CHAPTER 8
Risk and Rates of Return



Stand-alone risk
(This is the risk when we only invest in one thing.)
Portfolio risk
(This is the risk for well-diversified portfolios.)
Risk & return relationship


Capital Asset Pricing Model (“CAPM”)
Security Market Line (“SML”)
8-1
CHAPTER 8
Risk and Rates of Return
Suggestions for learning Chapter 8:



Read Chapter 8.
Know the Key Terms (ST-1 on page 260-261).
Work all 13 problems 8-1 through 8-13 (pages 262263).
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) / $1,000 = 10%.
8-3
What is investment risk?

Investment risk is related to the probability of earning a low
or negative actual return.

The greater the chance of lower than expected or negative
returns, the riskier the investment.

Two types of investment risk



Stand-alone risk (i.e., the risk when we only invest in one thing).
Portfolio risk (i.e., the risk for well-diversified portfolios).
Let’s take a look at the “stand-alone” investment risk for
Apple’s stock.
8-4
Excel spreadsheet calculation of average return
and standard deviation of returns using Apple
(“AAPL”) weekly returns (finance.google.com)

Step 1: Calculate weekly return using formula on slide 8-3

Step 2: Use Excel functions for average and “STDEVA”
Date
Close Price ($/share) Return
9-Apr-09
119.57
3.1%
3-Apr-09
115.99
8.6%
27-Mar-09
106.85
5.2%
20-Mar-09
101.59
5.9%
13-Mar-09
95.93
12.5%
6-Mar-09
85.3
-4.5%
27-Feb-09
89.31
-2.1%
20-Feb-09
91.2
-8.0%
13-Feb-09
99.16
-0.6%
6-Feb-09
99.72
expected return = mean = average
standard deviation (Excel function "STDEVA")
2.2%
6.6%
8-5
Apple Excel spreadsheet calculation (continued)

Note: the formula for sample standard deviation is
 1
2
 
( ri  r ) 

 ( n  1) i 1

n
_
1/ 2
where n is the number of weeks, r is the average return, and
the ri are the weekly returns.
(Don’t panic! I will not ask you to use this formula on the final.)
8-6
Probability distributions



A listing of all possible outcomes, and the probability of each occurrence.
This is an example of a “normal” distribution or a “bell-shaped” curve.
The area under the curve sums to 100%. We expect about 68% of the
returns will be in the range of 15% +/- one standard deviation.
Firm X
Firm Y
-70
0
15
Expected Rate of Return
100
Rate of
Return (%)
8-7
Probability distribution for Apple example:
Using weekly data, we calculated that average
return = 2.2%, standard deviation = 6.6%.



The area under the curve sums to 100%.
We expect about 68% of the returns will be in the range of the average
return +/- one standard deviation. Lower: 2.2% - 6.6% = - 4.4%.
Upper: 2.2% + 6.6% = 8.8%.
We expect 68% of the weeks, we will have a weekly return that is a
number between – 4.4% and + 8.8%.
Probability Distribution of Returns is
Area under the Curve.
68%
Return
-4.4%
2.2%
8.8%
“Expected” Return =
Average
8-8
Normal probability distribution (continued):
□ 68.2% probability is average return +/-1 standard dev.
□ 95.5% probability is average return +/-2 standard dev.
□ 99.7% probability is average return +/-3 standard dev.



The area under the curve sums to 100%.
We expect about 99.7% of the returns will be in the range of the average
return +/- 3 standard deviations. Lower: 2.2% - 3(6.6%) = – 19.7%.
Upper: 2.2% + 3(6.6)% = 21.9%.
We expect 99.7% of the weeks, we will have a weekly return that is a
number between – 19.7% and + 21.9%.
Probability Distribution of Returns is
Area under the Curve.
99.7%
Return
- 19.7%
2.2%
21.9%
“Expected” Return =
Average
8-9
Selected Realized Annual Returns,
1926 – 2007
Small-company stocks
Large-company stocks
L-T corporate bonds
L-T government bonds 5.8
U.S. Treasury bills
Average
Return
17.1%
12.3
6.2
3.8
Standard
Deviation
32.6%
20.0
8.4
9.2
3.1
Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation
Edition) 2008 Yearbook (Chicago: Ibbotson Associates, 2008), p28.
8-10
Comparing historical distributions
of 3 different investments
Prob.
T - bill
50/50 blend of
stocks and
USR
bonds
Small company
HT
stocks
0
3.8 9.8
17.1
Rate of Return (%)
8-11
Comments on standard
deviation as a measure of risk



