Chapter 9 - An Introduction to Asset Pricing Models

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Investment Analysis and Portfolio Management

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Chapter 9

mailto:[email protected]

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SAIF ULLAH, [email protected], +923216633271

Chapter 9 - An Introduction to Asset Pricing Models

Questions to be answered:

• What are the assumptions of the capital asset pricing model?

• What is a risk-free asset and what are its risk-return characteristics?

• What is the covariance and correlation between the risk-free asset and a risky asset or portfolio of risky assets?



• What is the expected return when you combine the risk-free asset and a portfolio of risky assets?

• What is the standard deviation when you combine the risk-free asset and a portfolio of risky assets?

• When you combine the risk-free asset and a portfolio of risky assets on the Markowitz efficient frontier, what does the set of possible portfolios look like?



• Given the initial set of portfolio possibilities with a risk-free asset, what happens when you add financial leverage (that is, borrow)?

• What is the market portfolio, what assets are included in this portfolio, and what are the relative weights for the alternative assets included?

• What is the capital market line (CML)?

• What do we mean by complete diversification?



• How do we measure diversification for an individual portfolio?

• What are systematic and unsystematic risk?

• Given the capital market line (CML), what is the separation theorem?

• Given the CML, what is the relevant risk measure for an individual risky asset?

• What is the security market line (SML) and how does it differ from the CML?



• What is beta and why is it referred to as a standardized measure of systematic risk?

• How can you use the SML to determine the expected (required) rate of return for a risky asset?

• Using the SML, what do we mean by an undervalued and overvalued security, and how do we determine whether an asset is undervalued or overvalued?



• What is an asset’s characteristic line and how do you compute the characteristic line for an asset?

• What is the impact on the characteristic line when you compute it using different return intervals (e.g., weekly versus monthly) and when you employ different proxies (i.e., benchmarks) for the market portfolio (e.g., the S&P 500 versus a global stock index)?



• What is the arbitrage pricing theory (APT) and how does it differ from the CAPM in terms of assumptions?

• How does the APT differ from the CAPM in terms of risk measure?


Capital Market Theory: An Overview

• Capital market theory extends portfolio theory and develops a model for pricing all risky assets

• Capital asset pricing model (CAPM) will allow you to determine the required rate of return for any risky asset


Assumptions of Capital Market Theory

1. All investors are Markowitz efficient investors who want to target points on the efficient frontier. – The exact location on the efficient frontier and,

therefore, the specific portfolio selected, will depend on the individual investor’s risk-return utility function.



2. Investors can borrow or lend any amount of money at the risk-free rate of return (RFR). – Clearly it is always possible to lend money at the

nominal risk-free rate by buying risk-free securities such as government T-bils. It is not always possible to borrow at this risk-free rate, but we will see that assuming a higher borrowing rate does not change the general results.



3. All investors have homogeneous expectations; that is, they estimate identical probability distributions for future rates of return.– Again, this assumption can be relaxed. As long

as the differences in expectations are not vast, their effects are minor.



4. All investors have the same one-period time horizon such as one-month, six months, or one year. – The model will be developed for a single

hypothetical period, and its results could be affected by a different assumption. A difference in the time horizon would require investors to derive risk measures and risk-free assets that are consistent with their time horizons.



5. All investments are infinitely divisible, which means that it is possible to buy or sell fractional shares of any asset or portfolio. – This assumption allows us to discuss

investment alternatives as continuous curves. Changing it would have little impact on the theory.



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. Again, relaxing this assumption modifies the results, but does not change the basic thrust.



7. There is no inflation or any change in interest rates, or inflation is fully anticipated.– This is a reasonable initial assumption, and it

can be modified.



8. Capital markets are in equilibrium.– This means that we begin with all investments

properly priced in line with their risk levels.



• Some of these assumptions are unrealistic

• Relaxing many of these assumptions would have only minor influence on the model and would not change its main implications or conclusions.

• Judge a theory on how well it explains and helps predict behavior, not on its assumptions.


Risk-Free Asset

• An asset with zero variance

• Zero correlation with all other risky assets

• Provides the risk-free rate of return (RFR)

• Will lie on the vertical axis of a portfolio graph


Risk-Free Asset

Covariance between two sets of returns is

n

1ijjiiij )]/nE(R-)][RE(R-[RCov

Because the returns for the risk free asset are certain,

0RF Thus 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. Similarly the correlation between any risky asset and the risk-free asset would be zero.


