Portfolio Risk and Return: Part II (Ch. 6)

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CHAPTER 6
PORTFOLIO RISK AND RETURN: PART II
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Venue
Date
FORMULAS FOR PORTFOLIO RISK AND
RETURN
N
R   
E
p
wi E
 Ri 
i 1
N

2
p


wi w j C o v (i , j )
i  1, j  1
N

wi  1
i 1
G iv e n : C o v ( i , j )   ij  i 
N
T hen: 
2
p


i 1

p


2
p
a n d C o v ( i, i )   i
2
j
N
wi  i 
2
2

i , j  1, i  j
w i w j  ij  i 
j
EXHIBIT 6-1 PORTFOLIO RISK AND
RETURN
Portfolio
X
Y
Z
Return
Standard deviation
Correlation between
Assets 1 and 2
X 
Weight in
Asset 1
25.0%
50.0
75.0
Weight in
Asset 2
75.0%
50.0
25.0
10.0%
20.0%
5.0%
10.0%
0.0
.25 .20   .75 .10 
2
2
2
2
Portfolio
Return
6.25%
7.50
8.75
Portfolio Standard Deviation
9.01%
11.18
15.21
 (. 25 )( 0 )(. 20 )(. 10 )  (. 75 )( 0 )(. 10 )(. 20 )  9 . 01 %
PORTFOLIO OF RISK-FREE AND RISKY
ASSETS
Combine
risk-free
asset and
risky asset
Capital
allocation line
(CAL)
Superimpose
utility curves
on the CAL
Optimal
Risky
Portfolio
EXHIBIT 6-2 RISK-FREE ASSET AND
PORTFOLIO OF RISKY ASSETS
DOES A UNIQUE OPTIMAL RISKY
PORTFOLIO EXIST?
Single
Optimal
Portfolio
Identical
Expectations
Different
Expectations
Different
Optimal
Portfolios
CAPITAL MARKET LINE (CML)
Expected Portfolio Return E (Rp)
EXHIBIT 6-3 CAPITAL MARKET LINE
CML
Points above the
CML are not
achievable
Efficient
frontier
E(Rm)
M
Individual
Securities
Rf
Standard Deviation of Portfolio
p
CML: RISK AND RETURN
E  R p   w1 R f   1  w1  E  R m 
 p   1  w1   m
By substitution, E(Rp) can be expressed in terms
of σp, and this yields the equation for the CML:
E Rp 
 E  Rm   R f 
 Rf  
 p
m


EXAMPLE 6-1 RISK AND RETURN ON THE
CML
Mr. Miles is a first time investor and wants to build a
portfolio using only U.S. T-bills and an index fund
that closely tracks the S&P 500 Index. The T-bills
have a return of 5%. The S&P 500 has a standard
deviation of 20% and an expected return of 15%.
1. Draw the CML and mark the points where the
investment in the market is 0%, 25%, 75%, and
100%.
2. Mr. Miles is also interested in determining the
exact risk and return at each point.
EXAMPLE 6-2 RISK AND RETURN OF A LEVERAGED
PORTFOLIO WITH EQUAL LENDING AND BORROWING
RATES
Mr. Miles decides to set aside a small part of his
wealth for investment in a portfolio that has greater
risk than his previous investments because he
anticipates that the overall market will generate
attractive returns in the future. He assumes that he
can borrow money at 5% and achieve the same
return on the S&P 500 as before: an expected return
of 15% with a standard deviation of 20%. Calculate
his expected risk and return if he borrows 25%, 50%,
and 100% of his initial investment amount.
SYSTEMATIC AND NONSYSTEMATIC RISK
Can be eliminated by diversification
Nonsystematic
Risk
Total Risk
Systematic
Risk
RETURN-GENERATING MODELS
ReturnGenerating
Model
Different
Factors
Estimate of
Expected
Return
GENERAL FORMULA FOR RETURNGENERATING MODELS
All models contain return
on the market portfolio as
a key factor
Factor weights or factor
loadings
E  Ri   R f 
k
  E F   
ij
j 1
j
i1
 E  Rm   R f  


Risk factors
k
  E F 
ij
j2
j
THE MARKET MODEL
E  R i   R f   i  E  R m   R f 
R i  R f   i  R m  R f   ei
R i   i   i R m  ei
Single-index model
The difference between
expected returns and realized
returns is attributable to an
error term, ei.
The market model: the intercept, αi, and
slope coefficient, βi, can be estimated by
using historical security and market
returns. Note αi = Rf(1 – βi).
CALCULATION AND INTERPRETATION OF
BETA
i 
i 
C ov  R i , R m 

