CHAPTER 8
Risk and Rates of Return

This chapter is most important and will
be emphasized in tests.
8-1
Chapter Objectives





Define and measure the expected rate of return of an
individual investment
Define and measure the riskiness of an individual
investment
Compare the historical relationship between risk and
rates of return in the capital markets
Explain how diversifying investments affect the
riskiness and expected rate of return of a portfolio or
combination of assets
Explain the relationship between an investor’s
required rate of return and the riskiness of the
investment
8-2
Investment returns
The rate of return on an investment can be
calculated as follows:
Return =
(Amount received – Amount invested)
________________________
Amount invested
For example, if $1,000 is invested and $1,100 is
returned after one year, the rate of return for this
investment is:
($1,100 - $1,000) / $1,000 = 10%.
8-3
What is investment risk?





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
The Chinese definition
Two types of investment risk


Stand-alone risk (when the return is analyzed in isolation.)
Portfolio risk (when the return is analyzed in a portfolio.)
8-4
PART I: Standard alone risk

The risk an investor would face if s/he
held only one asset.
8-5
Probability distributions


A listing of all possible outcomes, and the
probability of each occurrence.
Can be shown graphically.
Firm X
Firm Y
-70
0
15
Expected Rate of Return
100
Rate of
Return (%)
8-6
Which firm is more likely to have a return closer
to its expected value?


Firm X?
Firm Y?
8-7
Investor attitude towards risk



Risk aversion – assumes investors dislike risk and
require higher rates of return to encourage them
to hold riskier securities.
Who wants to be a millionaire?
Risk premium – the difference between the return
on a risky asset and less risky asset, which serves
as compensation for investors to hold riskier
securities.
8-8
Selected Realized Returns,
1926 – 2001
Small-company stocks
Large-company stocks
L-T corporate bonds
L-T government bonds 5.7
U.S. Treasury bills
Average
Return
17.3%
12.7
6.1
3.9
Standard
Deviation
33.2%
20.2
8.6
9.4
3.2
Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation
Edition) 2002 Yearbook (Chicago: Ibbotson Associates, 2002), 28.
8-9
The Value of an Investment of $1 in 1926
Index
1000
6402
S&P
Small Cap
Corp Bonds
Long Bond
T Bill
2587
64.1
48.9
10
16.6
1
0.1
1925
1940
Source: Ibbotson Associates
1955
1970
1985
2000
Year End
8-10
Percentage Return
Rates of Return 1926-2000
60
Common Stocks
Long T-Bonds
T-Bills
40
20
0
-20
Source: Ibbotson Associates
95
90
85
80
75
20
00
Year
70
65
60
55
50
45
40
35
26
-60
30
-40
8-11
Suppose there are 5 possible outcomes
over the investment horizon for the
following securities:
Economy
Prob.
T-Bill
HT
Coll
USR
MP
Recession
0.1
8.0%
-22.0%
28.0%
10.0%
-13.0%
Below avg
0.2
8.0%
-2.0%
14.7%
-10.0%
1.0%
Average
0.4
8.0%
20.0%
0.0%
7.0%
15.0%
Above avg
0.2
8.0%
35.0%
-10.0%
45.0%
29.0%
Boom
0.1
8.0%
50.0%
-20.0%
30.0%
43.0%
8-12
Why is the T-bill return independent of
the economy?




T-bills will return the promised 8%, regardless of
the economy.
T-bills are risk-free in the default sense of the
word.
Do T-bills promise a completely risk-free return?
No, T-bills do not provide a risk-free return, as
they are still exposed to inflation. Although, very
little unexpected inflation is likely to occur over
such a short period of time.
8-13
How do the returns of HT and Coll.
behave in relation to the market?


