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

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