Transcript Risk and Return: Extensions
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CHAPTER 5
Risk and Return: Portfolio Theory and Asset Pricing Models
Portfolio Theory
Capital Asset Pricing Model (CAPM)
Efficient frontier
Capital Market Line (CML)
Security Market Line (SML)
Beta calculation
Arbitrage pricing theory
Fama-French 3-factor model
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Portfolio Theory Ch 5, in-class, portfolio theory exercise
Suppose Asset A has an expected return of 10 percent and a standard deviation of 20 percent. Asset B has an expected return of 16 percent and a standard deviation of 40 percent. If the correlation between A and B is 0.4*, what are the expected return and standard deviation for a portfolio comprised of 30 percent Asset A and 70 percent Asset B?
n.b. this is .4, not .6, as in binder
Portfolio Expected Return rˆ P
w A rˆ A
( 1
w A ) rˆ B
0 .
3 ( 0 .
1 )
0 .
7 ( 0 .
16 )
0 .
142
14 .
2 %.
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Portfolio Standard Deviation
p
W A 2
2 A
( 1
W A ) 2
B 2
2 W A ( 1
W A )
AB
A
B
0 .
3
0 .
309 2 ( 0 .
2 2 )
0 .
7 2 ( 0 .
4 2 )
2 ( 0 .
3 )( 0 .
7 )( 0 .
4 )( 0 .
2 )( 0 .
4 )
Attainable Portfolios:
AB = 0.4
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AB = +0.4: Attainable Set of Risk/Return Combinations 20% 15% 10% 5% 0% 0% 10% 20% Risk,
p 30% 40%
Attainable Portfolios:
AB = +1
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AB = +1.0: Attainable Set of Risk/Return Combinations 20% 15% 10% 5% 0% 0% 10% 20% Risk,
p 30% 40%
Attainable Portfolios:
AB = -1
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AB = -1.0: Attainable Set of Risk/Return Combinations 20% 15% 10% 5% 0% 0% 10% 20% Risk,
p 30% 40%
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Attainable Portfolios with Risk-Free Asset (Expected risk-free return = 5%) Attainable Set of Risk/Return Combinations with Risk-Free Asset 15% 10% 5% 0% 0% 5% 10% Risk,
p 15% 20%
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Expected Portfolio Return, r p Efficient Set Feasible Set Feasible and Efficient Portfolios Risk,
p
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The feasible set of portfolios all represents portfolios that can be constructed from a given set of stocks.
An efficient portfolio is one that offers:
the most return for a given amount of risk, or
the least risk for a give amount of return.
The collection of efficient portfolios is called the efficient set or efficient frontier .
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Investor’s preferences
Add to this diagram information about the risk aversion of investors
I B2 I B1
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Expected Return, r p I A2 I A1 Optimal Portfolios Optimal Portfolio Investor B Optimal Portfolio Investor A Risk
p
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Indifference curves reflect an investor’s attitude toward risk as reflected in his or her risk/return tradeoff function. They differ among investors because of differences in risk aversion.
An investor’s optimal portfolio is defined by the tangency point between the efficient set and the investor’s indifference curve.
Different indifference curves reflect different investor’s risk aversion
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WHAT is CAPM?
An equilibrium model specifying the relationship between risk and required return on assets held in diversified portfolios.
RISK RETURN
(Systematic) RISK RETURN
If , then
Only one factor affects risk, and it is?
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What are the assumptions of the CAPM?
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Investors all think in terms of a single holding period .
All investors have identical expectations .
Investors can borrow or lend unlimited amounts at the risk-free rate .
(More...)
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All assets are perfectly divisible .
There are no taxes and no transactions costs .
All investors are price takers influence stock prices.
, that is, investors’ buying and selling won’t
Quantities of all assets are given and fixed.
What impact does r RF have on the efficient frontier?
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When a risk-free asset is added to the feasible set, investors can create portfolios that combine this asset with a portfolio of risky assets.
The straight line connecting r RF with M, the tangency point between the line and the old efficient set, becomes the new efficient frontier .