HT – Moves with the economy, and has
a positive correlation. This is typical.
Coll. – Is countercyclical with the
economy, and has a negative
correlation. This is unusual.
8-14
Return: Calculating the expected
return for each alternative
^
k  expected rate of return
^
n
k   k i Pi
i1
^
k HT  (-22.%) (0.1)  (-2%) (0.2)
 (20%) (0.4)  (35%) (0.2)
 (50%) (0.1)  17.4%
8-15
Summary of expected returns
for all alternatives
HT
Market
USR
T-bill
Coll.
Exp return
17.4%
15.0%
13.8%
8.0%
1.7%
HT has the highest expected return, and appears
to be the best investment alternative, but is it
really? Have we failed to account for risk?
8-16
Risk: Calculating the standard
deviation for each security
  Standard deviation
  Variance  2

n
 (k  k̂ ) P
i1
2
i
i
8-17
Standard deviation calculation
 
n

i1
^
(k i  k )2 Pi
(8.0 - 8.0) (0.1)  (8.0 - 8.0) (0.2)
  (8.0 - 8.0)2 (0.4)  (8.0 - 8.0)2 (0.2)
2
 (8.0 - 8.0) (0.1)
2
 T bills
 T bills  0.0%
 HT  20.0%
2




1
2
 C oll  13.4%
 USR  18.8%
 M  15.3%
8-18
Comparing standard deviations
Prob.
T - bill
USR
HT
0
8
13.8
17.4
Rate of Return (%)
8-19
Comments on standard
deviation as a measure of risk



Standard deviation (σi) measures
“total”, or stand-alone, risk.
The larger σi is, the lower the
probability that actual returns will
be closer to expected returns.
Larger σi is associated with a wider
probability distribution of returns.
8-20
Comparing risk and return
Security
Expected
return
8.0%
Risk, σ
17.4%
20.0%
Coll*
1.7%
13.4%
USR*
13.8%
18.8%
Market
15.0%
15.3%
T-bills
HT
0.0%
* Seem out of place.
8-21
(Not required) Coefficient of Variation (CV)
A standardized measure of dispersion about
the expected value, that shows the risk per
unit of return.
Std dev 
CV 
 ^
Mean
k
8-22
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.
8-23
In a portfolio…


The expected return is the weighted
average of each individual stock’s
expected return.
However, the portfolio standard
deviation is generally lower than the
weighted average of each individual
stock’s standard deviation.
8-24
Portfolio construction:
Risk and return
Assume a two-stock portfolio is created with
$50,000 invested equally in both HT and
Collections. That is, you invest 50% in HT
and 50% in Coll. What are the expected
returns and standard deviation for the 2stock portfolio?
8-25
Over the investment horizon, there are 5 possible
outcomes.
Security
HT
Coll*
Economy
Prob.
HT
Coll
Expected
return
Risk, σ
Recession
0.1
-22.0%
28.0%
17.4%
20.0%
Below avg
0.2
-2.0%
14.7%
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%
8-26
Calculating portfolio expected return
^
k p is a weighted average :
^
n
^
k p   wi k i
i1
^
k p  0.5 (17.4%)  0.5 (1.7%)  9.6%
8-27
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%
8-28
Calculating portfolio standard
deviation
 0.10 (3.0 - 9.6)

2

0.20
(6.4
9.6)

   0.40 (10.0 - 9.6) 2

  0.20 (12.5 - 9.6) 2

2

0.10
(15.0
9.6)

2
p








1
2
 3.3%
8-29
Comments on portfolio risk
measures




σp = 3.3% is much lower than the σi of either stock
(σHT = 20.0%; σColl. = 13.4%). This is not generally
true.
σp = 3.3% is lower than the weighted average of
HT and Coll.’s σ (16.7%). This is usually 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!
8-30
General comments about risk



Most stocks are positively correlated
with the market.
σ  35% for an average stock.
Combining stocks in a portfolio
generally lowers risk.
8-31
Returns distribution for two perfectly
negatively correlated stocks
Stock W
Stock M
Portfolio WM
25
25
25
15
15
15
0
0
0
-10
-10
-10
8-32
Returns distribution for two perfectly
positively correlated stocks
Stock M’
Stock M
Portfolio MM’
25
25
25
15
15
15
0
0
0
-10
-10
-10
8-33
Returns


