FINC4101
Investment Analysis
Instructor: Dr. Leng Ling
Topic: Asset Pricing Models
1
Learning objectives
1.
2.
3.
4.
5.
6.
7.
8.
9.
Distinguish between systematic and unsystematic risk.
Define the single index model and identify its
components.
Relate a security’s beta to its systematic risk.
Describe the Capital Asset Pricing Model (CAPM) and
list its assumptions.
Identify the implications, applications and limitations of
the CAPM.
Use the CAPM to compute an asset’s expected return.
Define the Security Market Line (SML).
Understand the concepts of fairly priced, underpriced
and overpriced.
Describe the implementation of the CAPM.
2
Concept Map
Portfolio
Theory
Foreign
Exchange
Asset
Pricing
FINC4101
Derivatives
Equity
Market
Efficiency
Fixed
Income
3
Why do we need an
asset pricing model?


An asset pricing model allows us to
figure out the required return of an asset.
Required rate of return can be used for:
1.
2.
3.
Valuation of assets.
Identification of attractive investments.
Capital budgeting.
4
Preparation for CAPM
 To
understand the CAPM, we need to
know the following:

Systematic vs. unsystematic risk

Single index model
5
Unsystematic risk
 Unsystematic
risk: uncertainty or variability
in returns that affects a specific asset
without affecting other assets.

Also known as unique risk, firm-specific risk,
diversifiable risk.

Sources of unsystematic risk: litigation, patents, R&D,
management style and philosophy, financial leverage,
etc
6
Systematic risk
 Systematic
risk: uncertainty or variability in
returns that affects all risky assets.

Also known as: market risk, nondiversifiable risk

Sources of systematic risk: fluctuation in the stock
market, business cycles, inflation rate, monetary
policy, exchange rates, wars, political unrest,
technological change etc.
7
Systematic vs unsystematic risk
Diversification works but has its limit.
 At the very most, diversification eliminates
unsystematic risk. When there is no
unsystematic risk, only systematic risk remains,
but that is not diversifiable.
 It is risk that investors bear in exchange for
enjoying the return from investing. After all,
bearing risk is a fundamental part of investment.
 Investors holding well-diversified portfolios will
only demand a risk premium for bearing the
systematic risk.

8
Graphical depiction of
systematic and unsystematic risks
Well-diversified
portfolio virtually
eliminates
unsystematic risk.
As the number
of stocks in the
portfolio
increases,
unsystematic
risk decreases.
9
Single-index model of security returns


Statistical model that estimates the systematic
and unsystematic risk of a security or portfolio.
Looks at “excess return”, denoted by “R”
What is a security’s excess return
Security’s HPR in excess of the risk free rate.
Excess return of security i, Ri
= security i’s HPR – risk-free rate
= ri – rf
The following discussion focuses on excess returns.
10
Anatomy of single-index model

Uses a broad index of securities (e.g., S&P500)
to represent systematic risk.

This broad index is called, “market index”,
“market factor”, or just “market” for short.

Market excess return is denoted RM .

The model says that a security’s excess return
consists of 3 parts:
11
ai
biRM
ei
R i = ai + b i R M + e i
Stock’s excess return if market factor is
neutral, i.e., market excess return = 0
Excess return due to movements in the
overall market.
bi = responsiveness of security’s return to
to market return.
Excess return due to unexpected events
relevant only to the security (firm-specific).
On average, ei is assumed to be 0. That is,
E(ei) = 0.
12
Using the single-index model to
decompose security risk(p.168)
Variance of Ri
= Systematic risk + Firm-specific risk
= Variance(biRM) + Variance(ei)
=
bi2 sM2
+
s2(ei)
Variance of market
excess return.
13
Graphical representation of
single-index model: Ri = ai + biRM + ei
Scatter diagram
Security
Characteristic Line
14
Measuring the importance of
systematic risk
2 is the ratio of systematic risk to total risk.
= Systematic Risk/Total Risk
= bi2 sM2 /[ bi2 sM2 + s2(ei) ]
2 ranges from 0 to 1

As 2



Systematic risk becomes a bigger part of total risk.
Points on scatter diagram lie close to regression line.
As 2


 1,
 0,
Unsystematic risk becomes a bigger part of total risk.
Points on scatter diagram lie away from regression line.
15
Different types of securities
16
Relevant risk measure for
diversified investors
In highly diversified portfolios, unsystematic risk
can be virtually eliminated and thus becomes
irrelevant.
 Only systematic risk remains in diversified
portfolios.
 In measuring security risk for diversified
investors, we focus on the security’s systematic
risk, i.e., bi2 sM2 .
 Equivalently, we can use bi as the measure of
the security’s systematic risk.