Combining a Risk-Free Asset with a Risky Portfolio

Expected return

the weighted average of the two returns

))E(RW-(1(RFR)W)E(R iRFRFport

This is a linear relationship



Standard deviation

The expected variance for a two-asset portfolio is

211,22122

22

21

21

2port rww2ww)E(

Substituting the risk-free asset for Security 1, and the risky asset for Security 2, this formula would become

iRFiRF iRF,RFRF22

RF22

RF2port )rw-(1w2)w1(w)E(

Since we know that the variance of the risk-free asset is zero and the correlation between the risk-free asset and any risky asset i is zero we can adjust the formula

22RF

2port )w1()E( i



Given the variance formula22

RF2port )w1()E( i

22RFport )w1()E( i the standard deviation is

i)w1( RF

Therefore, the standard deviation of a portfolio that combines the risk-free asset with risky assets is the linear proportion of the standard deviation of the risky asset portfolio.



Since both the expected return and the standard deviation of return for such a portfolio are linear combinations, a graph of possible portfolio returns and risks looks like a straight line between the two assets.


Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier

)E( port

)E(R port Figure 9.1

RFR

M

C

AB

D


Risk-Return Possibilities with Leverage

To attain a higher expected return than is available at point M (in exchange for accepting higher risk)

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


Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier

)E( port

)E(R port

Figure 9.2

RFR

M

CML

Borrowing

Lending


The Market Portfolio

Because portfolio M lies at the point of tangency, it has the highest portfolio possibility line

Everybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML

Therefore this portfolio must include ALL RISKY ASSETS



Because the market is in equilibrium, all assets are included in this portfolio in proportion to their market value



Because it contains all risky assets, it is a completely diversified portfolio, which means that all the unique risk of individual assets (unsystematic risk) is diversified away


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 and can change over time


Examples of Macroeconomic Factors Affecting Systematic Risk

• Variability in growth of money supply

• Interest rate volatility

• Variability in

industrial production

corporate earnings

and cash flow


How to Measure Diversification

All portfolios on the CML are perfectly positively correlated with each other and with the completely diversified market Portfolio M

A completely diversified portfolio would have a correlation with the market portfolio of +1.00


Diversification and the Elimination of Unsystematic RiskThe purpose of diversification is to reduce the

standard deviation of the total portfolio

This assumes that imperfect correlations exist among securities

As you add securities, you expect the average covariance for the portfolio to decline

How many securities must you add to obtain a completely diversified portfolio?


Diversification and the Elimination of Unsystematic Risk

Observe what happens as you increase the sample size of the portfolio by adding securities that have some positive correlation


Number of Stocks in a Portfolio and the Standard Deviation of Portfolio Return

Figure 9.3Standard Deviation of Return

Number of Stocks in the Portfolio

Standard Deviation of the Market Portfolio (systematic risk)

Systematic Risk

Total Risk

Unsystematic (diversifiable) Risk


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

How an investor gets to a point on the CML is based on financing decisions

Risk averse investors will lend part of the portfolio at the risk-free rate and invest the remainder in the market portfolio



Investors preferring more risk might borrow funds at the RFR and invest everything in the market portfolio



The decision of both investors is to invest in portfolio M along the CML

(the investment decision)



The decision to borrow or lend to obtain a point on the CML is a separate decision based on risk preferences

(financing decision)



Tobin refers to this separation of the investment decision from the financing decision the separation theorem


A Risk Measure for the CML

Covariance with the M portfolio is the systematic risk of an asset

The Markowitz portfolio model considers the average covariance with all other assets in the portfolio

The only relevant portfolio is the M portfolio



Together, this means the only important consideration is the asset’s covariance with the market portfolio



Because all individual risky assets are part of the M portfolio, an asset’s rate of return in relation to the return for the M portfolio may be described using the following linear model:

Miiiit RbaRwhere:

