Chapter 7 An Introduction to Asset Pricing Models
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Transcript of Chapter 7 An Introduction to Asset Pricing Models
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Chapter 7An Introduction to
Asset Pricing Models
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Recall:The Portfolio Management
Process
1. Policy statement (road map)- Focus: Investor’s short-term and long-term needs, familiarity with capital market history, and expectations
2. Examine current and projected financial, economic, political, and social conditions - Focus: Short-term and intermediate-term expected conditions to use in constructing a specific portfolio
3. Implement the plan by constructing the portfolio - Focus: Meet the investor’s needs at the minimum risk levels
4. Feedback loop: Monitor and update investor needs, environmental conditions, portfolio performance
Exhibit 2.2
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A. Capital Market Theory: assumptions
1. All investors are Markowitz efficient investors who want to target points on the efficient frontier.
The exact location on the efficient frontier and the specific portfolio selected will depend on the individual investor’s utility function.
Background for Capital Market Theory
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2. Investors can borrow or lend any amount of money at the risk-free rate of return (RFR).
It is always possible to lend money at the nominal risk-free rate by buying risk-free securities.It is not always possible to borrow at this risk-free rate, but later we will see that a higher borrowing rate does not change the general results.
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3. All investors have homogeneous expectations; that is, they estimate identical probability distributions for future rates of return.
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4. All investors have the same one-period time horizon such as one-month, six months, or one year.
A difference in the time horizon would require investors to derive risk measures and risk-free assets that are consistent with their time horizons.
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5. All investments are infinitely divisible, which means that it is possible to buy or sell fractional shares of any asset or portfolio.
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6. There are no taxes or transaction costs involved in buying or selling assets.
This is a reasonable assumption in many instances. Neither pension funds nor religious groups have to pay taxes, and the transaction costs for most financial institutions are less than 1 percent on most financial instruments.
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7. There is no inflation or any change in interest rates, or inflation is fully anticipated.
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8. Capital markets are in equilibrium.
This means that we begin with all investments properly priced in line with their risk levels.
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B. Risk-Free AssetAn asset with zero standard deviationZero correlation with all other risky
assetsProvides the risk-free rate of return
(RFR)Will lie on the vertical axis of a return-
risk graph
Background for Capital Market Theory
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Recall: Covariance between asset returns
n)]-E(R)][R-E(R[RCovn
ijjiiij
1
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Because the returns for the risk free asset (suppose “i”=“RF”) are certain
0RFσThus Ri = E(Ri), and Ri - E(Ri) = 0
Consequently, the covariance of the risk-free asset with any risky asset or portfolio will always equal zero.
01
n)]-E(R)][R-E(R[RCovn
ijjiiij
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Combining a risk-free asset with a risky portfolio
Expected return computation:
))E(R-w((RFR)w)E(R iRFRFport 1
w: weight
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Risk computation:The expected variance for a two-asset portfolio is
21212122
22
21
21
2 2 σσrwwσwσwσ ,port
iRFRF,iRFRFiRFRFRFport σσ)r-w(wσ)w(σwσ 121 22222 222 1 iRFport σ)w(σ
When asset 1 is a risk-free asset:
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Given the variance formula
222 1 iRFport σ)w(σ
iRFiRFport σ)w(σ)w(σ 11 22
the standard deviation is
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Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios
portσ
)E(Rport
Exhibit 7.1
RFR
M
C
AB
D
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Risk-Return Possibilities with Leverage
To attain a higher expected return than is available at point M:Either invest along the efficient frontier
beyond point M, such as point D.Or, add leverage to the portfolio by
borrowing money at the risk-free rate and investing in the risky portfolio at point M.
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portσ
)E(Rport
Exhibit 7.1
RFR
M
C
AB
D
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portσ
)E(Rport
Exhibit 7.2
RFR
M
CML
Borrowing
Lending
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The Market PortfolioPortfolio M
lies at the point of tangencyhas the highest portfolio
possibility line
Everybody will want to invest in Portfolio M, then borrow or lend at risk-free rate. (The combination will lie on CML.)M must include all risky assets
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The Market PortfolioBecause it contains all risky
assets, it is a completely diversified portfolio.All the unique risk of individual
assets (unsystematic risk) is diversified away.
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Systematic Risk
Only systematic risk remains in the market portfolio.Systematic risk is the variability in all
risky assets caused by macroeconomic variables.Systematic risk can be measured by
the standard deviation of returns of the market portfolio.
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Standard Deviation of the Market Portfolio (systematic risk)
Exhibit 7.3Standard Deviationof Return
Number of Stocks in the Portfolio
Systematic Risk
Total Risk
Unsystematic (diversifiable) Risk
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The CML and the Separation Theorem
The CML leads all investors to invest in the M portfolio.Individual investors should differ in
position on the CML depending on risk preferences.
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How an investor gets to a point on the CML is based on financing decisions.
Risk lovers might borrow funds at the RFR and invest everything in the market portfolio.Risk averse investors will lend part of the portfolio at the risk-free rate and invest the remainder in the market portfolio.
