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Risk, Return, Portfolio
Theory and CAPM
Where does the discount
rate (for stock valuation)
come from? TIP
If you do not understand
anything,
ask me!
So far,
We have taken the discount rate as given.
We have also learned the factors that
determine the discount rate in bond
valuation.
What is the appropriate discount rate in
stock valuation?
2
Topics
Risk and returns
75 Years of Capital Market History
How to measuring risk
Individual security risk
Portfolio risk
Diversification
Unique risk
Systematic risk or market risk
Measure market risk: beta
CAPM
3
The Value of an Investment of $1 in 1926
6402
S&P
Small Cap 2587
1000 Corp Bonds
Long Bond
T Bill
64.1
Index
48.9
10
16.6
1
0.1
1925 1940 1955 1970 1985 2000
Source: Ibbotson Associates
Year End
4
The Value of an Investment of $1 in 1926
Real returns
S&P
Small Cap
1000
Corp Bonds 660
Long Bond
T Bill 267
Index
10 6.6
5.0
1 1.7
0.1
1925 1940 1955 1970 1985 2000
Source: Ibbotson Associates
Year End
5
Percentage Return Rates of Return 1926-2000
60 Common Stocks
Long T-Bonds
T-Bills
40
20
0
-20
-40
-60
26
30
35
40
45
50
55
60
65
70
75
80
85
90
95
00
20
Year
Source: Ibbotson Associates
6
Selected Realized Returns,
1926 – 2001
Average Standard
Return Deviation
Small-company stocks 17.3% 33.2%
Large-company stocks 12.7 20.2
L-T corporate bonds 6.1 8.6
L-T government bonds 5.7 9.4
U.S. Treasury bills 3.9 3.2
Source: Ibbotson Associates.
7
Risk premium
Risk aversion – in Finance we assume investors dislike
risk, so when they invest in risky securities, they require a
higher expected rate of return to encourage them to bear
the risk.
The risk premium is the difference between the expected
rate of return on a risky security and the expected rate of
return on a risk-free security, e.g., T-bills.
Over the last century, the average risk premium is about
7% for stocks.
8
Measuring Risks
How to measure the risk of a security?
Stand-alone risk: when the return is analyzed
in isolation. This provides a starting point.
Portfolio risk: when the return is analyzed in
a portfolio. This is what matters in reality
when people hold portfolios.
9
PART I:
Standard alone risk
The risk an investor would face if s/he held
only one asset.
Investment risk is related to the probability of
earning a low or negative actual return.
The greater the chance of lower than expected or
negative returns, the riskier the investment.
The greater the range of possible events that can
occur, the greater the risk
10
Probability distributions
Which firm is more likely to have a return closer to its
expected value?
A listing of all possible outcomes, and the
probability of each occurrence.
Can be shown graphically.
Firm X
Firm Y
Rate of
-70 0 15 100 Return (%)
Expected Rate of Return
11
Measuring Risk
In financial markets, we use the volatility of a
security return to measure its risk.
Variance – Weighted average value of squared
deviations from mean.
Standard Deviation – Squared root of variance.
12
Some basic concepts
Some basic formula for Expectation and
Variance
Let X be a return of a security in the next
period. Then we have
N N
X E[ X ] p(i) X (i) Var[ X ] p(i)( X (i) X ) 2
i 1 i 1
13
Investment alternatives
Economy Prob. T-Bill HT Coll USR MP
Recession 0.1 8.0% -22.0% 28.0% 10.0% -13.0%
Below 0.2 8.0% -2.0% 14.7% -10.0% 1.0%
avg
Average 0.4 8.0% 20.0% 0.0% 7.0% 15.0%
Above 0.2 8.0% 35.0% -10.0% 45.0% 29.0%
avg
Boom 0.1 8.0% 50.0% -20.0% 30.0% 43.0%
14
Return: Calculating the expected
return for each alternative
^
k ex pectedrate of return
^ n
k k i Pi
i1
^
k HT (-22.%) (0.1) (-2%) (0.2)
(20%) (0.4) (35%) (0.2)
(50%) (0.1) 17.4%
15
Risk: Calculating the standard
deviation for each alternative
Standard deviation
Variance 2
n
ˆ)2 Pi
(k i k
i1
16
Standard deviation calculation
n ^
i1
(k i k ) 2 Pi
1
(8.0 - 8.0) (0.1) (8.0 - 8.0) (0.2)
2 2 2
T bills (8.0 - 8.0)2 (0.4) (8.0 - 8.0)2 (0.2)
2
(8.0 - 8.0) (0.1)
T bills 0.0% Coll 13.4%
HT 20.0% USR 18.8%
M 15.3%
17
Comparing risk and return
Security Expected return Risk, σ
T-bills 8.0% 0.0%
HT 17.4% 20.0%
Coll 1.7% 13.4%
USR 13.8% 18.8%
Market 15.0% 15.3%
18
Comments on standard
deviation as a measure of risk
Standard deviation (σi) measures “total”, or
stand-alone, risk.