Rit = return for asset i during period t

ai = constant term for asset i

bi = slope coefficient for asset i

RMt = return for the M portfolio during period t

= random error term


Variance of Returns for a Risky Asset

)Rba(Var)Var(R Miiiit )(Var)Rb(Var)a(Var Miii

)(Var)Rb(Var0 Mii

risk icunsystemator portfoliomarket the

torelatednot return residual theis )(Var

risk systematicor return market to

related varianceis )Rb(Var that Note Mii


The Capital Asset Pricing Model: Expected Return and Risk

• The existence of a risk-free asset resulted in deriving a capital market line (CML) that became the relevant frontier

• An asset’s covariance with the market portfolio is the relevant risk measure

• This can be used to determine an appropriate expected rate of return on a risky asset - the capital asset pricing model (CAPM)


The Capital Asset Pricing Model: Expected Return and Risk

• CAPM indicates what should be the expected or required rates of return on risky assets

• This helps to value an asset by providing an appropriate discount rate to use in dividend valuation models

• You can compare an estimated rate of return to the required rate of return implied by CAPM - over/ under valued ?


The Security Market Line (SML)

• The relevant risk measure for an individual risky asset is its covariance with the market portfolio (Covi,m)

• This is shown as the risk measure

• The return for the market portfolio should be consistent with its own risk, which is the covariance of the market with itself - or its variance:

2m


Graph of Security Market Line (SML)

)E(R i

Figure 9.5

RFR

imCov2m

mR

SML


The Security Market Line (SML)The equation for the risk-return line is

)Cov(RFR-R

RFR)E(R Mi,2M

Mi

RFR)-R(Cov

RFR M2M

Mi,

2M

Mi,Cov

We then define as beta

RFR)-(RRFR)E(R Mi i

)( i


Graph of SML with Normalized Systematic Risk

)E(R i

Figure 9.6

)Beta(Cov 2Mim/0.1

mR

SML

0

Negative Beta

RFR


Determining the Expected Rate of Return for a Risky Asset

The expected rate of return of a risk asset is determined by the RFR plus a risk premium for the individual asset

The risk premium is determined by the systematic risk of the asset (beta) and the prevailing market risk premium (RM-RFR)

RFR)-(RRFR)E(R Mi i



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)-(RRFR)E(R Mi i

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%



In 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

A superior investor must derive value estimates for assets that are consistently superior to the consensus market evaluation to earn better risk-adjusted rates of return than the average investor


Identifying Undervalued and Overvalued Assets

Compare the required rate of return to the expected rate of return for a specific risky asset using the SML over a specific investment horizon to determine if it is an appropriate investment

Independent estimates of return for the securities provide price and dividend outlooks


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

Table 9.1


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

Table 9.2


Plot of Estimated Returnson SML Graph

Figure 9.7)E(R i

Beta0.1

mRSML

0 .20 .40 .60 .80 1.20 1.40 1.60 1.80-.40 -.20

.22 .20 .18 .16 .14 .12 Rm .10 .08 .06 .04 .02

AB

C

D

E


Calculating Systematic Risk: The Characteristic Line

The systematic risk input of an individual asset is derived from a regression model, referred to as the asset’s characteristic line with the model portfolio:

tM,iiti, RRwhere: 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-R

2M

Mi,Cov

i

error term random the


Scatter Plot of Rates of ReturnFigure 9.8

RM

RiThe characteristic line is the regression line of the best fit through a scatter plot of rates of return


The Impact of the Time IntervalNumber of observations and time interval used in

regression vary

Value Line Investment Services (VL) uses weekly rates of return over five years

Merrill Lynch, Pierce, Fenner & Smith (ML) uses monthly return over five years

There is no “correct” interval for analysis

Weak relationship between VL & ML betas due to difference in intervals used

Interval effect impacts smaller firms more


The Effect of the Market ProxyThe 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– Large proportion of the total market value of

U.S. stocks– Value weighted series


Weaknesses of Using S&P 500as the Market Proxy

– Includes only U.S. stocks – The theoretical market portfolio should include

U.S. and non-U.S. stocks and bonds, real estate, coins, stamps, art, antiques, and any other marketable risky asset from around the world


Comparing Market ProxiesCalculating Beta for Coca-Cola using Morgan

Stanley (M-S) World Equity Index and S&P 500 as market proxies results in a 1.27 beta when compared with the M-S index, but a 1.01 beta compared to the S&P 500

The difference is exaggerated by the small sample size (12 months) used, but selecting the market proxy can make a significant difference

