F520 Asset Valuation and Strategy
Overview
Risk and Return
F520 – Portfolio Concepts
1
Overview of Market Participants and
Financial Innovation
• What Types of Risk does a Corporation or a
Financial Intermediary Encounter?
F520 – Portfolio Concepts
2
Overview (Cont.)
• How can Financial Products or Intermediaries
reduce these risks
F520 – Portfolio Concepts
3
Risk and Return - Outline
• How is the return on an asset affected by the
risk of the asset?
• How do we measure risk and return on an asset?
– Unique Risk
(diversifiable, unsystematic, residual, or specific)
– Market Risk
(undiversifiable, systematic, or covariance)
• Constructing Portfolios -- How do we measure
risk and return on a portfolio of assets?
• Choosing Stocks -- Development of the Efficient
Frontier and use of Indifference Curves
F520 – Portfolio Concepts
4
Outline - Cont.
• More on Systematic Risk
 Beta
 The Capital Asset Pricing Model (CAPM)
 Security Market Line (SML)
•
•
•
•
Obtaining Estimates of Beta
Uses of Beta
Tests of the Capital Asset Pricing Line and Beta.
Arbitrage Pricing Theory (APT),
an alternative to CAPM
F520 – Portfolio Concepts
5
Measuring Risk - Single Period
r
1


1
P
P
P

D
0
0
P1 = the market value at the end of the interval
P0 = the market value at the beginning of the interval
D = the cash distributions during the interval
F520 – Portfolio Concepts
6
Measuring Return - Multiple Periods
Arithmetic
N
^
Ra 
R
i 1
i
N
• Assumes no reinvestment of cash flows at
the end of each period
F520 – Portfolio Concepts
7
Measuring Return - Multiple Periods
Geometric
R  [(1  R
t
)(1  R p 2)(1  R p 3)...(1  R pN )]
1/ N
p1
1
• Also referred to as Time-Weighted Rate of Return
• Assumes reinvestment of cash flows at the end of each
period.
F520 – Portfolio Concepts
8
Measuring Return - Multiple Periods
Internal Rate of Return
C3
C N  VN
C1
C2
V0 



...

1
2
3
N
(1  RD ) (1  RD ) (1  RD )
(1  RD )
• Also referred to as Dollar-Weighted Rate of Return
• Allows additions and withdrawals
• When no further additions or withdrawals occur and all
dividends are reinvested, the Geometric and the IRR will
yield the same
F520 – Portfolio Concepts
9
Comparing Return Calculations
Without Dividend (Income) Cash Flows
Growth of
$1 investment
assuming
Dividend Return reinvestment
1.00
2.00
0 100%
1.00
-50%
Example of Return Calculations:
Period Price
10
0
20
1
10
2
Arithmetic Return:
Geometric Return
IRR without reinvestment
IRR with reinvestment
25.00%
0.00%
0.00%
0.00%
IRR Cash Flows
Cash flows
Shares
Cash flows
for IRR with
owned with
for IRR no
reinvestment reinvestment reinvestment
-10
1
-10
0
1.00
0
10
1.00
10
F520 – Portfolio Concepts
10
Comparing Return Calculations
With Dividend (Income) Cash Flows
Example of Return Calculations:
Period Price
0
10
1
18
2
9
Growth of
$1 investment
assuming
Dividend Return reinvestment
1.00
2 100%
2.00
-50%
1.00
Arithmetic Return:
Geometric Return
IRR without reinvestment
IRR with reinvestment
25.00%
0.00%
5.39%
0.00%
IRR Cash Flows
Cash flows
Shares
Cash flows
for IRR no
owned with for IRR with
reinvestment reinvestment reinvestment
-10
1
-10
2
1.11
0
9
1.11
10
F520 – Portfolio Concepts
11
Measuring Total Risk
Variance of actual returns
1 n
Variance(r ) 

N  1 t 1
r r 
2
t
• Measures of the dispersion of returns
• Standard Deviation (STD)  Variance
Standard deviation measures dispersion in percents
F520 – Portfolio Concepts
12
Historical Returns, Standard Deviations, and
Frequency Distributions: 1926-2009
F520 – Portfolio Concepts
13
Example Frequency Distribution
• Frequency distribution is a histogram of yearly returns
Goal: Select the lowest risk portfolio
•
•
•
•
•
•
0% stock, 100% bond
20% stock, 80% bond
40% stock, 60% bond
60% stock, 40% bond
80% stock, 20% bond
100% stock, 0% bond
F520 – Portfolio Concepts
15
Constructing Portfolios
• Investors seek to maximize the expected return from their
investment given some level of risk, or
• Investors seek to minimize the risk they are exposed to
given some target expected return.
F520 – Portfolio Concepts
16
Constructing Portfolios
Portfolio Return
• Expected Return of a Portfolio equals the weighted
average return on the portfolio
Rp = wa * Ra + wb * Rb
wa
wb
Ra
Rb
=
=
=
=
weight of asset a
weight of asset b
Expected return of asset a
Expected return of asset b
• General Formula
R
p

