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Assumptions of Capital Market Theory ßAll investors are Markowitz efficient investors who choose investments on the basis of expected return and risk. ßInvestors can borrow or lend any amount of money at the riskfree rate of return (RFR). ßAll investors have homogeneous expectations; that is, they estimate identical probability distributions for future rates of return. ßAll investors have the same one-period time horizon, such as one-month, six months, or one year.
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Chapter 9
Dr. A. DeMaskey
An Introduction to Asset Pricing Models
Innovative Financial Instruments
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
All investors are Markowitz efficient investors who choose investments on the basis of expected return and risk.
Investors can borrow or lend any amount of money at the riskfree rate of return (RFR).
All investors have homogeneous expectations; that is, they estimate identical probability distributions for future rates of return.
All investors have the same one-period time horizon, such as one-month, six months, or one year.
Assumptions of Capital Market Theory
All investments are infinitely divisible, which means that it is possible to buy or sell fractional shares of any asset or portfolio.
There are no taxes or transaction costs involved in buying or selling assets.
There is no inflation or any change in interest rates, or inflation is fully anticipated.
Capital markets are in equilibrium; that is, we begin with all investments properly priced in line with their risk levels.
Assumptions of Capital Market Theory
Some of these assumptions are unrealisticRelaxing 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.
Riskfree AssetProvides the risk-free rate of return (RFR)An asset with zero variance and standard
deviationZero correlation with all other risky assetsCovariance between two sets of returns isWill lie on the vertical axis of a portfolio
graph
Combining a Riskfree Asset with a Risky Portfolio
Expected return:
The expected variance for a two-asset portfolio:
Because the variance of the riskfree asset is zero and the correlation between the riskfree asset and any risky asset i is zero, this simplifies to:
Combining a Risk-Free Asset with a Risky Portfolio
Given the variance formula:
The standard deviation is:
Therefore, the standard deviation of a portfolio that combines the riskfree asset with risky assets is the linear proportion of the standard deviation of the risky asset portfolio.
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 DOr, add leverage to the portfolio by borrowing
money at the riskfree rate and investing in the risky portfolio at point M
The Market PortfolioBecause portfolio M lies at the point of tangency, it
has the highest portfolio possibility lineEverybody will want to invest in Portfolio M and
borrow or lend to be somewhere on the CMLTherefore, this portfolio must include ALL RISKY
ASSETSSince the market is in equilibrium, all assets are included
in this portfolio in proportion to their market value.Since 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 RiskOnly systematic risk remains in the market
portfolioSystematic risk is the variability in all risky assets
caused by macroeconomic variablesSystematic risk can be measured by the standard
deviation of returns of the market portfolio and can change over time
Factors Affecting Systematic Risk
Variability in growth of money supplyInterest rate volatilityVariability in
How to Measure DiversificationAll 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 Risk
The 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?
Observe what happens as you increase the sample size of the portfolio by adding securities that have some positive correlation
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 riskfree rate and invest the remainder in the market portfolio
The CML and the Separation Theorem
Investors preferring more risk might borrow funds at the RFR and invest everything in the market portfolioThe decision of both investors is to invest in portfolio
M along the CMLThe decision to borrow or lend to obtain a point on the
CML is a separate decision based on risk preferences
Tobin refers to this separation of the investment decision from the financing decision as the separation theorem
A Risk Measure for the CMLCovariance with the M portfolio is the
systematic risk of an assetThe Markowitz portfolio model considers the
average covariance with all other assets in the portfolio
The only relevant portfolio is the M portfolioTogether, this means the only important
consideration is the asset’s covariance with the market portfolio
A Risk Measure for the CMLSince all individual risky assets are part of the M portfolio, an asset’s rate of return in relation to the return of 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 iRMt = 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
Note:Var(biRMi) is variance related to market return Var() is the residual return not related to the market portfolio
The Capital Asset Pricing Model: Expected Return and Risk
The existence of a riskfree 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
The estimated rate of return can also be compared to the required rate of return implied by CAPM to determine whether a risky asset is over- or undervalued
