11-1 Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin.
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Transcript of 11-1 Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin.
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
Return to Quick Quiz
11-3
Expected Returns
Expected Return on Stock A
E(RA) = ∑ (Probability for state of economy X expected
return for state of economy)
Return to Quick Quiz
11-4
Expected risk premium
Expected risk premium =
Expected return on risky investment
- Return on risk free investment
Return to Quick Quiz
11-5
Variance and Standard Deviation
• Variance and standard deviation measure the volatility of returns
Return to Quick Quiz
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.
Return to Quick Quiz
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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