Chapter 9 Dr. A. DeMaskey An Introduction to Asset Pricing Models Innovative Financial Instruments

Capital Market Theory: An Overview ßCapital market theory extends portfolio theory and develops a model for pricing all risky assets ßCapital asset pricing model (CAPM) will allow you to determine the required rate of return for any risky asset

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.

Assumptions of Capital Market Theory ßAll investments are infinitely divisible, which means that it is possible to buy or sell fractional shares of any asset or portfolio. ßThere are no taxes or transaction costs involved in buying or selling assets. ßThere is no inflation or any change in interest rates, or inflation is fully anticipated. ßCapital markets are in equilibrium; that is, we begin with all investments properly priced in line with their risk levels.

Assumptions of Capital Market Theory ßSome of these assumptions are unrealistic ßRelaxing many of these assumptions would have only minor influence on the model and would not change its main implications or conclusions. ßJudge a theory on how well it explains and helps predict behavior, not on its assumptions.

Riskfree Asset ßProvides the risk-free rate of return (RFR) ßAn asset with zero variance and standard deviation ßZero correlation with all other risky assets ßCovariance between two sets of returns is ßWill lie on the vertical axis of a portfolio graph

Combining a Riskfree Asset with a Risky Portfolio ßExpected return: ßThe expected variance for a two-asset portfolio: ßBecause 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 ßGiven the variance formula: ßThe standard deviation is: ßTherefore, 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 ßTo attain a higher expected return than is available at point M (in exchange for accepting higher risk) ßEither invest along the efficient frontier beyond point M, such as point D ßOr, add leverage to the portfolio by borrowing money at the riskfree rate and investing in the risky portfolio at point M

The Market Portfolio ßBecause portfolio M lies at the point of tangency, it has the highest portfolio possibility line ßEverybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML ßTherefore, this portfolio must include ALL RISKY ASSETS ßSince the market is in equilibrium, all assets are included in this portfolio in proportion to their market value. ßSince 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 ßOnly systematic risk remains in the market portfolio ßSystematic risk is the variability in all risky assets caused by macroeconomic variables ßSystematic risk can be measured by the standard deviation of returns of the market portfolio and can change over time

Factors Affecting Systematic Risk ßVariability in growth of money supply ßInterest rate volatility ßVariability in ß

How to Measure Diversification ßAll portfolios on the CML are perfectly positively correlated with each other and with the completely diversified market Portfolio M ßA completely diversified portfolio would have a correlation with the market portfolio of +1.00

Diversification and the Elimination of Unsystematic Risk ßThe purpose of diversification is to reduce the standard deviation of the total portfolio ßThis assumes that imperfect correlations exist among securities ßAs you add securities, you expect the average covariance for the portfolio to decline ßHow many securities must you add to obtain a completely diversified portfolio? ßObserve 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 ßThe CML leads all investors to invest in the M portfolio ßIndividual investors should differ in position on the CML depending on risk preferences ßHow an investor gets to a point on the CML is based on financing decisions ßRisk 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 ßInvestors preferring more risk might borrow funds at the RFR and invest everything in the market portfolio ßThe decision of both investors is to invest in portfolio M along the CML ßThe decision to borrow or lend to obtain a point on the CML is a separate decision based on risk preferences ßTobin refers to this separation of the investment decision from the financing decision as the separation theorem

A Risk Measure for the CML ßCovariance with the M portfolio is the systematic risk of an asset ßThe Markowitz portfolio model considers the average covariance with all other assets in the portfolio ßThe only relevant portfolio is the M portfolio ßTogether, this means the only important consideration is the asset’s covariance with the market portfolio

A Risk Measure for the CML Since 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: where: R it = return for asset i during period t a i = constant term for asset i b i = slope coefficient for asset i R Mt = return for the M portfolio during period t  = random error term

Variance of Returns for a Risky Asset Note: Var(b i R Mi ) 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 ßThe existence of a riskfree asset resulted in deriving a capital market line (CML) that became the relevant frontier ßAn asset’s covariance with the market portfolio is the relevant risk measure ßThis 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 ßCAPM indicates what should be the expected or required rates of return on risky assets ßThis helps to value an asset by providing an appropriate discount rate to use in dividend valuation models ßThe 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) ßThe relevant risk measure for an individual risky asset is its covariance with the market portfolio (Cov i,m ) ßThe return for the market portfolio should be consistent with its own risk, which is the covariance of the market with itself - or its variance:

The Security Market Line (SML) The equation for the risk-return line is given as: We then define as beta

Determining the Expected Rate of Return for a Risky Asset ßThe expected rate of return of a risky asset is determined by the RFR plus a risk premium for the individual asset ßThe risk premium is determined by the systematic risk of the asset (beta) and the prevailing market risk premium (R M -RFR)

