Chapter 6
Why Diversification Is a Good Idea
1
Lessons from
Evans and Archer
 Introduction
 Methodology
 Results
 Implications
 Words
of caution
2
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
3
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
4
Results
 Definitions
 General
results
 Strength in numbers
 Biggest benefits come first
 Superfluous diversification
5
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
6
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
7
General Results
Portfolio Variance
Number of Securities
8
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
9
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
10
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
11
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
12
Implications (cont’d)
 Owning
all possible securities would
require high commission costs
 It
is difficult to follow every stock
13
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
14
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
15
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
16
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
17
Markowitz Quadratic
Programming Problem
18
Concept of Dominance
 Dominance
is a situation in which investors
universally prefer one alternative over
another
• All rational investors will clearly prefer one
alternative
19
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
20
Concept of Dominance (cont’d)
Example (cont’d)
In the previous example, the B/C combination dominates the A/C
combination:
0.14
Expected Return
0.12
0.1
B/C combination
dominates A/C
0.08
0.06
0.04
0.02
0
0
0.005
0.01
0.015
Risk
0.02
0.025
0.03
21
Terminology
 Security
Universe
 Efficient frontier
 Capital market line and the market portfolio
 Security market line
 Expansion of the SML to four quadrants
 Corner portfolio
22
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
23
Efficient Frontier
 Construct
a risk/return plot of all possible
portfolios
• Those portfolios that are not dominated
constitute the efficient frontier
24
Efficient Frontier (cont’d)
Expected Return
No points plot above
the line
All portfolios
on the line
are efficient
100% investment in security
with highest E(R)
Points below the efficient
frontier are dominated
100% investment in minimum
variance portfolio
Standard Deviation
25
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
26
Efficient Frontier (cont’d)
Expected Return
C
B
Rf
A
Standard Deviation
27
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
28
Theorem
For any constant Rf on the returns axis, the
weights of the tangency portfolio B are:

