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7/30/2019 Lecture Ch.13 1
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Ch.13 Return, Risk, and theSecurity Market Line
Know how to calculate expected returns Understand the impact of diversification Understand the systematic risk principle Understand the security market line
Understand the risk-return trade-off Be able to use the Capital Asset Pricing Model
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Chapter Outline
Expected Returns and Variances Portfolios
Announcements, Surprises, and ExpectedReturns Risk: Systematic and Unsystematic
Diversification and Portfolio Risk Systematic Risk and Beta The Security Market Line
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Expected Returns
Expected returns are based on theprobabilities of possible outcomes
In this context, expected means average if the process is repeated many times
The expected return does not even have tobe a possible return
n
iii R p R E
1
)(
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Expected Returns Example 1 Suppose you have predicted the following returns for
stocks C and T in three possible states of nature.What are the expected returns?
State Probability C T Boom 0.3 0.15 0.25 Normal 0.5 0.10 0.20 Recession ??? 0.02 0.01
RC = .3(.15) + .5(.10) + .2(.02) = .099 = 9.9% RT = .3(.25) + .5(.20) + .2(.01) = .177 = 17.7%
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Variance and Standard Deviation
Variance and standard deviation still measurethe volatility of returns
You can use unequal probabilities for theentire range of possibilities
Weighted average of squared deviations
n
iii R E R p
1
22 ))((
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Variance and Standard Deviation Example
Consider the previous example. What is the varianceand standard deviation for each stock?
Stock C 2
= .3(.15-.099)2
+ .5(.1-.099)2
+ .2(.02-.099)2
=.002029 = .045
Stock T 2 = .3(.25-.177) 2 + .5(.2-.177) 2 + .2(.01-.177) 2 =
.007441 = .0863
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Quick Quiz I Consider the following information:
State Probability ABC, Inc. Boom .25 .15
Normal .50 .08 Slowdown .15 .04 Recession .10 -.03
What is the expected return? What is the variance? What is the standard deviation?
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Portfolios
A portfolio is a collection of assets An assets risk and return is important in how
it affects the risk and return of the portfolio The risk-return trade-off for a portfolio is
measured by the portfolio expected returnand standard deviation, just as with individualassets
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Example: Portfolio Weights
Suppose you have $15,000 to invest and youhave purchased securities in the followingamounts. What are your portfolio weights ineach security?
$2000 of ABC $3000 of DEF $4000 of GHI $6000 of JKL
ABC: 2/15 = .133 DEF: 3/15 = .2 GHI: 4/15 = .267 JKL: 6/15 = .4
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Portfolio Expected Returns
The expected return of a portfolio is the weightedaverage of the expected returns for each asset in theportfolio
You can also find the expected return by finding theportfolio return in each possible state and computingthe expected value as we did with individualsecurities
m
j j jP R E w R E
1)()(
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Example: Expected PortfolioReturns
Consider the portfolio weights computed previously.If the individual stocks have the following expectedreturns, what is the expected return for the
portfolio? ABC: 19.65% DEF: 8.96% GHI: 9.67% JKL: 8.13%
E(RP) = .133(19.65) + .2(8.96) + .267(9.67) + .4(8.13)= 10.24%
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Portfolio Variance
Compute the portfolio return for each state:RP = w1R1 + w2R2 + + wmRm
Compute the expected portfolio return usingthe same formula as for an individual asset
Compute the portfolio variance and standarddeviation using the same formulas as for anindividual asset
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Example: Portfolio Variance Consider the following information
Invest 60% of your money in Asset A State Probability A B Boom .5 70% 10% Bust .5 -20% 30%
What is the expected return and standard
deviation for each asset? What is the expected return and standard
deviation for the portfolio?
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Another Way to Calculate Portfolio Variance
Portfolio variance can also be calculated using thefollowing formula:
0.0529
0.10.45-1.004.06.020.010.40.20256.0
havewe1.00,-isBandAbetweenncorrelatiothat theAssuming
2
2
222
,22222
P
P
U LU LU LU U L LP CORR x x x x
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Different Correlation Coefficients(Corr=1)
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Different Correlation Coefficients(Corr=-1)
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Different Correlation Coefficients(Corr=0)
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Graphs of Possible Relationships BetweenTwo Stocks
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Diversification
There are benefits to diversification wheneverthe correlation between two stocks is lessthan perfect (p < 1.0)
If two stocks are perfectly positivelycorrelated, then there is simply a risk-returntrade-off between the two securities.
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Diversification Figure 13.4
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Terminology
Feasible set (also called the opportunity set) thecurve that comprises all of the possible portfoliocombinations
Efficient set the portion of the feasible set that onlyincludes the efficient portfolio (where the maximumreturn is achieved for a given level of risk, or wherethe minimum risk is accepted for a given level of return)
Minimum Variance Portfolio the possible portfoliowith the least amount of risk
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Quick Quiz II
Consider the following information State Probability X Z Boom .25 15% 10% Normal .60 10% 9% Recession .15 5% 10%
What is the expected return and standarddeviation for a portfolio with an investment of $6000 in asset X and $4000 in asset Y?
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Expected versus Unexpected Returns
Realized returns are generally not equal toexpected returns
There is the expected component and theunexpected component
At any point in time, the unexpected return can beeither positive or negative
Over time, the average of the unexpectedcomponent is zero
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Announcements and News
Announcements and news contain both anexpected component and a surprisecomponent
It is the surprise component that affects astocks price and therefore its return
This is very obvious when we watch how stockprices move when an unexpectedannouncement is made or earnings aredifferent than anticipated
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Systematic Risk
Risk factors that affect a large number of assets
Also known as non-diversifiable risk or marketrisk
Includes such things as changes in GDP,inflation, interest rates, etc.
