Topic 5: Capital Asset Pricing
Model
Purpose:
Review the highlights of Asset Pricing
Theory, focusing on the CAPM
Apply these concepts to portfolio
performance evaluation
-1-
Asset pricing models in general:
• Creating surrogate portfolios
• Using surrogate portfolios for arbitrage
– Goal: buy something cheap and sell
something just like it for a higher price
– Goal: hedge against market movements as
protection while trading on scarce
information
-2-
Betting Without Hedge
• Suppose you sell GM short, based on scarce
information
GM
Factor
Uncertain Future
Take Position
-3-
Betting Without Hedge
• Then market rises, carrying GM with it
GM
Factor
Uncertain Future
Take Position
-4-
Betting Without Hedge
• Then market rises, carrying GM with it
• GM takes hit, but you still lose
GM takes hit
GM
Uncertain Future
Factor
Take Position
-5-
A formal look at the assumptions necessary to
partition risk
n
Markowitz formula
Security 1
Security 1
Security 2
Security 3
n
n
s 2p = Â Â wi w j s i s j r ij
i=1 j=1
Security 2
Security 3
w12 s 12
2w1w2 s 1 s 2 r12
w22 s 22
2w1w3 s 1s 3r 13 2w2w3 s 2 s 3r 23
w32 s 32
-6-
Complex analysis becomes simple
n
Sharpe’
Sharpe’s approximation
n
s 2p = Â wi2 b i2 s m2
i=1
Security 1
Security 1
Security 2
Security 3
w12 b 12 s m2
Security 2
Security 3
w22 b 22 s m2
w32 b 32 s m2
-7-
Measuring systematic risk with a singleindex model (the CAPM)
• Regression analysis:
tool for measuring
interaction between
two variables related
by a linear function
y = a + bx + e
R j - R f = a j + b j ( Rm - Rf ) + e j
-8-
Coin Tossing Game
Heads: Up 20%
Tails: Down 10%
Start at 100
300.00
250.00
200.00
150.00
100.00
50.00
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
-9-
Another Coin Tossing Game
2 stocks, start at 100, tend to follow market except for epsilon
Stock A: Beta 1
Stock B: Beta 2
Market
Stock A
Stock B
500.00
450.00
400.00
350.00
300.00
250.00
200.00
150.00
100.00
50.00
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
- 10 -
Example of company-specific news
• DELL COMPUTER said it will meet
quarterly forecasts, bucking a gloomy
industry trend. This was a surprise to
many analysts, who had predicted a
shortfall. Shares rose 8.1%.
– Source: Wall Street Journal, Oct. 4, 2001
- 11 -
Partitioning total risk into systematic and
unsystematic elements: The Intuition
• Assignment: design a video game to mimic stock
market
• Step 1: Random move applies equally to all stocks
• Step 2: Assign response coefficient to each stock
– Average of all coefficients is one
• Step 3: Add unique random move for each stock
– Average of all these is zero
• Question: What happens if a player diversifies across
all the stocks in this game?
- 12 -
Criticism of the Model
• The model is very simple
• Two kinds of news
– Company-specific
– Global
• Question for discussion: Does the model
need additional factors, such as industry
effects or regional effects?
- 13 -
How does a single-index model
work?
• For the moment, let us suspend the
criticism and consider how we could
analyze this simple game
- 14 -
Discovering beta
• The characteristic
line
• The slope is the b
Ri - Rf
Rm - Rf
- 15 -
Estimating beta from historical data
• The meaning of r
Ri - Rf
Rm - Rf
- 16 -
Estimating beta from historical data
• The meaning of r
Ri - Rf
Rm - Rf
- 17 -
Estimating beta from historical data
r2 is the proportion of
uncertainty that is
explained when we
consider the
influence of the
market (this is the
proportion of risk
that is systematic)
after
before
- 18 -
Estimating beta from historical data
• The meaning of a
Market pressure
Ri - Rf
A
B
Rm - Rf
Market pressure
- 19 -
Which is riskier?
Ri - Rf
Ri - Rf
Rm - Rf
Rm - Rf
- 20 -
Which is realistic?
