Chapter 8
Risk and Rates of Return
8-1
The Risk-Return Trade-Off
•
Investors like returns and they dislike risk.
– The slope depends on the investor’s willingness to take
on risk.
•
If a company is investing in riskier projects, it must offer
its investors (bondholders and stockholders) higher
expected returns.
8-2
What is investment risk?
•
•
Risk refers to the chance that some unfavorable
event will occur.
Two types of investment risk
– Stand-alone risk: the risk an investor would face if he
or she held only this one asset.
•
•
– 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.
8-3
Probability Distributions
•
•
•
•
•
A listing of all possible outcomes, and the probability
of each occurrence
Expected rates of return, 𝑟 (“r hat”)
Historical, or past realized rates of return, 𝑟 (“r bar”)
Standard deviation, 𝜎 (sigma)
Coefficient of variation (CV)
8-4
Hypothetical Investment Alternatives
Economy
Recession
Prob. T-Bills
HT
Coll
0.1 5.5% -27.0% 27.0%
USR
MP
6.0% -17.0%
Below avg
0.2
5.5%
-7.0%
13.0% -14.0% -3.0%
Average
0.4
5.5%
15.0%
0.0%
Above avg
0.2
5.5%
30.0% -11.0% 41.0% 25.0%
Boom
0.1
5.5%
45.0% -21.0% 26.0% 38.0%
3.0%
10.0%
 High Tech moves with the economy, and has a positive
correlation. This is typical. Collections is countercyclical with
the economy, and has a negative correlation. This is unusual.
8-5
Calculating the Expected Return for HT
r̂  Expected rate of return
N
r̂   Piri
i1
r̂  (0.1)(-27%)  (0.2)(-7%)  (0.4)(15%)
 (0.2)(30%)  (0.1)(45%)
 12.4%
8-6
Summary of Expected Returns
T-bills
High Tech
US Rubber
Market Portfolio
Expected Return
5.5%
12.4%
9.8%
10.5%
High Tech has the highest expected return, and appears
to be the best investment alternative, but is it really?
Have we failed to account for risk?
8-7
Calculating Standard Deviation
  Standard deviation
  Variance  2

N
2
(
r

r̂
)
Pi

i 1
 Standard deviation (σi) measures total, or stand-alone,
risk. The larger σi is, the lower the probability that actual
returns will be close to expected returns.
8-8
Standard Deviation for Each Investment

N
2
(
r

r̂
)
Pi

i 1
(5.5  5.5) (0.1)  (5.5  5.5) (0.2)


 (5.5  5.5)2 (0.4)  (5.5  5.5)2 (0.2)
2



(
5
.
5

5
.
5
)
(0.1)


2
 T -bills
2
1/2
 T -bills  0.0%
σHT = 20%
σColl = 13.2%
σM = 15.2%
σUSR = 18.8%
8-9
Standard Deviations and Continuous Distributions
 Larger σi is associated with a wider probability distribution
of returns.
8-10
Using Historical Data to Measure Risk
•
•
•
We found the mean and standard deviation based on a
subjective probability distribution.
The historical 𝜎 is often used as an estimate of future risk,
because past results are often repeated in the future.
How far back in time should we go?
– Using a longer historical time series has the benefit of
giving more information.
– But, some of that information may be misleading if
you believe that the level of risk in the future is likely
to be very different than in the past.
8-11
Comparing Risk and Return
Security
T-bills
High Tech
Collections
US Rubber
Market
Expected Return, r̂
5.5%
12.4
1.0
9.8
10.5
Risk, 
0.0%
20.0
13.2
18.8
15.2
8-12
Coefficient of Variation (CV)
•
A standardized measure of dispersion about the
expected value, that shows the risk per unit of
return.
– The CV provides a more meaningful risk measure
when the expected returns on two alternatives
are not the same.
Standard deviation 
CV 

Expected return
r̂
8-13
Risk Rankings by Coefficient of Variation
T-bills
High Tech
Collections
US Rubber
Market
•
•
CV
0.0
1.6
13.2
1.9
1.4
Collections has the highest degree of risk per
unit of return.
High Tech, despite having the highest standard
deviation of returns, has a relatively average CV.
8-14
Risk Aversion and Required Returns
•
Suppose you want to invest your $1 million:
– A: 5% US Treasury bill
– B: Stock in R&D Enterprises whose value can be $2.1
million or 0 with the same probability.
Expected ending value - Cost
Expected rateof return
Cost
•
Most investors are risk-averse.
→ In a market dominated by risk-averse investors, riskier
securities compared to less risky securities must have
higher expected returns.
8-15
Risk in a Portfolio Context: The CAPM
•
•
Risk of stocks when they are held in portfolios: the
Capital Asset Pricing Model (CAPM)
– Most stocks are held in portfolios.
What is important is the return on portfolio and the
portfolio’s risk.
– Logically, then, the risk and return of an individual
stock should be analyzed in terms of how the
security affects the risk and return of the portfolio in
which it is held.
– Example of Pay Up Inc.
8-16
Calculating Portfolio Expected Return
•
Assume a two-stock portfolio is created with $50,000
invested in both High Tech and Collections.
– A portfolio’s expected return is a weighted average of
the returns of the portfolio’s component assets.
– Standard deviation is a little more tricky and requires
that a new probability distribution for the portfolio
returns be constructed.
r̂p isa weighted average :
r̂p 
N
w irˆi  0 .5(12 .4 %)  0 .5(1 .0 %)

i
 6 .7 %
1
8-17
An Alternative Method for Determining Portfolio
Expected Return
Economy
Recession
Below avg
Average
Above avg
Boom
Prob
0.1
0.2
0.4
0.2
0.1
HT
-27.0%
-7.0%
15.0%
30.0%
45.0%
Coll
27.0%
13.0%
0.0%
-11.0%
-21.0%
Port
0.0%
3.0%
7.5%
9.5%
12.0%
r̂p  0.10 (0.0%)  0.20 (3.0%)  0.40 (7.5%)
 0.20 (9.5%)  0.10 (12.0%)  6.7%
8-18
Calculating Portfolio Standard Deviation and CV
 0.10 (0.0 - 6.7) 

