Risk and Rates of Return



Stand-alone risk
Portfolio risk
Risk & return: CAPM / SML
5-1
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%.
5-2
What is investment risk?

Two types of investment risk




Stand-alone risk
Portfolio 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.
5-3
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 (%)
5-4
Selected Realized Returns,
1926 – 2001
Small-company stocks
Large-company stocks
L-T corporate bonds
L-T government bonds
U.S. Treasury bills
Average
Return
17.3%
12.7
6.1
5.7
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.
5-5
Investment alternatives
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%
5-6
Why is the T-bill return independent of
the economy? Do T-bills promise a
completely risk-free return?


T-bills will return the promised 8%, regardless of the
economy.
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.

T-bills are also risky in terms of reinvestment rate risk.

T-bills are risk-free in the default sense of the word.
5-7
Risks of Different Asset Classes
SECURITY
RISK
TREASURY BILLS
NO RISK IN NOMINAL RETURN
BUT UNCERTAIN REAL RETURN
TREASURY BONDS
AND INTEREST RATE RISK
CORPORATE BONDS
AND DEFAULT RISK
AND EVENT RISK
COMMON STOCKS
AND MARKET RISK
AND UNSYSTEMATIC RISK
5-8
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.
5-9
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%
5-10
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?
5-11
Risk: Calculating the standard
deviation for each alternative
  Standard deviation
  Variance  
2

n
2
(
k

k̂
)
Pi
 i
i1
5-12
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%
5-13
Comparing standard deviations
Prob.
T - bill
USR
HT
0
8
13.8
17.4
Rate of Return (%)
5-14
Comments on standard
deviation as a measure of risk




Standard deviation (σi) measures total, or standalone, 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.
Difficult to compare standard deviations, because
return has not been accounted for.
5-15
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.
5-16
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
5-17
Risk rankings,
by coefficient of variation
T-bill
HT
Coll.
USR
Market


CV
0.000
1.149
7.882
1.362
1.020
Collections has the highest degree of risk per unit of
return.
HT, despite having the highest standard deviation of
returns, has a relatively average CV.
5-18
Illustrating the CV as a
measure of relative risk
Prob.
A
B
0
Rate of Return (%)
σA = σB , but A is riskier because of a larger probability of
losses.
In other words, the same amount of risk (as measured by σ)
for less returns.
5-19
Investor attitude towards risk


Risk aversion – assumes investors dislike risk and
require higher rates of return to encourage them
to hold riskier securities.
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.
5-20
Portfolio construction:
Risk and return
Assume a two-stock portfolio is created with $50,000
invested in both HT and Collections.


Expected return of a portfolio is a weighted average
of each of the component assets of the portfolio.
Standard deviation is a little more tricky and requires
that a new probability distribution for the portfolio
returns be devised.
5-21
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%
5-22
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%
5-23
Calculating portfolio standard
deviation and CV
 0.10 (3.0 - 9.6)
 0.20 (6.4 - 9.6)2

 p   0.40 (10.0 - 9.6)2
 0.20 (12.5 - 9.6)2

2

0.10
(15.0
9.6)

2







1
2
 3.3%
3.3%
CVp 
 0.34
9.6%
5-24
Comments on portfolio risk
measures




σp = 3.3% is much lower than the σi of either stock
(σHT = 20.0%; σColl. = 13.4%).
σp = 3.3% is lower than the weighted average of
HT and Coll.’s σ (16.7%).
\ Portfolio provides average return of component
stocks, but lower than average risk.
Why? Negative correlation between stocks.
5-25
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.
5-26
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
5-27
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
5-28
Creating a portfolio:
Beginning with one stock and adding
randomly selected stocks to portfolio



σp decreases as stocks added, because they would
not be perfectly correlated with the existing portfolio.
Expected return of the portfolio would remain
relatively constant.
Eventually the diversification benefits of adding more
stocks dissipates (after about 10 stocks), and for
large stock portfolios, σp tends to converge to 
20%.
5-29
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
5-30
Breaking down sources of risk
Stand-alone risk = Market risk + Firm-specific risk


Market risk – portion of a security’s stand-alone risk
that cannot be eliminated through diversification.
Measured by beta.
Firm-specific risk – portion of a security’s stand-alone
risk that can be eliminated through proper
diversification.
5-31
Failure to diversify

If an investor chooses to hold a one-stock portfolio,
would the investor be compensated for the risk they
bear?