n
 wi R
i 1
i
– Weights must add to 1
w1 + w2 + ... + wn = 1
F520 – Portfolio Concepts
17
Constructing Portfolios
Portfolio Variance
• Two Asset Case
Var(Rp) = Var(wa * Ra + wb * Rb )
 w   w   2w w 
2
2
2
2
a
a
b
b
a
b
ab
• General Case
G
  w2g
g 1
–
–
–
–
G
  w2g
g 1
G
G
 g  g1 h1 wg wh  gh
2
for h  g
since 12 = 21, each covariance term is included in this equation twice.
i is the variance of asset i
gh is the covariance between asset g and asset h
G
G
 g  g1 h1 wg wh
2
p gh g h
F520 – Portfolio Concepts
where
p
gh



gh
g
h
18
Portfolio Variance
Using Correlation
p



gh
• Correlation
gh
g
h
is the covariance standardized by the standard deviation of
the two variables.
– p = 1, perfect positive correlation
– p = -1, perfect negative correlation
– p = 0, no correlation
• Two Asset Case
VAR( R p)  x 
2
2
a
a
 xb  b  2 xa xb
2
2
p 
ab
a
b
• General Case
G
2
VAR( R p )   w g
g 1
G
G
 g  g1 h1 wg wh
2
F520 – Portfolio Concepts
p gh g h
19
Input Data
A
B
Efficient Frontier
Correlation = 1
Efficeint Frontier (Corr = 1)
Return
12%
16%
Std. Dev.
10%
20%
Correlation
17.0%
1.00
Expected Return
16.0%
15.0%
14.0%
13.0%
12.0%
11.0%
10.0%
0.0%
3.0%
6.0%
9.0%
12.0%
15.0%
18.0%
21.0%
Standard Deviation



 xb  b  2 xa xb
p  xa  a
2
2
p
p
2
2
2
2
x a a  xb b
 x a a  x b b 

p 
ab
a
b
Correlation = 1
2
F520 – Portfolio Concepts
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Input Data
A
B
Efficient Frontier
Correlation = -1
Efficeint Frontier (Corr = -1)
Return
12%
16%
Std. Dev.
10%
20%
Correlation
17.0%
-1.00
16.0%
Expected Return
15.0%
14.0%
13.0%
12.0%
11.0%
10.0%
0.0%
3.0%
6.0%
9.0%
12.0%
15.0%
18.0%
21.0%
Standard Deviation
  x   x   2x x p  
  x   x   2x x  
  x a a  xb b
F520 – Portfolio Concepts
  x a a  xb b
2
2
2
2
2
p
a
a
b
b
2
2
2
2
2
p
a
a
b
b
2
a
a
b
b
ab
a
a
b
Correlation = -1
b
2
p
p
21
Input Data
A
B
Efficient Frontier
Correlation = 0
Efficeint Frontier (Corr = 0)
Return
12%
16%
Std. Dev.
10%
20%
Correlation
17.0%
0.00
Expected Return
16.0%
15.0%
14.0%
13.0%
12.0%
11.0%
10.0%
0.0%
3.0%
6.0%
9.0%
12.0%
15.0%
18.0%
21.0%
Standard Deviation
Correlation = -1
 p  x a  a  xb  b  2 x a xb
2
2
2
2
2
 p  x a  a  xb  b
2

p
2

2
x
2
2
2
a
a
p 
ab
a
Correlation = 1
Correlation = 0
b
2
 xb  b
2
2
F520 – Portfolio Concepts
22
Portfolio Diversification
Average annual
standard deviation (%)
49.2
Diversifiable risk
23.9
19.2
Nondiversifiable
risk
1
10
20
30
40
F520 – Portfolio Concepts
1000
Number of stocks
in portfolio
23
Efficient Frontier Conclusions
• The covariance of two assets is important in determining
the variance of a portfolio
• As long as assets are not perfectly correlated, combining
them in a portfolio reduces risk
• Systematic risk cannot be eliminated by diversification
because it is the covariance risk. Also called nondiversifiable or market risk, since it is primarily from
economy wide factors.
• Unsystematic risk (also called diversifiable risk, unique
risk, or firm specific risk) comes from circumstances
unique to the firm. This is why in a well diversified
portfolio, unique risk is unimportant.
F520 – Portfolio Concepts
24
Covariance – the key to diversification
Mathematical Example
• Assume a Special Case:
G
  wi2
i 1