Standard deviation (σi) measures total, or
stand-alone, risk.
The smaller the σi is, the bigger the
probability that actual returns will be closer
to expected returns.
But the larger the σi is, the wider the
probability distribution of returns. Bigger
probability that return will be better, but
also bigger probability return will be worse!
8-12
Coefficient of Variation (CV)
A standardized measure of dispersion about
the expected value, that shows the risk per
unit of return.
Standard deviation 
CV 

Expected return
rˆ
Note: this is a measure of STAND-ALONE risk,
but not Portfolio risk.
8-13
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 smaller returns.
8-14
Investor attitude towards risk



Risk aversion – assumes investors dislike risk and
require higher rates of return to encourage them
to hold riskier securities.
Most investors choosing between A and B would
prefer B. B has the lower CV.
Risk premium – the difference between the return
on a risky asset and a riskless asset, which serves
as compensation for investors to hold riskier
securities.
8-15
Consider these forecasts for 5 investment alternatives
(from Integrated Case, pages 266-237 of your text)
T-Bill
High
Tech
(HT)
Collections
USR
Market
Portfolio
(MP)
0.1
5.5%
-27.0%
27.0%
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%
3.0%
10.0%
Above avg
0.2
5.5%
30.0%
-11.0%
41.0%
25.0%
Boom
0.1
1.0
5.5%
45.0%
-21.0%
26.0%
38.0%
Forecast of
Economy
Probability
Recession
8-16
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 5.5%, regardless of the
economy.
No, T-bills do not provide a completely 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.
8-17
How do the returns of High Tech (HT) and Collections
(Coll.) behave in relation to the market?


HT – Moves with the economy, and has a positive
correlation. This is typical behavior for stocks.
Coll. – Is countercyclical with the economy, and
has a negative correlation. This is unusual.
Why might a company’s stock return move in the
opposite direction of the economy?
8-18
Calculate the expected return for High Tech given the
forecasts of economy, probabilities, etc., on slide 8-16.
r̂  Expected rate of return
r̂ 
N
 rP
i 1
i i
r̂  (-27%)(0.1)  (-7%)(0.2)  (15%)(0.4)
 (30%)(0.2)  (45%)(0.1)
 12.4%
8-19
Similarly, you could calculate the expected return for
each of the 5 Investments:
HT
Market
USR
T-bill
Coll.
Expected return
12.4%
10.5%
9.8%
5.5%
1.0%
HT has the highest expected return, and appears to be the
best investment alternative, but is it really?
We don’t know yet because we have failed to consider risk!
8-20
You could use the data in slide 8-16 to the standard
deviations for each investment. Here is the formula.
Don’t panic --- I won’t ask you to do this!
  Standard deviation
 
 
Variance 
2
N
2
(
r

r̂
)
Pi

i 1
This formula for standard deviation is different than the one we used on
slides 8-5 and 8-6 for Apple. There we had historical (real) data.
Here we only have forecasts of returns and probabilities.
8-21
Standard Deviation for Each Investment

N
 (r  r̂ ) P
i1
 T -bills
2
i
(5.5  5.5)2 (0.1)  (5.5  5.5)2 (0.2) 
 (5.5  5.5)2 (0.4)  (5.5  5.5)2 (0.2)


2
 (5.5  5.5) (0.1)


1/2
 T -bills  0.0%
σHT = 20%
σColl = 13.2%
σM = 15.2%
σUSR = 18.8%
8-22
Comparing Standard Deviations: This is similar to slide
8-11; but in that case, the standard deviations were
calculated for us from historical data for 1926 to 2007.
Prob.
T-bill
USR
HT
0
5.5 9.8
12.4
Rate of Return (%)
8-23
Comparing Risk and Return
Security
T-bills
HT
Coll*
USR*
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
*Seems out of place.
8-24
8-24
Risk Rankings by Coefficient of Variation (CV):
(We defined CV on slide 8-13.)
CV
T-bill
HT
Coll.
USR
Market


0.0
1.6
13.2
1.9
1.4
HT, despite having the highest standard deviation of returns, has
a relatively average CV.
Collections has the highest degree of risk per unit of return.
8-25
Portfolio Construction: Risk and Return




Assume you have chosen to create an investment portfolio
of 2 stocks.
You’ve chosen HT and Collections, and you will invest
$50,000 in each of these 2 stocks.
Your portfolio’s expected return is a weighted average of
the returns of the portfolio’s component assets (ie, the 2
stocks in this case). We’ll calculate it on the next slide.
The portfolio’s standard deviation is a little more tricky!
Although the portfolio provides the average return of
component stocks, it will have a lower risk than just an
average of the component stock risks.
Why? We have to take into account the correlation between
the different stock’s returns. And this will lower the risk.
8-26
Calculating the Portfolio’s expected return for
your 2-stock portfolio:
r̂p is a weighted average :
N
^
r̂p   w i r i
i1
r̂p  0.5 (12.4%)  0.5 (1.0%)  6.7%
8-27
Correlation coefficient (r)





