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EFFICIENTPORTFOLIOS&
EFFICIENTFRONTIERS
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EFFICIENTVS. INEFFICIENT
PORTFOLIOS
It is impossible to predict in advance which
portfolios will be the most efficient as this would
require knowing in advance asset classperformance and correlations.
A portfolio that has been diversified into a variety
of asset classes should be close to efficient over
the longer term, provided it is rebalancedregularly.
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3PORTFOLIOPROGRAMMING
INA NUTSHELL
Various portfolio combinations may result in a
given return
The investor wants to choose the portfolio
combination that provides the least amount of
variance
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4PORTFOLIOPROGRAMMING
INA NUTSHELL(CONTD)
Example
Assume the following statistics for Stocks A, B,
and C:
Stock A Stock B Stock C
Expected return .20 .14 .10
Standard deviation .232 .136 .195
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PORTFOLIOPROGRAMMING
INA NUTSHELL(CONTD)
Example (contd)
The correlation coefficients between the three
stocks are:
Stock A Stock B Stock C
Stock A 1.000
Stock B 0.286 1.000
Stock C 0.132 -0.605 1.000
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6
PORTFOLIOPROGRAMMING
INA NUTSHELL(CONTD)
Example (contd)
An investor seeks a portfolio return of 12%.
Which combinations of the three stocksaccomplish this objective? Which of thosecombinations achieves the least amount ofrisk?
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PORTFOLIOPROGRAMMING
INA NUTSHELL(CONTD)
Example (contd)
Solution: Two combinations achieve a 12% return:
1) 50% in B, 50% in C: (.5)(14%) + (.5)(10%) = 12%
2) 20% in A, 80% in C: (.2)(20%) + (.8)(10%) = 12%
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PORTFOLIOPROGRAMMING
INA NUTSHELL(CONTD)
Example (contd)
Solution (contd):Calculate the variance of the
B/C combination:
2 2 2 2 2
2 2
2
(.50) (.0185) (.50) (.0380)
2(.50)(.50)( .605)(.136)(.195)
.0046 .0095 .0080
.0061
p A A B B A B AB A Bx x x x
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PORTFOLIOPROGRAMMING
INA NUTSHELL(CONTD)
Example (contd)
Solution (contd):Calculate the variance of the
A/C combination:2 2 2 2 2
2 2
2
(.20) (.0538) (.80) (.0380)
2(.20)(.80)(.132)(.232)(.195)
.0022 .0243 .0019
.0284
p A A B B A B AB A Bx x x x
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PORTFOLIOPROGRAMMING
INA NUTSHELL(CONTD)
Example (contd)
Solution (contd):Investing 50% in Stock B and
50% in Stock C achieves an expected return of
12% with the lower portfolio variance. Thus,
the investor will likely prefer this combinationto the alternative of investing 20% in Stock A
and 80% in Stock C.
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11 CONCEPTOFDOMINANCE
Dominanceis a situation in which investors
universally prefer one alternative over another
All rational investors will clearly prefer one
alternative
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12
CONCEPTOFDOMINANCE
(CONTD)
A portfolio dominates all others if:
For its level of expected return, there is no other
portfolio with less risk
For its level of risk, there is no other portfolio
with a higher expected return
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CONCEPTOFDOMINANCE
(CONTD)
Example (contd)
In the previous example, the B/C combination dominates the
A/C combination:
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.005 0.01 0.015 0.02 0.025 0.03
Risk
Expe
cted
Return
B/C combination
dominates A/C
H M
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14
HARRYMARKOWITZ:
FOUNDEROFPORTFOLIO
THEORY
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15 INTRODUCTION
Harry Markowitzs Portfolio SelectionJournal of Financearticle (1952) set thestage for modern portfolio theory
The first major publication indicating theimportant of security return correlation inthe construction of stock portfolios
Markowitz showed that for a given level ofexpected return and for a given securityuniverse, knowledge of the covariance andcorrelation matrices are required
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16 EFFICIENTFRONTIER
Construct a risk/return plot of all possible
portfolios
Those portfolios that are not dominated
constitute the efficient frontier
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IMPROVINGTHEEFFICIENT
FRONTIER
Investors desire higher returns with lower risk. There is however a
limit to what can be achieved with a particular set of assets, that limit
is drawn on charts as the efficient frontier.
By adding more assets we can change the shape of the efficient
frontier. Assets carry two items of interest to us, their returns and
their correlation with the rest of the portfolio.