2
i
Cov(i,h) = 0
G G
   wi
i 1 h 1
wh 
ih
• As our portfolio gets large, the variances of the portfolio gets vary
small if all the covariances are 0.
n
 p   xi i
2
2
2
i 1
•
If all assets have weight Yn then x = 1 / n
•
2


2
2

1
j 

   j  
 p
2
i 1  n 
i 1
n


If the largest variance is V
2
n
n
p  
2
i 1
•
•
V
2
n
n

nV
2
n

V
n
As n gets large, this goes to zero.
Therefore, our portfolio choices are dominated by concern over the covariance
terms. In other words, well diversified investors need only price the risk
F520of
– Portfolio
25
associated with the covariance
assets. Concepts
Covariance the key to diversification
- Intuitive Example
# of Assets
in the
Portfolio
# of
# of Variance Covariance
Terms
terms
1
2
3
4
5
10
20
50
100
1
2
3
4
5
10
20
50
100
F520 – Portfolio Concepts
0
1
3
6
10
45
190
1225
4950
26
Conclusions on Covariance
• Question
What will the addition of this asset to my portfolio do to
my level of risk?
• Answer:
Look at the covariance of the asset with my portfolio,
rather than the variance.
F520 – Portfolio Concepts
27
Choosing Stocks
• Investors maximize their welfare by choosing the:
– Set of securities (investments) that maximize return for a given
level of risk.
– Set of securities (investments) that minimize risk for a given level
of return.
F520 – Portfolio Concepts
28
Input Data
Efficient Frontier
Correlation = 0
Efficeint Frontier
A
B
Return
6.5%
12%
Std. Dev.
7.1%
16%
Correlation
0.00
Expected Return
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
0.0%
3.0%
6.0%
9.0%
12.0%
15.0%
18.0%
Standard Dev iation
Correlation = 0
QU: How do Investors Choose a
F520 – Portfolio Concepts
Portfolio on the
Efficient Frontier?
29
Use Indifference Curves –
measures of investor risk aversion
Efficeint Frontier
Expected Return
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
0.0%
3.0%
6.0%
9.0%
12.0%
15.0%
18.0%
Standard Dev iation
Correlation = 0
QU: How Does this Change when a
F520 – Portfolio Concepts
Risk-free asset
is offered?
30
Investors can move to a higher
indifference curve – greater utility.
Efficeint Frontier
12.0%
Expected Return
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
0.0%
3.0%
6.0%
9.0%
12.0%
15.0%
18.0%
Standard Deviation
Correlation = 0
QU: Can you identify the important parts in the graph.
F520 – Portfolio Concepts
31
Important points on the graph.
AAL – Asset Allocation or
CML – Capital Market Line
Efficeint Frontier
Borrowing
12.0%
Expected Return
10.0%
Risk-free
rate
8.0%
Lending
Market
Portfolio
6.0%
4.0%
2.0%
0.0%
0.0%
3.0%
6.0%
9.0%
12.0%
15.0%
18.0%
Standard Deviation
Correlation = 0
QU: What is meant by two-fund separation?
F520 – Portfolio Concepts
32
Measuring Risk and Return for the CML
• The risk free asset has no variance and its return is known
with certainty (proxy – T-bill)
• Portfolio Return on CML
R x R x R
p
F
RF
m
m
• Portfolio Risk on CML
  2w w p  
VAR( R )  w 0  w   2w w 0 0 
VAR( R )  w 
STD ( R )  w 
Standard Deviation is a linear function
of the STD of the market portfolio
STD ( R )  w 
VAR( R p )  wa
2
p
p

2
2
b
2
2
2
2
a
a
b
b
2
2
b
b
p
p
 wb
a
2
b
2
2
b
b
a
a
b
b
a
ab
ab
a
b
b
b
F520 – Portfolio Concepts
33
Conclusions from Efficient Frontier and CML
• As long as there are only risky assets, it makes sense for
investors to hold a portfolio on the efficient frontier.
The existence of a risk-free asset changes this. The new
efficient frontier (called the capital market line) will
connect the risk free asset to some risky portfolio.
• The market portfolio (Rm) should be chosen because any
other security will lead to a lower return for a given level
of risk (Tangent portfolio).
• All investors will hold some combination of the risk-free
asset and the market portfolio, since this will maximize
their risk-return trade-off. (called two-fund separation)
• The CML portfolio chosen by an investor depends upon
their risk aversion
F520 – Portfolio Concepts
34
• The Capital Market Line (CML) is
Rp = Rf + slope (Standard Deviation)
 RM  RF 

