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2
Introduction Diversification of a portfolio is logically a
good idea
Virtually all stock portfolios seek to diversify in one respect or another
3
Carrying Your Eggs in More Than One Basket
Investments in your own ego The concept of risk aversion revisited Multiple investment objectives
4
Investments in Your Own Ego Never put a large percentage of investment
funds into a single security• If the security appreciates, the ego is stroked
and this may plant a speculative seed• If the security never moves, the ego views this
as neutral rather than an opportunity cost• If the security declines, your ego has a very
difficult time letting go
5
The Concept of Risk Aversion Revisited
Diversification is logical• If you drop the basket, all eggs break
Diversification is mathematically sound• Most people are risk averse• People take risks only if they believe they will
be rewarded for taking them
6
The Concept of Risk Aversion Revisited (cont’d)
Diversification is more important now• Journal of Finance article shows that volatility
of individual firms has increased
– Investors need more stocks to adequately diversify
7
Multiple Investment Objectives Multiple objectives justify carrying your
eggs in more than one basket• Some people find mutual funds “unexciting”• Many investors hold their investment funds in
more than one account so that they can “play with” part of the total
– E.g., a retirement account and a separate brokerage account for trading individual securities
9
Introduction Evans and Archer’s 1968 Journal of
Finance article• Very consequential research regarding portfolio
construction
• Shows how naïve diversification reduces the dispersion of returns in a stock portfolio
– Naïve diversification refers to the selection of portfolio components randomly
10
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
11
Results Definitions General results Strength in numbers Biggest benefits come first Superfluous diversification
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Definitions Systematic risk is the risk that remains after
no further diversification benefits can be achieved
Unsystematic risk is 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
13
Definitions (cont’d) 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 beta coefficients
15
Strength in Numbers Portfolio variance (total risk) declines as the
number of securities included in the portfolio increases• On average, a randomly selected ten-security
portfolio will have less risk than a randomly selected three-security portfolio
• Risk-averse investors should always diversify to eliminate as much risk as possible
16
Biggest Benefits Come 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
17
Superfluous Diversification Superfluous diversification refers to the
addition of unnecessary components to an already well-diversified portfolio• Deals with the diminishing marginal benefits of
additional portfolio components
• The benefits of additional diversification in large portfolio may be outweighed by the transaction costs
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Implications Very effective diversification occurs when
the investor owns only a small fraction of the total number of available securities• Institutional investors may not be able to avoid
superfluous diversification due to the dollar size of their portfolios
– Mutual funds are prohibited from holding more than 5 percent of a firm’s equity shares
19
Implications (cont’d) Owning all possible securities would
require high commission costs
It is difficult to follow every stock
20
Words of Caution Selecting securities at random usually gives
good diversification, but not always Industry effects may prevent proper
diversification Although naïve diversification reduces risk,
it can also reduce return• Unlike Markowitz’s efficient diversification
21
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
22
Quadratic Programming The Markowitz algorithm is an application
of quadratic programming• The objective function involves portfolio
variance
• Quadratic programming is very similar to linear programming
23
Portfolio Programming in A 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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Concept of Dominance Dominance is a situation in which investors
universally prefer one alternative over another• All rational investors will clearly prefer one
alternative
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Concept of Dominance (cont’d) 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
27
Concept of Dominance (cont’d)Example (cont’d)
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
Exp
ecte
d R
etu
rn
B/C combination dominates A/C
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Terminology Security Universe Efficient frontier Capital market line and the market portfolio Security market line Expansion of the SML to four quadrants Corner portfolio
29
Security Universe The security universe is the collection of all
possible investments• For some institutions, only certain investments
may be eligible
– E.g., the manager of a small cap stock mutual fund would not include large cap stocks
30
Efficient Frontier Construct a risk/return plot of all possible
portfolios• Those portfolios that are not dominated
constitute the efficient frontier
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Efficient Frontier (cont’d)
Standard Deviation
Expected Return100% 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
32
Efficient Frontier (cont’d) 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
34
Efficient Frontier (cont’d) 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
35
Theorem For any constant Rf on the returns axis, the
weights of the tangency portfolio B are:
11
1 1
, , where z is given by: [ ]
In other words, the weights of tangency portfolio B are the normalized
weights of portfolio z.
nfn n
j jj j
zzz V R
z z
36
Example with Rf=0 and Rf=6.5%
23456789101112131415161718
A B C D E F G H I J
MeanMean minus
Variance-covariance matrix returns constant0.400 0.030 0.020 0.000 0.06 -0.005 <-- =F6-$E$110.030 0.200 0.001 -0.060 0.05 -0.015 <-- =F7-$E$110.020 0.001 0.300 0.030 0.07 0.005 <-- =F8-$E$110.000 -0.060 0.030 0.100 0.08 0.015 <-- =F9-$E$11
Rf= 0.00 Rf= 0.065z B z B
0.1019 0.0540 -0.0101 -0.11630.5657 0.2998 -0.0353 -0.40670.1141 0.0605 0.0047 0.05441.1052 0.5857 0.1274 1.4687
Cells E13:E16 contain the array function =MMULT(MINVERSE(A6:D9),G6:G9). Cell F13 contains the function=E13/SUM($E$13:$E$16). This function is copied to cells F14:F16.