Ri - Rf
Ri - Rf
Rm - Rf
Rm - Rf
- 21 -
Security Market Line
– Market portfolio
– T-bill
L
SM
Return
• Linear relationship
between reward and risk
– Risk premium
proportional to beta
• Two things are on the
line:
M
kt
Rf
1
Beta
- 22 -
Practice
See problems 19 and 20 in the problem set
• Calculate the beta for a portfolio made from equal proportions of
securities with individual betas of .75, 1.0, and 1.25
• Calculate the beta for a portfolio with $200 invested in stock A,
$300 in stock B, and $500 in stock C. Betas for the individual
stocks are .75 for stock A, 1.0 for stock B, and 1.25 for stock C
Beta = (.2*.75) + (.3*1) + (.5*1.25) = 1.075
- 23 -
Practice
See problems 21 and 22 in the problem set
• When the T-Bill rate is 9% and the expected return for the market
portfolio is 12%, what is the opportunity cost of capital for an
investment with beta of 1.5?
• Assume the T-Bill rate is 10% and the expected return on the
market portfolio is 18%. An investment is twice as risky as the
market portfolio, in terms of its systematic risk. What is the
opportunity cost of capital for this investment?
- 24 -
Estimating beta from subjective inputs
• Estimate s from
confidence interval
• Estimate r from
coefficient of
determination:
bbii=(s
=(sii/s
/smm )r
)rim
im
r2 = proportion of risk
arising from systematic
factors
- 25 -
Evaluating an investment from subjective inputs
Assume the following:
• 95% confidence interval for stock
ranges from -30% to +70%
• 95% confidence interval for
market portfolio: -10% to +30%
• 75% of risk is diversifiable
• T-bill rate is 5%
bbii=(s
=(sii/s
/smm )r
)rim
im
bbi=
(.25/ .10 )* .5
i= (.25/ .10 )* .5
== 1.25
1.25
OCC
OCC
=5%
=5%++1.25(10%
1.25(10%-5%)
-5%)
==11.25%
11.25%
E(R)
E(R)==20%,
20%,so
soaccept
accept
- 26 -
Practice
See problem 23 in the problem set
•
An initial public offering (IPO) under analysis by Ajax Pension Fund is in
a relatively new technology, and 75% of its risk is considered to be
unsystematic. The 95% confidence interval for its rate of return ranges
from 4% to 40%, and the probability distribution is symmetric normal.
The expected return for the market portfolio is being quoted at 15% with
standard deviation of 5%. Estimate the beta for the IPO, and then
recommend whether to accept or reject.
E(R) =
22%
Beta = (9%/5%) * .5 = 0.9
Conclusion: This IPO has above-average reward with below-average risk, so
is attractive
OCC = 5% + 0.9(15% - 5%) = 14%
- 27 -
Practice
See problem 24 in the problem set
•
Another IPO under analysis by Ajax Pension Fund is in an established
technology, and only 19% of its risk is considered to be unsystematic. The
95% confidence interval for its rate of return ranges from -5% to 35%,
and the probability distribution is symmetric normal. The expected
return for the market portfolio is being quoted at 15% with standard
deviation of 5%. Estimate the beta for this IPO, and then recommend
whether to accept or reject.
E(R) =
15%
Beta = (10%/5%) * .9 = 1.8
Conclusion: This IPO has average reward with above-average risk, so is not
attractive
OCC = 5% + 1.8(15% - 5%) = 23%
- 28 -
Practice
See problem 25 in the problem set
•
Large State Pension Fund recently installed an evaluation system centered
on the capital asset pricing model. Data for actual outcomes of several
managers and accompanying market conditions during the evaluation
period are given below. See if you can identify any possible problem areas
in the fund managers’ performance.
Manager
1
1
2
2
3
3
Fund
A
B
C
D
E
F
bj
0.4
1.1
0.8
1.7
0.5
1.3
Rj
12%
18%
13%
15%
13%
25%
RTbill
9%
10%
10%
9%
9%
10%
Rmkt
14%
15%
15%
13%
14%
15%
- 29 -
Practice
See problems 26 through 28 in the problem set
• If you were certain that the market was going to rise, what sort of
beta would you prefer for your portfolio?
• If you were certain that the market was going to drop, what sort of
beta would you prefer for your portfolio?
• If you thought the market might be turning up, but still recognize
the possibility for a downturn, what sort of beta would you prefer
for your portfolio?
- 30 -
What’s wrong with WACC?
L
SM
Return
• Project A is above
WACC, but below
SML
• Project B is below
WACC, but above
SML
• WACC would
wrongly reject B and
accept A
B
A
WACC
Rf
Beta
WACC Ignores Risk
- 31 -
• Empirical SML has higher
intercept than theoretical
SML
• Empirical SML has a more
gradual slope than the
theoretical SML (there is
less additional reward for
added risk, in practice
compared with theory)
• Evidence motivates search
for improved Arbitrage
Pricing Theories
Return
Empirical evidence concerning the
CAPM
Theory
Actual
Beta
- 32 -