bài giảng investment analysis and management chapter 07

Portfolio Theory
Chapter 7
Charles P. Jones, Investments: Analysis and
Management,
Tenth Edition, John Wiley & Sons
Prepared by
G.D. Koppenhaver, Iowa State University

7-1


Investment Decisions



Involve uncertainty
Focus on expected returns




Estimates of future returns needed to
consider and manage risk

Goal is to reduce risk without affecting
returns



Accomplished by building a portfolio
Diversification is key

7-2


Dealing With Uncertainty






Risk that an expected return will not be
realized
Investors must think about return
distributions, not just a single return
Probabilities weight outcomes




Should be assigned to each possible
outcome to create a distribution
Can be discrete or continuous

7-3


Calculating Expected
Return


Expected value






The single most likely outcome from a
particular probability distribution
The weighted average of all possible return
outcomes
Referred to as an ex ante or expected
return
m

E(R )  R ipri
i1

7-4


7-5


Calculating Risk


Variance and standard deviation used
to quantify and measure risk








Measures the spread in the probability
distribution
Variance of returns: σ² = (Ri - E(R))²pri
Standard deviation of returns:
σ =(σ²)1/2
Ex ante rather than ex post σ relevant

7-6


7-7


Portfolio Expected Return


Weighted average of the individual
security expected returns


Each portfolio asset has a weight, w, which
represents the percent of the total portfolio
value

n

E(R p )   w iE(R i )
i1

7-8


Example 7-4
Consider a three-stock portofolio consisting of stock G, H and I
with expected returns of 12%, 20% and 17%, respectively. Assume
that 50% of investable fund is invested in security G, 30% in H, and
20% in I. The expected return on this portofolio is :
E(Rp) : 0,5(12%) + 0,3(20%) + 0,2(17%)
: 15,4%

7-9


Portfolio Risk






Portfolio risk not simply the sum of
individual security risks
Emphasis on the risk of the entire
portfolio and not on risk of individual

securities in the portfolio
Individual stocks are risky only if they
add risk to the total portfolio

7-10


Portfolio Risk


Measured by the variance or standard
deviation of the portfolio’s return


Portfolio risk is not a weighted average of
the risk of the individual securities in the
portfolio
n

  wi
i1
2
p

2
i

7-11

Risk Reduction in
Portfolios




Assume all risk sources for a portfolio of
securities are independent
The larger the number of securities the
smaller the exposure to any particular
risk




“Insurance principle”

Only issue is how many securities to
hold
7-12


Risk Reduction in
Portfolios


Random diversification








Diversifying without looking at relevant
investment characteristics
Marginal risk reduction gets smaller and
smaller as more securities are added

A large number of securities is not
required for significant risk reduction
International diversification benefits

7-13


Portfolio Risk and Diversification
p %
Portfolio risk

35

20

Market Risk

0
10

20

30

40

......

Number of securities in portfolio

100+


Markowitz Diversification


Non-random diversification






Active measurement and management of
portfolio risk
Investigate relationships between portfolio
securities before making a decision to
invest
Takes advantage of expected return and
risk for individual securities and how

security returns move together

7-15


Measuring Portfolio Risk


Needed to calculate risk of a portfolio:


Weighted individual security risks






Calculated by a weighted variance using the
proportion of funds in each security
For security i: (wi × i)2

Weighted comovements between returns




Return covariances are weighted using the
proportion of funds in each security
For securities i, j: 2wiwj × ij

7-16


Correlation Coefficient



Statistical measure of association
 mn = correlation coefficient between
securities m and n





 mn = +1.0 = perfect positive correlation
 mn = -1.0 = perfect negative (inverse)
correlation
 mn = 0.0 = zero correlation

7-17


Correlation Coefficient


When does diversification pay?


With perfectly positive correlated
securities?





Risk is a weighted average, therefore there is no
risk reduction

With zero correlation correlation securities?
With perfectly negative correlated
securities?

7-18


Covariance


Absolute measure of association




Not limited to values between -1 and +1
Sign interpreted the same as correlation
Correlation coefficient and covariance are
related by the following equations:
m

 AB  [R A ,i  E(R A )][R B,i  E(R B )]pri
i 1

 AB  AB  A  B
7-19


Calculating Portfolio Risk


Encompasses three factors






Variance (risk) of each security
Covariance between each pair of securities
Portfolio weights for each security

Goal: select weights to determine the
minimum variance combination for a
given level of expected return

7-20


Calculating Portfolio Risk



Generalizations




the smaller the positive correlation between
securities, the better
Covariance calculations grow quickly




n(n-1) for n securities

As the number of securities increases:




The importance of covariance relationships
increases
The importance of each individual security’s risk
decreases
7-21


Simplifying Markowitz
Calculations



Markowitz full-covariance model






Requires a covariance between the returns
of all securities in order to calculate
portfolio variance
n(n-1)/2 set of covariances for n securities

Markowitz suggests using an index to
which all securities are related to
simplify

7-22


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7-23