The correlation coefficient is a measure of the degree of
relationship between 2 variables, such as the forecast return
of company A and the forecast return of company B.
Positively correlated stocks’ rates of return tend to move in
the same direction.
Negatively correlated stocks’ rates of return tend to move in
opposite directions.
Perfectly correlated stocks’ rates of return move exactly
together or exactly opposite.
Correlation coefficients range from -1 (means perfectly
negatively correlated) to +1 (means perfectly positively
correlated).
8-28
Returns Distribution for Two Perfectly
Negatively Correlated Stocks (ρ = -1.0)
8-29
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
8-30
Risk reduction through “diversification”




Combining stocks that are NOT perfectly positively correlated
will reduce portfolio risk.
We call this diversification.
The risk of a portfolio is reduced as the number of stocks in
the portfolio increases.
The lower the positive correlation of each stock we add to the
portfolio, the lower the risk.
Example: Which 2-stock portfolio has a lower risk?
Exxon and Chevron (r = 0.9) or Exxon and P&G (r = 0.4)
8-31
Creating a Portfolio: Beginning with One Stock and
Adding Randomly Selected Stocks to Portfolio


Portfolio risk (σp) decreases as stocks are added (if they are
not perfectly correlated with the existing portfolio).
The diversification benefits of adding more stocks decreases
(after about 10 stocks), and for large stock portfolios, σp
tends to converge to  20%. (This 20% number is the
approximate standard deviation of the annual return on the
portfolio. See slide 8-10: this is about the number we saw
for the standard deviation for large stock annual returns
historically from 1926 to 2007.)
8-32
Illustrating Diversification Effects of a
Stock Portfolio
8-33
Classifying the 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”. (Aka, systematic risk or non-diversifiable risk.)


Examples: interest rates, savings rates, dollar strength.
Diversifiable risk – portion of a security’s stand-alone risk that
can be eliminated through proper diversification. (Aka, firmspecific or un-systematic risk.)

Examples: CEO will embezzle, hurricane will hit refinery.
8-34
What if you choose not to diversify when you
buy stock?




You will have more risk than those who hold that same stock in
well-diversified portfolios.
You will be trading at the same prices as those who hold the
stock in well-diversified portfolios.
Your return will equal their return for that stock, but YOU will be
taking more risk.
Bottom line: you will not be compensated for having taken
unnecessary, diversifiable risk.
8-35
Beta

Beta is commonly used to indicate how risky a stock is when
the stock is held as part of a well-diversified portfolios.

If you hold lots of investments, the stock’s beta is a better
measure of risk than the stock’s standard deviation or CV.

Beta measures a stock’s volatility relative to the market.





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.
Betas are often calculated by regressions on historical data.
8-36
Illustrating the calculation of beta:
Use historical data on returns of the stock and the
“market” (e.g., S&P 500 index). Beta is the slope!
_
Stock’s
return
ri
20
.
15
.
10
Year
1
2
3
rM
15%
-5
12
5
-5
.
0
-5
-10
ri
18%
-10
16
Market’s
return
5
10
15
_
20
rM
Regression line:
^
r = -2.59 + 1.44 r^
i
M
Beta =
1.44
8-37
Beta Coefficients (b) for HT, Coll, and T-Bills
from our Integrated Case
40
HT: b = 1.32
ri
20
T-bills: b = 0
-20
0
20
40
rM
Coll: b = -0.87
-20
8-38
Comparing Expected Returns and Beta
Coefficients
Security
HT
Market
USR
T-Bills
Coll.
Expected Return
12.4%
10.5
9.8
5.5
1.0
Beta
1.32
1.00
0.88
0.00
-0.87
Riskier securities have higher returns, so the rank
order is OK.
8-39
Capital Asset Pricing Model (CAPM) and Beta

The CAPM Model links risk (beta) and required returns.

CAPM states that there is an equation called the Security Market Line
(SML):
ri = rRF + (rM – rRF) bi



In words: A stock’s required return (ri) equals the risk-free return (rRF ) plus
the stock’s beta (bi) multiplied by the market’s risk premium (rM – rRF).
In other words: for an investor to add a stock with beta (bi) to a welldiversified portfolio, the stock should be priced so that it is expected to
have a return of (ri) as calculated by the above equation.
Primary conclusion: The relevant riskiness of a stock is its contribution to
the riskiness of a well-diversified portfolio!
8-40
We can write this SML relationship in
2 different ways:



1)
ri = rRF + (rM – rRF) bi
2)
ri = rRF + (RPM) bi
Use the 1st way if you want to work with market return and
risk-free return.
Use the 2nd way if you specifically want to work with market
risk premium (RPM).
Market risk premium (RPM) = (rM – rRF), so it’s easy to use
either equation.
8-41
Let’s say a little more about the market
risk premium.




