Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 1 Chapter 13: Capital Market Equilibrium Objective The Theory of the CAPM Use of CAPM in benchmarking Using CAPM to determine correct rate for discounting

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 2 Chapter 13 Contents 13.1 The Capital Asset Pricing Model in Brief 13.2 Determining of the Risk Premium on the Market Portfolio 13.3 Beta and Risk Premiums on Individual Securities 13.4 Using the CAPM in Portfolio Selection 13.5 Valuation & Regulating Rates of Return 13.6 Modifications and Alternatives to the CAPM

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 3 Introduction CAPM is a theory about equilibrium prices in the markets for risky assetsCAPM is a theory about equilibrium prices in the markets for risky assets It is important because it providesIt is important because it provides –a justification for the widespread practice of passive investing called indexing –a way to estimate expected rates of return for use in evaluating stocks and projects

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall The Capital Asset Pricing Model in Brief Developed in the 1960’s by Sharp, and independently by Lintner, and MossinDeveloped in the 1960’s by Sharp, and independently by Lintner, and Mossin It answers the questionIt answers the question –What would equilibrium risk premiums be if people had the same set of forecasts of expected returns, risk, and correlationspeople had the same set of forecasts of expected returns, risk, and correlations all chose their portfolios according the principles of efficient diversificationall chose their portfolios according the principles of efficient diversification

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 5 So what’s wrong with  - analysis The assumptions of the last chapter appeared fully acceptableThe assumptions of the last chapter appeared fully acceptable –In fact it may appear to be pedantic to mention them at all Why develop a new model for risk-return if the present model ain’t broke?Why develop a new model for risk-return if the present model ain’t broke?

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 6  -analysis: Estimation We did not spell it out, but if you recall the mnemonic for obtaining the portfolio volatility in the  -model, (given n- shares in the portfolio,) we neededWe did not spell it out, but if you recall the mnemonic for obtaining the portfolio volatility in the  -model, (given n- shares in the portfolio,) we needed –n-means (no problem) –n-standard deviations (no problem) –n*(n-1)/2 correlations (? problem)

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 7  -analysis: Estimation All parameters need estimation, and there are n*(n+1)/2 + n parametersAll parameters need estimation, and there are n*(n+1)/2 + n parameters Assume a portfolio of, say, 2,000 shares represent the market, then we need to estimate more than 2,000,000 parameters, most of which are correlationsAssume a portfolio of, say, 2,000 shares represent the market, then we need to estimate more than 2,000,000 parameters, most of which are correlations

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 8  -analysis: Estimation Recall that when you estimate parameters, it is done with only a given level of confidenceRecall that when you estimate parameters, it is done with only a given level of confidence Confidence improves with the number of observationsConfidence improves with the number of observations In practice the parameters have time dependence, so old data introduces errorIn practice the parameters have time dependence, so old data introduces error For 2,000 shares, and a 99% confidence, about 20,000 parameters will be in errorFor 2,000 shares, and a 99% confidence, about 20,000 parameters will be in error

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 9  -analysis: Estimation –The errors may, or may not, be significant to your investment decision, but their existence calls for further analysis –In any case, the data collection, verification, and processing, is a significant use of analytical resources

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 10  -analysis: Wishes After we have the estimated parameters, finding the optimal portfolio requires quadratic programming, and this again requires heavy use of computational resourcesAfter we have the estimated parameters, finding the optimal portfolio requires quadratic programming, and this again requires heavy use of computational resources –The problem is similar to knowing the position and velocity of every star in the Milky Way, and attempting to predict their futures by computing individual interactions

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 11  -analysis: Guidance Principles for Simplification An important principle of financial modeling is to create equations that capture the key factors parsimoniouslyAn important principle of financial modeling is to create equations that capture the key factors parsimoniously Another important principle is to attempt to develop simple modelsAnother important principle is to attempt to develop simple models –Linear models are then preferred to quadratic models

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 12 The Astrophysics of Finance –In the Milky Way problem, an astronomer should specify exactly what needs to be predicted, and give attention to the variables that most affect it –So, if he wants to know when the next star will come close enough to Sol to disturb the Oort cloud then –close stars need individual analysis –distant stars may be treated homogeneously

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 13 Specifying the Model In the last chapter we examined diversifying a homogenous portfolio, and we observed that there were two kinds of riskIn the last chapter we examined diversifying a homogenous portfolio, and we observed that there were two kinds of risk –diversifiable or individual risk –Nondiversifiable or market risk

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 14 Specifying the Model We also observed that in the limit as the number of securities becomes large, we obtained the formulaWe also observed that in the limit as the number of securities becomes large, we obtained the formula –This formula tells us that the correlations are of crucial importance in the relationship between a portfolio risk and the stock risk