R  R 
 M 
p
F
p
• The CML is a linear relationship between the efficient
portfolio’s standard deviation and its expected return.
QU: Can we transform the CML to another measure of
risk which only accounts for systematic risk?
F520 – Portfolio Concepts
35
SML, Beta, and CAPM
• The CML shows that all investors must hold a combination
of the risk-free asset and the market portfolio to maximize
their utility.
Furthermore, it shows that their is a linear relationship
between risk and return.
Knowing that two points make a line, let’s form the SML
by plotting these points.
Return
Rm
Rf
F520 – Portfolio0.0
Concepts
1.0
Beta
36
Security Market Line
Return
Rm
Rf
0.0
1.0
Beta
• Ri = Rf + (Rm - Rf)
• Where (Rm - Rf) is the slope of the line
• Beta measures the risk of a stock in regards to the market
portfolio (similar to the average stock).
F520 – Portfolio Concepts
37
Understanding Beta and
Calculating Portfolio Betas
• Beta measures the relative volatility of stock i with the
market portfolio.
• The beta of a portfolio is the market value weighted
average of the betas in the portfolio.
n
B x B
p
i 1
i
i
F520 – Portfolio Concepts
38
Example: Portfolio Beta Calculations
Stock
(1)
Market Portfolio
Value Weights
(2)
(3)
Haskell Mfg. $ 6,000
Cleaver, Inc.
4,000
Rutherford Co. 2,000
50%
33%
17%
Portfolio
$12,000
100%
F520 – Portfolio Concepts
Beta
(4) (3) x (4)
0.90
1.10
1.30
0.450
0.367
0.217
1.034
39
Beta, Expected Return and the Choice of
Projects (Stock)
• The concept that all assets must lie on the SML can also be
Shown through an arbitrage argument. Consider Assets A,
B, C, and D below. What will happen to the prices and
expected returns of these assets in a competitive market
using diversification techniques to eliminate all
unsystematic risk?
Return
B
Rm
A
Rf
C
D
0.0
1.0
Beta
QU: How do I set up a trade
to take advantage of this
“mis-pricing”?
F520 – Portfolio Concepts
40
Hedge Fund Example
• How should I invest in these securities to take advantage of
my expectations in returns relative to the required return.
(Think about a hedge fund.)
Return
B
CML = 5+B(6)
Rm
A
Rf
C
D
0.0
1.0
Beta
Beta E(Return) Req. Ret
A
0.6
8.6
5+.6*6 = 8.6
B
0.8
12.0
5+.8*6 = 9.8
C
1.4
D
0.6
F520 – Portfolio Concepts
10 5+1.4*6 = 13.4
4
5+.6*6 = 8.6
41
Hedge Fund Example
• Some may think of having a net investment of zero, but look at the
returns with market movements. None of our securities moved closer
to efficiency in the example below. They each just followed the
market as their risk would suggest.
A
B
C
D
Beta E(Return)
0.6
8.6
0.8
12
1.4
10
0.6
4
Req.
Portfolio Market
Ret
Invest Beta
+10%
8.6
0
6.00%
9.8 2000
0.40
8.00%
13.4 -1000
(0.35) 14.00%
8.6 -1000
(0.15)
6.00%
4000
(0.10) -1.00%
Absolute
Profit Market
(Loss) -10%
$ -6.00%
$ 160
-8.00%
$(140) -14.00%
$ (60) -6.00%
$ (40)
1.00%
Profit
(Loss)
$ $ (160)
$ 140
$
60
$
40
• How can we reduce our market risk while still taking a position on our
expectations?
F520 – Portfolio Concepts
42
Hedge Fund Example
• How can we reduce our market risk while still taking a position on our
expectations?
A
B
C
D
Beta E(Return)
0.6
8.6%
0.8
12.0%
1.4
10.0%
0.6
4.0%
Req.
S=-1 Net
Portfolio Market
Ret
Invest L=+1 Position Beta
+10%
8.6
0
0.00%
9.8 2000
1
2,000
0.40
8.00%
13.4
500
-1
(500)
(0.17) -14.00%
8.6 1500
-1 (1,500)
(0.23) -6.00%
4000
0
0.00
0.00%
Profit
Market
(Loss)
-10%
$
0.00%
$
160
-8.00%
$
(70) 14.00%
$
(90) 6.00%
$
0
0.00%
• Wb*Bb + Wc*Bc + Wd*Bd = 0 [no market risk]
• Wb + Wc + Wd = 0 [no investment for arbitrage]
• Having a portfolio beta of zero immunizes the portfolio from the
market changes, and allows us to profit only from the unsystematic
movements in prices, which is where one would find “mis-pricing”.
• Remember this still has risk (betas could be incorrect, our estimates of
