INVESTMENTS
Fourth Edition
Risk
and
Risk Aversion
Chapter 2
INVESTMENTS
Fourth Edition
W
1
= 150 Profit = 50
W
2
= 80 Profit = -20
p = .6
1-p = .4
100
Risky Inv.
Risk Free T-bills Profit = 5
Risk Premium = 17
Risky Investments
with Risk-Free Investment
INVESTMENTS
Fourth Edition
Asset Returns
? Asset returns over a given period are often uncertain.
1
0
11
0
011
?
+
=
?+
=
P
PD
P
PPD
R
P0 is the price at the beginning of period
P1 is the price at the end of the period--Uncertain
D1 is the dividend at the end of the period—Uncertain
So return on a asset is a random variable, characterized by:
? All possible outcomes
? Probability of each outcome
INVESTMENTS
Fourth Edition
Expected rate of return on a investment
1
)(
0
1010
0
0110
0
?
+
=
?+
=
P
PEDE
P
PPDE
RE
Expected rate of return compensates for time-
value and risk
premiumRiskRRE
f
+=)(
0
For example:
%221
100
122
)(
0
0110
0
=?=
?+
=
P
PPDE
RE
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Fourth Edition
Variance on a investment
? Fro example
1176.0
%)22%20(4.0%)22%50(6.0)(
222
=
??×+?×=Rσ
3429.0
%)22%20(4.0%)22%50(6.0)(
22
=
??×+?×=Rσ
INVESTMENTS
Fourth Edition
Measuring expected return and risk--
example
? Moments of return distributions
5
5
5
5
5
20
15
14.5
5
5
5
6
5
-10
-5
-5.5
R0(%)
R1
R2
R3
Asset 0
Asset 1
Asset 2
Asset 3
1/31/31/3probabilityAsset
Mean321state
Between asset 0 and 1,which one would you choose?
Between asset 1 and 2……
Between asset 2 and 3……
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Fourth Edition
These returns have the following
moments
-0.578.165R3
08.165R2
012.255R1
00.005R0(%)
SkewnessSt DMean
{ } xofDStExxESkewness ./][(
3/1
3
?=
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Fourth Edition
? Investor’s view of risk
- Risk Averse
- Risk Neutral
- Risk Seeking
? Utility
? Utility Function
U = E ( R) - .005 A σ
2
A measures the degree of risk aversion
Risk Aversion & Utility
INVESTMENTS
Fourth Edition
To measure risk, we need to know how investors react to
uncertainty—Key assumption of most model
? Higher mean in return is preferred:
? ER
? Higher standard deviation in return is disliked
? Investor care only about mean and st. D
? Investor don’t care about higher moments, such as
skewness
? Under the assumption 1-4. Std give a measure of
risk
? In general, other moments may matter.
2
)( ERRE ?=σ
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Fourth Edition
Risk Aversion and Value:
Using the Sample Investment
U = E ( R ) - .005 A σ
2
= .22 - .005 A (34%)
2
Risk Aversion A Value
High 5 -6.90
34.6
Low 1 16.22
T-bill = 5%
INVESTMENTS
Fourth Edition
Dominance Principle
1
23
4
Expected Return
Variance or Standard Deviation
? 2 dominates 1; has a higher return
? 2 dominates 3; has a lower risk
? 4 dominates 3; has a higher return
INVESTMENTS
Fourth Edition
Utility and Indifference Curves
? Represent an investor’s willingness to trade-
off return and risk
? Example
Exp Ret St Deviation U=E ( R ) - .005Aσ
2
10 20.0 2
15 25.5 2
20 30.0 2
25 33.9 2
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Fourth Edition
Indifference Curves
Expected Return
Standard Deviation
Increasing Utility
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Fourth Edition
Expected Return
Rule 1 : The return for an asset is the
probability weighted average return in all
scenarios.
∑
=
s
sRsRE )()Pr()(
INVESTMENTS
Fourth Edition
Variance of Return
Rule 2: The variance of an asset’s return is the
expected value of the squared deviations
from the expected return.
])()()[Pr(
2
2
∑
?
=
s
REsRs
σ
INVESTMENTS
Fourth Edition
Return on a Portfolio
Rule 3: The rate of return on a portfolio is a weighted average
of the rates of return of each asset comprising the portfolio,
with the portfolio proportions as weights.
R
p
= W
1
R
1
+W
2
R
2
W
1
= Proportion of funds in Security 1
W
2
= Proportion of funds in Security 2
R
1
=( Expected) return on Security 1
R
2
=( Expected) return on Security 2
2211
ERWERWER +=
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Fourth Edition
Portfolio Risk with Risk-Free Asset
Rule 4: When a risky asset is combined with a risk-
free asset, the portfolio standard deviation equals
the risky asset’s standard deviation multiplied by
the portfolio proportion invested in the risky asset.
σσ
riskyassetriskyassetp
w
×=
INVESTMENTS
Fourth Edition
Rule 5: When two risky assets with variances σ
1
2
and σ
2
2
, respectively, are combined into a
portfolio with portfolio weights w
1
and w
2
,
respectively, the portfolio variance is given by
σ
p
2
= w
1
2
σ
1
2
+ w
2
2
σ
2
2
+ 2W
1
W
2
Cov(R
1
R
2
)
Cov(R
1
R
2
) = Covariance of returns for
Security 1 and Security 2
Portfolio Risk