Determining the Expected Rate of Return for a Risky Asset ßIn equilibrium, all assets and all portfolios of assets should plot on the SML ßAny security with an estimated return that plots above the SML is underpriced ßAny security with an estimated return that plots below the SML is overpriced ßTo 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 ßCompare 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 ßIndependent 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: where: R i,t = the rate of return for asset i during period t R M,t = the rate of return for the market portfolio M during t  = the random error term

The Impact of the Time Interval ßNumber of observations and time interval used in regression vary ßValue Line Investment Services (VL) uses weekly rates of return over five years ßMerrill Lynch, Pierce, Fenner & Smith (ML) uses monthly return over five years ßWeak relationship between VL & ML betas due to difference in intervals used ßThere is no “correct” interval for analysis ßInterval effect impacts smaller firms more

The Effect of the Market Proxy ßChoice of market proxy is crucial ßProper measure must include all risky assets ßStandard & Poor’s 500 Composite Index is most often used ßLarge proportion of the total market value of U.S. stocks ßValue weighted series ßWeaknesses

Arbitrage Pricing Theory (APT) ßCAPM is criticized because of the difficulties in selecting a proxy for the market portfolio as a benchmark ßAn alternative pricing theory with fewer assumptions was developed: ßArbitrage Pricing Theory

Assumptions of Arbitrage Pricing Theory (APT) ßCapital markets are perfectly competitive ßInvestors always prefer more wealth to less wealth with certainty ßThe stochastic process generating asset returns can be presented as K factor model

Assumptions of CAPM That Were Not Required by APT ßAPT does not assume: ßA market portfolio that contains all risky assets, and is mean-variance efficient ßNormally distributed security returns ßQuadratic utility function

Arbitrage Pricing Theory (APT) For i = 1 to N where: R i = return on asset i during a specified time period E i = expected return for asset i b ik = reaction in asset i’s returns to movements in a common factor  k = a common factor with a zero mean that influences the returns on all assets  i = a unique effect on asset i’s return that, by assumption, is completely diversifiable in large portfolios and has a mean of zero N = number of assets

Arbitrage Pricing Theory (APT)  Multiple factors,  k, expected to have an impact on all assets: ßInflation ßGrowth in GNP ßMajor political upheavals ßChanges in interest rates ßAnd many more…. ßContrast with CAPM’s insistence that only beta is relevant

Arbitrage Pricing Theory (APT) ßB ik determine how each asset reacts to this common factor ßEach asset may be affected by growth in GNP, but the effects will differ ßIn 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) ßAPT assumes that, in equilibrium, the return on a zero-investment, zero-systematic-risk portfolio, is zero when the unique effects are diversified away ßThe expected return on any asset i (E i ) can be expressed as:

Arbitrage Pricing Theory (APT) Where: 0 = the expected return on an asset with zero systematic risk where 0 = E 0 1 = the risk premium related to each of the common factors, with i = 1 to k b i = 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: b oj = 0) is 3% ( 3 = 0.03)

Example of Two Stocks and a Two-Factor Model b x1 = the response of asset X to changes in the rate of inflation is 0.50 (b x1 = 0.50) b y1 = the response of asset Y to changes in the rate of inflation is 2.00 (b y1 = 2.00) b x2 = the response of asset X to changes in the growth rate of real GNP is 1.50 (b x2 = 1.50) b y2 = the response of asset Y to changes in the growth rate of real GNP is 1.75 (b y2 = 1.75)

Example of Two Stocks and a Two-Factor Model =.03 + (.01)b i1 + (.02)b i2 E x =.03 + (.01)(0.50) + (.02)(1.50) =.065 = 6.5% E y =.03 + (.01)(2.00) + (.02)(1.75) =.085 = 8.5%

Empirical Tests of the APT ßStudies by Roll and Ross and by Chen support APT by explaining different rates of return with some better results than CAPM ßReinganum’s study did not explain small-firm results ßDhrymes and Shanken question the usefulness of APT because it was not possible to identify the factors

Summary ßWhen you combine the riskfree asset with any risky asset on the Markowitz efficient frontier, you derive a set of straight-line portfolio possibilities ßThe dominant line is tangent to the efficient frontier ßReferred to as the capital market line (CML) ßAll investors should target points along this line depending on their risk preferences

Summary ßAll investors want to invest in the risky portfolio, so this market portfolio must contain all risky assets ßThe investment decision and financing decision can be separated ßEveryone wants to invest in the market portfolio ßInvestors finance based on risk preferences

Summary ßThe relevant risk measure for an individual risky asset is its systematic risk or covariance with the market portfolio ßOnce 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 ßAssuming 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 ßThe Arbitrage Pricing Theory (APT) model makes simpler assumptions, and is more intuitive, but test results are mixed at this point

The Internet Investments Online