2 
 0.20 (3.0 - 6.7) 
p   0.40 (7.5 - 6.7)2 


 0.20 (9.5 - 6.7)2 


2
 0.10 (12.0 - 6.7) 
2
1
2
 3.4%
3.4%
CVp 
 0.51
6.7%
8-19
Portfolio Risk Measures
•
•
•
•
•
σp = 3.4% is much lower than the σi of either stock (σHT
= 20.0%; σColl = 13.2%).
σp = 3.4% is lower than the weighted average of High
Tech and Collections’ σ (16.6%).
Therefore, the portfolio provides the average return of
component stocks, but lower than the average risk.
Why? Diversification lowers the portfolio’s risk.
The tendency of two variables to move together is
called correlation, and the correlation coefficient, 𝜌
(rho), measures this tendency. (−1 ≤ 𝜌 ≤ 1)
8-20
Creating a Portfolio: Beginning with One Stock and
Adding Randomly Selected Stocks to Portfolio
•
•
•
σp decreases as stocks are 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 40 to 50 stocks).
8-21
Breaking Down Sources of Risk
Stand-alone risk = Market risk + Diversifiable risk
•
•
Market risk: portion of a security’s stand-alone risk
that cannot be eliminated through diversification.
Measured by beta.
Diversifiable risk: portion of a security’s standalone risk that can be eliminated through proper
diversification.
8-22
Failure to Diversify
•
If an investor chooses to hold a one-stock portfolio
(doesn’t diversify), would the investor be
compensated for the extra risk they bear?
– NO!
– Stand-alone risk is not important to a well-diversified
–
–
investor.
Rational, risk-averse investors are concerned with σp,
which is based upon market risk.
No compensation should be earned for holding
unnecessary, diversifiable risk.
8-23
Risk in a Portfolio Context: The Beta Coefficient
•
•
How do we measure a stock’s relevant risk in a
portfolio context?
The risk that remains once a stock is in a diversified
portfolio is its contribution to the riskiness of the
portfolio, and that risk can be measured by the extent
to which the stock moves up or down with the
market, beta coefficient, b.
– Without a crystal ball to predict the future, analysts are
forced to rely on historical data. A typical approach to
estimate beta is to run a regression of the security’s
past returns against the past returns of the market.
8-24
Comments on Beta
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•
•
•
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.
The beta of a security can be negative, if the correlation
between Stock i and the market is negative (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.
Most stocks have betas in the range of 0.5 to 1.5.
8-25
The Relationship between Risk and Rates of Return
•
•
•
•
•
•
•
•
According to the CAPM, beta is the most appropriate
measure of a stock’s relevant risk. The next issue is:
For a given level of risk as measured by beta, what
rate of return is required to compensate the investor?
𝑟𝑖 = expected rates of return on the ith stock.
𝑟𝑖 = required rate of return. In equilibrium, 𝑟𝑖 = 𝑟𝑖 .
𝑟𝑖 = realized rate of return.
𝑏𝑖 = beta coefficient of the ith stock.
𝑟𝑀 = required rate of return on the market portfolio.
𝑅𝑃𝑀 = 𝑟𝑀 − 𝑟𝑅𝐹 = risk premium on the market.
𝑅𝑃𝑖 = 𝑟𝑖 − 𝑟𝑅𝐹 = risk premium on the ith stock.
8-26
The Relationship between Risk and Rates of Return
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Example: Beta and the risk premium
E(rA )  20%, b A  1.6;E(rB )  16%, b B  1.2;rRF  8%
Reward-to-risk ratio must be the same for all the assets
in the market.
– If one asset has twice as much systematic risk as
•
another asset, its risk premium will simply be twice
as large.
Because all the assets have the same reward-to-risk,
they all must plot on the same line. This line is called
the security market line (SML).
8-34
The Security Market Line (SML): Calculating
Required Rates of Return
SML: ri = rRF + (rM – rRF)bi = rRF + (RPM)bi
•
•
Assume the yield curve is flat and that rRF = 5.5% and
RPM = rM  rRF = 10.5%  5.5% = 5.0%.
Market risk premium is the 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.
8-28
Expected vs. Required Returns
b
r
12.4% 1.32 12.1% Undervalued (r̂  r)
10.5
1.0 10.5 Fairly valued (r̂  r)
9.8
0.88
9.9 Overvalued (r̂  r)
5.5
0
5.5 Fairly valued (r̂  r)
1.0 -0.87
1.15 Overvalued (r̂  r)
r̂
High Tech
Market
US Rubber
T-bills
Collections
8-29
Some Concerns about Beta and the CAPM
•
•
A number of recent studies have raised concerns about
the validity of the CAPM.
Researchers and practitioners are developing models
with more explanatory variables than just beta. In
these multivariable models, risk is assumed to be
caused by a number of different factors.
ri = rRF + (rM – rRF)bi + ???
•
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.
8-30