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Portfolios Combining the Risk Free Asset and the Market Portfolio Expected Portfolio Return Z .
B M .
r M A .
The Capital Market Line (CML) r RF Slope =?
M Risk,
p
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Portfolio M must contain every asset in exact proportion to that asset’s fraction of total market value of all securities
What is the Capital Market Line?
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The Capital Market Line (CML) is all linear combinations of the risk-free asset and Portfolio M.
Portfolios below the CML are inferior.
The CML defines the new efficient set.
All investors will choose a portfolio on the CML.
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The CML Equation ^ p = r RF + r M - r RF
M
p .
Intercept Slope Risk measure
What does the CML tell us?
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The expected rate of return on any efficient portfolio is equal to the risk-free rate plus a risk premium .
The optimal portfolio for any investor is the point of tangency between the CML and the investor’s indifference curves.
Expected Return, r p r M r R R .
I 2 I 1 M .
r RF
R 0 =original portfolio
R
M R = Optimal Portfolio CML Risk,
p
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What is the Security Market Line (SML)?
The CML gives the risk/return relationship for efficient portfolios .
The Security Market Line (SML), also part of the CAPM, gives the risk/return relationship for individual stocks .
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The SML Equation
The measure of risk used in the SML is the beta coefficient of company i,
β
i .
The SML equation:
r
i
= r
RF
+ (RP
M
) β
i
Recall how are betas calculated?
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Run a regression line of past returns on Stock i versus returns on the market.
The regression line is called the characteristic line .
The slope coefficient of the characteristic line is defined as the beta coefficient .
.
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-5 Illustration of beta calculation _ r i 20 15 .
.
Year 1 r M 15% 2 -5 r i 18% -10 10 3 12 16 5 0 5 10 15 20 _ r M r i = -2.59 + 1.44 k M -10
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Method of Calculation
Analysts use a computer with statistical or spreadsheet software to perform the regression.
At least 3 year’s of monthly returns or 1 year’s of weekly returns are used.
Many analysts use 5 years of monthly returns. (More...)
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If beta = 1.0, stock is average risk.
If beta > 1.0, stock is riskier than average.
If beta < 1.0, stock is less risky than average.
Most stocks have betas in the range of 0.5 to 1.5.
Interpreting Regression Results
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The R 2 measures the percent of a stock’s variance that is explained by the market. The typical R 2 is:
0.3 for an individual stock
over 0.9 for a well diversified portfolio
Interpreting Regression Results (Continued)
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The 95% confidence interval shows the range in which we are 95% sure that the true value of beta lies. The typical range is:
from about 0.5 to 1.5 for an individual stock
from about .92 to 1.08 for a well diversified portfolio
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What is the relationship between stand alone, market, and diversifiable risk.
2 j
2 j = b 2 j
2 M +
e 2 j .
= variance = stand-alone risk of Stock j.
b 2 j
2 M
e 2 j = market risk of Stock j.
= variance of error term = diversifiable risk of Stock j.
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What are two potential tests that can be conducted to verify the CAPM?
Beta stability tests
Tests based on the slope of the SML
Tests of the SML indicate:
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A more-or-less linear relationship between realized returns and market risk.
Slope is less than predicted.
Irrelevance of diversifiable risk specified in the CAPM model can be questioned.
(More...)
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Betas of individual securities are not good estimators of future risk.
Betas of portfolios of 10 or more randomly selected stocks are reasonably stable.
Past portfolio betas are good estimates of future portfolio volatility.
Are there problems with the CAPM tests?
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Yes .
Richard Roll questioned whether it was even conceptually possible to test the CAPM.
Roll showed that it is virtually impossible to prove investors behave in accordance with CAPM theory.
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What are our conclusions regarding the CAPM?
It is impossible to verify.
Recent studies have questioned its validity.
Investors seem to be concerned with both market risk and stand-alone risk. Therefore, the SML may not produce a correct estimate of r i .
(More...)
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CAPM/SML concepts are based on expectations , yet betas are calculated using historical data. A company’s historical data may not reflect investors’ expectations about future riskiness .