17
Capital Asset Pricing Model (CAPM)
 Equilibrium
model that underlies all
modern financial theory
 Derived using principles of diversification
with simplified assumptions
 Markowitz, Sharpe, Lintner and Mossin
are researchers credited with its
development
 Theoretical model based on a list of
simplifying assumptions
18
Assumptions of the CAPM (1)
1.
2.
3.
4.
5.
6.
7.
Individual investors are price takers
Single-period investment horizon
Investments are limited to traded financial
assets
No taxes, and transaction costs
Information is costless and available to all
investors
Investors are rational mean-variance optimizers
Homogeneous expectations
19
Implications of the CAPM
1.
All investors will hold the same portfolio
for risky assets – market portfolio (M).

2.
Market portfolio contains all securities. The
proportion of each security is its market value as a
percentage of total market value of all securities.
Market portfolio will be on the efficient
frontier and will be the tangency portfolio.

Capital Allocation Line is now called the Capital
Market Line (CML).
20
Implications of the CAPM (cont’d)
3.
Risk premium on the market depends on:


4.
Average risk aversion of all market
participants.
Variance of the market portfolio.
Risk premium on an individual security
depends on:
 Its beta, b

Market portfolio risk premium
21
Why all investors hold the market
portfolio?
All investors
 Use identical mean-variance analysis
 Apply it to the same set of securities
 Have the same time horizon
 Use the same security analysis
 Have identical tax consequences
Arrive at the same efficient frontier and tangency
portfolio.
22
The Efficient Frontier and the
Capital Market Line
23
Risk premium of market portfolio
E(rM) – rf = A* x
Risk aversion of
average investor
(sM)2
SD of the
return on
the market
portfolio.
Expected return:
E(rM) = rf + (A* x
2
sM )
24
Risk premium & expected return
of single security
 For
security i, risk premium:
E(ri) – rf = bi [ E(rM) – rf ]
Security i’s expected return:
E(ri) = rf + bi [ E(rM) – rf ]
Expected return-beta relationship
 Formulas also hold for a portfolio of
securities.
25
Portfolio beta, bP
 The
beta of a portfolio, P,
Weighted average of the individual asset
betas.
 Weights are the portfolio proportions.

bP = (w1 x b1) + (w2 x b2) + … + (wN x bN)
Proportion of
portfolio invested in
asset 1.
Asset 1’s beta.
26
Problem involving portfolio beta

Suppose the market risk premium is 8%. What is
the risk premium of a portfolio invested 25% in
Coca-Cola and 75% in BellSouth. Coca-Cola
has a beta of 0.85 and BellSouth has a beta of
1.2.

Verify that portfolio beta is 1.1125
Verify that risk premium is 8.9%


Suppose you now invest 20% in the risk-free
asset, 25% in Coca-Cola and the rest in
BellSouth. What is the portfolio beta and risk
premium?
27
Questions
Are the following statements true or false? Explain.
a) Stocks with a beta of zero offer an expected
rate of return of zero.
b) The CAPM implies that investors require a
higher return to hold highly volatile securities.
c) You can construct a portfolio with a beta of
0.75 by investing 0.75 of the budget in T-bills
and the remainder in the market portfolio.
28
More questions
1. What
is the beta of a portfolio with E(rP) =
20%, if rf = 5% and E(rM) = 15%.
2. Assume
both portfolios A and B are well
diversified, and E(rA) = 14% and E(rB) =
14.8%. If the economy has only one factor,
and bA = 1.0 while bB = 1.1, what must be
the risk-free rate?
29
Security market line (SML)
Expected
return
SML
M
E(rM)
Rise
Slope = rise / run
Risk-free
rate, rf
= (E(rM) – rf)/ 1
Run
0
= market risk premium
1
b
30
Security market line (SML)
1.
2.
Graphical representation of the expected
return-beta relationship.
Provides a benchmark for evaluating
investment performance.