over- and under-pricing could be incorrect).
• Controlling for market movements, you expect prices of securities with
expected returns that are higher relative to the required return to
increase and lower expected returns to decrease.
F520 – Portfolio Concepts
43
Hedge Fund Example
• The prior example showed no profit, because we assume that the
returns on the stock were exactly equal to their expected return based
on the market return and their beta. What is the hedge fund correctly
predicted over and undervalued stocks?
Exp Ret,
A
B
C
D
Beta E(Return)
0.6
8.6%
0.8
12.0%
1.4
10.0%
0.6
4.0%
•
•
•
•
Req.
S=-1 Net
Portfolio Mkt
Actual
Ret
Invest L=+1 Position Beta
+10%
Return
8.6
0
0.00%
0.00%
9.8 2000
1
2,000
0.40
8.00% 10.00%
13.4
500
-1
(500)
(0.17) -14.00% -12.00%
8.6 1500
-1 (1,500)
(0.23) -6.00% -4.00%
4000
0
0.00
0.00%
Profit
(Loss)
$
$
200
$
(60)
$
(60)
$
80
Stock B is undervalued (Exp Ret > Req Ret), so we purchased a long position. Based on
a market return of 10%, we expected it to increase 8% (market * beta), but our hedge
fund model prediction was correct, adding 2%, so we made a net 10%.
Stock C is overvalued (Exp Ret < Req Ret), so we took a short position. Based on a
market return of 10%, we expected it to increase 14% (market * beta), but our hedge
fund model prediction was correct, reducing it by 2% for a net increase of 12%. Since
we were short, we lost 12%.
Stock D is overvalued (Exp Ret < Req Ret), so we took a short position. Based on a
market return of 10%, we expected it to increase 6% (market * beta), but our hedge fund
model prediction was correct, reducing it by 2% for a net increase of 4%. Since we
were short, we lost 4%.
Our portfolio has 0 beta and made money.
F520 – Portfolio Concepts
44
Hedge Fund Example
• What is the market had decreased in value?
A
B
C
D
Beta E(Return)
0.6
8.6%
0.8
12.0%
1.4
10.0%
0.6
4.0%
•
•
•
•
•
Req.
S=-1 Net
Portfolio Exp Ret, Actual
Ret
Invest L=+1 Position Beta
Mkt -10% Return
8.6
0
0.00%
0.00%
9.8 2000
1
2,000
0.40
-8.00% -6.00%
13.4
500
-1
(500)
(0.17) 14.00% 16.00%
8.6 1500
-1 (1,500)
(0.23)
6.00%
8.00%
4000
0
0.00
0.00%
Profit
(Loss)
$
$ (120)
$
80
$
120
$
80
Stock B is undervalued (Exp Ret > Req Ret), so we purchased a long position. Based on
a market return of -10%, we expected it to decrease 8% (market * beta), but our hedge
fund model prediction was correct, adding 2%, so we made a lost 6%.
Stock C is overvalued (Exp Ret < Req Ret), so we took a short position. Based on a
market return of -10%, we expected it to decrease 14% (market * beta), but our hedge
fund model prediction was correct, reducing it by 2% for a net decrease of 16%. Since
we were short, we made 16%.
Stock D is overvalued (Exp Ret < Req Ret), so we took a short position. Based on a
market return of -10%, we expected it to decrease 6% (market * beta), but our hedge
fund model prediction was correct, reducing it by 2% for a net decrease of 8%. Since
we were short, we made 8%.
Our portfolio has 0 beta and made money.
As long as our hedge fund model to predict over and under-valued stocks is correct, we
make money in either an up or a down market.
F520 – Portfolio Concepts
45
Uses of Beta
• Discount rates in capital budgeting
• Discount rates for pricing assets (stocks)
• Utilities often base rates on the rate of return investors
demand.
• Cost of capital calculations
• QU: What does the SML tell about the risk that managers
should be concerned with when choosing a real asset
investment (specifically a capital budgeting decision)?
F520 – Portfolio Concepts
46
Estimating Beta – Characteristic Line
• Ri = Rf + (Rm - Rf)
• rearranging terms
Ri = Rf + *Rm - *Rf
Ri = (1- ) Rf + * Rm
• Characteristic Line (also called market model)
Ri = ά + * Rm + eit
• Where
 im
Bi 