Other models are being developed that will one day replace the CAPM, but it still provides a good framework for thinking about risk and return.
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When using the CAPM, there are choices that have to be made:
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CAPM: COMMENTS ON ESTIMATING Required rates of return
Under the CAPM, required rate of return (or cost of equity) depends on the risk free rate, the market risk premium, and the Beta, as shown in the Security market Line
k i = k rf + (k m -k rf )
i
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Risk free rate: short term or long term?
Long term used because (1)equities are long term instruments, and (2)there is less volatility in long term rate.
Market risk premium: ex-post or ex ante?
Ex-post from Ibbotson
Ex-ante from IBES (Will vary over time).
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Estimating Betas: historical or anticipated?
Historical from characteristic line: assumes that future will reflect the past.
Adjusted Betas (Marshall Blume): Betas tend to move towards 1 over time. Math. procedure used to estimate this.
Fundamental Betas: Adjustment process to include (1)financial leverage (2)sales volatility, etc. Adjusted continuously.
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Choices in estimating Betas
Historical periods of different length: one, two, ...five,... years?
Different holding periods: daily, weekly, monthly, annual returns?
More frequent--more data, but also more noise.
What is used to represent k m ?
S&P500, NYSE index, Wilshire index or what?
TWO CALCULATIONS OF BETAS
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Merrill Lynch (& Yahoo!Finance)
k m : S&P500
B: Pure Historical
Data: Monthly(60 observations)
Value Line
k m : NYSE index
B: Adjusted
Data: Weekly:(260 observations)
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CONCLUSION
One can calculate Betas in many ways; depending on the method used, different betas may result from SML: k i = k rf + (k m -k rf )B
With luck, different Betas from different sources will be close.
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What is the difference between the CAPM and the Arbitrage Pricing Theory (APT)?
The CAPM is a single factor model.
The APT proposes that the relationship between risk and return is more complex and may be due to multiple factors such as GDP growth, expected inflation, tax rate changes, and dividend yield.
Required Return for Stock i under the APT
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r i = r RF + (r 1 - r RF )
1 + ... + (r j - r RF )
+ (r 2 j .
- r RF )
2 r j = required rate of return on a portfolio sensitive only to economic Factor j.
j = sensitivity of Stock i to economic Factor j.
What is the status of the APT?
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The APT is being used for some real world applications.
Its acceptance has been slow because the model does not specify what factors influence stock returns.
More research on risk and return models is needed to find a model that is theoretically sound, empirically verified, and easy to use.
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Fama-French 3-Factor Model
Fama and French propose three factors:
The excess market return, r M -r RF .
the return on, S, a portfolio of small firms (where size is based on the market value of equity) minus the return on B, a portfolio of big firms. This return is called r SMB , for S minus B.
Fama-French 3-Factor Model (Continued)
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the return on, H, a portfolio of firms with high book-to-market ratios (using market equity and book equity) minus the return on L, a portfolio of firms with low book-to-market ratios. This return is called r HML , for H minus L.
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Required Return for Stock i under the Fama-French 3-Factor Model r i = r RF + (r M - r RF )b i + (r SMB )c i + (r HMB )d i b i = sensitivity of Stock i to the market return.
c j = sensitivity of Stock i to the size factor.
d j = sensitivity of Stock i to the book to-market factor.
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Required Return for Stock i: b i =0.9, r RF =6.8%, the market risk premium is 6.3%, c i =-0.5, the expected value for the size factor is 4%, d i =-0.3, and the expected value for the book-to-market factor is 5%. r i = r RF + (r M - r RF )b i + (r SMB )c i + (r HMB )d i r i = 6.8% + (6.3%)(0.9) + (4%)(-0.5) + (5%)(-0.3) = 8.97%
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CAPM Required Return for Stock i CAPM: r i = r RF + (r M - r RF )b i r i = 6.8% + (6.3%)(0.9) = 12.47% Fama-French (previous slide): r i = 8.97%
THE END, thank god .
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