3.
Given an investment’s beta, the SML tells us the
return we should require or demand from this
investment.
“Fairly priced” assets plot exactly on the SML.
“Underpriced” assets plot above the SML.
“Overpriced” assets plot below the SML.
31
Fairly priced security







Suppose stock A is currently priced at $45. Everyone agrees that its
year-end price will be $50 and a dividend of $2.02 will be paid then.
Stock A’s beta is 1.2, the market portfolio’s expected return is 14%
and the risk-free rate is 6%.
Given the stock price and future cash flows, expected HPR = (50 +
2.02 – 45)/45 = 0.156 or 15.6%
CAPM’s expected return = 6 + 1.2[14 – 6] = 15.6% !
Stock A is FAIRLY PRICED. Its current price leads to an expected
return that is EXACTLY equal to the expected return indicated by the
CAPM.
So, CAPM says that $45 is the “correct” price for A given it’s risk.
This stock plots exactly on the SML.
32
Underpriced security






Now supposed everything stays the same, BUT stock A
is now priced at $44.46.
Given the stock price and future cash flows, expected
HPR = (50 + 2.02 – 44.46)/44.46 = 0.17 or 17%
But CAPM says stock A should provide 15.6% return
and it should be worth $45.
Stock A is UNDERPRICED. It’s price of $44.46 is LESS
than the CAPM price of $45.
Also, A’s expected return given its price is 17% which is
MORE than the CAPM expected return of 15.6%
Stock A now plots ABOVE the SML.
33
Overpriced security






Now supposed everything stays the same, BUT stock A
is now priced at $46.04.
Given the stock price and future cash flows, expected
HPR = (50 + 2.02 – 46.04)/46.04 = 0.13 or 13%
But CAPM says stock A should provide 15.6% return
and it should be worth $45.
Stock A is OVERPRICED. It’s price of $46.04 is MORE
than the CAPM price of $45.
Also, A’s expected return given its price is 13% which is
LESS than the CAPM expected return of 15.6%
Stock A now plots BELOW the SML.
34
All together now
(1)
(2)
(3)
Current Expected CAPM says
price
HPR
expected
return
should be
(4)
Stock is
(5)
$45
15.6%
15.6%
Fairly priced
0%
$44.46
17%
15.6%
Underpriced
1.4%
$46.04
13%
15.6%
Overpriced
-2.6%
a, Alpha
= (2) – (3)
Alpha: Difference between expected HPR and the
return predicted by the CAPM.
35
Fairly priced, underpriced,
overpriced
When stock A is
UNDERPRICED, it
plots above the
SML.
When stock A is
fairly priced, it plots
exactly on the SML.
13
When stock A is
OVERPRICED, it plots
below the SML.
36
Capital Allocation Line (CAL)
CAL becomes CML when the market portfolio is used as the
risky portfolio
37
Capital Market Line (CML) vs.
Security Market Line (SML)
CML
 Graphs expected
return of market
portfolio against
standard deviation.
SML
 Graphs expected
return of individual
asset against beta.
Individual assets,
portfolios
 Beta is the valid risk
measure

Efficient portfolios
 SD is the valid risk
measure.

38
Questions

a)
b)
c)
d)
The SML depicts:
A security’s expected return as a function of its
systematic risk.
The market portfolio as the optimal portfolio of
risky securities.
The relationship between a security’s return
and the return on an index.
The complete portfolio as a combination of the
market portfolio and the risk-free asset.
39
Applications of the CAPM
1.
Identification of attractive investments.

2.
Valuation of assets.

3.
CAPM expected return is used as discount rate to
find the value of assets.
Capital budgeting.

4.
Positive alpha / underpriced assets.
“Hurdle rate” for project under consideration.
Rate-setting for utilities.

Rate of return that a regulated utility should be
allowed to earn.
40
Identifying attractive investment (1)

Karen Kay, a portfolio manager at Collins Asset
Management, is using the CAPM for making
recommendations to her clients. Her research
department has developed the information shown in the
following exhibit
Forecasted return S.D.
Beta
Stock X
14%
36%
0.8
Stock Y
17
25
1.5
Market index
14
15
1.0
Risk-free rate
5
41
Identifying attractive investment (2)
a)
b)
Calculate expected return and alpha for
each stock.
Identify and justify which stock would be
more appropriate for an investor who
wants to:
Add this stock to a well-diversified equity
portfolio.
ii. Hold this stock as a single-stock portfolio.
i.
42
Valuation of assets (1)
 The
market price of a security is $40. its
expected rate of return is 13%. The riskfree rate is 7%, and the market risk
premium is 8%. What will the market price
of the security be if its beta doubles (and
all other variables remain unchanged)?
Assume the stock is expected to pay a
constant dividend in perpetuity.
43
Valuation of assets (2)