37
Graphically:Finding Envelope Portfolios
0%
2%
4%
6%
8%
10%
12%
0% 10% 20% 30% 40% 50% 60% 70% 80% 90%
Portfolio standard deviation
Po
rtfo
lio
me
an r
etu
rn
Rf
B, the tangency portfolio given Rf
Zero beta portfolio
38
What is the zero-beta portfolio? The zero beta portfolio P0 is the portfolio
determined by the intersection of the frontier with a horizontal line originating from the constant Rf selected.
Property: whatever Rf we choose, we always have Cov(B,P0)=0
(Notice, however, that the location of B and P0 will depend on the value selected for Rf)
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Note that the last proposition is true even if the risk-free rate (i.e. a riskless security) doesn’t exist in the economy.
The way the tangency portfolio B was determined also remains valid even if there is no riskless rate in the economy.
All one has to do is replace Rf by a chosen constant c. The mathematics of the last propositions will remain valid.
40
Fisher Black zero beta CAPM (1972)
For a chosen constant c on the vertical axis of returns, the tangency portfolio B can be computed, and for ANY portfolio x we have a linear relationship if we regress the returns of x on the returns of B:
Moreover, c is the expected rate of return of a portfolio P0 whose covariance with B is zero.
2( ) [ ( ) ] where ( , ) /x x B x BE R c E R c Cov x B
0 0( ) and ( , ) 0Pc E R Cov P B
41
Fisher Black zero beta CAPM (Cont’d)
The name “zero beta” stems from the fact that the covariance between P0 and B is zero, since a zero covariance implies that the beta of P0 with respect to B is zero too.
If a riskless asset DOES exist in the economy, however, we can replace the constant c in Black’s zero beta CAPM by Rf
and the portfolio B is the market portfolio.
42
Capital Market Line and the Market Portfolio
The tangent line passing from the risk-free rate through point B is the capital market line (CML)• When the security universe includes all possible
investments, point B is the market portfolio– It contains every risky assets in the proportion of its
market value to the aggregate market value of all assets– It is the only risky assets risk-averse investors will hold
43
Capital Market Line and the Market Portfolio (cont’d)
Implication for investors:• Regardless of the level of risk-aversion, all
investors should hold only two securities:– The market portfolio– The risk-free rate
• Conservative investors will choose a point near the lower left of the CML
• Growth-oriented investors will stay near the market portfolio
44
Capital Market Line and the Market Portfolio (cont’d)
Any risky portfolio that is partially invested in the risk-free asset is a lending portfolio
Investors can achieve portfolio returns greater than the market portfolio by constructing a borrowing portfolio
45
Capital Market Line and the Market Portfolio (cont’d)
Standard Deviation
Expected Return
Rf A
B
C
46
Security Market Line The graphical relationship between
expected return and beta is the security market line (SML)• The slope of the SML is the market price of
risk
• The slope of the SML changes periodically as the risk-free rate and the market’s expected return change
48
123456789101112131415161718192021
A B C D E F G H I J K L M N O
AMR BS GE HR MO UK SP500 Regressing the means on the betas:1974 -0.3505 -0.1154 -0.4246 -0.2107 -0.0758 0.2331 -0.2647 Intercept 0.07659 <-- =INTERCEPT(B15:G15,B16:G16)1975 0.7083 0.2472 0.3719 0.2227 0.0213 0.3569 0.3720 Slope 0.054509 <-- =SLOPE(B15:G15,B16:G16)1976 0.7329 0.3665 0.2550 0.5815 0.1276 0.0781 0.2384 R-squared 0.279317 <-- =RSQ(B15:G15,B16:G16)1977 -0.2034 -0.4271 -0.0490 -0.0938 0.0712 -0.2721 -0.07181978 0.1663 -0.0452 -0.0573 0.2751 0.1372 -0.1346 0.06561979 -0.2659 0.0158 0.0898 0.0793 0.0215 0.2254 0.1844 Mean Beta1980 0.0124 0.4751 0.3350 -0.1894 0.2002 0.3657 0.3242 0.203227 1.4820381981 -0.0264 -0.2042 -0.0275 -0.7427 0.0913 0.0479 -0.0491 0.053134 1.0839761982 1.0642 -0.1493 0.6968 -0.2615 0.2243 0.0456 0.2141 0.150107 1.3106731983 0.1942 0.3680 0.3110 1.8682 0.2066 0.2640 0.2251 0.152874 1.299112
0.102546 0.262246Mean 0.2032 0.0531 0.1501 0.1529 0.1025 0.1210 0.1238 0.120996 0.493903Beta 1.4820 1.0840 1.3107 1.2991 0.2622 0.4939 1.0000
=SLOPE(B4:B13,$H$4:$H$13)=COVAR(B4:B13,$H$4:$H$13)/VARP($H$4:$H$13)
THE SECURITY MARKET LINE--A SIMPLE EXAMPLE
49
Notice that we obtained very poor results. The R-squared is only 27.93% !