The larger the σi , the lower the probability
that actual returns will be closer to expected
returns, that is : the larger the stand-alone
risk.
19
PART II:
Risk in a portfolio context
Portfolio risk is more important because
in reality no one holds just one single
asset.
The risk & return of an individual security
should be analyzed in terms of how this
asset contributes the risk and return of the
whole portfolio being held.
20
Portfolio
A portfolio is a set of securities and can be
regarded as a security.
If you invest W dollars in a portfolio of n
securities, let Wi be the money invested in
security i, then the portfolio weight on
stock i is n
Wi
xi , with property xi 1
W i 1
21
Example 1
Suppose that you want to invest $1,000 in a
portfolio of IBM and GE. You spend $200
on IBM and the other $800 on GE. What is
the portfolio weight on each stock?
xIBM 200 / 1000 0.2
xGE 800 / 1000 0.8
22
Portfolio return and risk
of two stocks
Expected Portfolio Return (x1r1) ( x 2 r2 )
Portfolio Variance x σ x σ 2( x 1x 2ρ 12σ 1σ 2 )
2
1
2
1
2
2
2
2
23
In a portfolio…
In English, we say:
The expected return = weighted
average of each stock’s expected
return.
But the portfolio standard deviation is
<= the weighted average of each
stock’s standard deviation.
I Mathematics, we say:
24
(Not required) Suppose a portfolio is
made up of x1 shares of stock 1 and x2
shares of stock 2.
E ( x1r1 x2 r2 ) x1E (r1 ) x2 E (r2 )
2 ( x1r1 x2 r2 ) Var ( x1r1 x2 r2 ) x12Var (r1 ) x2 2Var (r2 )
2 x1 x2 Var (r1 )Var (r2 )
x12 12 x2 2 2 2 2 x1 x2 1 2
( x1 1 x2 2 ) 2 because 1 1
Thus ( x1r1 x2 r2 ) x1 1 x2 2
We say that a portfolio ' s standard deviation is a convex function
of its components ' standard deviations.
25
Suppose you invest 50% in HT and 50% in Coll.
What are the expected returns and standard
deviation for the 2-stock portfolio?
Economy Prob. HT Coll
Security Expected return Risk, σ Recession 0.1 -22.0% 28.0%
HT 17.4% 20.0%
Below avg 0.2 -2.0% 14.7%
Coll* 1.7% 13.4%
Average 0.4 20.0% 0.0%
Above avg 0.2 35.0% -10.0%
Boom 0.1 50.0% -20.0%
26
Calculating portfolio expected
return
^
k p is a weighted average :
^ n ^
k p wi k i
i1
^
k p 0.5 (17.4%) 0.5 (1.7%) 9.6%
27
An alternative method for
determining portfolio expected
return
Economy Prob. HT Coll Port.
Recession 0.1 -22.0% 28.0% 3.0%
Below avg 0.2 -2.0% 14.7% 6.4%
Average 0.4 20.0% 0.0% 10.0%
Above avg 0.2 35.0% -10.0% 12.5%
Boom 0.1 50.0% -20.0% 15.0%
^
k p 0.10 (3.0%) 0.20 (6.4%) 0.40 (10.0%)
0.20 (12.5%) 0.10 (15.0%) 9.6%
28
Calculating portfolio
standard deviation
1
0.10 (3.0 - 9.6) 2
2
0.20 (6.4 - 9.6)2
0.40 (10.0 - 9.6)2
p 3.3%
0.20 (12.5 - 9.6)2
0.10 (15.0 - 9.6)2
29
Comments on portfolio risk
measures
σp = 3.3% is lower than the weighted average of HT
and Coll.’s σ (16.7%). This is true so long as the two
stocks’ returns are not perfectly positively
correlated.
