11-1

Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

11-2

Expected Returns

• Expected returns are based on the probabilities of possible states of the economy

– To simplify, assume states of the economy:• Boom• Recession

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11-3

Expected Returns

Expected Return on Stock A

E(RA) = ∑ (Probability for state of economy X expected

return for state of economy)

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11-4

Expected risk premium

Expected risk premium =

Expected return on risky investment

- Return on risk free investment

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11-5

Variance and Standard Deviation

• Variance and standard deviation measure the volatility of returns

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11-6

Variance and Standard Deviation

Variance = Weighted average of squared deviations

Ơ2 = ∑ [(Return for state of economy –

Expected return for stock)2 x Probability for state of economy]

Standard deviation = square root of variance

11-7

Variance and Standard Deviation

Variance and Standard deviation calculated differently than Chapter 10

Chapter 10: historical / actual events

Chapter 11: projections

11-8

Portfolios

• Portfolio = collection of assets

• Stock weights in a portfolio:

stock value divided by total portfolio value

(% of portfolio invested in each asset)

• The risk-return trade-off for a portfolio is measured by:

• portfolio expected return • standard deviation

11-9

Portfolio Expected Returns

• The expected return of a portfolio is the weighted average of the expected returns for each asset in the portfolio

∑ (Portfolio weight X expected return for individual stock)

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11-10

Portfolio RiskVariance & Standard Deviation

• Portfolio standard deviation is NOT a weighted average of the standard deviation of the component securities’ risk– If it were, there would be no benefit to

diversification.

11-11

Portfolio Variance• Compute portfolio return for each state of

economy:∑ (portfolio weight x expected return for individual stock for that state of economy)

• Compute the overall expected portfolio ∑ (portfolio weight X expected return for individual stock)

• Compute the portfolio variance and standard deviation

Ơ2 = ∑ (Portfolio Return for state of economy – Expected return for portfolio)2 x Probability for state of economy

Standard deviation = square root of variance

11-12

Announcements, News and Efficient markets

• Announcements and news contain both expected and surprise components

• The surprise component affects stock prices • Efficient markets result from investors trading

on unexpected news– The easier it is to trade on surprises, the more

efficient markets should be

• Efficient markets involve random price changes because we cannot predict surprises

11-13

Returns

• Total Return = Expected return + unexpected return

R = E(R) + U• Unexpected return (U) = Systematic

portion (m) + Unsystematic portion (ε)

• Total Return = Expected return E(R) + Systematic portion m

+ Unsystematic portion ε= E(R) + m + ε

11-14

Systematic Risk

• Factors that affect a large number of assets

• “Non-diversifiable risk”

• “Market risk”

• Examples: changes in GDP, inflation, interest rates, etc.

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11-15

Unsystematic Risk

• = Diversifiable risk• Risk factors that affect a limited number of

assets• Risk that can be eliminated by combining

assets into portfolios• “Unique risk”• “Asset-specific risk”• Examples: labor strikes, part shortages,

etc.Return to Quick Quiz

11-16

The Principle of Diversification

• Diversification can substantially reduce risk without an equivalent reduction in expected returns– Reduces the variability of returns– Caused by the offset of worse-than-

expected returns from one asset by better-than-expected returns from another

• Minimum level of risk that cannot be diversified away = systematic portion

11-17

Standard Deviations of Annual Portfolio ReturnsTable 11.7

11-18

Portfolio Conclusions

• As more stocks are added, each new stock has a smaller risk-reducing impact on the portfolio

p falls very slowly after about 40 stocks are included – The lower limit for p ≈ 20% = M.

Forming well-diversified portfolios can eliminate about half the risk of owning a single stock.

11-19

Total Risk = Stand-alone Risk

Total risk = Systematic risk + Unsystematic risk– The standard deviation of returns is a measure

of total risk

• For well-diversified portfolios, unsystematic risk is very small

Total risk for a diversified portfolio is essentially equivalent to the systematic risk

11-20

Systematic Risk Principle

• There is a reward for bearing risk• There is no reward for bearing risk

unnecessarily

• The expected return (market required return) on an asset depends only on that asset’s systematic or market risk.

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11-21

Market Risk for Individual Securities

• The contribution of a security to the overall riskiness of a portfolio

• Relevant for stocks held in well-diversified portfolios

• Measured by a stock’s beta coefficient

• Measures the stock’s volatility relative to the market

11-22

Interpretation of beta

• If = 1.0, stock has average risk

• If > 1.0, stock is riskier than average

• If < 1.0, stock is less risky than average

• Most stocks have betas in the range of 0.5 to 1.5

• Beta of the market = 1.0

• Beta of a T-Bill = 0

11-23

Beta Coefficients for Selected CompaniesTable 11.8

11-24

Example: Work the Web

• Many sites provide betas for companies

• Yahoo! Finance provides beta, plus a lot of other information under its profile link

• Click on the Web surfer to go to Yahoo! Finance– Enter a ticker symbol and get a basic quote– Click on key statistics– Beta is reported under stock price history

11-25

Quick Quiz:Total vs. Systematic Risk

• Consider the following information: Standard Deviation Beta

Security C 20% 1.25

Security K 30% 0.95

• Which security has more total risk?

