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CHAPTER 9
Risk and Rates of Return

Stand-alone risk (statistics review)

Portfolio risk (investor view) -diversification important

Risk & return: CAPM/SML (market
equilibrium)
Risk is viewed primarily from the
stockholder perspective





Management cares about risk because
stockholders care about risk.
If stockholders like or dislike something about a
company (like risk) it affects the stock price.
Risk affects the discount rate for future returns -directly affecting the multiple (P/E ratio)
Thus, the concern is still about the stock price.
Stockholders have portfolios of investments –
they have stock in more than just one company
and a great deal of flexibility in which stocks they
buy.
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What is investment risk?



Investment risk pertains to the
uncertainty regarding the rate of return.
Especially when it is less than the
expected (mean) return.
The greater the chance of low or
negative returns, the riskier the
investment.
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Return = dividend + capital gain or loss



Dividends are relatively stable
Stock price changes (capital gains/losses)
are the major uncertain component
There is a range of possible outcomes and
a likelihood of each -- a probability
distribution.
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Expected Rate of Return

The mean value of the probability
distribution of possible returns

It is a weighted average of the
outcomes, where the weights are the
probabilities
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Expected Rate of Return
(k hat)
k̂  p1k1  p2 k 2  ...  pn k n
n
  pi k i
i 1
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Investment Alternatives
Economy
Prob.
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
1.0
T-bill
HT
Coll
USR
MP
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Why is the T-bill
return independent
of the economy?
Will return be 8%
regardless of the economy?
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Do T-bills really promise a
completely risk-free return?
No, T-bills are still exposed to
the risk of inflation.
However, not much unexpected
inflation is likely to occur over a
relatively short period.
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Do the returns of HT and Coll.
move with or counter to the
economy?
 High Tech: With. Positive correlation.
Typical.
 Collections: Countercyclical.
Negative correlation. Unusual.
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Calculate the expected rate of
return for each alternative:
^
k = expected rate of return
n
k =  k i pi
^
i=1
^
kHT = (-22%)0.1 + (-2%)0.20
+ (20%)0.40 +
(35%)0.20
+ (50%)0.1 = 17.4%
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Calculate others on your own
^
k
HT
17.4%
Market
15.0
USR
13.8
T-bill
8.0
Coll.
1.7
HT appears to be the best, but is it really?
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What’s the standard deviation
of returns for each alternative?
 = standard deviation
 =
Variance
n
=
 (k
i =1
=
 k̂) pi
2
i

2
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Normal Distribution
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In a sample of observations




One often assumes that data are from
an approximately normally distributed
population. then
about 68.26% of the values are at
within 1 standard deviation away from
the mean,
95.46% of the values are within two
standard deviations and
99.73% lie within 3 standard
deviations.
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=
n
^ 2P
 (k i  k)
i
i=1
 HT = [- 22 - 17.42 0.1 + - 2 - 17.42 0.2  20 - 17.42 0.4 + 35 - 17.42 0.20
 50  17.42 0.1]0.5  [403]0.5  20.0748599
T-bills = 0.0%.
HT = 20.0%.
Coll = 13.4%.
USR = 18.8%.
M = 15.3%.
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

Standard deviation (i)
measures total, or standalone, risk.
The larger the i , the lower
the probability that actual
returns will be close to the
expected return.
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Expected Returns vs. Risk:
Security
High Tech
Market
US Rubber
T-bills
Collections
Expected
return
17.4%
15.0
13.8*
8.0
1.7*
*Return looks low relative to 
Risk, 
20.0
15.3
18.8*
0.0
13.4*
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Coefficient of variation (CV):
Standardized measure of dispersion
about the expected value:
Std dev