Perfect correlation means the returns of two stocks
will move exactly in same rhythm.
Portfolio provides average return of component
stocks, but lower than average risk.
The more negatively correlated the two stocks, the
more dramatic reduction in portfolio standard
deviation. 30
Returns distribution for two perfectly
negatively correlated stocks (ρ = -1.0)
Stock W Stock M Portfolio WM
25 25 25
15 15 15
0 0 0
-10 -10 -10
31
Returns distribution for two perfectly
positively correlated stocks (ρ = 1.0)
Stock M Stock M’ Portfolio MM’
25 25 25
15 15 15
0 0 0
-10 -10 -10
32
General comments about risk
Most stocks are positively correlated
with the market (ρk,m 0.65).
σ 35% for an average stock.
Combining stocks in a portfolio
generally lowers risk.
33
Total Risk
A stock’s realized return is often different from
its expected return.
Total return= expected return + unexpected
return
Unexpected return=systematic portion +
unsystematic portion
Thus: Total return= expected return +systematic
portion + unsystematic portion
Total risk (stand-alone risk)= systematic portion
+ unsystematic portion
34
Systematic Risk
Total risk (stand-alone risk)= systematic portion
+ unsystematic portion
The systematic portion will be affected by
factors such as changes in GDP, inflation,
interest rates, etc.
This portion is not diversifiable because the
factor will affect all stocks in the market.
Such risk factors affect a large number of stocks.
Also called Market risk, non-diversifiable risk,
beta risk.
35
Unsystematic Risk
Total risk (stand-alone risk)= systematic portion
+ unsystematic portion
This portion is affected by factors such as labor
strikes, part shortages, etc, that will only affect a
specific firm, or a small number of firms.
Also called diversifiable risk, firm specific risk.
36
Diversification
Portfolio diversification is the investment in
several different classes or sectors of stocks.
Diversification is not just holding a lot of
stocks.
For example, if you hold 50 internet stocks,
you are not well diversified.
37
Creating a portfolio:
Beginning with one stock and adding
randomly selected stocks to portfolio
σp decreases in general as stocks added.
Expected return of the portfolio would
remain relatively constant.
Diversification can substantially reduce the
variability of returns with out an equivalent
reduction in expected returns.
Eventually the diversification benefits of
adding more stocks dissipates (after about 10
stocks), and for large stock portfolios, σp
tends to converge to 20%.
38
Illustrating diversification
effects of a stock portfolio
p (%)
Company-Specific Risk
35
Stand-Alone Risk, p
20
Market Risk
0
10 20 30 40 2,000+
# Stocks in Portfolio
39
Breaking down sources of
total risk
Total risk= systematic portion (market risk) + unsystematic
portion (firm-specific risk)
Market risk (systematic risk, non-diversifiable risk, beta risk) –
portion of a security’s stand-alone risk that cannot be
eliminated through diversification. It is affected by
economy-wide sources of risk that affect the overall stock
market.
Firm-specific risk (unsystematic risk, diversifiable risk,
idiosyncratic risk) – portion of a security’s stand-alone risk
that can be eliminated through proper diversification.
If a portfolio is well diversified, unsystematic is very
small. Rational, risk-averse investors are just concerned
with portfolio standard deviation σp, which is based
upon market risk. That is: investor care little about a
stock’s firm–specific risk.
40
How do we measure a tock’s
systematic (market) risk? Beta
Measures a stock’s market risk, and shows a
stock’s volatility relative to the market (i.e.,
degree of co-movement with the market
return.)
Indicates how risky a stock is if the stock is
held in a well-diversified portfolio.
41
Calculating betas
Run a regression of past returns of a security
against past market returns.
Market return is the return of market
portfolio.
Market Portfolio - Portfolio of all assets in the
economy. In practice, a broad stock market
index, such as the S&P Composite, is used to
represent the market.
The slope of the regression line (called the
security’s characteristic line) is defined as the
beta for the security.