2
m
0.026250
0.02250


 i ,m  i  m

2
m

 i ,m  i
m
0.70  0.25  0.15
0.02250

0.70  0.25
0.15
 1.17
EXHIBIT 6-6 BETA ESTIMATION USING A PLOT
OF SECURITY AND MARKET RETURNS
Ri
Slope = β [Beta]
Rm
Market Return
CAPITAL ASSET PRICING MODEL (CAPM)
Beta is the primary
determinant of expected return
E  R i   R f   i  E  R m   R f 
E  R i   3%  1.5  9%  3%   12.0%
E  R i   3%  1.0  9%  3%   9.0%
E  R i   3%  0.5  9%  3%   6.0%
E  R i   3%  0.0  9%  3%   3.0%
ASSUMPTIONS OF THE CAPM
Investors are risk-averse, utility-maximizing, rational
individuals.
Markets are frictionless, including no transaction costs or
taxes.
Investors plan for the same single holding period.
Investors have homogeneous expectations or beliefs.
All investments are infinitely divisible.
Investors are price takers.
EXHIBIT 6-7 THE SECURITY MARKET LINE
(SML)
E(Ri)
Expected Return
SML
E(Rm)
M
βi = βm
Slope = Rm – Rf
Rf
1.0
Beta
The SML is a
graphical
representation
of the CAPM.
PORTFOLIO BETA
Portfolio beta is the weighted sum of the betas of the
component securities:
N
p 
 w
i
i
 (0 .4 0  1 .5 0 )  (0 .6 0  1 .2 0 )  1 .3 2
i 1
The portfolio’s expected return given by the CAPM is:
E  R p   R f   p  E  R m   R f 
E  R p   3 %  1 .3 2  9 %  3 %   1 0 .9 2 %
APPLICATIONS OF THE CAPM
Estimates of
Expected
Return
CAPM
Applications
Security
Selection
Performance
Appraisal
PERFORMANCE EVALUATION: SHARPE RATIO
AND TREYNOR RATIO
S h arp e ratio 
T reyn o r ratio 
Rp  Rf
p
Rp  Rf
p
PERFORMANCE EVALUATION: MSQUARED (M2)
Sharpe Ratio
•Identical rankings
•Expressed in
percentage terms
M
2
 Rp  Rf

m
p
  Rm  R f

PERFORMANCE EVALUATION: JENSEN’S
ALPHA
 p  R p   R f   p  Rm  R f  


EXHIBIT 6-8 MEASURES OF PORTFOLIO
PERFORMANCE EVALUATION
Manager
Ri
σi
βi
E(Ri)
X
Y
Z
M
Rf
10.0%
11.0
12.0
9.0
3.0
20.0%
10.0
25.0
19.0
0.0
1.10
0.70
0.60
1.00
0.00
9.6%
7.2
6.6
9.0
3.0
Sharpe Treynor
Ratio
Ratio
0.35
0.064
0.80
0.114
0.36
0.150
0.32
0.060
–
–
M2
αi
0.65%
9.20
0.84
0.00
–
0.40%
3.80
5.40
0.00
0.00
EXHIBIT 6-11 THE SECURITY
CHARACTERISTIC LINE (SCL)
Ri – Rf
Excess Security Return
Excess
Returns
Jensen’s
Alpha
[Beta]
Rm – Rf
Excess Market Return
EXHIBIT 6-12 SECURITY SELECTION
USING SML
SML
15
%
A ( = 11%, β = 0.5)
B ( = 12%, β = 1.0)
Rf = 5%

= 1.0
Beta
Overvalued
Return on Investment
C ( = 20%, β = 1.2)
Undervalued
Ri
DECOMPOSITION OF TOTAL RISK FOR A
SINGLE-INDEX MODEL
R i  R f  i R m  R f
  ei
T o ta l va ria n ce = S yste m a tic va ria n ce + N o n syste m a tic va ria n ce
 i   i  m   e i  2C o v  R m , e i
2
2
2
2
Zero

2
2
2
  i  m   ei
EXHIBIT 6-13 DIVERSIFICATION WITH
NUMBER OF STOCKS
Non-Systematic Variance
Variance
Total
Variance
Variance of Market Portfolio
Systematic Variance
1
5
10
20
Number of Stocks
30
WHAT SHOULD THE RELATIVE WEIGHT
OF SECURITIES IN THE PORTFOLIO BE?
Higher Alpha
→Higher Weight
Inform ation ratio 
Greater NonSystematic Risk
→Lower Weight
i
 ei
2
LIMITATIONS OF THE CAPM
Theoretical
Practical
• Single-factor model
• Single-period model
•
•
•
•
•
Market portfolio
Proxy for a market portfolio
Estimation of beta
Poor predictor of returns
Homogeneity in investor expectations
EXTENSIONS TO THE CAPM:
ARBITRAGE PRICING THEORY (APT)
Risk Premium for
Factor 1
E  R p   R F   1 p ,1 
Sensitivity of the
Portfolio to Factor 1
  K  p ,K
FOUR-FACTOR MODEL
Value
Anomaly
E  R it    i   i , M K T M K Tt   i , SM B SM B t   i , H M L H M L t   i ,U M D U M D t
Size
Anomaly
SUMMARY
• Portfolio risk and return
• Optimal risky portfolio and the capital market line
(CML)
• Return-generating models and the market model
• Systematic and non-systematic risk
• Capital asset pricing model (CAPM) and the
security market line (SML)
• Performance measures
• Arbitrage pricing theory (APT) and factor models

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