-L by
either borrowing or lending at the rate RF,
depending on their preferences
• Risky portfolios are not tailored to each
individual’s taste
• The separation theorem argues that the
tailoring process is inappropriate
Implications of Portfolio Selection
• Investors should focus on risk that cannot be
managed by diversification
• Total risk =

Systematic (non-diversifiable) risk
+

Non-systematic (diversifiable) risk
Systematic risk
• Systematic risk (unavoidable)



Variability in a security’s total returns directly
associated with economy-wide events
Common to virtually all securities
E.g., interest rate risk, market risk, and inflation
risk
Non-Systematic Risk
• Non-Systematic Risk


Variability of a security’s total return not related to
general market variability
Diversification decreases this risk
• The relevant risk of an individual stock is its
contribution to the riskiness of a well-diversified
portfolio
Portfolios rather than individual assets most
important
Recent Canadian research suggests that 70 or more
stocks are required to obtain a well diversified
portfolio

Portfolio Risk and Diversification
p %
Total risk
35
Diversifiable
Risk
20
Systematic Risk
0
10
20
30
40
......
Number of securities in portfolio
100+
Appendix 8-A: Modern Portfolio Theory
and the Portfolio Management Process
• Demonstrated by the impact on regulations
governing the investment behaviour of professional
money managers
• Require them to adhere to “prudence, loyalty,
reasonable administrative cost, and diversification”
• Portfolio management process:




Designing an investment policy
Developing and implementing an asset mix
Monitoring the economy, the markets, and the client
Adjusting the portfolio and measuring performance