• Which security has more systematic risk?

• Which security should have the higher expected return?

11-26

Beta and the Risk Premium

• Risk premium = E(R ) – Rf

• The higher the beta, the greater the risk premium should be

• Can we define the relationship between the risk premium and beta so that we can estimate the expected return?– YES!

11-27

SML and Equilibrium

Figure 11.4

11-28

Reward-to-Risk Ratio

• Reward-to-Risk Ratio:

• = Slope of line on graph• In equilibrium, ratio should be the same for all

assets• When E(R) is plotted against β for all assets, the

result should be a straight line

i

fi RRE

)(

11-29

Market Equilibrium

• In equilibrium, all assets and portfolios must have the same reward-to-risk ratio

• Each ratio must equal the reward-to-risk ratio for the market

M

fM

A

fA )RR(ER)R(E

11-30

Security Market Line

• The security market line (SML) is the representation of market equilibrium

• The slope of the SML = reward-to-risk ratio:

(E(RM) – Rf) / M

• Slope = E(RM) – Rf = market risk premium

– Since of the market is always 1.0

11-31

The SML and Required Return

• The Security Market Line (SML) is part of the Capital Asset Pricing Model (CAPM)

Rf = Risk-free rate (T-Bill or T-Bond)

RM = Market return ≈ S&P 500

RPM = Market risk premium = E(RM) – Rf

E(Ri) = “Required Return”

iMfi

ifMfi

RPR)R(E

R)R(ER)R(E

11-32

Capital Asset Pricing Model

• The capital asset pricing model (CAPM) defines the relationship between risk and return

E(RA) = Rf + (E(RM) – Rf)βA

• If an asset’s systematic risk () is known, CAPM can be used to determine its expected return

11-33

SML exampleExpected vs Required Return

Stock E(R) Beta Req RA 14% 1.3 13.4% UndervaluedB 10% 0.8 11.1% Overvalued

Assume: Market Return = 12.0%Risk-free Rate = 7.5%

ifMfi RRERRE )()(

11-34

Factors Affecting Required Return

• Rf measures the pure time value of money

• RPM = (E(RM)-Rf) measures the reward for bearing systematic risk

i measures the amount of systematic risk

ifMfi R)R(ER)R(E

11-35

Portfolio Beta

βp = Weighted average of the Betas of the assets in the portfolio

Weights (wi) = % of portfolio invested in asset i

n

1iiip w

11-36

Covariance of Returns

• Measures how much the returns on two risky assets move together.

ibbi

aaab

ab

pRERRER

baCov

ii)()(

),(

11-37

Covariance vs. Variance of Returns

iaaaai

a

aaa

pRERRER

aVar

ii)()(

)(2

2

ibbi

aaab

ab

pRERRER

baCov

ii)()(

),(

11-38

Correlation Coefficient• Correlation Coefficient = ρ (rho)

• Scales covariance to [-1,+1]– -1 = Perfectly negatively correlated – 0 = Uncorrelated; not related– +1 = Perfectly positively correlated

ba

abab

11-39

Two-Stock Portfolios

• If = -1.0– Two stocks can be combined to form a

riskless portfolio

• If = +1.0 – No risk reduction at all

• In general, stocks have ≈ 0.65– Risk is lowered but not eliminated

• Investors typically hold many stocks

11-40

Covariance & Correlation Coefficient

Covariance

State (i) p(i) E(R) Dev L E(R) Dev U Dev*Dev x p(i)Recession 0.5 -20% -45% 30% 10% -4.5% -0.0225

Boom 0.5 70% 45% 10% -10% -4.5% -0.02251.0

25% 20%45% 10%

-4.50%-1.00

Expected Return

CovarianceCorrelation Coefficient

Stock L Stock U

Standard Deviation

ibbi

aaab

ab

pRERRER

baCov

ii)()(

),(

ba

abab

11-41

of n-Stock Portfolio

n

i

n

jijjip

ij

n

i

n

jjijip

ww

ww

1 1

2

1 1

2

Subscripts denote stocks i and j i,j = Correlation between stocks i and j σi and σj =Standard deviations of stocks i and j σij = Covariance of stocks i and j

ba

abab

11-42

Portfolio Risk-n Risky Assets

n

i

n

jijjip ww

1 1

2

i j for n=21 1 w1w111 = w1

212

1 2 w1w212

2 1 w2w121

2 2 w2w222 = w222

2

p2

= w121

2 + w222

2 + 2w1w2 12

11-43

p2

= w121

2 + w222

2 + 2w1w2 12

Portfolio Risk-2 Risky Assets

i j for n=2

1 1 w1w111 = w121

2 = (.50)(.50)(45)(45)

1 2 w1w212 = (.50)(.50)(-.045)

2 1 w2w121 = (.50)(.50)(-.045)

2 2 w2w222 = w222

2 = (.50)(.50)(10)(10)

p2

= w121

2 + w222

2 + 2w1w2 12 = 0.030625

11-44

Portfolio Variance & Standard Dev

Stock PF % σL 50% 45%U 50% 10%

Covariance -4.50%Portfolio Variance 0.030625

17.50%Portfolio Standard Dev

n

i

n

jijjip ww

1 1

2

Portfolio Risk

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