42
Illustrating the calculation of beta
(security’s characteristic line)
_
ki
20 . Year kM ki
15 . 1
2
15%
-5
18%
-10
10 3 12 16
5
_
-5 0 5 10 15 20
kM
-5 Regression line:
. -10
^
^ = -2.59 + 1.44 k
k i M
43
Security Character Line
What does the slope of SCL mean?
Beta
What variable is in the horizontal line?
Market return.
The steeper the line, the more sensitive the
stock’s return relative to the market return, that
is, the greater the beta.
44
Comments on beta
A stock with a Beta of 0 has no systematic risk
A stock with a Beta of 1 has systematic risk equal
to the “typical” stock in the marketplace
A stock with a Beta greater than 1 has systematic
risk greater than the “typical” stock in the
marketplace
A stock with a Beta less than 1 has systematic risk
less than the “typical” stock in the marketplace
The market return has a beta=1 (why?).
Most stocks have betas in the range of 0.5 to 1.5.
45
Can the beta of a security be
negative?
Yes, if the correlation between Stock i and
the market is negative.
If the correlation is negative, the regression
line would slope downward, and the beta
would be negative.
However, a negative beta is very rare.
A stock with negative beta will give your
higher return in recession and hence is
more valuable to investors, thus required
rate of return is lower.
46
Beta coefficients for HT, Coll,
and T-Bills
_
ki HT: β = 1.30
40
20
T-bills: β = 0
_
-20 0 20 40 kM
Coll: β = -0.87
-20
47
Comparing expected return and beta
coefficients
Security Exp. Ret. Beta
HT 17.4% 1.30
Market 15.0 1.00
USR 13.8 0.89
T-Bills 8.0 0.00
Coll. 1.7 -0.87
Riskier securities have higher returns, so the rank
order is OK.
48
Until now...
We argued that well-diversified investors
only cares about a stock’s systematic risk
(measured by beta).
The higher the systematic risk, the higher
the rate of return investors will require to
compensate them for bearing the risk.
This extra return above risk free rate that
investors require for bearing the non-
diversifiable risk of a stock is called risk
premium.
49
Beta and risk premium
That is: the higher the systematic risk (measured
by beta), the greater the reward (measured by risk
premium).
risk premium =expected return - risk free rate.
In equilibrium, all stocks must have the same
reward to systematic risk ratio.
For any stock i and stock m:
(ri-rf)/beta(i)=(rm-rf)/beta(m)
50
The higher the beta, the
higher the risk premium.
The above equation should hold for any two
securities. It should also hold if m stands for
market portfolio.
(ri –rf ) / (rM – rf)= βi /1
Thus, we have
ri = rf + (rM – rf) βi
You’ve got CAPM!
Yes, this ri decides the discount rate in stock
valuation.
51
Capital Asset Pricing Model
(CAPM)
Model based upon concept that a
stock’s required rate of return is equal
to the risk-free rate of return plus a risk
premium that reflects the riskiness of
the stock after diversification.
52
Calculating required rates of
return according to CAPM
ri = rf + (rM – rf) βi
Assume rf = 8% and rM = 15%.
The market (or equity) risk premium is rM – rf
= 15% – 8% = 7%.
If a stock has a beta=1.5, how much is its
required rate of returns?
53
Risk-Free Rate
Required rate of return for risk-less
investments
Typically measured by yield on 90 days
U.S. Treasury Bills.
54
What is the market risk
premium (rm-rf)?
Additional return over the risk-free
rate needed to compensate investors
for assuming an average amount of
systematic risk.
Its size depends on the perceived risk
of the stock market and investors’
degree of risk aversion.
Historically between 4% and 8%.
55
Comparing expected return
and beta coefficients
Security Exp. Ret. Beta
HT 17.4% 1.30
Market 15.0 1.00
USR 13.8 0.89
T-Bills 8.0 0.00
Coll. 1.7 -0.87
Assume kRF = 8% and kM = 15%.
The market risk premium is kM – kRF = 15% – 8%
Please find the require rates of return for each
security.
