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Chapter 8
Risk and Return
2nd .
4 Enter
CE/C (4 decimal places)
The rate of return on an investment is calculated
Return = Amount received - Amount invested
Amount invested
If $1000 were invested and $1100 was received from the
investment one year later
Return = 1100 – 1000 / 1000
= 10%
Risk versus Return

The quantification of risk and return is a crucial aspect of modern finance. The
return one expects than the more risk one must assume.
Expected return - weighted average of the distribution of possible returns in
the future.
Variance of returns - a measure of the dispersion of the distribution of possible
returns in the future.
State of the
economy
Probability
of state
Boom
Bust

A.
0.40
0.60
1.00
Return on Return on
asset A
asset B
30%
-10%
-5%
25%
Expected returns
^
rA =
0.40 x (30) + 0.60 x (-10) = 6 = 6%
^
rB =
0.40 x (-5) + 0.60 x (25) = 13 = 13%
^
ri = ∑ Pi ri
1

B.
Variances
Var(rA) = σA2 =0.40 x (30 - 6)2 + 0.60 x (-10 - 6)2 = 384
Var(rB) = σB2 =0.40 x (-5 - 13)2 + 0.60 x (25 -13)2 = 216
Var(ri) =

C.
σ i2
^
= ∑ (ri – r)2Pi
Standard deviations
SD(rA) = σA = (384)1/2 = 19.6 = 19.6%
SD(rB) = σB = (216)1/2 = 14.7 = 14.7%
1.
Expected return
A stock’s expected return has the following distribution:
States of
PROBABILITY OF State
RATE OF RETURN
the Economy
Weak
0.1
(50%)
Below average
0.2
(5)
Average
0.4
16
Above average
0.2
25
Strong
0.1
60
1.0
Calculate the stock's expected return, standard deviation,
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Portfolio Expected Returns


Portfolio weights: put 50% in Asset A and 50% in Asset B:
State of the
Probability
economy of state
Return
on A
Return
on B
Return on
portfolio
Boom
0.40
30%
-5%
12.5%
Bust
0.60
1.00
-10%
25%
7.5%
A.
^
rP =
0.40 x (12.5) + 0.60 x (7.5) = 9.5 = 9.5%
^
^
rP = ∑ wi ri



B.
C.
wi = % invested in security
Var(rP)= 0.40 x (12.5 – 9.5)2 + 0.60 x (7.5 – 9.5)2 = 6
SD(rP) = σp = (6) 1/2 = 2.45 = 2.45%
^
^
^
rP
=
.50 x rA + .50 x rB = 9.5%
Var (rP)


BUT:
.50 x Var(rA) + .50 x Var(rB)

Consider the following information:
State of Prob. of State Stock A
Economy of Economy Return
Stock B
Return
Stock C
Return
Boom
Bust
18%
2%
26%
- 2%
0.65
0.35
14%
8%

What is the expected return on an equally weighted portfolio of these three
stocks?

Expected returns on the equally-weighted portfolio
^
boom: rp = (14 + 18 + 26)/3 = 19.33%
^
bust: rp = (8 + 2 + -2)/3 = 2.67%
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so the overall portfolio expected return must be
^
rp = .65(19.33) + .35(2.67) = 13.5%

What is the variance of a portfolio invested 25 percent in A, 25 percent in B,
and 50 percent in C?

Variance of portfolio returns
boom: rp
=
.25(14) + .25(18) + .50(26) = 21%
bust: rp
=
.25(8) + .25(2) + .50(-2) = 1.5%
rp
=
.65(21) + .35(15) = 14.175%
So
σ 2p =

.65(21 – 14.175)2 + .35(15 – 14.175)2 = 30.515
Total Stand Alone Risk = σi2= Market Risk + Firm Specific Risk
Market Risk – Risk of Security that cannot be diversified away – Measures by
beta. Also called Systematic Risk
Firm Specific Risk – Portion of Security’s risk that can be diversified away. Also
called unsystematic risk
Standard Deviations of Annual Portfolio Returns
( 3)
(2)
Ratio of Portfolio
Average Standard
Standard Deviation to
# of Stocks Deviation of Annual
Standard Deviation
in Portfolio Portfolio Returns
of a Single Stock
1
10
50
100
300
500
1,000
49.24%
23.93
20.20
19.69
19.34
19.27
19.21
1.00
0.49
0.41
0.40
0.39
0.39
0.39
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Beta = measure degree to which security’ s returns move with the market –
This risk cannot be diversified away.
Betamarket = 1.0 Beta for security < 1.0 it is less volatile than the market
Beta for security > 1.0 it is more volatile than the market
Company
Coefficients (Betai)
Exxon
IBM
Wal-Mart
General Motors
Microsoft
IBM
Harley-Davidson
0.80
0.95
1.10
1.05
1.10
1.15
1.65
Required Returns for individual securities and portfolios – measured
with Security Market Line
Security Market Line (SML): r i= rrf + (rm - rrf) bi
The SML is called the Capital Asset Pricing Model (CAPM)
2. Over the last 7 decades, the historic market risk premium on large firm
common stocks has been about 9% (Market Return of 14% less a Risk
Free rate of 5%). Assume the risk-free rate is 5%. GTX Corp. has a beta
of .85. What return should you require from an investment in GTX?
5% + (9%  .85) = 12.65%
5% + [(14% - 5%) .85] = 12.65%
rGTX =
rGTX =
3.
Expected & required
rates of return
Assume that the risk-free rate is 5 percent and the market risk premium is 6
percent. What is the
expected return for the overall stock market? What is the required rate of return
on a stock that has
a beta of 1.2?
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4.
Assume that the risk-free rate is 6 percent and the expected return on the market
is 13 percent.
Required rate
What is the required rate of return on a stock that has a beta of 0.7?
rate of return
5.
Beta & required
rate of return
A stock has a required return of 11 percent. The risk- free rate is 7 percent, and
the market risk
premium is 4 percent.
a. What is the stock's beta?
b. If the market risk premium increases to 6 percent, what will happen to the
stock's required rate of return? Assume the risk-free rate and the stock's beta
remain unchanged.
Portfolio Beta
Stock
(1)
Invested
(2)
Weights
(3)
Haskell Mfg. $ 6,000
Cleaver, Inc.
4,000
Rutherford Co. 2,000
Portfolio
$12,000
50%
33%
17%
100%
Beta
(4)
0.90
1.10
1.30
(3) x (4)
0.450
0.367
0.217
1.034
bp = ∑ wi bi
6.
Portfolio beta
7.
Portfolio required
return
An individual has $35,000 invested in a stock that has a beta of 0.8 and
$40,000 invested in a
stock with a beta of 1.4. If these are the only two investments in her portfolio,
what is her port- folio's beta?
Suppose you are the money manager of a $4 million investment fund. The fund
consists of 4
stocks with the following investments and betas:
STOCK
A
B
C
D
INVESTMENT
$ 400,000
600,000
1,000,000
2,000,000
BETA
1.50
(0.50)
1.25
0.75
If the market’s required return is 14% and the risk free rate is 6%, what is the fund’s required return?
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