Risk and Risk Aversion
BKM Ch 6
Zvi Wiener
tel: 02-588-3049
Fall-02
[email protected]
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
Investments
Risk - Uncertain Outcomes
W1 = 150 Profit = 50
W = 100
1-p = 40%
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 =
.6 (150-122)2 + .4(80=122)2 = 1,176,000
s = 34.293
Zvi Wiener
BKM Ch 6
slide 2
Risky Investments
with Risk-Free Investment
W1 = 150 Profit = 50
Risky Inv.
100
1-p = .4
Risk Free T-bills
W2 = 80 Profit = -20
Profit = 5
Risk Premium = 17
Zvi Wiener
BKM Ch 6
slide 3
Risk Aversion & Utility
Investor’s view of risk
Risk Averse
Risk Neutral
Risk Seeking
Utility
Utility Function
U = E ( r ) - .005 A s 2
A measures the degree of risk aversion
Zvi Wiener
BKM Ch 6
slide 4
Risk Aversion and Value:
Using the Sample Investment
U = E ( r ) - .005 A s 2
=
.22 - .005 A (34%) 2
Risk Aversion
High
Low
Zvi Wiener
A
Value
5
-6.90
3
4.66
1
16.22
BKM Ch 6
T-bill = 5%
slide 5
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
Zvi Wiener
BKM Ch 6
slide 6
Utility and Indifference Curves
Represent an investor’s willingness to tradeoff return and risk.
Example
Exp Ret
10
15
20
25
Zvi Wiener
St Deviation U=E ( r ) - .005As2
20.0
2
25.5
2
30.0
2
33.9
2
BKM Ch 6
slide 7
Indifference Curves
Expected Return
Increasing Utility
Standard Deviation
Zvi Wiener
BKM Ch 6
slide 8
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
Zvi Wiener
BKM Ch 6
slide 9
Variance of Return
Rule 2: The variance of an asset’s return is the
expected value of the squared deviations
from the expected return.
s = s P(s)[ r (s)  E (r )]
2
2
Zvi Wiener
BKM Ch 6
slide 10
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.
rp = W1r1 + W2r2
W1 = Proportion of funds in Security 1
W2 = Proportion of funds in Security 2
r1 = Expected return on Security 1
r2 = Expected return on Security 2
Zvi Wiener
BKM Ch 6
slide 11
Portfolio Risk with Risk-Free
Asset
Rule 4: When a risky asset is combined with a riskfree asset, 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
Zvi Wiener
BKM Ch 6
slide 12
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 + 2W1W2 Cov(r1r2)
Cov(r1r2) = Covariance of returns for
Security 1 and Security 2
Zvi Wiener
BKM Ch 6
slide 13
Home Assignment
Required:
• problems 1, 2, 3, 4, (3rd ed).
• problems 1, 2, 3, 4, (5th ed).
• Appendix b:1 – submit !
• closely follow financial news!
Zvi Wiener
BKM Ch 6
slide 14