NO!
Stand-alone risk is not important to a welldiversified investor.
Rational, risk-averse investors are concerned with
σp, which is based upon market risk.
There can be only one price (the market return) for
a given security.
No compensation should be earned for holding
5-32
unnecessary, diversifiable risk.
Stocks have two types of Risk


Market risk

Or systematic risk or undiversifiable risk

Affects all stocks
Unique risk

Or unsystematic risk or diversifiable risk or specific risk or residual
risk

Affects individual stocks or small groups of stocks

Unique risk of different firms unrelated

Eliminated by diversification
5-33
Unique or Unsystematic risk



A drug trial showing that beta blockers increase risk of cancer
in older people will affect Pfizer’s stock
 But has no affect on shares of GM or IBM
A strike at a single gm plant will affect only GM and perhaps its
suppliers and competitors
A hot summer will increase demand for air conditioners
 But won’t affect the demand for computers
5-34
Market or Systematic Risk


All firms affected by economy and exposed to market risk
 Example: surprise in rate of growth in GNP
Market risk cannot be diversified away
5-35
Advantages to Diversification


Single stock
 Exposed to market risk and unique risk
Diversified portfolio
 Only exposed to market risk
 Major uncertainty is whether market will rise or fall
 Most of benefits of diversification achieved with 10 - 20
stocks
5-36
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.
Primary conclusion: The relevant riskiness of a stock
is its contribution to the riskiness of a well-diversified
portfolio.
5-37
Beta


Measures a stock’s market risk, and shows a stock’s
volatility relative to the market.
Indicates how risky a stock is if the stock is held in a
well-diversified portfolio.
5-38
Calculating betas


Run a regression of past returns of a security against
past returns on the market.
The slope of the regression line (sometimes called
the security’s characteristic line) is defined as the
beta coefficient for the security.
5-39
Illustrating the calculation of beta
_
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
5-40
Comments on beta

If beta = 1.0, the security is just as risky as the
average stock.

If beta > 1.0, the security is riskier than average.

If beta < 1.0, the security is less risky than average.

Most stocks have betas in the range of 0.5 to 1.5.
5-41
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.
5-42
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
5-43
Comparing expected return
and beta coefficients
Security
HT
Market
USR
T-Bills
Coll.
8.0
Exp. Ret.
17.4%
15.0
13.8
1.7
0.00
Beta
1.30
1.00
0.89
-0.87
Riskier securities have higher returns, so the rank order is
OK.
5-44
The Security Market Line (SML):
Calculating required rates of return
SML: ki = kRF + (kM – kRF) βi


Assume kRF = 8% and kM = 15%.
The market (or equity) risk premium is RPM = kM – kRF
= 15% – 8% = 7%.
5-45
What is the market risk premium?



Additional return over the risk-free rate needed to
compensate investors for assuming an average
amount of 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.
5-46
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%
5-47
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)
5-48
Illustrating the
Security Market Line
SML: ki = 8% + (15% – 8%) βi
ki (%)
SML
.
..
HT
kM = 15
kRF = 8
-1
.
Coll.
. T-bills
0
USR
1
2
Risk, βi
5-49
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
5-50
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%
5-51
Factors that change the SML

What if investors raise inflation expectations by 3%,
what would happen to the SML?
ki (%)
D I = 3%
SML2
SML1
18
15
11
8
Risk, βi
0
0.5
1.0
1.5
5-52
Factors that change the SML
What if investors’ risk aversion increased, causing
the market risk premium to increase by 3%, what
would happen to the SML?

ki (%)
D RPM = 3%
SML2
SML1
18
15
11
8
Risk, βi
0
0.5
1.0
1.5
5-53
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.
5-54
More thoughts on the CAPM

Investors seem to be concerned with both market
risk and total risk. 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.
5-55