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, which is the covariance of the market with itself - or its variance:
2m
The Security Market Line (SML)
The equation for the risk-return line is given as:
)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
Determining the Expected Rate of Return for a Risky Asset
The expected rate of return of a risky 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
Determining the Expected Rate of Return for a Risky Asset
In equilibrium, all assets and all portfolios of assets should plot on the SMLAny security with an estimated return that plots above the
SML is underpricedAny security with an estimated return that plots below the
SML is overpricedTo earn better risk-adjusted rates of return than the
average investor, a superior investor must derive value estimates for assets that are consistently superior to the consensus market evaluation
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
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
2iMCov
M
i
= the random error term
The Impact of the Time Interval
Number 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
Weak relationship between VL & ML betas due to difference in intervals used
There is no “correct” interval for analysis Interval effect impacts smaller firms more
The Effect of the Market Proxy
Choice of market proxy is crucialProper measure must include all risky
assetsStandard & Poor’s 500 Composite Index is
most often usedLarge proportion of the total market value of
U.S. stocksValue weighted seriesWeaknesses
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)
Capital markets are perfectly competitiveInvestors always prefer more wealth to less
wealth with certaintyThe stochastic process generating asset
returns can be presented as K factor model
Assumptions of CAPMThat Were Not Required by APT
APT does not assume: A market portfolio that contains all risky assets,
and is mean-variance efficientNormally distributed security returns Quadratic utility function
Arbitrage Pricing Theory (APT)
ikikiiiiii bbbER ...21
For i = 1 to N where:Ri = return on asset i during a specified time periodEi = expected return for asset ibik = reaction in asset i’s returns to movements in a common factork = a common factor with a zero mean that influences the returns on all assetsi = a unique effect on asset i’s return that, by assumption, is completely diversifiable in large portfolios and has a mean of zeroN = number of assets
Arbitrage Pricing Theory (APT)
Multiple factors, k, expected to have an impact on all assets:InflationGrowth in GNPMajor political upheavalsChanges in interest ratesAnd many more….
Contrast with CAPM’s insistence that only beta is relevant
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 applying the theory, the factors are not identified
Similar to the CAPM in that the unique effects (i) are independent and will be diversified away in a large portfolio
Arbitrage Pricing Theory (APT)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:
ikkiii bbbE ...22110
Arbitrage Pricing Theory (APT)
Where:0 = the expected return on an asset with zero systematic risk
where 0 = E0
1 = the risk premium related to each of the common factors,
with i = 1 to kbi = pricing relationship between the risk premium and asset i
Example of Two Stocks and a Two-Factor Model
1 = changes in the rate of inflation. The risk premium
related to this factor is 1% for every 1% change in the rate (1 = 0.1)
2 = percent growth in real GNP. The average risk premium
related to this factor is 2% for every 1% change in the rate (2 = 0.02)
3 = the rate of return on a zero-systematic-risk asset (zero
beta: boj = 0) is 3% (3 = 0.03)
Example of Two Stocks and a Two-Factor Model
bx1 = the response of asset X to changes in the rate of inflation is 0.50 (bx1 = 0.50)by1 = the response of asset Y to changes in the rate of inflation is 2.00 (by1 = 2.00)bx2 = the response of asset X to changes in the growth rate of real GNP is 1.50 (bx2 = 1.50)by2 = the response of asset Y to changes in the growth rate of real GNP is 1.75 (by2 = 1.75)
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 APTStudies 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
SummaryWhen you combine the riskfree asset with any risky
asset on the Markowitz efficient frontier, you derive a set of straight-line portfolio possibilities
The dominant line is tangent to the efficient frontierReferred to as the capital market line (CML)All investors should target points along this line
depending on their risk preferences
SummaryAll investors want to invest in the risky portfolio, so
this market portfolio must contain all risky assetsThe investment decision and financing decision can be
separatedEveryone wants to invest in the market portfolioInvestors finance based on risk preferences
SummaryThe relevant risk measure for an individual risky
asset is its systematic risk or covariance with the market portfolioOnce 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
SummaryAssuming 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
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.comwww.barra.comwww.stanford.edu/~wfsharpe.comwww.wsharpe.com
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