Here are the computations from page 303:


Computation of Beta of Coca-Colawith Selected Indexes Table 9.3

Return S&P 500 M-S World Coca-ColaS&P M-S S&P M-S Coca- RS&P - E(RS&P) RM-S - E(RM-S) RKO - E(RKO)

Date 500 World 500 World Cola ( 1 ) ( 2 ) ( 3 ) ( 4 )a ( 5 )b

Dec-97 970.43 933.60Jan-98 980.28 961.50 1.02 2.99 (2.91) -1.16 1.08 -3.40 3.94 -3.69Feb-98 1049.34 1025.30 7.04 6.64 5.98 4.87 4.73 5.49 26.73 25.96Mar-98 1101.75 1067.40 4.99 4.11 8.47 2.82 2.20 7.97 22.49 17.55Apr-98 1111.75 1059.30 0.91 -0.76 1.93 -1.27 -2.66 1.43 -1.81 -3.82May-98 1090.82 1061.80 -1.88 0.24 3.29 -4.06 -1.67 2.80 -11.35 -4.67Jun-98 1133.84 1085.70 3.94 2.25 9.09 1.77 0.35 8.59 15.21 2.98Jul-98 1120.67 1082.70 -1.16 -0.28 (5.85) -3.34 -2.18 -6.35 21.17 13.84Aug-98 957.98 937.10 -14.52 -13.45 (19.10) -16.69 -15.35 -19.60 327.10 300.87Sep-98 1017.01 952.40 6.16 1.63 (11.52) 3.99 -0.27 -12.01 -47.91 3.27Oct-98 1098.67 1032.20 8.03 8.38 17.25 5.86 6.47 16.75 98.07 108.43Nov-98 1163.63 1097.60 5.91 6.34 3.70 3.74 4.43 3.20 11.97 14.19Dec-98 1229.23 1150.00 5.64 4.77 (4.37) 3.46 2.87 -4.87 -16.87 -13.97

2.17 1.90 0.50 Total = 448.74 460.93Standard Deviation 6.18 5.63 9.87

CovKO ,S&P= 448.74/ 12 = 37.39 VarS&P = St.Dev.S&P2

= 38.19 BetaKO,S&P= 0.98 AlphaKO ,S&P= -1.63

CovKO ,M-S= 460.93/ 12 = 38.41 VarM-S = St.Dev.M-S2

= 31.70 BetaKO,M-S= 1.21 AlphaKO ,M-S= -1.81

Correlation coef.KO ,S&P= 0.61 Correlation coef.KO ,M-S= 0.69

aColumn (4) is equal to column (1) multiplied by column (3) bColumn (5) is equal to column (2) multiplied by column (3)

Index

Average


Arbitrage Pricing Theory (APT)

• CAPM is criticized because of the difficulties in selecting a proxy for the market portfolio as a benchmark

• An alternative pricing theory with fewer assumptions was developed:

• Arbitrage Pricing Theory


Assumptions of Arbitrage Pricing Theory (APT)

1. Capital markets are perfectly competitive

2. Investors always prefer more wealth to less wealth with certainty

3. The stochastic process generating asset returns can be presented as K factor model (to be described)


Assumptions of CAPMThat Were Not Required by APTAPT does not assume

• A market portfolio that contains all risky assets, and is mean-variance efficient

• Normally distributed security returns • Quadratic utility function



For i = 1 to N where:

= return on asset i during a specified time period

ikikiiiitt bbbER ...21

Ri





= expected return for asset i


Ri

Ei






= reaction in asset i’s returns to movements in a common factor


Ri

Ei

bik







= a common factor with a zero mean that influences the returns on all assets


Ri

Ei

bik

k








= a unique effect on asset i’s return that, by assumption, is completely diversifiable in large portfolios and has a mean of zero


Ri

Ei

bik

ki








= a unique effect on asset i’s return that, by assumption, is completely diversifiable in large portfolios and has a mean of zero

= number of assets


Ri

Ei

bik

ki

N



Multiple factors expected to have an impact on all assets:

k



Multiple factors expected to have an impact on all assets:– Inflation

k



Multiple factors expected to have an impact on all assets:– Inflation– Growth in GNP

k



Multiple factors expected to have an impact on all assets:– Inflation– Growth in GNP– Major political upheavals

k



Multiple factors expected to have an impact on all assets:– Inflation– Growth in GNP– Major political upheavals– Changes in interest rates

k



Multiple factors expected to have an impact on all assets:– Inflation– Growth in GNP– Major political upheavals– Changes in interest rates– And many more….

k



Multiple factors expected to have an impact on all assets:– Inflation– Growth in GNP– Major political upheavals– Changes in interest rates– And many more….