CV =
=
Mean

k
Shows risk per unit of return.
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Portfolio Risk & Return
Assume a two-stock portfolio
with $50,000 in HighTech and
$50,000 in Collections.
Calculate kp and p.
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Portfolio Expected Return, k^p
^
kp is a weighted average:
n
k̂p =  w i k̂ i .
i =1
^
kp = 0.5(17.4%) + 0.5(1.7%) = 9.6%
^
^
^
kp is between kHT and kCOLL.
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Alternative Method:
Estimated Return
Economy
Prob.
HT
Coll.
Port.
Recession
0.10
-22.0%
28.0%
3.0%
Below avg.
0.20
-2.0
14.7
6.4
Average
0.40
20.0
0.0
10.0
Above avg.
0.20
35.0
-10.0
12.5
Boom
0.10
50.0
-20.0
15.0
^
kp = (3.0%)0.1 + (6.4%)0.20 + (10.0%)0.4
+ (12.5%)0.20 + (15.0%)0.1 = 9.6%
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3.0 - 9.6 0.1  6.4  9.6 0.20



2
2
 P =  10.0  9.6 0.4  12.5  9.6 0.20


2
 15.0  9.6 0.1

2
2
1/ 2
= 3.3%
3.3%
CVP =
= 0.34.
9.6%
Note: you can work the variance calculation in either
decimal or percentage
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 p = 3.3% is much lower than that of




either stock (20% and 13.4%).
p = 3.3% is also lower than avg. of HT
and Coll, which is 16.7%.
Portfolio provides avg. return but
lower risk.
Reason: diversification.
Negative correlation is present
between HT and Coll but is not
required to have a diversification
effect
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General Statements about risk:


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Most stocks are positively
correlated. rk,m  0.65.
You still get a lot of
diversification effect at .65
correlation
35% for an average stock.
Combining stocks generally
lowers risk.
What would happen to the
riskiness of a 1-stock
portfolio as more randomly
selected stocks were added?

p would decrease because the
added stocks would not be
perfectly correlated
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p %
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Company Specific risk
Total Risk, P
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Market Risk
0
10
20
30
40
......
1500+
# stocks in portfolio
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

As more stocks are added, each
new stock has a smaller riskreducing impact.
p falls very slowly after about 40
stocks are included. The lower
limit for p is about 20% = M .
Decomposing Risk—Systematic (Market) and
Unsystematic (Business-Specific) Risk


Fundamental truth of the investment world
– The returns on securities tend to move up and
down together
• Not exactly together or proportionately
Events and Conditions Causing Movement in
Returns
– Some things influence all stocks (market risk)
• Political news, inflation, interest rates, war, etc.
– Some things influence only particular firms
(business-specific risk)
• Earnings reports, unexpected death of key
executive, etc.
– Some things affect all companies within an
industry
• A labor dispute, shortage of a raw material
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Total = Market + Firm specific
risk
risk
risk
Market risk is that part of a
security’s risk that cannot be
eliminated by diversification.
Firm-specific risk is that part
of a security’s risk which can
be eliminated with
diversification.
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
By forming portfolios, we
can eliminate nearly half
the riskiness of individual
stocks (35% vs. 20%).
(actually35% vs. 20% is a
43%reduction)
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CAPM -- Capital Asset
Pricing Model
If you chose to hold a onestock portfolio and thus are
exposed to more risk than
diversified investors, would
you be compensated for all
the risk you bear?
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


NO!
Stand-alone risk as
measured by a stock’s 
or CV is not important to
well-diversified investors.
Rational, risk averse
investors are concerned
with p , which is based
on market risk.



Beta measures a stock’s
market risk. It shows a
stock’s volatility relative
to the market.
Beta shows how risky a
stock is when the stock is
held in a well-diversified
portfolio.
The higher beta, the
higher the expected rate
of return.
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How are betas calculated?


Run a regression of past
returns on Stock i versus
returns on the market.
The slope coefficient is
the beta coefficient.
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Illustration of beta = slope:
Regression 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
ki = -2.59 + 1.44 kM
kM
Find beta:
 Statistics program or spreadsheet
regression
 Find someone’s estimate of beta
for a given stock on the web
 Generally use weekly or monthly
returns, with at least a year of data
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If beta = 1.0, average risk stock. (The
‘market’ portfolio has a beta of 1.0.)
If beta > 1.0, stock riskier than
average.
If beta < 1.0, stock less risky than
average.
Most stocks have betas in the range
of 0.5 to 1.5.
Some ag. related companies have
betas less than 0.5
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


=1, get the market expected
return
<1, earn less than the market
expected return
>1, get an expected return
greater than the market
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Can a beta be negative?



