Download Expected Return Standard Deviation Increasing Utility

Risk and Risk Aversion
Chapter 6
McGraw-Hill/Irwin
Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.
Suppose that your initial wealth is $100.
Assume two possible outcome exists,
final wealth is is $150 with probability of
0.60. Otherwise it is $80 with probability
of 0.40 a less favorable outcome.
6-2
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 =
.6 (150-122)2 + .4(80-122)2 = 1,176,000
s = 34.293
6-3
This is risky because std dev (34.293)  22
(122-100)
T-bills are one alternative to the risky
portfolio, 1 year t-bill offers a rate of return
5%. There is a certain $5 profit.
Exp profit: 22
Exp marginal or incremental profit of the risky
portfolio over investing in safe t-bills is
22-5=17 (risk premium)
6-4
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
Risk Premium = 17
6-5
Risk Aversion & Utility
Investor’s view of risk
Risk Averse
Risk Neutral
Risk Seeking
6-6
Risk averse investors are willing to
consider only risk-free assets or
speculative prospects with only positive
risk premiums.
Risk neutral investors judge risky
prospects just by their expected returns.
6-7
Utility Function
Investor’s preferences toward the exp. return and risk may be
expressed by the utility function that is higher for higher exp.
returns and lower for higher risks.
More risk-averse investors will apply greater penalties for risk.
The greater the risk, the larger the penalty.
We can formalize the risk-penalty system. We will assume that
each investor can assign a utility score. Portfolios receive higher
utility scores for higher exp. returns and lower score for higher
risk (volatility).
Utility Function
U = E ( r ) - .005 A s 2
A measures the degree of risk aversion
6-8
Risk Aversion and Value:
U = E ( r ) - .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%
6-9
Problem 1
A portfolio has an expected rate of
return of 20% and standard deviation of
20%. Bills offer a sure rate of return of
7%. Which investment alternative will be
chosen by an investor A = 4? What if
A=8
6-10
Solution 1
For A=4
Utility of Risky portfolio;
U = 20 – (.005x4x202)=12%
Utility for bills;
U = 7 – (.005x4x0) = 7%
The investors will prefer the risky portfolio
to bills.
6-11
For A=8
Utility of Risky portfolio;
U = 20 – (.005x8x202)=4%
Utility for bills;
U = 7 – (.005x8x0) = 7%
The more risk averse investor prefers the
risk free alternative.
6-12
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
6-13
Utility and Indifference Curves
Represent an investor’s willingness to
trade-off return and risk.
Example
Exp Ret
10
15
20
25
St Deviation U=E ( r ) - .005As2
20.0
10-.005x4x400 = 2
25.5
15-.005x4x650 = 2
30.0
20-.005x4x900 = 2
33.9
25-.005x4x1150=2
6-14
Indifference Curves
Expected Return
Increasing Utility
Standard Deviation
6-15
A review of Portfolio Mathematics
Example: Suppose that the value of the Best Candy
stock is sensitive to the price of sugar. In years when
world sugar crops are low, the price of sugar
increases and Best Candy suffers considerable
losses.
Normal Year for Sugar
Abnormal Year
Bullish
Bearish
Stock Mrk Stock Mrk
Sugar Crisis
Probability
0.5
0.3
0.2
Rate of Return 25%
10%
-25%
6-16
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
E(rbast ) = (.5  25)  (.3 10)  .2(25) = 10.5%
6-17
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
=357
2
2
s Best
= .5 (25  10.5) 2  .3 (10  10.5) 2  .2 (25  10.5) 2
s = 357 = 18.9%
6-18
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
6-19
Suppose you are going to invest 50% of
your money into Best Candy stock and
remainder in T-bills, which yield 5% rate
of return.
E(rp)= .5 E(rbest) + .5 rbills
= .5 x 10.5 + .5 x 5= 7.75%
6-20
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.
s p = wriskyasset  s riskyasset
σp= .5 σBest =.5 x 18.9 = 9.45%
6-21
Our portfolio analyst discovers that during years of sugar
shortage, Sugarcane Co reaps unusual profits and its stock
price increases (soars). A scenario analysis of Sugarcane’s
stock;
Normal Year for Sugar Abnormal Year
Bullish
Bearish
Stock Mrk Stock Mrk Sugar Crisis
Probability
0.5
0.3
0.2
Rate of Return
1%
-5%
35%
E(rcane)= 6%
σcane= 14.73%
6-22
Sugarcane offers excellent hedging potential
for holders of Best because its stock return is
highest when Best’s return is lowest-during a
sugar crisis.
Suppose you split your investment between
Best and Sugarcane. The rate of return for
each scenario is the average rate of return on
Best and Sugarcane.
6-23
Normal Year for Sugar Abnormal Year
Bullish
Bearish
Sugar
Stock Mrk Stock Mrk Crisis
Probability 0.5
0.3
0.2
ROR
13%
2,5%
5%
E(rhedged)= 8.35%
σhedged= 4.83%
6-24
3 Alternatives
Portfolio:
E(r)
σ
1. All in Best Candy 10.5% 18.50%
2. Half in T-bills
7.75
9.45
3. Half in Sugarcane
8.25
4.83
Third alternative has higher expected
return and lower risk than the second
one
6-25
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
6-26
Covariance measures how much the return on two
risky assets move in tandem. A positive covariance
means that assets move together. A negative
covariance means that they vary inversely.
E(rbest)= 10.5% E(rcane)=6%
Cov(rbase, rcane) = .5 (25-10.5) (1-6) + .3 (10-10.5)
(-5-6) +.2 (-25-10.5) (35-6)
= -240.5
Sugarcane’s returns move inversely with Best’s.
6-27
s p2 = w12s 12  w22s 22  2w1 w2Cov(r1 , r2 )
Correlation coefficient: scales the covariance between to a value
between -1 (perfectly negative correlation) and +1(perfectly
positive correlation).
 B,S =
Cov B , S
s Bs S
=
 240.5
= .86
18.9  14.73
Risk of portfolio consists of two assets.
s p2 = w12s 12  w22s 22  2w1 w2Cov(r1 , r2 )
σp2= (0.54)2(-18.5)2+ (.50)2 (-14.73)2+ 2.05 x .5 x (-240.5)=
23.3
σp= 4.83%
6-28
Problem
Suppose that the distribution of the
Sugurcane stock were as follows;
Bullish
Bearish
Sugar
Stock Mrk Stock Mrk Crisis
Probability 0.5
0.3
0.2
ROR
7%
-5%
20%
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A. What would be its correlation with Best?
B. Is Sugarcane stock a useful hedge asset
now?
C. Calculate the portfolio rate of return for each
scenario and the standard deviation of the
portfolio from the scenario returns. The
evaluate standard deviation using Rule 5.
D. Are the two methods of computing portfolio
standard deviation consistent?
6-30
MIDTERM QUESTION 2
Calculate the correlation coefficients
and covariances between each of your
stocks in your portfolio.
6-31