Contrast with CAPM insistence that only beta is relevant

k


Arbitrage Pricing Theory (APT)Bik determine how each asset reacts to this

common factor

Each asset may be affected by growth in GNP, but the effects will differ

In application of the theory, the factors are not identified

Similar to the CAPM, the unique effects are independent and will be diversified away in a large portfolio

i



• APT assumes that, in equilibrium, the return on a zero-investment, zero-systematic-risk portfolio is zero when the unique effects are diversified away

• The expected return on any asset i (Ei) can be expressed as:



where:

= the expected return on an asset with zero systematic risk where

ikkiii bbbE ...22110

0

01 EEi

00 E1 = the risk premium related to each of the common

factors - for example the risk premium related to interest rate risk

bi = the pricing relationship between the risk premium and asset i - that is how responsive asset i is to this common factor K


Example of Two Stocks and a Two-Factor Model= changes in the rate of inflation. The risk premium

related to this factor is 1 percent for every 1 percent change in the rate

1)01.( 1

= percent growth in real GNP. The average risk premium related to this factor is 2 percent for every 1 percent change in the rate

= the rate of return on a zero-systematic-risk asset (zero beta: boj=0) is 3 percent

2)02.( 2

)03.( 3 3


Example of Two Stocks and a Two-Factor Model= the response of asset X to changes in the rate of

inflation is 0.501xb

)50.( 1 xb

= the response of asset Y to changes in the rate of inflation is 2.00 )50.( 1 yb1yb

= the response of asset X to changes in the growth rate of real GNP is 1.50

= the response of asset Y to changes in the growth rate of real GNP is 1.75

2xb

2yb)50.1( 2 xb

)75.1( 2 yb


Example of Two Stocks and a Two-Factor Model

= .03 + (.01)bi1 + (.02)bi2 Ex = .03 + (.01)(0.50) + (.02)(1.50)

= .065 = 6.5%

Ey = .03 + (.01)(2.00) + (.02)(1.75)

= .085 = 8.5%

22110 iii bbE


Empirical Tests of the APT• Studies by Roll and Ross and by Chen

support APT by explaining different rates of return with some better results than CAPM

• Reinganum’s study did not explain small-firm results

• Dhrymes and Shanken question the usefulness of APT because it was not possible to identify the factors


Summary

• When you combine the risk-free asset with any risky asset on the Markowitz efficient frontier, you derive a set of straight-line portfolio possibilities


Summary

• The dominant line is tangent to the efficient frontier– Referred to as the capital market line

(CML)

– All investors should target points along this line depending on their risk preferences


Summary• All investors want to invest in the risky

portfolio, so this market portfolio must contain all risky assets– The investment decision and financing decision

can be separated

– Everyone wants to invest in the market portfolio

– Investors finance based on risk preferences


Summary• The relevant risk measure for an

individual risky asset is its systematic risk or covariance with the market portfolio– Once you have determined this Beta

measure and a security market line, you can determine the required return on a security based on its systematic risk


Summary

• Assuming security markets are not always completely efficient, you can identify undervalued and overvalued securities by comparing your estimate of the rate of return on an investment to its required rate of return


Summary

• The Arbitrage Pricing Theory (APT) model makes simpler assumptions, and is more intuitive, but test results are mixed at this point


The InternetInvestments Online

www.valueline.com

www.barra.com

www.stanford.edu/~wfsharpe.com

http://www.valueline.com/

http://www.barra.com/

http://www.stanford.edu/~wfsharpe.com


End of Chapter 9–An Introduction to Asset Pricing Models


Future topicsChapter 10

• Extensions of Capital Asset Pricing Model

• Relaxation of Assumptions– Effect on SML and CML

• Empirical Tests of the Theory

Documents

Chapter 9 - An Introduction to Asset Pricing Models