Answer: Yes, if the
correlation between ki and kM
is negative.
Then in a “beta graph” the
regression line will slope
downward.
Negative beta -- rare
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ki
HT
b = 1.29
40
b=0
20
T-bills
-20
0
-20
20
40
kM
Coll
b = -0.86
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Security
HighTech
Expected
Return
17.4%
Risk
(Beta)
1.29
“Market”
15.0
1.00
US Rubber
13.8
0.68
T-bills
8.0
0.00
Collections
1.7
-0.86
Riskier securities have higher
returns, so the rank order is O.K.
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Given the beta of a stock, a theoretical
required rate of return can be
calculated.

The Security Market Line (SML) is used.

SML: ki = kRF + (kM - kRF)bi
MRP
MRP= market risk premium
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ki = kRF + (kM - kRF)bi
For Term Projects (2013)
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
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

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Use KRF = 3.0%; this is a bit more than the
current 10 year treasury rate of 2.75%.
Sometimes analysts use a shorter term rate
and short term treasuries are still extremely
low, but we are going to use 3.0%.
Use MRP = 5%. This is MRP, not KM.
The historical average MRP is about 5%.
Find your own beta from the web
On Yahoo Finance look up your company and
then the “key statistics” tab on the left will
give you their beta
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The Bottom Line on Riskfree Rates
Using a long term government rate (even on a coupon bond) as the riskfree
rate on all of the cash flows in a long term analysis will yield a close
approximation of the true value. For short term analysis, it is entirely
appropriate to use a short term government security rate as the riskfree rate.
The riskfree rate that you use in an analysis should be in the same currency
that your cashflows are estimated in.
• In other words, if your cashflows are in U.S. dollars, your riskfree rate has to
be in U.S. dollars as well.
• If your cash flows are in Euros, your riskfree rate should be a Euro riskfree
rate.
The conventional practice of estimating riskfree rates is to use the
government bond rate, with the government being the one that is in
control of issuing that currency. In US dollars, this has translated into
using the US treasury rate as the riskfree rate. In May 2009, for
instance, the ten-year US treasury bond rate was 3.5%.
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Use the SML to calculate
the required returns (for the example)
SML: ki = kRF + (kM - kRF)bi .



Assume kRF = 8%.
^
Note that kM = kM is 15%.
MRP = kM - kRF = 15% - 8% = 7%
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Required rates of return:
kHT
= 8.0% + (15.0% - 8.0%) 1.29
= 8.0% + (7%)1.29
= 8.0% + 9.0%
= 17.0%
kM
kUSR
kTbill
kColl
=
=
=
=
8.0% + (7%)1.00
8.0% + (7%)0.68
8.0% + (7%)0.00
8.0% + (7%)(-0.86)
=
=
=
=
15.0%
12.8%
8.0%
2.0%
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Calculate beta for a portfolio with
50% HT and 50% Collections:
Portfolio Beta
bP =
weighted average of the
betas of the stocks in the portfolio
=
0.5(bHT) + 0.5(bColl)
=
0.5(1.29) + 0.5(-0.86)
=
0.22 .
Weights are the proportions invested
in each stock.
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The required return on the HT/Coll.
portfolio is:
kP
=
=
Weighted average k
0.5(17%) + 0.5(2%)
=
9.5% .
Or use SML for the portfolio:
kP
=
=
=
kRF + (kM - kRF) bP
8.0% + (15.0% - 8.0%) (0.22)
8.0% + 7%(0.22)
=
9.5% .
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Using Beta—The Capital Asset Pricing
Model (CAPM)


The CAPM helps us determine how stock prices
are set in the market
Developed in 1950s and 1960s by Harry
Markowitz and William Sharpe
The CAPM's Approach
People won't invest unless a stock's expected
return is at least equal to their required return
The CAPM attempts to explain how investors'
required returns are determined
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Has the CAPM been verified through
empirical tests?

Not completely. Because
statistical tests have
problems which make
verification almost
impossible.
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
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)b + ?
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
Also, CAPM/SML concepts are
based on expectations, yet
betas are calculated using
historical data. A company’s
historical data may not reflect
investors’ expectations about
future riskiness.