A stock’s realized return is often different
from its expected return.
Total return= expected return + unexpected
return

Unexpected return=systematic portion +
unsystematic portion

Total risk (stand-alone risk)= systematic
portion of risk + unsystematic portion of risk
8-34
Systematic Risk



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, nondiversifiable risk, beta risk.
8-35
Unsystematic Risk


This unsystematic 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.
8-36
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.
8-37
Creating a portfolio:
Beginning with one stock and adding
randomly selected stocks to portfolio




σp decreases as stocks added, because stocks usually
would not be perfectly correlated with the existing
portfolio.
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%.
8-38
Illustrating diversification effects of
a stock portfolio
p (%)
35
Company-Specific Risk
Stand-Alone Risk, p
20
Market Risk
0
10
20
30
40
2,000+
# Stocks in Portfolio
8-39
Breaking down sources of
total risk (stand-alone risk)
Stand-alone risk = Market risk + Firm-specific risk



Market risk (systematic risk, non-diversifiable risk) –
portion of a security’s stand-alone risk that cannot be
eliminated through diversification.
Firm-specific risk (unsystematic risk, diversifiable 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.
8-40
Failure to diversify

If an investor chooses to hold just one stock in her/his portfolio
(exposed to more risk than a diversified investor), would the
investor be compensated for the firm-specific risk ?
 NO!
 Firm-specific risk is not important to a well-diversified investor.
 Suppose investor A and B all want to buy a stock of IBM.
Investor A’s portfolio is not no well-diversified and will demand
a higher return from the stock than investor B would demand.
That is: A is willing to pay a lower price to buy the IBM stock.
 B can offer to buy at a higher price because the same stock
appears not as risky to B.
 B will get the stock. The expected return on IBM will reflect
the degree of systematic risk of stock IBM in B’s portfolio.
8-41
So,



Rational, risk-averse investors are
concerned with σp, which is based upon
market risk.
No compensation should be earned for
holding unnecessary, diversifiable risk.
Only systematic risk will be compensated.
8-42
How do we measure systematic risk?
Beta




Measures a stock’s market risk, and shows a
stock’s volatility relative to the market (i.e.,
the degree of co-movement with the market
return.)
Indicates how risky a stock is if the stock is
held in a well-diversified portfolio.
Measure of a firm’s market risk or the risk
that remains after diversification
Beta will decide a stock’s required rate of
return.
8-43
Calculating betas


Run a regression of past returns of a
security against past market returns.
(Market return is the weighted average
of all stocks’ returns at a certain time.)
The slope of the regression line (called
the security’s characteristic line) is
defined as the beta coefficient for the
security.
8-44
Illustrating the calculation of beta
(security’s characteristic line)
_
ki
20
.
15
.
10
Year
1
2
3
kM
15%
-5
12
ki
18%
-10
16
5
-5
.
0
-5
-10
5
10
15
_
20
kM
Regression line:
^
^
k = -2.59 + 1.44 k
i
M
8-45
Security Character Line





What does the slope of SML 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.
8-46
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.
Most stocks have betas in the range of 0.5 to 1.5.
8-47
Can the beta of a security be
negative?




