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Risk and Risk Aversion
Chapter 6
McGraw-Hill/Irwin
Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.
Risk - Uncertain Outcomes
W1 = 150 Profit = 50
W = 100
1-p = .4
W2 = 80 Profit = -20
E(W) = pW1 + (1-p)W2 = 6 (150) + .4(80) = 122
s2 = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2 =
0.6 (150-122)2 + 0.4(80-122)2 = 1,176,000
s = 34.293
6-2
Risky Investments with Risk-Free
W1 = 150 Profit = 50
Risky Inv.
100
1-p = .4
Risk Free T-bills
W2 = 80 Profit = -20
Profit = 5
Expected profit on risky is: 0.6*50 + 0.4*(-20) = 22
Risk Premium = 22 – 5 = 17 (on a 100 investment)
6-3
Risk Aversion & Utility
Investor’s view of risk
Risk Averse – must be paid to take a fair bet
Risk Neutral – is indifferent to a fair bet
Risk Seeking – will pay you to take a fair bet
Utility – a measure of gain from taking an
investment (rather than measuring $)
Utility Function (many possible)
U = E ( r ) - 0.005 A s 2
A = measure of the degree of risk Aversion
0.005 is a scaling factor to make the range of A more
pleasing when expressing the inputs as percent
rather than decimals
6-4
Certainty Equivalent
A Certainty Equivalent is the amount
you would take for sure, in place of a
risky bet.
It is simply the value of the utility, for the
function noted above
If the certainty equivalent is lower than
the risk free rate, the investor should
choose the risk free asset
A risk lover would have a negative value
for A in the Utility formula
6-5
Risk Aversion and Value:
U = E (r) ― 0.005 A s 2
= 22 ― .005 A (34) 2
Risk Aversion A
Value
High
5
-6.90
3
4.66
Low
1
16.22
T-bill = 5%
For T-bill: U = E(r) = 5
6-6
Mean Variance Dominance
D dominates C if:
E(rD) ≥ E(rC)
and σD ≤ σC
And at least one inequality is strict
(i.e. two equal things cannot
dominate one another)
6-7
Dominance Principle
Expected Return
4
2
3
1
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
6-8
Utility and Indifference Curves
Represent an investor’s willingness
to trade-off return and risk.
Example (for A=4)
Exp Ret St Deviation U=E ( r ) - .005As2
10
20.0
2
15
25.5
2
20
30.0
2
25
33.9
2
6-9
Indifference Curves
Expected Return
Note: Where the indifference
curve hits the y axis is the
certain equivalent
Increasing Utility
Standard Deviation
6-10
Expected Return
Rule 1 : The return for an asset is the
probability weighted average return in
all scenarios.
E (r ) =  P( s )r ( s )
s
6-11
Variance of Return
Rule 2: The variance of an asset’s return
is the expected value of the squared
deviations from the expected return.
P
(
s
)[
r
(
s
)

E
(
r
)
]

=
s
s
2
2
6-12
Return on a Portfolio
Rule 3: The expected rate of return on a portfolio
is a weighted average of the expected rates of
return of each asset comprising the portfolio,
with the portfolio proportions as weights.
Example: A two asset portfolio
E(rp ) = w1E(r1) + w2E(r2)
w1 = Proportion of funds in Security 1
w2 = Proportion of funds in Security 2
E(r1) = Expected return on Security 1
E(r2) = Expected return on Security 2
Note: w1 + w2 = 1
6-13
Portfolio Risk with Risk-Free Asset
Rule 4: When a risky asset is combined with a
risk-free asset, Rule 5 shows that the portfolio
standard deviation equals the risky asset’s
standard deviation multiplied by the portfolio
proportion invested in the risky asset.
s p = wriskyasset  s riskyasset
6-14
Covariance and Correlation
Covariance measures the tendency for two
random variable to move together
For example, it makes sense that the returns
on Ford and the returns on GM will usually
move together (Why?)
Calculated like a variance, but for two assets
cov( x, y) = s xy =  P(s)[ x  x ][ y  y ]
6-15
Covariance and Correlation
Covariance can be positive or negative and
from plus to minus infinity, so often one
chooses to standardize it to the -1 though 1
interval using a measure called covariance
Corr ( x, y) =  xy =
Cov( x, y)
s xs y
6-16
Portfolio Risk
Rule 5: When two risky assets with variances s12
and s22, respectively, are combined into a
portfolio with portfolio weights w1 and w2,
respectively, the portfolio variance is given by:
sp2 = w12s12 + w22s22 + 2w1ww2 Cov(r1r2)
Cov(r1r2) = Covariance of returns for
Security 1 and Security 2
6-17