The market risk premium is the additional return over the
risk-free rate needed to compensate investors for assuming
an average amount of risk (i.e., beta = 1).
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.
Look at our SML equation:

if bi = 1, then ri = rM.
ri = rRF + (rM – rRF) bi
8-42
Example of a Security Market Line: ri = rRF + (RPM) bi .
Here, we’ve assumed the T-bill return = 5.5% and the market risk
premium = 5.0%.
SML: ri = 5.5% + (5.0%) bi
ri (%)
.
rM = 10.5
rRF = 5.5
-1
SML
Market portfolio
.
0
T-bills
1
2
Risk, bi
8-43
Another Example of the Security Market Line
(SML): Calculating Required Rates of Return for
the 5 Investments in the Integrated Case
SML: ri = rRF + (rM – rRF)bi
ri = rRF + (RPM)bi

Assume T-bill yield (rRF) = 5.5% and RPM = 5.0%.
SML: ri = 5.5% + (5.0%) bi
8-44
8-44
Calculating Required Rates of Return using the
SML and the betas on slide 8-39:
rHT
= 5.5% + (5.0%)(1.32)
= 5.5% + 6.6%
= 12.10%
rM
= 5.5% + (5.0%)(1.00)
= 10.50%
rUSR
= 5.5% +(5.0%)(0.88)
=
9.90%
rT-bill
= 5.5% + (5.0)(0.00)
=
5.50%
rColl
= 5.5% + (5.0%)(-0.87) =
1.15%
8-45
Comparing the Expected Returns (slide 8-39)
vs. the Required Returns we just calculated:
Expected
Returns
HT
Market
USR
T-bills
Coll.

r̂
12.4%
10.5
9.8
5.5
1.0
Required
Returns
r
12.1%
10.5
9.9
5.5
1.15
Undervalued ( r̂ >r)
Fairly valued ( r̂ =r)
Overvalued ( r̂ < r)
Fairly valued ( r̂ = r)
Overvalued ( r̂ < r)
What we are trying to say here is that expected returns were based
on some forecast of probabilities and returns. If we believe them,
and if the prices in the market reflect SML pricing (ie, required
returns), then we should buy undervalued HT.
8-46
Continued from the last slide: The small boxes
indicate our “expected” returns. If fairly valued ,
the small boxes would lie exactly on the SML line.
SML: ri = 5.5% + (5.0%)bi
ri (%)
SML
.
.
.
HT
rM = 10.5
rRF = 5.5
-1
.
Coll.
.
0
T-bills
USR
1
Risk, bi
2
8-47
8-47
A New 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.
bP = wHTbHT + wCollbColl
bP = 0.5(1.32) + 0.5(-0.87)
bP = 0.225
8-48
Example Continued: Calculate portfolio
required returns for this 2-stock portfolio


The required return of a portfolio is the weighted
average of each of the stock’s required returns.
rP = wHTrHT + wCollrColl
rP = 0.5(12.10%) + 0.5(1.15%)
rP = 6.625%
Or, using the portfolio’s beta, CAPM can be used
to solve for expected return. From the last slide:
rP = rRF + (RPM)bP
rP = 5.5% + (5.0%)(0.225)
rP = 6.625%
8-49
Factors That Change the SML
What if investors raise inflation expectations by 3%, what
would happen to the SML?

ri (%)
ΔI = 3%
SML2
SML1
13.5
10.5
8.5
5.5
Risk, bi
0
0.5
1.0
1.5
8-50
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?

ri (%)
ΔRPM = 3%
SML2
SML1
13.5
10.5
5.5
Risk, bi
0
0.5
1.0
1.5
8-51
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.
8-52
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 ri.
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.
8-53
Pages 258-259: Key Ideas





Trade-off between risk and return: most of us want higher
expected return if we take on higher expected risk.
Diversification is crucial. Huge risk: many people invest high
% of their savings in the stock of the company they work for.
What if the company goes bankrupt or stock gets slammed?
Real returns matter – what did you make after inflation?
Many people believe that over the long run, stocks are a good
investment. Similarly, if you need your money back over the
short term, stocks can be extremely risky. Implications for
folks nearing retirement? Rules of thumb re age and % of
your money invested in stocks versus safer bets.
Past returns are no guarantee of future performance!
8-54