Modern Portfolio Theory

Evolution of Modern Portfolio Theory
 By
 MUSHTAQ AHMAD SHAH
 Research scholar
 Department of Management studies
 Guru Ghasidas vishwavidalaya

 Efficient Frontier
 Single Index Model
 Capital Asset Pricing Model (CAPM)
 Arbitrage Pricing Theory (APT)

 Efficient Frontier
 Markowitz, H. M., “Portfolio Selection,” Journal of
Finance (December 1952).
 Rather than choose each security individually,
choose portfolios that maximize return for given
levels of risk (i.e., those that lie on the efficient
frontier). Problem: When managing large numbers
of securities, the number of statistical inputs
required to use the model is tremendous. The
correlation or covariance between every pair of
securities must be evaluated in order to estimate
portfolio risk.

(Continued)
 Single Index Model
 Sharpe, W. F., “A Simplified Model of Portfolio
Analysis,” Management Science (January 1963).
 Substantially reduced the number of required
inputs when estimating portfolio risk. Instead
of estimating the correlation between every pair
of securities, simply correlate each security with
an index of all of the securities included in the
analysis.

(Continued)
 Capital Asset Pricing Model (CAPM)
 Sharpe, W. F., “Capital Asset Prices: A Theory of Market
Equilibrium Under Conditions of Risk,” Journal of
Finance (September 1964).
 Instead of correlating each security with an index of
all securities included in the analysis, correlate each
security with the efficient market value weighted
portfolio of all risky securities in the universe (i.e., the
market portfolio). Also, allow investors the option of
investing in a risk-free asset.

(Continued)
 Arbitrage Pricing Theory (APT)
 Ross, S. A., “The Arbitrage Theory of Capital Asset Pricing,”
Journal of Economic Theory (December 1976).
 Instead of correlating each security with only the
market portfolio (one factor), correlate each security
with an appropriate series of factors (e.g., inflation,
industrial production, interest rates, etc.).

Markowitz’s Contribution
 Harry Markowitz’s “Portfolio Selection” Journal of
Finance article (1952) set the stage for modern
portfolio theory
 The first major publication indicating the important of
security return correlation in the construction of stock
portfolios
 Markowitz showed that for a given level of expected
return and for a given security universe, knowledge of
the covariance and correlation matrices are required
8

Harry Markowitz Model
•Harry Max Markowitz (born August 24,
1927) is an American economist.
•He is best known for his pioneering work
in Modern Portfolio Theory.
•Harry Markowitz put forward this model
in 1952.
• Studied the effects of asset risk, return,
correlation and diversification on probable
investment portfolio returns
Harry Markowitz
Essence of Markowitz Model
1. An investor has a certain amount of capital he wants to invest over a single time
horizon.
2. He can choose between different investment instruments, like stocks, bonds,
options, currency, or portfolio. The investment decision depends on the future
risk and return.
3. The decision also depends on if he or she wants to either maximize the yield or
minimize the risk

Essence of Markowitz Model
1. Markowitz model assists in the selection of the most efficient by analysing
various possible portfolios of the given securities.
2. By choosing securities that do not 'move' exactly together, the HM model shows
investors how to reduce their risk.
3. The HM model is also called Mean-Variance Model due to the fact that it is
based on expected returns (mean) and the standard deviation (variance) of the
various portfolios.
Diversification and Portfolio Risk
PortfolioRisk
Number of Shares
5 10 15 20
Total
Risk
S
R
USR
p p deviationstandardthe
SR: Systematic Risk
USR: Unsystematic Risk

Assumptions
 An investor has a certain amount of capital he wants to
invest over a single time horizon.
 He can choose between different investment
instruments, like stocks, bonds, options, currency, or
portfolio.
 The investment decision depends on the future risk
and return.
 The decision also depends on if he or she wants to
either maximize the yield or minimize the risk.
 The investor is only willing to accept a higher risk if he
or she gets a higher expected return.

Efficient Frontier
 Construct a risk/return plot of all possible
portfolios
Those portfolios that are not dominated
constitute the efficient frontier
12

Efficient Frontier (cont’d)
13
Standard Deviation
Expected Return
100% investment in security
with highest E(R)
100% investment in minimum
variance portfolio
Points below the efficient
frontier are dominated
No points plot above
the line
All portfolios
on the line
are efficient

 When a risk-free investment is available, the shape of the
efficient frontier changes
 The expected return and variance of a risk-free rate/stock
return combination are simply a weighted average of the
two expected returns and variance
 The risk-free rate has a variance of zero
14

15
Standard Deviation
Expected Return
Rf
A
B
C

 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
16

Need for Sharpe Model
In Markowitz model a number of co-variances have to
be estimated.
If a financial institution buys 150 stocks, it has to
estimate 11,175 i.e., (N2 – N)/2 correlation
co-efficients.
Sharpe assumed that the return of a security is linearly
related to a single index like the market index.