2
m
 = covariance (Ri, Rm) / Var (Rm)
• Based on the market model, we can also break down an
assets total risk into systematic and unsystematic
components.
Total Risk = 2i = 2i 2m + 2ei
F520 – Portfolio Concepts
47
Differences in Beta Calculations
•
•
•
•
Merrill Lynch – 5 years of monthly returns
Value Line – 5 years of weekly returns
Historic Beta – Calculated with only the raw return data
Adjusted Beta – Begins with a firms historic beta and
makes an adjustment for the expected future movement
towards one. (Beta has been found to gradually approach
1 over time)
• Fundamental Beta – Adjusts historic betas for variables
such as financial leverage, sale volatility, etc.
F520 – Portfolio Concepts
48
Data For Beta Calculation – Lilly Stock
Calculations in yellow, WRETD = Value weighted return,
DATE
DIVAMT
PRC
CFACPR
Adj Prc
Adj Div
Return
31-Jan-95
65.875
4 16.46875
0
28-Feb-95
0.645
67
4
16.75
0.16125 0.026869
31-Mar-95
73.125
4 18.28125
0 0.091418
28-Apr-95
74.75
4
18.6875
0 0.022222
31-May-95
0.645
74.625
4 18.65625
0.16125 0.006957
30-Jun-95
78.5
4
19.625
0 0.051926
31-Jul-95
78.25
4
19.5625
0 -0.003185
31-Aug-95
0.645
81.875
4 20.46875
0.16125 0.054569
29-Sep-95
89.875
4 22.46875
0
0.09771
31-Oct-95
96.625
4 24.15625
0 0.075104
30-Nov-95
0.685
99.5
4
24.875
0.17125 0.036843
29-Dec-95
0
56.25
2
28.125
0 0.130653
31-Jan-96
57.25
2
28.625
0 0.017778
29-Feb-96
0.3425
60.625
2
30.3125
0.17125 0.064934
29-Mar-96
65
2
32.5
0 0.072165
30-Apr-96
59.125
2
29.5625
0 -0.090385
31-May-96
0.3425
64.25
2
32.125
0.17125 0.092474
28-Jun-96
65
2
32.5
0 0.011673
31-Jul-96
56
2
28
0 -0.138462
30-Aug-96
0.3425
57.25
2
28.625
0.17125 0.028438
30-Sep-96
64.5
2
32.25
0 0.126638
31-Oct-96
70.5
2
35.25
0 0.093023
29-Nov-96
0.3425
76.5
2
38.25
0.17125 0.089965
31-Dec-96
73
2
36.5
0 -0.045752
31-Jan-97
87.125
2
43.5625
0 0.193493
28-Feb-97
0.36
87.375
2
43.6875
0.18 0.007001
31-Mar-97
82.25
2
41.125
0 -0.058655
30-Apr-97
87.875
2
43.9375
0 0.068389
30-May-97
0.36
93
2
46.5
0.18 0.062418
30-Jun-97
109.3125
2 54.65625
0 0.175403
31-Jul-97
113
2
56.5
0 0.033734
29-Aug-97
0.36
104.625
2 F520
52.3125
0.18
-0.070929
– Portfolio Concepts
30-Sep-97
121
2
60.5
0 0.156511
31-Oct-97
0
67.0625
1
67.0625
0 0.108471
VWRETD
0.0396228
0.0269823
0.0248828
0.0341465
0.0308407
0.0406674
0.0093427
0.0363905
-0.011145
0.042971
0.0153999
0.0280874
0.0160582
0.0112021
0.0251253
0.0267221
-0.007659
-0.05339
0.0322224
0.0529918
0.0139377
0.0657296
-0.011362
0.0530395
-0.000889
-0.044396
0.0424802
0.071263
0.0441994
0.0763158
-0.036456
0.0580094
-0.034116
49
Data For Beta Calculation – Lilly Stock
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.089765
R Square
0.008058
Adjusted R Square
-0.006318
Standard Error
0.09864
Observations
71
0.008057732
Percent of total variation
explained =
MS(reg)/SS(total)
Total
Percent
Systematic Risk
7.6811E-05
0.00806 R-squared
Unsystematic Risk
0.00945575
0.99194 1- R-squared
Total Risk
0.00953256
0.00953256
1 Should add to 1
due to the degrees of freedom in
regression these are not exactly
equal to each other
ANOVA
df
Regression
Residual
Total
Intercept
X Variable 1
1
69
70
SS
0.00545357
0.671358409
0.676811978
Coefficients Standard Error
0.027862
0.01243276
0.196282
0.262175336
MS
F
Significance F
0.00545357 0.56049986 0.456603
0.009729832
t Stat
P-value Lower 95% Upper 95% Lower 95.0%
2.241034391 0.02824458
0.00306 0.05266491 0.00305957
0.748665388 0.45660343 -0.326744 0.71930692 -0.3267437
Upper 95.0%
0.052664912
0.719306923
Alpha
Beta estimate
RESIDUAL OUTPUT
Observation
Predicted Y
1 0.035639
Residuals
SS(residuals)
-0.008770407
7.692E-05
F520 – Portfolio Concepts
50
Assumptions of the CAPM (SML)
• Assumptions about investor behavior
– Investors use only two measures to determine their strategy,
expected return and risk,
– Investors will choose portfolios as a risk reduction technique,
– Investors make investment decisions over some single-period
investment horizon,