(Sorry I use K and r interchangeably)
56
Calculating required rates of
return
kHT = 8.0% + (15.0% - 8.0%)(1.30)
= 8.0% + (7.0%)(1.30)
= 8.0% + 9.1% = 17.10%
kM = 8.0% + (7.0%)(1.00) = 15.00%
kUSR = 8.0% + (7.0%)(0.89) = 14.23%
kT-bill = 8.0% + (7.0%)(0.00) = 8.00%
kColl = 8.0% + (7.0%)(-0.87)= 1.91%
57
Expected vs. Required
returns
^
k k
^
HT 17.4% 17.1% Undervalue d (k k)
^
Market 15.0 15.0 Fairly val ued (k k)
^
USR 13.8 14.2 Overvalued (k k)
^
T - bills 8.0 8.0 Fairly val ued (k k)
^
Coll. 1.7 1.9 Overvalued (k k)
58
Expected v. required rate of
return
In short run, there might be mis-valued stocks
and expected return may be different from the
required return. In the long run and in an efficient
market , expected returns = required returns.
If many people believe that the a stock’s expected
return is higher than required return (stock is
undervalued), they would bid for that stock,
pushing up the stock price, hence lowering the
expected return, until market competition will
lead to: expected returns = required returns.
59
Security Market Line (SML): ri = rf +
(rM – rf) βi SML (redline) is a
graphical representation of CAPM
SML: ki = 8% + (15% – 8%) βi
ki (%)
SML
HT .
kM = 15 ..
kRF = 8 . T-bills USR
-1
. 0 1 2
Risk, βi
Coll.
60
Security Market Line
What does the slope of SML mean?
Market risk premium= kM- kRF
What variable is in the horizontal line?
Beta
Which stock is over valued from previous graph?
Coll.
61
CAPM—Efficient frontier (won’t
test)
In addition to saying that ri = rf + (rM –
rf)βi , CAPM also says each investor
should hold a combination of the market
portfolio and risk free security.
62
CAPM—Efficient frontier
(won’t test)
If an investor has a preference that is decided by
the expected return and the standard deviation
of a portfolio, then he or she will choose a
portfolio that has the highest expected return
given the standard deviation of the portfolio.
The set of these portfolios are called the efficient
portfolio frontier.
63
CAPM—Efficient frontier (won’t
test) Example with two stocks
Expected Returns and Standard Deviations vary
given different weighted combinations of the stocks
Expected Return (%)
Reebok
35% in Reebok
S Coca Cola
Standard Deviation
64
Efficient Frontier with n
stocks (Won’t test)
A
Expected Return (%)
S
Standard Deviation
65
Efficient Frontier with a risk-
free security (won’t test)
•Lending or Borrowing at the risk free rate (r f) will change the
efficient frontier when there are only risky securities
Expected Return (%) T A
rf
S
Standard Deviation
66
CAPM—Efficient frontier
(won’t test)
The red line from previous graph gives the
efficient frontier if a risk free security is
available.
Each investor should hold a portfolio on the red
line from previous graph.
That is: each investor should invest in a S&P
composite index and some T-bills (bonds). This
perhaps is too strong a prediction of CAPM.
67
How to get portfolio beta: Equally-
weighted two-stock portfolio
Create a portfolio with 50% invested in
HT and 50% invested in Collections.
The beta of a portfolio is the weighted
average of each of the stock’s betas.
βP = wHT βHT + wColl βColl
βP = 0.5 (1.30) + 0.5 (-0.87)
βP = 0.215
68
Calculating portfolio required
returns
The required return of a portfolio is the weighted
average of each of the stock’s required returns.
kP = wHT kHT + wColl kColl
kP = 0.5 (17.1%) + 0.5 (1.9%)
kP = 9.5%
Or, using the portfolio’s beta, CAPM can be used
to solve for expected return.
kP = kRF + (kM – kRF) βP
kP = 8.0% + (15.0% – 8.0%) (0.215)
kP = 9.5%
69
Verifying the CAPM
empirically
The CAPM has not been verified
completely.
Statistical tests have problems that make
verification almost impossible.
Some argue that there are additional risk
factors, other than the market risk
premium, that must be considered.
70
More thoughts on the
CAPM
Investors seem to be concerned with both
market risk and other risk factors. Therefore,
the SML may not produce a correct estimate of
ki.
ki = kRF + (kM – kRF) βi + ???
CAPM/SML concepts are based upon
expectations, but betas are calculated using
historical data. A company’s historical data
may not reflect investors’ expectations about
future riskiness.
71
On the midterm
Please prepare for the scantron, pencil,
eraser, and calculator.
There are 20 multiple choice problems to
be finished in 1.25 hours.
72
