Chapter 6

McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.

Chapter 6Chapter 6Chapter 6Chapter 6

Risk and Risk AversionRisk and Risk Aversion

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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.

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W = 100

W1 = 150 Profit = 50

W2 = 80 Profit = -20

p = .6

1-p = .4

E(W) = pW1 + (1-p)W2 = 6 (150) + .4(80) = 122

2 = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2 =

.6 (150-122)2 + .4(80-122)2 = 1,176,000

Risk - Uncertain Outcomes

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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: 22Exp marginal or incremental profit of the risky portfolio over investing in safe t-bills is 22-5=17 (risk premium)

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W1 = 150 Profit = 50

W2 = 80 Profit = -20

p = .6

1-p = .4100

Risky Inv.

Risk Free T-bills Profit = 5

Risk Premium = 17

Risky Investments with Risk-Free

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Investor’s view of riskRisk Averse

Risk Neutral

Risk Seeking

Risk Aversion & Utility

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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.

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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 2

A measures the degree of risk aversion

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Risk Aversion and Value:

U = E ( r ) - .005 A 2

= .22 - .005 A (34%) 2

Risk Aversion A Value

High 5 -6.90

3 4.66

Low 1 16.22 T-bill = 5%

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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

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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.

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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.

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Dominance Principle

1

2 3

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

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Utility and Indifference Curves

Represent an investor’s willingness to trade-off return and risk.Example

Exp Ret St Deviation U=E ( r ) - .005A2

10 20.0 10-.005x4x400 = 215 25.5 15-.005x4x650 = 220 30.0 20-.005x4x900 = 225 33.9 25-.005x4x1150=2

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Indifference Curves

Expected Return

Standard Deviation

Increasing Utility

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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 YearBullish BearishStock Mrk Stock Mrk Sugar Crisis

Probability 0.5 0.3 0.2Rate of Return 25% 10% -25%

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Expected Return

Rule 1 : The return for an asset is the probability weighted average return in all scenarios.

s

srsPrE )()()(

%5.10)25(2.)103(.)255(.)( bastrE

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Variance of Return

Rule 2: The variance of an asset’s return is the expected value of the squared deviations from the expected return.

])()()[( 22

srEsrsP

2222 )5.1025(2.)5.1010(3.)5.1025(5. Best

=357

%9.18357

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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

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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%

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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.

σp= .5 σBest =.5 x 18.9 = 9.45%

riskyassetriskyassetp w

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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 YearBullish BearishStock Mrk Stock Mrk Sugar Crisis

Probability 0.5 0.3 0.2Rate of Return 1% -5% 35%

E(rcane)= 6%σcane= 14.73%

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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.

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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%

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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

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Rule 5: When two risky assets with variances 12

and 22, respectively, are combined into a

portfolio with portfolio weights w1 and w2, respectively, the portfolio variance is given by:

p2

= w121

2 + w222

2 + 2W1W2 Cov(r1r2)

Cov(r1r2) = Covariance of returns for Security 1 and Security 2

Portfolio Risk

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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)

= -240.5Sugarcane’s returns move inversely with Best’s.

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Correlation coefficient: scales the covariance between to a value between -1 (perfectly negative correlation) and +1(perfectly positive correlation).

Risk of portfolio consists of two assets.

σp2= (0.54)2(-18.5)2+ (.50)2 (-14.73)2+ 2.05 x .5 x (-240.5)= 23.3

σp= 4.83%

86.73.149.18

5.240,,

SB

SBSB

Cov

),(2 212122

22

21

21

2 rrCovwwwwp ),(2 212122

22

21

21

2 rrCovwwwwp ),(2 212122

22

21

21

2 rrCovwwwwp

),(2 212122

22

21

21

2 rrCovwwwwp

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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?

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MIDTERM QUESTION 2

Calculate the correlation coefficients and covariances between each of your stocks in your portfolio.