Single Index Model
 Casual observation of the stock prices over a period of
time reveals that most of the stock prices move with
the market index.
 When the Sensex increases, stock prices also tend to
increase and vice – versa.
 This indicates that some underlying factors affect the
market index as well as the stock prices.

Stock prices are related to the market index and this
relationship could be used to estimate the return of
stock.
Ri = ai + bi Rm + ei
where Ri — expected return on security i
ai — intercept of the straight line or alpha co-efficient
bi — slope of straight line or beta co-efficient
Rm — the rate of return on market index
ei — error term

“What Is Beta and How Is It
Calculated?”

Beta
 A “coefficient measuring a stock’s relative volatility”
 Beta measures a stock’s sensitivity to overall market
movements
Source:UBS Warburg Dictionary of Finance and Investment Terms

 In practice, Beta is measured by comparing changes in
a stock price to changes in the value of the S&P 500
index over a given time period
 The S&P 500 index has a beta of 1

A Generic Example
 Stock XYZ has a beta of 2
 The S&P 500 index increases in value by 10%
 The price of XYZ is expected to increase 20% over the
same time period

Beta can be Negative
 Stock XYZ has a beta of –2
 The S&P 500 index INCREASES in value by 10%
 The price of XYZ is expected to DECREASE 20% over
the same time period

 If the beta of XYZ is 1.5 …
 And the S&P increases in value by 10%
 The price of XYZ is expected to increase 15%

 A beta of 0 indicates that changes in the market index
cannot be used to predict changes in the price of the
stock
 The company’s stock price has no correlation to
movments in the market index

Company Beta
AMGN 0.82
BRK.B 0.73
C 1.37
XOM 0.10
MSFT 1.80
MWD 2.19
NOK 2.05
PXLW 1.93
TXN 1.70
VIA.B 1.39
Source: taken from yahoo.finance.com, except PXLW from bloomberg.com

 If beta is a measure of risk, then investors who hold
stocks with higher betas should expect a higher return
for taking on that risk
 What does this remind you of?

Beta and Risk
 Beta is a measure of volatility
 Volatility is associated with risk

How to Calculate Beta
Beta = Covariance(stock price, market index)
Variance(market index)
**When calculating, you must compare the percent
change in the stock price to the percent change in the
market index**

How to Calculate Beta
 Easily calculated using Excel and Yahoo! Finance
 Use COVAR and VARP worksheet functions
 An example:
Calculate the beta of Citigroup stock over the 5-yr time
period from Jan. 1, 1997 – Dec. 31, 2001

Risk
 Systematic risk = bi
2 × variance of market index
= bi
2 m
2
Unsystematic risk= Total variance – Systematic risk
ei
2 = i
2 – Systematic risk
Thus the total risk= Systematic risk + Unsystematic risk
= bi
2 m
2 + ei
2

Portfolio Variance
2N N
2 2 2 2
p i i m i i
i=1 i=1
σ = x β + x e
    
         
 
where
σ2
p = variance of portfolio
σ2
m = expected variance of market index
e2
i= Unsystematic risk
xi = the portion of stock i in the portfolio

Example
 The following details are given for x and y companies’
stocks and the Sensex for a period of one year. Calculate the
systematic and unsystematic risk for the companies stock.
If equal amount of money is allocated for the stocks , then
what would be the portfolio risk ?
 X stock Y stock Sensex
 Average return 0.15 0.25 0.06
 Variance of return 6.30 5.86 2.25
 Βeta 0.71 0.27

Company X
= bi
2 m
2 = ( 0.71)2 x 2.25 = 1.134
ei
2 = i
2 – Systematic risk = 6.3 – 1.134 =5.166
Total risk= Systematic risk + Unsystematic risk
= bi
2 m
2 + ei
2 = 1.134 + 5.166 = 6.3

Company Y
= bi
2 m
2 = ( 0.27)2 x 2.25 = 0.1640
ei
2 = i
2 – Systematic risk = 5.86 – 1.134 =5.166

 σ2
p = [ ( .5 x .71 + .5 x .27)2 2.25 ] + [ ( .5)2 (5.166) + (.5 )2 ( 5.696) ]
 = [ ( .355 + .135 )2 2.25 ] + [ ( 1.292 + 1.424 ) ]
 = 0.540 + 2.716
 = 3.256

Sharpe’s optimal portfolio
i f
i
R R
β

The selection of any stock is directly related to its
excess return to beta ratio.
where Ri = the expected return on stock i
Rf = the return on a risk less asset
bi = Systematic risk

Capital Asset Pricing Model (CAPM)
 Introduction
 Systematic and unsystematic risk
 Fundamental risk/return relationship revisited
40

Introduction
 The Capital Asset Pricing Model (CAPM) is a
theoretical description of the way in which the market
prices investment assets
 The CAPM is a positive theory
41

Systematic and
Unsystematic Risk
 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)
42

CAPM
u The more risk you carry, the greater the expected
return:
43
( ) ( )
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
E R
b
b
    




% %
%
%

CAPM (cont’d)
 The CAPM deals with expectations about the future
 Excess returns on a particular stock are directly related
to:
 The beta of the stock
 The expected excess return on the market
44

