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Chapter 8
AN INTRODUCTION TO
PORTFOLIO MANAGEMENT
Chapter 8 Questions
What do we mean be risk aversion, and what
evidence indicates that investors are
generally averse to risk?
What are the basic assumptions behind the
Markowitz portfolio theory?
What do we mean by risk, and what are some
of the measures of risk used in investments?
How does one compute the expected rate of
return for an individual risky asset or a
portfolio of assets?
Chapter 8 Questions
How does one compute the standard
deviation of rates of return for an
individual risky asset?
What do we mean by the covariance
between rates of return, and how is it
computed?
What is the relationship between
covariance and correlation?
Chapter 8 Questions
What is the formula for the standard deviation
for a portfolio of risky assets, and how does it
differ from the standard deviation of an
individual risky asset?
Given the formula for the standard deviation
of a portfolio, why and how does one diversify
a portfolio?
What happens to the standard deviation of a
portfolio when we change the correlation
between the assets in the portfolio?
Chapter 8 Questions
What is the risk-return efficient frontier
of risky assets?
Is it reasonable for alternative investors
to select different portfolios from the
portfolios on the efficient frontier?
What determines which portfolio on the
efficient frontier is selected by an
individual investor?
Background
Assumptions
As an investor you want to maximize the
returns for a given level of risk.
Your portfolio includes all of your assets, not
just financial assets
The relationship between the returns for
assets in the portfolio is important.
A good portfolio is not simply a collection of
individually good investments.
Risk Aversion
Portfolio theory assumes that investors
are averse to risk
Given a choice between two assets with
equal expected rates of return, risk
averse investors will select the asset
with the lower level of risk
It also means that a riskier investment
has to offer a higher expected return or
else nobody will buy it
Are investors risk
averse?
The popularity of insurance of various
types attests to risk aversion
Yield on bonds increase with risk
classifications from AAA to AA to A….,
indicating that investors require risk
premiums as compensation
Experimental psychology also confirms
that humans tend to be risk averse
Are investors always
risk averse?
Risk preference may
have to do with amount
of money involved risking only small
amounts.
Trips to the casino
might seem to refute
risk aversion, but
realize that gaming is
best thought of as
entertainment, not
investing
Definition of Risk
One definition: Uncertainty of future
outcomes
Alternative definition: The probability of
an adverse outcome
We will discuss several measures of
risk that are used in developing portfolio
theory
Markowitz Portfolio
Theory
Derives the expected rate of return for a
portfolio of assets and an expected risk
measure
Markowitz demonstrated that the variance of
the rate of return is a meaningful measure of
portfolio risk under reasonable assumptions
The portfolio variance formula shows how to
effectively diversify a portfolio
Markowitz Portfolio
Theory
Assumptions
Investors consider each investment
alternative as being presented by a
probability distribution of expected returns
over some holding period.
Investors minimize one-period expected
utility, and their utility curves demonstrate
diminishing marginal utility of wealth.
Investors estimate the risk of the portfolio on
the basis of the variability of expected returns.
Markowitz Portfolio
Theory
Assumptions
Investors base decisions solely on expected
return and risk, so their utility curves are a
function of expected return and the expected
variance (or standard deviation) of returns
only.
For a given risk level, investors prefer higher
returns to lower returns. Similarly, for a given
level of expected returns, investors prefer
less risk to more risk.
Markowitz Portfolio
Theory
Under these five assumptions, a single
asset or portfolio of assets is efficient if
no other asset or portfolio of assets
offers higher expected return with the
same (or lower) risk, or lower risk with
the same (or higher) expected return.
Alternative Measures of
Risk
Variance or standard deviation of
expected return (Main focus)
Based on deviations from the mean return
 Larger values indicate greater risk

Other measures
Range of returns
 Returns below expectations


Semivariance – measures deviations only
below the mean
Expected Rates of
Return
Individual risky asset (Chapter 2)


Weighted average of all possible returns
Probabilities serve as the weights
Portfolio


Weighted average of expected returns (Ri) for the
individual investments in the portfolio
Percentages invested in each asset (wi) serve as
the weights
E(Rport) = S wi Ri
Variance & Standard
Deviation of Returns
Individual Investment (Chapter 2)
Standard deviation is the positive
square root of the variance
Both measures are based on deviations
of each possible return (Ri) from the
expected return (E(R))
Variance:
s2 = SPi(Ri-E(R))2
Variance & Standard
Deviation of Returns
Before calculating the portfolio variance and
standard deviation, several other measures
need to be understood
Covariance


