Transcript Chapter 13
Chapter 13
RETURN RISK AND THE
SECURITY MARKET
LINE
Chapter Outline
Expected Returns and Variances of a
portfolio
Announcements, Surprises, and Expected
Returns
Risk: Systematic and Unsystematic
Diversification and Portfolio Risk
Systematic Risk and Beta
The Security Market Line (SML)
Expected Returns (1)
Expected return = return on a risky asset
expected in the future
Expected returns are based on the probabilities
of possible outcomes
Expected means average if the process is
repeated many timesn
E ( R) pi Ri
i 1
2
Expected Returns (2)
Probability
Boom
Normal
Recession
•R
A
0.2
0.4
Expected return
Stock A Stock B
20%
15%
10%
8%
-5%
2%
=
•RB =
•If the risk-free rate = 3.2%, what is the risk premium
for each stock?
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Variance and Standard Deviation (1)
Unequal probabilities can be used for the
entire range of possibilities
Weighted average of squared deviations
n
σ 2 pi ( Ri E ( R))2
i 1
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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
Portfolio = a group of assets held by an
investor
The risk-return trade-off for a portfolio is
measured by the portfolio expected return and
standard deviation, just as with individual assets
Portfolio weights = Percentage of a
portfolio’s total value in a particular asset
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Portfolio Weights
Suppose 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 D
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Portfolio Expected Returns (1)
The expected return of a portfolio is the
weighted average of the expected returns for
each asset in the portfolio
m
E ( RP ) w j E ( R j )
j 1
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 information
Invest 60% of your money in Asset A
◦ State Probability A
◦ Boom
.5
70%
◦ Recession .5
-20%
1.
2.
B
10%
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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Solution:
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Another Way to Calculate Portfolio
Variance
Portfolio variance can also be calculated
using the following formula:
x x 2xL xU CORRL,U LU
2
P
2
L
2
L
2
U
2
U
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?
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Solution:
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Different Correlation Coefficients (1)
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Different Correlation Coefficients (2)
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Different Correlation Coefficients(3)
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Possible Relationships between Two Stocks
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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 riskreturn trade-off between the two
securities.
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Diversification (2)
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Expected vs. Unexpected Returns
Total return = Expected return + Unexpected return
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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Announcements and News
Announcements 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
Announcement = Expected part + Surprise
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Systematic Risk
Risk 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 Risk
Risk 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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Returns
Unexpected 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
Principle of diversification = spreading
an investment across a number of assets
eliminates some, but not all of the risk
Diversification
assets
is not just holding a lot of
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The Principle of Diversification
Diversification 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
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Portfolio Diversification (1)
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Portfolio Diversification (2)
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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 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 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 Risk
Beta (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
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High and Low Betas
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Total vs. Systematic Risk
Consider the following information:
◦ Security A
◦ Security B
1.
2.
3.
Standard Deviation
20%
30%
Beta
1.25
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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Solution:
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Portfolio Betas
Consider the previous example with the
following four securities
◦ Security
◦A
◦B
◦C
◦D
Weight
.133
.2
.267
.4
Beta
3.69
0.64
1.64
1.79
What is the portfolio beta?
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Beta and the Risk Premium
The 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. Riskfree rate = 8%.
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Portfolio Expected Returns and Betas
Rf
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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 Result
The reward-to-risk ratio must be the
same for all assets in the market
E ( RA ) R f
A
E ( RM R f )
M
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-torisk 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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Security Market Line (2)
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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
A
3.6
B
.7
C
1.7
D
1.9
Expected Return
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Factors Affecting Expected Return
Time value of money – measured by the
risk-free rate
Reward for bearing systematic risk –
measured by the market risk premium
Amount of systematic risk – measured by
beta
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