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