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Optimal Portfolio Choices on the CML
Exhibit 7.4
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A Risk Measure for the CML
Covariance with the M portfolio is the systematic risk of an asset.Because all individual risky assets are part of
the M portfolio, an asset return in relation to the M return can be shown as:
εRbaR Mtiiit where: Rit = return for asset i during period tai = constant term for asset ibi = slope coefficient for asset iRMt = return for the M portfolio during period t
ε= random error term
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Variance of Returns for a Risky Asset
)()(0
)()()(
VarRbVar
VarRbVaraVar
ε)RbVar(a)Var(R
Mti
Mtii
Mtiiit
risk. icunsystemator portfoliomarket the
torelated NOTreturn residual theis )(Var
risk. systematicor return market to
related varianceis )Rb(Var that Note Mti
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The Security Market Line (SML)
The relevant risk measure for an individual risky asset is its covariance with the market portfolio (Covi,m)
The return for the market portfolio should be consistent with its own risk:
2m
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Graph of Security Market Line (SML)
)E(Ri
Exhibit 7.5
RFR
i,mCov2m
mR
SML
Market Portfolio
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The Security Market Line (SML)
The equation for the risk-return line is
)(2
,
2
RFRRCov
RFR
)(Covσ
-RFRRRFR)E(R
MM
Mi
i,MM
Mi
iM
i,M
σ
Cov
2
-RFR)(RβRFR)E(R Mii
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Graph of SML with Normalized Systematic Risk
Negative Beta
)E(RiExhibit 7.6
)/σBeta(Cov Mi,m20.1
mR
SML
0
RFR
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Determining the Expected Rate of Return for a Risky Asset
Assume: RFR = 6% (0.06) RM = 12% (0.12)
Implied market risk premium = 6% (0.06)
Stock Beta
A 0.70B 1.00C 1.15D 1.40E -0.30
-RFR)(RβRFR)E(R Mii E(RA) = 0.06 + 0.70 (0.12-0.06) = 0.102 = 10.2%E(RB) = 0.06 + 1.00 (0.12-0.06) = 0.120 = 12.0%E(RC) = 0.06 + 1.15 (0.12-0.06) = 0.129 = 12.9%E(RD) = 0.06 + 1.40 (0.12-0.06) = 0.144 = 14.4%E(RE) = 0.06 + -0.30 (0.12-0.06) = 0.042 = 4.2%
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Determining the Expected Rate of Return for a Risky
AssetIn equilibrium, all assets and all portfolios of assets should plot on the SML.Any security with an estimated return
that plots above the SML is underpriced.Any security with an estimated return
that plots below the SML is overpriced.
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Price, Dividend, and Rate of Return Estimates
Stock (Pi) Expected Price (Pt+1) (Dt+1) of Return (Percent)
A 25 27 0.50 10.0 %B 40 42 0.50 6.2C 33 39 1.00 21.2D 64 65 1.10 3.3E 50 54 0.00 8.0
Current Price Expected Dividend Expected Future Rate
Exhibit 7.7
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Comparison of Required Rate of Return to Estimated Rate of
Return
Stock Beta E(Ri) Estimated Return Minus E(Ri) Evaluation
A 0.70 10.2% 10.0 -0.2 Properly ValuedB 1.00 12.0% 6.2 -5.8 OvervaluedC 1.15 12.9% 21.2 8.3 UndervaluedD 1.40 14.4% 3.3 -11.1 OvervaluedE -0.30 4.2% 8.0 3.8 Undervalued
Required Return Estimated Return
Exhibit 7.8
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Plot of Estimated Returns on SML Graph
-.40 -.20
Exhibit 7.9)E(Ri
Beta0.1
SML
0 .20 .40 .60 .80 1.20 1.40 1.60 1.80
.22 .20 .18 .16 .14 .12 Rm .10 .08 .06 .04 .02
AB
C
D
E
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Calculating Systematic Risk:
The Characteristic LineεRβαR M,tiii,t
where: Ri,t = the rate of return for asset i during period tRM,t = the rate of return for the market portfolio M during t
miii R-β Rα 2/ Mi,Mi Cov error term random the
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Scatter Plot of Rates of Return
Exhibit 7.10
RM
RiThe characteristic line is the regression line of the best fit through a scatter plot of rates of return.
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The Effect of the Market Proxy
The market portfolio of all risky assets must be represented in computing an asset’s characteristic line.Standard & Poor’s 500
Composite Index is most often used.
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Investment Alternatives When the Cost of Borrowing is Higher than the Cost of
Lending
Exhibit 7.14
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SML with a Zero-Beta Portfolio
Exhibit 7.15
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SML with Transaction Costs
Exhibit 7.16
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Effects of Taxes
b
icgbei P
TDivTPPATRE
)1()1)(())((
Ri(AT): after-tax returnPe: ending pricePb: beginning priceTcg: capital gain taxDiv: dividendTi: income tax
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Empirical Tests of the CAPM
Stability of Beta:betas for individual stocks are not stable, but portfolio betas are reasonably stable.Further, the larger the portfolio of stocks and longer the period, the more stable the beta of the portfolio.
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Relationship Between Systematic Risk and
ReturnEffect of Skewness
investors prefer stocks with high positive skewness that provide an opportunity for very large returns
Effect of size, P/E, and leveragesize, and P/E have an inverse
impact on returns after considering the CAPM.
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Effect of Book-to-Market Value
Negative relationship between size and average returnPositive relation between B/M
and return
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Differential Performance Based on an Error in Estimating Systematic
Risk
Exhibit 7.19
βT: true beta
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Differential SML Based on Measured Risk-Free Asset and Proxy Market Portfolio
Exhibit 7.20
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Differential SML Using Market Proxy That is Mean-Variance Efficient
Exhibit 7.21