Measures the extent to which two variables move
together
For two assets, i and j, the covariance of rates of
return is defined as:
Covij = E{[Ri,t - E(Ri)][Rj,t - E(Rj)]}
Variance & Standard
Deviation of Returns
Correlation coefficient


Values of the correlation coefficient (r) go from -1
to +1
Standardized measure of the linear relationship
between two variables
rij = Covij/(sisj)
Covij= covariance of returns for securities i and j
si= standard deviation of returns for security i
sj= standard deviation of returns for security j
Portfolio Standard
Deviation Formula
s port 
n
w s
i 1
2
i
n
2
i
n
  w i w j Cov ij
i 1 i 1
where :
s port  the standard deviation of the portfolio
Wi  the weights of the individual assets in the portfolio, where
weights are determined by the proportion of value in the portfolio
s i2  the variance of rates of return for asset i
Cov ij  the covariance between th e rates of return for assets i and j,
where Cov ij  rijs is j
Portfolio Standard
Deviation Calculation
The portfolio standard deviation is a function
of:


The variances of the individual assets that make
up the portfolio
The covariances between all of the assets in the
portfolio
The larger the portfolio, the more the impact
of covariance and the lower the impact of the
individual security variance
Implications for
Portfolio Formation
Assets differ in terms of expected rates of
return, standard deviations, and correlations
with one another


While portfolios give average returns, they give
lower risk
Diversification works!
Even for assets that are positively correlated,
the portfolio standard deviation tends to fall
as assets are added to the portfolio
Implications for
Portfolio Formation
Combining assets together with low
correlations reduces portfolio risk more
The lower the correlation, the lower the
portfolio standard deviation
 Negative correlation reduces portfolio risk
greatly
 Combining two assets with perfect
negative correlation reduces the portfolio
standard deviation to nearly zero

Estimation Issues
Results of portfolio analysis depend on
accurate statistical inputs
Estimates of
Expected returns
 Standard deviations
 Correlation coefficients


With 100 assets, 4,950 correlation estimates
Estimation risk refers to potential errors
Estimation Issues
With assumption that stock returns can
be described by a single market model,
the number of correlations required
reduces to the number of assets
Single index market model:
R i  a i  bi R m   i
bi = the slope coefficient that relates the returns for security i
to the returns for the aggregate stock market
Rm = the returns for the aggregate stock market
The Efficient Frontier
The efficient frontier represents that set
of portfolios with the maximum rate of
return for every given level of risk, or the
minimum risk for every level of return
Frontier will be portfolios of investments
rather than individual securities

Exceptions being the asset with the highest
return and the asset with the lowest risk
Efficient Frontier and
Alternative Portfolios
E(R)
Efficient
Frontier
A
B
C
Standard Deviation of Return
The Efficient Frontier
and Portfolio Selection
Any portfolio that plots “inside” the efficient
frontier (such as point C) is dominated by
other portfolios

For example, Portfolio A gives the same expected
return with lower risk, and Portfolio B gives greater
expected return with the same risk
Would we expect all investors to choose the
same efficient portfolio?

No, individual choices would depend on relative
appetites return as opposed to risk
The Efficient Frontier
and Investor Utility
An individual investor’s utility curve specifies
the trade-offs she is willing to make between
expected return and risk
Each utility curve represent equal utility;
curves higher and to the left represent greater
utility (more return with lower risk)
The interaction of the individual’s utility and
the efficient frontier should jointly determine
portfolio selection
The Efficient Frontier
and Investor Utility
The optimal portfolio has the highest
utility for a given investor
It lies at the point of tangency between
the efficient frontier and the utility curve
with the highest possible utility
Selecting an Optimal
Risky Portfolio
E(R port )
U3’
U2’
U1’
Y
U3
X
U2
U1
E(s port )
Investor Differences and
Portfolio Selection
A relatively more conservative investor
would perhaps choose Portfolio X

On the efficient frontier and on the highest
attainable utility curve
A relatively more aggressive investor
would perhaps choose Portfolio Y

On the efficient frontier and on the highest
attainable utility curve