– Homogenous expectations with respect to asset returns, variances,
and correlations
• Assumptions about capital markets
– Perfect competition,
– No transaction costs
• -No bid-ask spreads, -No commissions, -No information costs, -No
taxes, -No regulation, and -all assets are marketable
– Investors can borrow and lend at the risk-free rate.
F520 – Portfolio Concepts
51
Test of the CAPM (SML)
• Clearly the assumptions are unrealistic, but the true test of a
model comes from answering two questions
– Does the model change when the assumptions are changed?
– How well does the model predict?
• Empirical Findings
– There is a significant positive relationship between realized returns and
systematic risk. However, the slope is usually less than predicted by the
CAPM.
– The relationship between risk and return appears to be linear. No evidence of
curvature has been found.
– Tests assessing the importance of company specific risk after controlling for
market risk are inconclusive. Econometrically controlling for market risk given
its high correlation with total risk is difficult.
– The CAPM should be valid for all assets; however, bonds do not track along the
SML.
– Betas of individual stocks are not stable over time; however, betas for portfolios
are stable over time.
F520 – Portfolio Concepts
52
Anomalies with using the CAPM
• Small firm effect
• Price-to-Book Ratios (Growth versus value stocks)
• January effect
F520 – Portfolio Concepts
53
Common Question:
When using CAPM [Ri=Rf+i(Rm – Rf)],
what is the Risk Premium (Rm – Rf)
What is the Rf you are using?
Should you use Large or Small Stocks?
Should you use arithmetic or geometric returns?
F301_CH12-54
Can CAPM be used for bond?
•
(August 9, 2013 data)
Lehman Index (ticker = AGG)
http://www.ishares.com/product_info/fund/overview/AGG.htm
Effective Duration
Average Yield to Maturity
5.05 years
2.16%
http://finance.yahoo.com/q/rk?s=AGG+Risk
Beta (against Standard Index)
R-squared (against Standard Index)
•
1.01 Yahoo
98.88
Yahoo betas are 5-years
Lehman 1-3 year Treasury Bond Fund (ticker = SHY)
http://www.ishares.com/product_info/fund/overview/SHY.htm
Effective Duration 1.86
Average Yield to Maturity 0.32%
http://finance.yahoo.com/q/rk?s=SHY+Risk
Beta (against Standard Index)
R-squared (against Standard Index)
0.14 Yahoo
24.31
What is the standard index in this case?
– So what is beta in this case?
1.86 / 5.05 = 0.36, compare to Beta?
F520 – Portfolio Concepts
55
Can CAPM be used for bond?
•
Lehman 7-10 Year Treasury Bond Fund (ticker = IEF)
http://www.ishares.com/product_info/fund/overview/IEF.htm
Effective Duration 7.48
Average Yield to Maturity 2.29%
http://finance.yahoo.com/q/rk?s=IEF+Risk
Beta (against Standard Index)
R-squared (against Standard Index)
1.70 Yahoo
69.52
– So what is beta in this case?
7.48 / 5.05 = 1.48, compare to Beta?
•
Lehman 20+ Year Treasury Bond Fund (ticker = TLT)
http://www.ishares.com/product_info/fund/overview/TLT.htm
Effective Duration 16.43
Average Yield to Maturity 3.61%
http://finance.yahoo.com/q/rk?s=TLT
Beta (against Standard Index)
R-squared (against Standard Index)
3.37 Yahoo
53.81
– So what is beta in this case?
16.43 / 5.05 = 3.25, compare to Beta?
F520 – Portfolio Concepts
56
Cont.
•
•
The concept of Beta, used by Yahoo Finance and MSN Money for bonds is not
the same concept of beta referred to in stocks. When a bond index is used as
the standard index, we obtain a relative measure of duration. When a stock
index is used, we obtain the traditional measure of systematic risk.
When using public betas, identify the index used to interpret the concept of
beta reported. For many companies/funds, they state a “Standard Index”, to
properly interpret the measures, you must clearly identify the index. (MSN
Money provides identification, Yahoo does not.)
–
–
For Ishare Austria Fund:
http://investing.money.msn.com/investments/etf-management?symbol=ewo
For Ishare Japan Fund:
http://investing.money.msn.com/investments/etf-management?symbol=ewj
