LO1
Expected Returns 13.1
• Expected returns are based on the
probabilities of possible outcomes
• In this context, “expected” means average
if the process is repeated many times
• The “expected” return does not even have
to be a possible return
n
E ( R)   pi Ri
i 1
© 2013 McGraw-Hill Ryerson Limited
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LO1
Expected Returns – Example 1
• Suppose you have predicted the following
returns for stocks C and T in three possible
states of nature. What are the expected returns?
•
•
•
•
State
Boom
Normal
Recession
Probability
0.3
0.5
???
C
0.15
0.10
0.02
T
0.25
0.20
0.01
• RC = .3(.15) + .5(.10) + .2(.02) = .099 = 9.9%
• RT = .3(.25) + .5(.20) + .2(.01) = .177 = 17.7%
© 2013 McGraw-Hill Ryerson Limited
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LO1
Expected Returns – Example 1
continued
• This example can also be done in a
spreadsheet
• Click on the Excel link to see this
© 2013 McGraw-Hill Ryerson Limited
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LO1
Variance and Standard
Deviation
• Variance and standard deviation still
measure the volatility of returns
• You can use unequal probabilities for the
entire range of possibilities
• Weighted average of squared deviations
n
σ 2   pi ( Ri  E ( R)) 2
i 1
© 2013 McGraw-Hill Ryerson Limited
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LO1
Variance and Standard Deviation – Example 1
• Consider the previous example. What is
the variance and standard deviation for
each stock?
• Stock C
• 2 = .3(.15-.099)2 + .5(.1-.099)2 + .2(.02.099)2 = .002029
•  = .045
• Stock T
• 2 = .3(.25-.177)2 + .5(.2-.177)2 + .2(.01.177)2 = .007441
•  = .0863
© 2013 McGraw-Hill Ryerson Limited
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LO1
Variance and Standard Deviation –
Example continued
• This can also be done in a spreadsheet
• Click on the Excel icon to see this
© 2013 McGraw-Hill Ryerson Limited
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Quick Quiz I
LO1
• Consider the following information:
•
•
•
•
•
State
Boom
Normal
Slowdown
Recession
Probability
.25
.50
.15
.10
ABC, Inc.
.15
.08
.04
-.03
• What is the expected return?
• What is the variance?
• What is the standard deviation?
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LO1
Portfolios 13.2
• A portfolio is a collection of assets
• An asset’s risk and return is important in
how it affects the risk and return of the
portfolio
• The risk-return trade-off for a portfolio is
measured by the portfolio expected return
and standard deviation, just as with
individual assets
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Example: Portfolio Weights
LO1
• Suppose you have $15,000 to invest and
you have purchased securities in the
following amounts. What are your portfolio
weights in each security?
•
•
•
•
$2000 of ABC
$3000 of DEF
$4000 of GHI
$6000 of JKL
•ABC: 2/15 = .133
•DEF: 3/15 = .2
•GHI: 4/15 = .267
•JKL: 6/15 = .4
© 2013 McGraw-Hill Ryerson Limited
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LO1
Portfolio Expected Returns
• The expected return of a portfolio is the weighted
average of the expected returns for each asset in
the portfolio
m
E ( RP )   w j E ( R j )
j 1
• You can also find the expected return by finding
the portfolio return in each possible state and
computing the expected value as we did with
individual securities
© 2013 McGraw-Hill Ryerson Limited
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LO1
Example: Expected Portfolio
Returns
• Consider the portfolio weights computed
previously. If the individual stocks have the
following expected returns, what is the expected
return for the portfolio?
•
•
•
•
ABC: 19.65%
DEF: 8.96%
GHI: 9.67%
JKL: 8.13%
• E(RP) = .133(19.65) + .2(8.96) + .267(9.67) +
.4(8.13) = 10.24%
• Click the Excel icon for an example
© 2013 McGraw-Hill Ryerson Limited
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LO1
Portfolio Variance
• Compute the portfolio return for each state:
RP = w1R1 + w2R2 + … + wmRm
• Compute the expected portfolio return
using the same formula as for an individual
asset
• Compute the portfolio variance and
standard deviation using the same
formulas as for an individual asset
© 2013 McGraw-Hill Ryerson Limited
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LO1
Example: Portfolio Variance
• Consider the following information
•
•
•
•
Invest 60% of your money in Asset A
State Probability A
B
Boom .5
70%
10%
Bust .5
-20%
30%
• What is the expected return and standard
deviation for each asset?
• What is the expected return and standard
deviation for the portfolio?
© 2013 McGraw-Hill Ryerson Limited
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LO1
Portfolio Variance Example
continued
• This can also be done in a spreadsheet
• Click on the Excel icon to see this
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LO1
Another Way to Calculate
Portfolio Variance
• Portfolio variance can also be calculated using
the following formula:
 P2  xL2 L2  xU2  U2  2 xL xU CORR L ,U  L U
Assuming that the correlatio n between A and B is - 1.00, we have
 P2  0.6 2  0.2025  0.4 2  0.01  2  0.6  0.4  -1.00 0.45  0.1
 P2  0.0529
© 2013 McGraw-Hill Ryerson Limited
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Quick Quiz II
LO1
• Consider the following information
•
•
•
•
State
Boom
Normal
Recession
Probability
.25
.60
.15
X
15%
10%
5%
Z
10%
9%
10%
• What is the expected return and standard
deviation for a portfolio with an investment
of $6,000 in asset X and $4,000 in asset
Z?
© 2013 McGraw-Hill Ryerson Limited
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LO1
Arbitrage Pricing Theory (APT) 13.8
• Similar to the CAPM, the APT can handle
multiple factors that the CAPM ignores
• Unexpected return is related to several market
factors
E( R)  RF  E( R1  RF )  1  E( R2  RF )  2  ...  E( RK  RF )   K
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