The risk-free rate is 4%. Suppose that the
expected return required by the market for a
portfolio with a beta of 1.0 is 12%. According to
the CAPM:
A.
What is the expected return on the market portfolio?
What would be the expected return on a zero-beta
stock?
Suppose you consider buying a share of stock at a
price of $40. the stock is expected to pay a dividend of
$3 next year and to sell then for $41. The stock’s beta
is -0.5. is the stock overpriced or underpriced?
B.
C.
44
Capital budgeting (1)
Texaco is considering a new oil rig in the Gulf of
Mexico. The business plan forecasts an internal
rate of return of 17% on the investment.
Research shows that the beta of similar projects
is a whopping 2.2! The risk-free rate is 4% and
the market risk premium is 8%. Compute the
hurdle rate for the project.
 Should the project be accepted?

45
Capital budgeting (2)

You are an analyst at Goldman Sachs. You are covering
a company which is considering a project with the
following net after-tax cash flows : an outlay of $20 mil
right now (t= 0), $10 mil at the end of each of the first
nine years and $20 mil at the end of year 10. The project
is terminated after 10 years. The project’s beta is 1.7.
Assuming rf = 9% and E(rM) = 19%, what is the project’s
NPV?
46
Implementing the CAPM (1)
The CAPM can be used in several ways.
To apply it, we need to calculate a security’s beta.
There are two major problems in calculating beta:
1.
Market portfolio of all assets is unobservable.
2.
The CAPM gives a relationship between
expected return and beta. Expected returns are
unobservable.
How do we get round these problems?
47
Implementing the CAPM (2)
Solutions:
1. Use an actual portfolio, e.g., S&P 500
stock index, as a proxy for the theoretical
market portfolio. The proxy represents
systematic risk in the economy.
2. Use realized (i.e., past) returns instead of
expected returns to estimate beta.
.
48
Implementing the CAPM (3)
 Use
linear regression to estimate the
single-index model:
ri - rf = ai + bi(RM-rf) + ei
Excess
return of
stock i
Excess return
when index
portfolio has 0
excess return.
Excess return on
stock index (e.g.,
S&P500).
Excess
return
due to
firmspecific
events.
Period can be month, week, day.
49
Implementing the CAPM (4)
 If
we express the single-index model in
terms of expectations, we get:
E(ri) – rf = ai + bi [ E(rM) – rf ]
Recall the CAPM equation:
E(ri) – rf = bi [ E(rM) – rf ]
=> The CAPM predicts that ai = 0.
50
Example of estimating beta
1.
Collect and process data (12/2000-12/2005)




2.
Monthly prices of Coca-Cola (KO)
Monthly index values of S&P 500 (^GSPC)
Get monthly risk-free rate from Prof Ken French’s
website:
http://mba.tuck.dartmouth.edu/pages/faculty/ken.fren
ch/data_library.html
Compute monthly excess returns of KO and ^GSPC.
Use Excel’s Data Analysis tool to estimate the
regression.
51
Multifactor asset pricing models

CAPM says that the market portfolio is the only
source of systematic risk.

But there can be multiple sources of systematic
risk, e.g., fluctuation in interest rates, fluctuation
in energy prices, uncertainty about inflation, etc.

Multifactor asset pricing models try to do better
than the CAPM by using more than one
systematic risk factor to explain security returns.
52
Summary






Asset pricing models tell us how to figure a risky asset’s
expected return.
The CAPM is such a model. It says that there is only one
source of systematic risk – the market.
In the CAPM world, b is the measure of risk for individual
securities.
CAPM says that the riskier the security, the higher the
required/expected return for holding that asset. SML
depicts the relationship between b and expected return.
CAPM can be used in a number of financial applications.
Newer asset pricing models use more than one
systematic risk factor.
53
Practice 4
 Chapter
6: 20, 21.
 Chapter 7:
1,4,6,7,9,10,12abcd,13--19, 21,24,25
CFA 6,7,8.
54
Homework 4
1.
2.
You already have a very well-diversified portfolio of
common stock with a beta of 2, and you want to add
stock AAA or BBB into the portfolio. AAA’s standard
deviation is 0.2300 and its beta is 4. BBB’s standard
deviation is 0.6500 and its beta is 0.5. How will the
addition affect the portfolio’s risk?
If we add more factors into the one-factor asset pricing
model, what happens to the required returns, increase
or decrease? If the expected future cash flows of the
financial securities remain the same, will that increase
or decrease the current market price? Why?
55
Homework 5
Regression to find stock Beta.
56