However, the math of the CAPM is undoubtedly true.
How then can CAPM fail in the real world? Possible explanations are that true asset
returns distributions are unobservable, individuals have non-homogenous expectations, the market portfolio is unobservable, the riskless rate is ambiguous, markets are not friction-free.
50
Efficient Portfolios Showing the S&P 500 Portfolio
0%
5%
10%
15%
20%
25%
30%
0% 5% 10% 15% 20% 25%
Sigma
Mea
n r
etu
rn
S&P 500
51
Using “Artificial Market Portfolio”474849505152535455565758596061626364656667686970
D E F G H I J K LANNUAL RETURNS ON SIX ASSETS AND "MARKET" PORTFOLIO
"Market" AMR BS GE HR MO UK portfolio
1974 -0.3505 -0.1154 -0.4246 -0.2107 -0.0758 0.2331 0.35601975 0.7083 0.2472 0.3719 0.2227 0.0213 0.3569 0.38451976 0.7329 0.3665 0.2550 0.5815 0.1276 0.0781 0.01891977 -0.2034 -0.4271 -0.0490 -0.0938 0.0712 -0.2721 0.15581978 0.1663 -0.0452 -0.0573 0.2751 0.1372 -0.1346 0.08721979 -0.2659 0.0158 0.0898 0.0793 0.0215 0.2254 0.20701980 0.0124 0.4751 0.3350 -0.1894 0.2002 0.3657 -0.11211981 -0.0264 -0.2042 -0.0275 -0.7427 0.0913 0.0479 0.23401982 1.0642 -0.1493 0.6968 -0.2615 0.2243 0.0456 0.55501983 0.1942 0.3680 0.3110 1.8682 0.2066 0.2640 0.3918
Mean 0.2032 0.0531 0.1501 0.1529 0.1025 0.1210 0.2278Beta with respect to "market"
0.7938 -0.4647 0.3484 0.3716 -0.0504 0.1043 1.0000
=SLOPE(E50:E59,$L$50:$L$59)
Regressing the means on the betasIntercept 0.1086 <-- =INTERCEPT(E61:J61,E63:J63)Slope 0.1193 <-- =SLOPE(E61:J61,E63:J63)R-squared 1.0000 <-- =RSQ(E61:J61,E63:J63)
52
We obtained a perfect 100% R-squared this time !
The reason is that when portfolio returns are regressed on their betas with respect to an efficient portfolio, an exact linear relationship holds.
53
Expansion of the SML to Four Quadrants
There are securities with negative betas and negative expected returns• A reason for purchasing these securities is their
risk-reduction potential– E.g., buy car insurance without expecting an
accident
– E.g., buy fire insurance without expecting a fire
55
Diversification and Beta 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
56
Capital Asset Pricing Model Introduction Systematic and unsystematic risk Fundamental risk/return relationship
revisited
57
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
58
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)
59
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
E R
60
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
61
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
62
CAPM (cont’d) CAPM assumptions (cont’d):
• Investors look only one period ahead
• Everyone is equally adept at analyzing securities and interpreting the news
63
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
64
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
65
Stationarity of Beta Beta is not stationary
• Evidence that weekly betas are less than monthly betas, especially for high-beta stocks
• Evidence that the stationarity of beta increases as the estimation period increases
The informed investment manager knows that betas change
66
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
67
Using A Scatter Diagram to Measure Beta
Correlation of returns Linear regression and beta Importance of logarithms Statistical significance
68
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
69
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
70
Importance of Logarithms Taking the logarithm of returns reduces the
impact of outliers• Outliers distort the general relationship
• Using logarithms will have more effect the more outliers there are
71
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
73
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
74
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
75
The APT Model General representation of the APT model:
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
76
APT1 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
ored for clarity purposes)
77
Replicating the Randomness
Let’s try to replicate the random component of security A by forming a portfolio with the following weights:
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
78
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).
79
Therefore the fixed components MUST BE THE SAME for security A and the portfolio of factors created, otherwise unlimited profits would be possible.
So we have:
1 1 2 2 3 3 1 2 3 f
f 1 1 f 2 2 f 2 3 f
( ) ( ) ( ) ( ) (1- )R
Rearranging terms yields:
( ) R [ ( ) R ] [ ( ) R ] [ ( ) R ]
E R E R E R E R
E R E R E R E R
80
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