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Unsystematic Risk
Risk factors that affect a limited number of assets
Also known as unique risk and asset-specificrisk
Includes such things as labor strikes,shortages, etc.
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Returns
Total Return = expected return + unexpectedreturn
Unexpected return = systematic portion +unsystematic portion
Therefore, total return can be expressed asfollows: Total Return = expected return +systematic portion + unsystematic portion
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Diversification
Portfolio diversification is the investment inseveral different asset classes or sectors
Diversification is not just holding a lot of assets
For example, if you own 50 internet stocks,you are not diversified
However, if you own 50 stocks that span 20different industries, then you are diversified
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Portfolio Diversification
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The Principle of Diversification Diversification can substantially reduce the
variability of returns without an equivalentreduction in expected returns
This reduction in risk arises because worsethan expected returns from one asset areoffset by better than expected returns fromanother
However, there is a minimum level of risk thatcannot be diversified away and that is thesystematic portion
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Figure 13.6 PortfolioDiversification
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Diversifiable (Unsystematic) Risk
The risk that can be eliminated by combiningassets into a portfolio
Synonymous with unsystematic, unique orasset-specific risk
If we hold only one asset, or assets in thesame industry, then we are exposing ourselvesto risk that we could diversify away
The market will not compensate investors forassuming unnecessary risk
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Total 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 Consequently, the total risk for a diversified
portfolio is essentially equivalent to thesystematic risk
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Systematic Risk Principle
There is a reward for bearing risk There is not a reward for bearing risk
unnecessarily The expected return on a risky asset depends
only on that assets systematic risk sinceunsystematic risk can be diversified away
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Measuring Systematic Risk
How do we measure systematic risk? We use the beta coefficient to measure systematic
risk
What does beta tell us? A beta of 1 implies the asset has the same
systematic risk as the overall market A beta < 1 implies the asset has less systematic
risk than the overall market A beta > 1 implies the asset has more systematic
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High and Low Betas
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Beta Coefficients for Selected Companies
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Total versus 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?
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Example: Portfolio Betas Consider the previous example with the
following four securities Security Weight Beta ABC .133 3.69 DEF .2 0.64 GHI .267 1.64
JKL .4 1.79 What is the portfolio beta? .133(3.69) + .2(.64) + .267(1.64) + .4(1.79) =
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Beta and the Risk Premium
Remember that the risk premium = expectedreturn risk-free rate
The higher the beta, the greater the riskpremium should be
Can we define the relationship between therisk premium and beta so that we canestimate the expected return?
YES!
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Portfolio Expected Returns and Betas
R f
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Reward to Risk Ratio: Definition and
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Reward-to-Risk Ratio: Definition andExample
The reward-to-risk ratio is the slope of the lineillustrated in the previous example
Slope = (E(RA) Rf ) / ( A 0)
Reward-to-risk ratio for previous example =(20 8) / (1.6 0) = 7.5 What if an asset has a reward-to-risk ratio of 8
(implying that the asset plots above the line)? What if an asset has a reward-to-risk ratio of 7
(implying that the asset plots below the line)?
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Market Equilibrium
In equilibrium, all assets and portfolios musthave the same reward-to-risk ratio and theyall must equal the reward-to-risk ratio for themarket
M
f M
A
f A R R E R R E
)()(
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Security Market Line
The security market line (SML) is therepresentation of market equilibrium
The slope of the SML is the reward-to-riskratio: (E(RM) Rf ) / M
But since the beta for the market is ALWAYSequal to one, the slope can be rewritten
Slope = E(RM) Rf = market risk premium
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Security Market Line
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The Capital Asset Pricing Model (CAPM)
The capital asset pricing model defines therelationship between risk and return
E(RA) = Rf + A(E(RM) Rf ) If we know an assets systematic risk, we can
use the CAPM to determine its expectedreturn
This is true whether we are talking aboutfinancial assets or physical assets
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Factors Affecting Expected Return
Pure time value of money measured by therisk-free rate
Reward for bearing systematic risk measuredby the market risk premium
Amount of systematic risk measured by beta
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Example - CAPM
Consider the betas for each of the assets givenearlier. If the risk-free rate is 4.5% and the market riskpremium is 8.5%, what is the expected return foreach?
Security Beta Expected Return
ABC 3.69 4.5 + 3.69(8.5) = 35.865%
DEF .64 4.5 + .64(8.5) = 9.940%
GHI 1.64 4.5 + 1.64(8.5) = 18.440%
JKL 1.79 4.5 + 1.79(8.5) = 19.715%
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Quick Quiz What is the difference between systematic and
unsystematic risk? What type of risk is relevant for determining the
expected return? What is the SML? What is the CAPM? What is the
APT? Consider an asset with a beta of 1.2, a risk-free rate
of 5% and a market return of 13%. What is the reward-to-risk ratio in equilibrium? What is the expected return on the asset?
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Summary There is a reward for bearing risk Total risk has two parts: systematic risk and
unsystematic risk Unsystematic risk can be eliminated through
diversification Systematic risk cannot be eliminated and is rewarded
with a risk premium Systematic risk is measured by the beta The equation for the SML is the CAPM, and it
determines the equilibrium required return for agiven level of risk