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EFFICIENTFRONTIER
(CONTD)
Standard Deviation
Expected Return100% investment in security
with highest E(R)
100% investment in minimumvariance portfolio
Points below the efficient
frontier are dominated
No points plot above
the line
All portfolios
on the line
are efficient
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EFFICIENTFRONTIER
(CONTD)
The farther you move to the left on the efficient
frontier, the greater the number of securities in
the portfolio
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EFFICIENTFRONTIER
(CONTD)
The efficient frontier with a risk-free rate:
Extends from the risk-free rate to point B
The line is tangent to the risky securities efficient
frontier
Follows the curve from point B to point C
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21
LESSONSFROM
EVANSANDARCHER
Introduction
Methodology
Results
Implications
Words of caution
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22 INTRODUCTION
Evans and Archers 1968Journal of Financearticle
Very consequential research regarding portfolioconstruction
Shows how nave diversificationreduces the
dispersion of returns in a stock portfolio
Nave diversification refers to the selection ofportfolio components randomly
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23 METHODOLOGY
Used computer simulations:
Measured the average variance of portfolios of
different sizes, up to portfolios with dozens of
components
Purpose was to investigate the effects of portfolio
size on portfolio risk when securities are
randomly selected
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24 DEFINITIONS
Systematic riskis the risk that remains after no
further diversification benefits can be achieved
Unsystematic riskis the part of total risk that is
unrelated to overall market movements and can
be diversified
Research indicates up to 75 percent of total risk is
diversifiable
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25 DEFINITIONS(CONTD)
Investors are rewarded only for systematic risk
Rational investors should always diversify
Explains why beta (a measure of systematic risk)
is important
Securities are priced on the basis of their betacoefficients
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26 GENERALRESULTS
Number of Securities
Portfolio Variance
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BIGGESTBENEFITSCOME
FIRST
Increasing the number of portfolio components
provides diminishing benefits as the number of
components increases
Adding a security to a one-security portfolio
provides substantial risk reduction
Adding a security to a twenty-security portfolio
provides only modest additional benefits
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28 DIVERSIFICATIONANDBETA
Beta measures systematic risk
Diversification does not mean to reduce beta
Investors differ in the extent to which they will
take risk, so they choose securities with different
betas
E.g., an aggressive investor could choose a
portfolio with a beta of 2.0
E.g., a conservative investor could choose a
portfolio with a beta of 0.5
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MODERNPORTFOLIOTHEORY
Modern portfolio theory(MPT) is a theoryof investment which attempts to maximizeportfolio expected return for a given amount ofportfolio risk, or equivalently minimize risk for agiven level of expected return, by carefullychoosing the proportions of various assets.Although MPT is widely used in practice in thefinancial industry and several of its creators won
a Nobel memorial prize for the theory, in recentyears the basic assumptions of MPT have beenwidely challenged by fields such as behavioraleconomics.
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MATHEMATICALMODEL
MPT was developed in the 1950s through the early
1970s and was considered an important advance in
the mathematical modeling of finance. Since then,
many theoretical and practical criticisms have been
leveled against it. These include the fact that financial
returns do not follow a Gaussian distribution or
indeed any symmetric distribution, and that
correlations between asset classes are not fixed but
can vary depending on external events (especially incrises). Further, there is growing evidence that
investors are not rational and markets are
not efficient.
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In 1952, Harry Markowitz published a formal
portfolio selection model in The Journal of
Finance. He continued to develop and publishresearch on the subject over the next twenty
years, eventually winning the 1990 Nobel Prize
in Economics for his work on the efficient frontier
and other contributions to modern portfolio
theory.
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EXAMPLE
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33 DIVERSIFICATIONANDBETA
Beta measures systematic risk
Diversification does not mean to reduce beta
Investors differ in the extent to which they will
take risk, so they choose securities with different
betas
E.g., an aggressive investor could choose a
portfolio with a beta of 2.0
E.g., a conservative investor could choose a
portfolio with a beta of 0.5
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34 INTRODUCTION
The Capital Asset Pricing Model (CAPM)is a
theoretical description of the way in which the
market prices investment assets
The CAPM is apositivetheory
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SYSTEMATICAND
UNSYSTEMATICRISK
Unsystematic risk can be diversified and is
irrelevant
Systematic risk cannot be diversified and is
relevant
Measured by beta
Beta determines the level of expected return on a
security or portfolio (SML)
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36 CAPM
The more risk you carry, the greater the expected
return:
( ) ( )
where ( ) expected return on security
risk-free rate of interest
beta of Security
( ) expected return on the market
i f i m f
i
f
i
m
E R R E R R
E R i
R
i
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