Standard Index is MSCI EAFE NDTR_D
EAFE stands for Europe, Australasia, and Far East. The index has stocks from 21 developed
markets, excluding the U.S. and Canada.
For Ishare S&P Small Cap 600 Index, (uses S&P500)
http://investing.money.msn.com/investments/etf-management?symbol=ijr
For Ishare NAREIT Industrial/Office Index Fund (uses MSCI World)
http://investing.money.msn.com/investments/etf-management?symbol=fnio
F520 – Portfolio Concepts
57
Multifactor CAPM
• Multi-Factor CAPM
E(Ri) = Rf + i,M[E(RM) - Rf] + i,f1[E(Rf1) - Rf]+
i,f2[E(Rf2) - Rf] +…+ i,fn[E(Rfn) - Rf]
• By rearranging terms we get the multiple
regression typically used.
E(Ri) = Rf + i,M*E(RM) - i,M*Rf + i,f1*E(Rf1) - i,f1*Rf +
i,f2*E(Rf2) - i,f2*Rf +…+ i,fn*E(Rfn) - i,fn*Rf
E(Rit) =  + i,Mt*E(RMt) + i,f1*E(Rf1t) + i,f2*E(Rf2t) +…+
i,fn*E(Rfnt) + eit
where
 = Rf - i,M*Rf - i,f1*Rf - i,f2*Rf -…- i,fn*Rf
Rf = Riskfree Rate
Rf1 = Expected Return on factor 1
F520 – Portfolio Concepts
58
Arbitrage Pricing Theory (APT), an
alternative to the CAPM
• E(Ri) = Rf + i,f1[E(Rf1) - Rf] + i,f2[E(Rf2) - Rf] +…+
i,fn[E(Rfn) - Rf]
• By rearranging terms we get the multiple regression typically
used.
E(Rit) = + i,f1*E(Rf1t) + i,f2*E(Rf2t) +…+ i,fn*E(Rfnt) + eit
where
 = Rf - i,f1*Rf - i,f2*Rf -…- i,fn*Rf
Rf = Risk-free Rate
Rf1 = Expected Return on factor 1
F520 – Portfolio Concepts
59
Assumptions of APT
• APT assumes returns are a function of several factors, not
just one as in the CAPM
• Suggested factors (Roll & Ross 1983)
–
–
–
–
Index of Industrial Production,
Changes in the default risk premium on bonds,
Changes in the yield curve,
Unanticipated inflation
• Other factors frequently considered
– Factors for size
– Factors for book-to-market value
F520 – Portfolio Concepts
60
Principles to Take Away
from the APT and CAPM
• Investing has two dimensions, risk and return.
• It is inappropriate to look at the risk of an individual asset
when deciding whether it should be included in a portfolio.
What is important is how the inclusion of an asset into a
portfolio will affect risk of the portfolio (covariance and/or
beta must be considered).
• Risk can be divided into two categories, systematic and
unsystematic
• Investors should only be concerned about systematic risks.
F520 – Portfolio Concepts
61
Commonly used Portfolio Performance Criteria
are based on the Efficient Frontier or CAPM concepts
Global Tech Fund: Return: +37.2%, Beta: 1.29, Std. Dev 25%, Riskfree = 5.0%, RiskPremium = 6.0%
• Sharpe Ratio
= (Rp – Rf)/σp
Efficeint Frontier
12.0%
Expected Return
10.0%
= (37.2 – 5)/25 = 1.29
8.0%
6.0%
4.0%
• Treynor Ratio
= (Rp – Rf)/Bp
2.0%
0.0%
0.0%
3.0%
6.0%
9.0%
12.0%
15.0%
18.0%
Standard Deviation
Correlation = 0
= (37.2 – 5)/1.29 = 25.0%
• Jensen’s alpha (αp)
Return
R
= Rp – CAPM
= Rp – [Rf+Bp(RM – Rf)]
m
= 37.2 – [5+1.29(6)] = 24.5%
Rf
0.0
F520 – Portfolio Concepts
1.0
Beta
62
M-Squared Measure (Modigliani and Modigliani)
Global Tech Fund: Return: +37.2%, Beta: 1.29, Std. Dev 25%, Riskfree = 5.0%, RiskPremium = 6.0%
This fund has a beta of 1.29, substantially greater than the market beta of 1.0. To
compare it to the market, we must determine what portion can be invested in
the risk-free rate and what portion invested in the Global Tech Fund to have
the same risk as the market.
Let x = the percent invested in the Global Tech Fund, subsequently (1-x) is the
percent in the risk-free asset.
Note that the beta of the risk-free asset is equal to zero.
(1-x)(0) + x(1.29) = 1.0
Solve for x.
x = 1/1.29 = .78 portion of the portfolio in the Global Tech Fund
1-x = .22 portion invested in the risk-free asset
Now calculate your risk-adjusted return:
The Global Tech Fund earned 37.2% and the risk-free asset over this 3-year period
earned 5%. The proportions in each asset are calculated above.
.22(5%) + .78(37.2%) = 30.1%
This value can be compared to what the market earned during this period, since it
has a beta of 1.
F520 – Portfolio Concepts
63