Yes, if the correlation between Stock i and
the market is negative (i.e., ρi,m < 0).
If the correlation is negative, the
regression line would slope downward,
and the beta would be negative.
However, a negative beta is highly
unlikely.
A stock that will give your higher return in
recession is generally more valuable to
investors, thus required rate of return is
lower.
8-48
Beta coefficients for
HT, Coll, and T-Bills
40
_
ki
HT: β = 1.30
20
T-bills: β = 0
-20
0
20
40
_
kM
Coll: β = -0.87
-20
8-49
Comparing expected return
and beta coefficients
Security
HT
Market
USR
T-Bills
Coll.
Exp. Ret.
17.4%
15.0
13.8
8.0
1.7
Beta
1.30
1.00
0.89
0.00
-0.87
Riskier securities have higher returns, so the
rank order is OK.
8-50
Until now…



We have argued that well-diversified investors
only cares about a stock’s systematic risk
(measured by beta).
The higher the systematic risk (nondiversifiable 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 nondiversifiable risk of a stock is called risk
premium.
8-51
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.
E(Ki-Krf)/Beta(i)=E(Km-Krf)/Beta(m)
8-52
The higher the beta, the
higher the risk premium.

Market beta=1
(ki –kRF ) / (kM – kRF)= βi /1
Thus, we have
ki = kRF + (kM – kRF) βi

You’ve got CAPM!



8-53
Capital Asset Pricing Model
(CAPM)
ki = kRF + (kM – kRF) βi
 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.
8-54
The Security Market Line (SML):
Calculating required rates of return
SML: ki = kRF + (kM – kRF) βi
SML is a graphical representation of CAPM
 Assume kRF = 8% and kM = 15%.
 The market (or equity) risk premium is
kM – kRF = 15% – 8% = 7%.
 If a stock has a beta=1.5, how much is
its required rate of returns?
8-55
Risk-Free Rate


Required rate of return for risk-less
investments
Typically measured by U.S. Treasury Bill
Rate
8-56
What is the market risk premium?



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.
Varies from year to year, but most
estimates suggest that it ranges between
4% and 8% per year.
8-57
Comparing expected return
and beta coefficients
Security
HT
Market
USR
T-Bills
Coll.
Exp. Ret.
17.4%
15.0
13.8
8.0
1.7
Beta
1.30
1.00
0.89
0.00
-0.87
Assume kRF = 8% and kM = 15%.
The market (or equity) risk premium is kM – kRF = 15%
– 8% = 7%.
Please find the require rates of return for each security.

8-58
Calculating required rates of return





kHT
kM
kUSR
kT-bill
kColl
=
=
=
=
=
=
=
8.0%
8.0%
8.0%
8.0%
8.0%
8.0%
8.0%
+
+
+
+
+
+
+
(15.0% - 8.0%)(1.30)
(7.0%)(1.30)
9.1%
= 17.10%
(7.0%)(1.00) = 15.00%
(7.0%)(0.89) = 14.23%
(7.0%)(0.00) = 8.00%
(7.0%)(-0.87) = 1.91%
8-59
Expected vs. Required returns
^
k
HT
Market
USR
T - bills
Coll.
k
17.4% 17.1%
15.0
13.8
8.0
1.7
15.0
14.2
8.0
1.9
^
Undervalued (k  k)
^
Fairly valued (k  k)
^
Overvalued (k  k)
^
Fairly valued (k  k)
^
Overvalued (k  k)
8-60
If market is fully efficient






Then there are no under-valued or over- valued stocks.
And the expected returns should be equal to required returns.
If many people believe that the a stock’s expected return is higher
than required return, they would bid for that stock, pushing up the
stock price, hence lowering the expected return.
Market competition will lead to:
expected returns = required returns.
In short run, there might be mis-valued stocks and expected return
may be different from the required return. In the long run, expected
returns = required returns.
An analogy: Gravity will lead to sea level be flat in the long run, but
you will seldom see a totally flat sea water level.
8-61
Security Market Line (SML)
SML: ki = 8% + (15% – 8%) βi
ki (%)
SML
.
..
HT
kM = 15
kRF = 8
-1
.
Coll.
. T-bills
0
USR
1
2
Risk, βi
8-62
Security Market Line

What does the slope of SML mean?

Market risk premium= kM- kRF

What variable is in the horizontal line?

Beta
8-63
An example:
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
8-64
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%
8-65
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.
ki = kRF + (kM – kRF) βi + ???
8-66