# Calculate Expected Rate of Return on Stockholders' by yek12436

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CHAPTER 8
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                    2

stockholder perspective
   Management cares about risk because
   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
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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)

ˆ
k  p1k1  p2 k 2  ... pn k n
n
  pi k i
i 1
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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

1.0
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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.
10
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.
11

Calculate the expected rate of
return for each alternative:
^ = expected rate of return
k

n

k =  k i pi
^

i=1

^ = (-22%)0.1 + (-2%)0.20
kHT
+ (20%)0.40 +
(35%)0.20
+ (50%)0.1 = 17.4%
12

^
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     =    
2

n
=  (ki  k i
ˆ )2 p
i =1
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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.
16

n
=               ^2
 (k i  k) Pi
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 stand-
alone, 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           Expected       Risk, 
return
High Tech           17.4%          20.0
Market               15.0          15.3
US Rubber            13.8*         18.8*
T-bills               8.0           0.0
Collections           1.7*         13.4*

*Return looks low relative to 
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Coefficient of variation (CV):

Standardized measure of dispersion

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.
21

^
Portfolio Return,kp
^
kp is a weighted average:

n
ˆ          ˆ
kp =  w i k 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
1/ 2
2                 2

                                      
 P =  10.0  9.6 0.4  12.5  9.6 0.20
2                  2

                                      
 15.0  9.6 0.1
2
                                      


= 3.3%

3.3%
CVP =      = 0.34.
9.6%
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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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   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.
26
What would happen to the
riskiness of a 1-stock
portfolio as more randomly

   p would decrease because the
perfectly correlated
27

p %
35    Company Specific risk

Total Risk, P
20

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 risk-
reducing impact.
   p falls very slowly after about 40
stocks are included. The lower
limit for p is about 20% = M .
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Decomposing Risk—Systematic (Market) and

   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
• 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.
31

   By forming portfolios, we
can eliminate nearly half
the riskiness of individual
stocks (35% vs. 20%).
   (actually35% vs. 20% is a
43%reduction)
32

CAPM -- Capital Asset
Pricing Model

If you chose to hold a one-
stock portfolio and thus are
exposed to more risk than
diversified investors, would
you be compensated for all
the risk you bear?
33

   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       34

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.
35

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.
36
Illustration of beta = slope:

Regression line

ki    20                      .
15                  .           Year
1
kM
15%
ki
18%
2     -5    -10
10                               3     12    16
5

-5     0      5      10       15      20           kM
-5
ki = -2.59 + 1.44 kM
.     -10
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Find beta:

   “By Eye.” Plot points, draw in
„regression‟ line, get slope as
b = Rise/Run. The “rise” is the
difference in ki , the “run” is the
difference in kM . For example,
how much does ki increase or
decrease when kM increases
from 0% to 10%?
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 Computer: statistics program or
 Calculator. Enter data points, and
calculator does least squares
regression: ki = a + bkM = -2.59 +
1.44kM . r = corr. coefficient = 0.997.
 Find someone‟s estimate of beta
for a given stock on the web
 In the real world, we would use
weekly or monthly returns, with at
least a year of data, and would
always use a computer.
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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
40

   =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?
correlation between ki and kM
is negative.
   Then in a “beta graph” the
regression line will slope
downward.
   Negative beta -- rare
42

ki
b = 1.29
HT
40

b=0
20
T-bills

-20    0    20              40
kM

Coll
-20             b = -0.86
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Expected           Risk
Security           Return           (Beta)
HighTech           17.4%             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

45

ki = kRF + (kM - kRF)bi
46

For Term Projects

   Use KRF = 2.5%; this is the 10 year treasury
rate. Often it is argued to use a shorter
term rate, but we are going to use 2.5%.
   Use MRP = 5%
   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
47

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%
48

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       =   8.0% + (7%)1.00      =   15.0%
kUSR     =   8.0% + (7%)0.68      =   12.8%
kTbill   =   8.0% + (7%)0.00      =   8.0%
kColl    =   8.0% + (7%)(-0.86)   =   2.0%
49

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
52

Has the CAPM been verified through
empirical tests?

   Not completely. Because
statistical tests have
problems which make
verification almost
impossible.
53

   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 + ?
54

   Also, CAPM/SML concepts are
based on expectations, yet
betas are calculated using
historical data. A company‟s
historical data may not reflect