CAPM (cont’d)
CAPM assumptions:
 Variance of return and mean return are all investors care
about
 Investors are price takers, they cannot influence the
market individually
 All investors have equal and costless access to
information
 There are no taxes or commission costs
45

CAPM (cont’d)
 CAPM assumptions (cont’d):
 Investors look only one period ahead
 Everyone is equally adept at analyzing securities and
interpreting the news
46

SML and CAPM
If you show the security market line with excess
returns on the vertical axis, the equation of the SML
is the CAPM
 The intercept is zero
 The slope of the line is beta
47

Note on the CAPM Assumptions
Several assumptions are unrealistic:
 People pay taxes and commissions
 Many people look ahead more than one period
 Not all investors forecast the same distribution
Theory is useful to the extent that it helps us learn
more about the way the world acts
 Empirical testing shows that the CAPM works
reasonably well
48

Stationarity of Beta
Beta is not stationary
 Evidence that weekly betas are less than monthly betas,
especially for high-beta stocks
 Evidence that the stationary of beta increases as the
estimation period increases
The informed investment manager knows that betas
change
49

Equity Risk Premium
Equity risk premium refers to the difference in the
average return between stocks and some measure of
the risk-free rate
 The equity risk premium in the CAPM is the excess
expected return on the market
 Some researchers are proposing that the size of the
equity risk premium is shrinking
50

Using A Scatter Diagram to measure
Beta
 Correlation of returns
 Linear regression and beta
 Importance of logarithms
 Statistical significance
51

Correlation of Returns
Much of the daily news is of a general economic nature
and affects all securities
 Stock prices often move as a group
 Some stock routinely move more than the others
regardless of whether the market advances or declines
 Some stocks are more sensitive to changes in economic
conditions
52

Linear Regression and Beta
 To obtain beta with a linear regression:
 Plot a stock’s return against the market return
 Use Excel to run a linear regression and obtain the
coefficients
 The coefficient for the market return is the beta statistic
 The intercept is the trend in the security price returns that is
inexplicable by finance theory
53

Statistical Significance
Published betas are not always useful numbers
 Individual securities have substantial unsystematic risk
and will behave differently than beta predicts
 Portfolio betas are more useful since some unsystematic
risk is diversified away
54

Arbitrage Pricing Theory
u APT background
u The APT model
u Comparison of the CAPM and the APT
55

APT Background
 Arbitrage pricing theory (APT) states that a number
of distinct factors determine the market return
 Roll and Ross state that a security’s long-run return is a
function of changes in:
 Inflation
 Industrial production
 Risk premiums
 The slope of the term structure of interest rates
56

APT Background (cont’d)
Not all analysts are concerned with the same set of
economic information
 A single market measure such as beta does not capture
all the information relevant to the price of a stock
57

The APT Model
General representation of the APT model:
58
1 1 2 2 3 3 4 4( )
where actual return on Security
( ) expected return on Security
sensitivity of Security to factor
unanticipated change in factor
A A A A A A
A
A
iA
i
R E R b F b F b F b F
R A
E R A
b A i
F i
    




%
%

APT
59
1 1 2 2 3 3
1 1 1 2 2 2 3 3 3
1 1 2 2 3 3 1 1 2 2 3 3
Fixed Random
(Notice that the security index "A" has been ign
( )
( ) [ ( )] [ ( )] [ ( )]
( ) ( ) ( ) ( )
R E R F F F
R E R R E R R E R R E R
R E R E R E R E R R R R
b b b 
b b b 
b b b b b b 
    
       
       
1 4 4 4 4 4 4 2 4 4 4 4 4 4 3 1 4 44 2 4 4 43
ored for clarity purposes)

Replicating the Randomness
 Let’s try to replicate the random component of
security A by forming a portfolio with the following
weights:
60
1 1 2 2 3 3 1 2 3 f
1 2 3 f 1 1 2 2 3 3
Fixed Random
on , on , on , and finally 1- on R
We get the following return (for this portfolio of factors):
(1- )R
R R R
R R R R
b b b b b b
b b b b b b 
 
      
1 4 44 2 4 4 43 1 4 44 2 4 4 43

Key Point in Reasoning
 Since we were able to match the random
components exactly, the only terms that differ
at this point are the fixed components.
 But if one fixed component is larger than the
other, arbitrage profits are possible by investing
in the highest yielding security (either A or the
portfolio of factors) while short-selling the
other (being “long” in one and “short” in the
other will assure an exact cancellation of the
random terms).
61

Comparison of the
CAPM and the APT
 The CAPM’s market portfolio is difficult to
construct:
 Theoretically all assets should be included (real estate,
gold, etc.)
 Practically, a proxy like the S&P 500 index is used
 APT requires specification of the relevant
macroeconomic factors
62

Comparison of the
CAPM and the APT (cont’d)
The CAPM and APT complement each other rather than
compete
 Both models predict that positive returns will result
from factor sensitivities that move with the market and
vice versa
63

Modern Portfolio Theory

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