return risk and the security market line chapter 13
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Chapter OutlineExpected Returns and Variances
of a portfolioAnnouncements, Surprises, and
Expected ReturnsRisk: Systematic and
UnsystematicDiversification and Portfolio RiskSystematic Risk and BetaThe Security Market Line (SML)
Expected Returns (1)
Expected returns are based on the probabilities of possible outcomes
Expected means average if the process is repeated many times
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Expected return = return on a risky asset expected in the future
Expected Returns (2)
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Probability Expected returnStock A Stock B
Boom 0.2 20% 15%Normal 0.4 10% 8%
Recession -5% 2%
• RA =
• RB =
• If the risk-free rate = 3.2%, what is the risk premium for each stock?
Variance and Standard Deviation (1)
Unequal probabilities can be used for the entire range of possibilities
Weighted average of squared deviations
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Variance and Standard Deviation (2)
Consider the previous example. What is the variance and standard deviation for each stock?
Stock A
Stock B
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Portfolios
The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation, just as with individual assets
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Portfolio = a group of assets held by an investor
Portfolio weights = Percentage of a portfolio’s total value in a particular asset
Portfolio WeightsSuppose you have $ 20,000 to invest
and you have purchased securities in the following amounts. What are your portfolio weights in each security?◦$5,000 of A
◦$9,000 of B
◦$5,000 of C
◦$1,000 of D8
Portfolio Expected Returns (1)The expected return of a portfolio is
the weighted average of the expected returns for each asset in the portfolio
You can also find the expected return by finding the portfolio return in each possible state and computing the expected value
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Expected Portfolio Returns (2)Consider the portfolio weights
computed previously. If the individual stocks have the following expected returns, what is the expected return for the portfolio?◦A: 19.65%◦B: 8.96%◦C: 9.67%◦D: 8.13%
E(RP) =
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Portfolio Variance (1)
Steps:1. Compute the portfolio return for
each state:RP = w1R1 + w2R2 + … + wnRn
2. Compute the expected portfolio return using the same formula as for an individual asset
3. Compute the portfolio variance and standard deviation using the same formulas as for an individual asset
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Portfolio Variance (2)Consider the following
informationInvest 60% of your money in Asset
A◦State Probability A B
◦Boom .5 70%10%
◦Recession .5 -20%30%
1. What is the expected return and standard deviation for each asset?
2. What is the expected return and standard deviation for the portfolio?
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Another Way to Calculate Portfolio VariancePortfolio variance can also be
calculated using the following formula:
Correlation is a statistical measure of how 2 assets move in relation to each other
If the correlation between stocks A and B = -1, what is the standard deviation of the portfolio?
ULULULUULLP CORRxxxx ,22222 2
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Diversification (1)There are benefits to
diversification whenever the correlation between two stocks is less than perfect (p < 1.0)
If two stocks are perfectly positively correlated, then there is simply a risk-return trade-off between the two securities.
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Expected vs. Unexpected Returns
Expected return from a stock is the part of return that shareholders in the market predict (expect)
The unexpected return (uncertain, risky part):◦At any point in time, the unexpected
return can be either positive or negative
◦Over time, the average of the unexpected component is zero
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Total return = Expected return + Unexpected return
Announcements and NewsAnnouncements and news
contain both an expected component and a surprise component
It is the surprise component that affects a stock’s price and therefore its return
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Announcement = Expected part + Surprise
Systematic RiskRisk factors that affect a large
number of assets
Also known as non-diversifiable risk or market risk
Examples: changes in GDP, inflation, interest rates, general economic conditions
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Unsystematic RiskRisk factors that affect a limited
number of assets
Also known as diversifiable risk and asset-specific risk
Includes such events as labor strikes, shortages.
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ReturnsUnexpected return = systematic
portion + unsystematic portion
Total return can be expressed as follows:
Total Return = expected return + systematic portion + unsystematic portion
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Effect of Diversification Portfolio diversification is the
investment in several different asset classes or sectors
Diversification is not just holding a lot of assets
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Principle of diversification = spreading an investment across a number of assets eliminates some, but not all of the risk
The Principle of DiversificationDiversification can substantially
reduce the variability of returns without an equivalent reduction in expected returns
Reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another
There is a minimum level of risk that cannot be diversified away and that is the systematic portion 28
Diversifiable (Unsystematic) Risk
The risk that can be eliminated by combining assets into a portfolio
If we hold only one asset, or assets in the same industry, then we are exposing ourselves to risk that we could diversify away
The market will not compensate investors for assuming unnecessary risk
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Total RiskThe 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 the systematic risk
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Systematic Risk Principle
There is a reward for bearing risk
There is no reward for bearing risk unnecessarily
The expected return (and the risk premium) on a risky asset depends only on that asset’s systematic risk since unsystematic risk can be diversified away
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Measuring Systematic RiskBeta (β) is a measure of systematic risk
Interpreting beta:◦β = 1 implies the asset has the same
systematic risk as the overall market◦β < 1 implies the asset has less
systematic risk than the overall market
◦β > 1 implies the asset has more systematic risk than the overall market 34
Total vs. Systematic RiskConsider the following
information: Standard Deviation
Beta◦Security A 20%
1.25◦Security B 30% 0.95
1. Which security has more total risk?
2. Which security has more systematic risk?
3. Which security should have the higher expected return?
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Portfolio BetasConsider the previous example
with the following four securities◦Security Weight Beta◦A .1333.69
◦B .2 0.64◦C .2671.64
◦D .4 1.79What is the portfolio beta?
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Beta and the Risk PremiumThe higher the beta, the greater
the risk premium should be
The relationship between the risk premium and beta can be graphically interpreted and allows to estimate the expected return
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Consider a portfolio consisting of asset A and a risk-free asset. Expected return on asset A is 20%, it has a beta = 1.6. Risk-free rate = 8%.
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Reward-to-Risk Ratio:
The reward-to-risk ratio is the slope of the line illustrated in the previous slide◦Slope = (E(RA) – Rf) / (A – 0)◦Reward-to-risk ratio =
If an asset has a reward-to-risk ratio = 8?
If an asset has a reward-to-risk ratio = 7?
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The Fundamental ResultThe reward-to-risk ratio must be
the same for all assets in the market
If one asset has twice as much systematic risk as another asset, its risk premium is twice as large
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Security Market Line (1)
The security market line (SML) is the representation of market equilibrium
The slope of the SML is the reward-to-risk ratio: (E(RM) – Rf) / M
The beta for the market is always equal to one, the slope can be rewritten
Slope = E(RM) – Rf = market risk premium
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The Capital Asset Pricing Model (CAPM)
The capital asset pricing model defines the relationship between risk and return
E(RA) = Rf + A(E(RM) – Rf)
If we know an asset’s systematic risk, we can use the CAPM to determine its expected return
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CAPM
Consider the betas for each of the assets given earlier. If the risk-free rate is 4.5% and the market risk premium is 8.5%, what is the expected return for each?Security Beta Expected Return
A 3.6
B .7
C 1.7
D 1.9
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