 z
 n1 ,

 zj
 j 1

zn 
, n

z

j 
j 1

where z is given by: z  V 1[   R f ]
In other words, the weights of tangency portfolio B are the normalized
weights of portfolio z.
29
Example with Rf=0 and Rf=6.5%
A
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
B
C
Variance-covariance matrix
0.400
0.030
0.020
0.030
0.200
0.001
0.020
0.001
0.300
0.000
-0.060
0.030
Rf=
0.00
z
0.1019
0.5657
0.1141
1.1052
D
E
0.000
-0.060
0.030
0.100
Rf=
B
0.0540
0.2998
0.0605
0.5857
0.065
z
-0.0101
-0.0353
0.0047
0.1274
F
G
Mean
returns
0.06
0.05
0.07
0.08
Mean
minus
constant
-0.005
-0.015
0.005
0.015
B
-0.1163
-0.4067
0.0544
1.4687
H
I
J
<-- =F6-$E$11
<-- =F7-$E$11
<-- =F8-$E$11
<-- =F9-$E$11
Cells E13:E16 contain the array function
=MMULT(MINVERSE(A6:D9),G6:G9).
Cell F13 contains the function
=E13/SUM($E$13:$E$16). This function
is copied to cells F14:F16.
30
Finding Envelope Portfolios
Graphically:
12%
10%
Portfolio mean return
B, the tangency portfolio
given Rf
8%
6%
Rf
Zero beta portfolio
4%
2%
0%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
Portfolio standard deviation
31
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)
32
 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.
33
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:
E ( Rx )  c   x [ E ( RB )  c]
where
 x  Cov( x, B) /  B2
 Moreover,
c is the expected rate of return of a
portfolio P0 whose covariance with B is zero.
c  E ( RP0 ) and Cov( P0 , B)  0
34
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.
35
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
36
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
37
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
38
Capital Market Line and the
Market Portfolio (cont’d)
Expected Return
C
B
Rf
A
Standard Deviation
39
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
40
Security Market Line (cont’d)
Expected Return
E(R)
Market Portfolio
Rf
1.0
Beta
41
THE SECURITY MARKET LINE--A SIMPLE EXAMPLE
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
Mean
Beta
AMR
-0.3505
0.7083
0.7329
-0.2034
0.1663
-0.2659
0.0124
-0.0264
1.0642
0.1942
BS
-0.1154
0.2472
0.3665
-0.4271
-0.0452
0.0158
0.4751
-0.2042
-0.1493
0.3680
GE
-0.4246
0.3719
0.2550
-0.0490
-0.0573
0.0898
0.3350
-0.0275
0.6968
0.3110
HR
-0.2107
0.2227
0.5815
-0.0938
0.2751
0.0793
-0.1894
-0.7427
-0.2615
1.8682
MO
-0.0758
0.0213
0.1276
0.0712
0.1372
0.0215
0.2002
0.0913
0.2243
0.2066
UK
0.2331
0.3569
0.0781
-0.2721
-0.1346
0.2254
0.3657
0.0479
0.0456
0.2640
Market
-0.2647
0.3720
0.2384
-0.0718
0.0656
0.1844
0.3242
-0.0491
0.2141
0.2251
0.2032
1.4820
0.0531
1.0840
0.1501
1.3107
0.1529
1.2991
0.1025
0.2622
0.1210
0.4939
0.1238
1.0000
=SLOPE(B4:B13,$H$4:$H$13)
=COVAR(B4:B13,$H$4:$H$13)/VARP($H$4:$H$13)
Regressing the means on the betas:
Intercept
0.0766 <-- =INTERCEPT(B15:G15,B16:G16)
Slope
0.0545 <-- =SLOPE(B15:G15,B16:G16)
R-squared
0.2793 <-- =RSQ(B15:G15,B16:G16)
42
 Notice
that we obtained very poor results.
The R-squared is only 27.93% !
 However, the math of the CAPM is
undoubtedly true.
 How then can CAPM fail in the real world?
 Possible explanations are that true asset
returns distributions are unobservable,
individuals have non-homogenous
expectations, the market portfolio is
unobservable, the riskless rate is
ambiguous, markets are not friction-free.
43
Using “Artificial Market Portfolio”
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
ANNUAL RETURNS ON SIX ASSETS AND "MARKET" PORTFOLIO
"Market"
AMR
BS
GE
HR
MO
UK
portfolio
-0.3505
-0.1154
-0.4246
-0.2107
-0.0758
0.2331
0.3560
0.7083
0.2472
0.3719
0.2227
0.0213
0.3569
0.3845
0.7329
0.3665
0.2550
0.5815
0.1276
0.0781
0.0189
-0.2034
-0.4271
-0.0490
-0.0938
0.0712
-0.2721
0.1558
0.1663
-0.0452
-0.0573
0.2751
0.1372
-0.1346
0.0872
-0.2659
0.0158
0.0898
0.0793
0.0215
0.2254
0.2070
0.0124
0.4751
0.3350
-0.1894
0.2002
0.3657
-0.1121
-0.0264
-0.2042
-0.0275
-0.7427
0.0913
0.0479
0.2340
1.0642
-0.1493
0.6968
-0.2615
0.2243
0.0456
0.5550
0.1942
0.3680
0.3110
1.8682
0.2066
0.2640
0.3918
Mean
0.2032
0.0531
Beta with respect to "market"
0.7938
-0.4647
0.1501
0.1529
0.1025
0.1210
0.2278
0.3484
0.3716
-0.0504
0.1043
1.0000
=SLOPE(E50:E59,$L$50:$L$59)
Regressing the means on the betas
Intercept
0.1086 <-- =INTERCEPT(E61:J61,E63:J63)
Slope
0.1193 <-- =SLOPE(E61:J61,E63:J63)
R-squared
1.0000 <-- =RSQ(E61:J61,E63:J63)
44
 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.
45
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
46
Security Market Line (cont’d)
Expected Return
Securities with Negative
Expected Returns
Beta
47
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
48
Capital Asset Pricing Model
 Introduction
 Systematic
and unsystematic risk
 Fundamental risk/return relationship
revisited
49
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
50
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)
51
CAPM
 The
more risk you carry, the greater the
expected return:
E ( Ri )  R f   i  E ( Rm )  R f 
where E ( Ri )  expected return on security i
R f  risk-free rate of interest
 i  beta of Security i
E ( Rm )  expected return on the market
52
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
53
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
54
CAPM (cont’d)
 CAPM
assumptions (cont’d):
• Investors look only one period ahead
• Everyone is equally adept at analyzing
securities and interpreting the news
55
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
56
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
57
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
58
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
59
Using A Scatter Diagram to
Measure Beta
 Correlation
of returns
 Linear regression and beta
 Importance of logarithms
 Statistical significance
60
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
61
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
62
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
63
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
64
Arbitrage Pricing Theory
 APT
background
 The APT model
 Comparison of the CAPM and the APT
65
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
66
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
67
The APT Model
 General
representation of the APT model:
RA  E ( RA )  b1 A F1  b2 A F2  b3 A F3  b4 A F4
where RA  actual return on Security A
E ( RA )  expected return on Security A
biA  sensitivity of Security A to factor i
Fi  unanticipated change in factor i
68
APT
R  E ( R )  1 F1   2 F2  3 F3  
R  E ( R )  1[ R1  E ( R1 )]   2 [ R2  E ( R2 )]  3[ R3  E ( R3 )]  
R  E ( R )  1 E ( R1 )   2 E ( R2 )  3 E ( R3 )  1R1   2 R2   3 R3  
Fixed
Random
(Notice that the security index "A" has been ignored for clarity purposes)
69
Replicating the Randomness
 Let’s
try to replicate the random component
of security A by forming a portfolio with
the following weights:
1 on R1 ,  2 on R2 , 3 on R3 , and finally 1-1   2  3 on R f
We get the following return (for this portfolio of factors):
R  (1-1   2  3 )R f  1 R1   2 R2  3 R3  
Fixed
Random
70
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).
71
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:

E ( R)  1E ( R1 )   2 E ( R2 )  3 E ( R3 )  (1-1   2  3 )R f
Rearranging terms yields:
E ( R)  R f  1[ E ( R1 )  R f ]   2 [ E ( R2 )  R f ]   2 [ E ( R3 )  R f ]
72
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
73
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
74