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 15 Specifying the Model In the homogenous model, we saw that there was individual- and market-riskIn the homogenous model, we saw that there was individual- and market-risk Assume that each equity’s return is the composition of two random variables:Assume that each equity’s return is the composition of two random variables: –one associated with the market’s return –one associated with the company-specific return

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 16 Specifying the Model: Assumptions Company-specific return on any stock xCompany-specific return on any stock x –is not correlated to the company-specific return on any other stock y –is correlated with the market return The risk-free rate is constant during the investment the periodThe risk-free rate is constant during the investment the period

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 17 Assumptions –Investors forecasts agree with respect to expectations, standard deviations, and correlations of the returns of risky securities –Therefore all investors hold risky assets in the same relative proportions –Investors behave optimally In equilibrium, prices adjust so that aggregate demand for each security is equal to its supplyIn equilibrium, prices adjust so that aggregate demand for each security is equal to its supply

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 18 Market Portfolio Since every investor’s relative holdings of the risky security is the same, the only way the asset market can clear is if those optimal relative proportions are the proportions in which they are valued in the market placeSince every investor’s relative holdings of the risky security is the same, the only way the asset market can clear is if those optimal relative proportions are the proportions in which they are valued in the market place Market PortfolioMarket Portfolio

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 19 CML and the CAPM CAPM says that in equilibrium, any investor’s relative holding of risky assets will be the same as in the market portfolioCAPM says that in equilibrium, any investor’s relative holding of risky assets will be the same as in the market portfolio Depending on their risk aversions, different investors hold portfolios with different mixes of riskless asset and the market portfolioDepending on their risk aversions, different investors hold portfolios with different mixes of riskless asset and the market portfolio

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 20 CAPM Formula

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 21 The Capital Market Line

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 22 Active v. Passive Management CAPM implies that, on average, the performances of active portfolio managers is equal to that of passive managers employing just the market portfolio and the risk-free securityCAPM implies that, on average, the performances of active portfolio managers is equal to that of passive managers employing just the market portfolio and the risk-free security Diligent managers do outperform passive managers, but only to the degree that their diligence is rewardedDiligent managers do outperform passive managers, but only to the degree that their diligence is rewarded

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 23 Reward Only for Market Risk The risk premium on any individual security is proportional only to its contribution to the risk of the market portfolio, and does not depend on its stand-alone riskThe risk premium on any individual security is proportional only to its contribution to the risk of the market portfolio, and does not depend on its stand-alone risk Investors are rewarded only for bearing market riskInvestors are rewarded only for bearing market risk

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall Determining the Risk Premium on the Market Portfolio CAPM states thatCAPM states that –the equilibrium risk premium on the market portfolio is the product of variance of the market,  2 Mvariance of the market,  2 M weighted average of the degree of risk aversion of holders of risk, Aweighted average of the degree of risk aversion of holders of risk, A

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 25 Comment –CAPM explains the difference between the riskless interest rate and the expected rate of return on the market portfolio, but not their absolute levels –The absolute level of the equilibrium expected rate of return on the market portfolio is determined by such factors as – –expected productivity – –household inter-temporal preferences for consumption

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 26 Example: To Determine ‘A’

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall Beta and Risk Premiums on Individual Securities –If risk is defined as that measure such that as it increases, a risk-averse investor would have to be compensated by a larger expected return in order that she would continue to hold it in her optimal portfolio, then the measure of a security’s risk is its beta,   tells you how much the security’s rate of return changes when the return on the market portfolio changes

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 28 Comment:  = 1 A security with a  = 1 on average rises and falls with the marketA security with a  = 1 on average rises and falls with the market –a 10% (say) unexpected rise (fall) in the market return premium will, on average, result in a 10% rise (fall) in the security’s return premium

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 29 Comment:   1 A security with a   1 on average rises and falls more than the marketA security with a   1 on average rises and falls more than the market –With a  = 1.3, a 10% (say) unexpected rise (fall) in the market return premium will, on average, result in a 13% rise (fall) in the security’s return premium Such a security is said to be aggressiveSuch a security is said to be aggressive

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 30 Comment:   1 A security with a   1 on average rises and falls less than the marketA security with a   1 on average rises and falls less than the market –With a  = 0.7, a 10% (say) unexpected rise (fall) in the market return premium will, on average, result in a 7% rise (fall) in the security’s return premium Such a security is said to be defensiveSuch a security is said to be defensive

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 31 CAPM Risk Premium on any Asset According the the CAPM, in equilibrium, the risk premium on any asset is equal the product ofAccording the the CAPM, in equilibrium, the risk premium on any asset is equal the product of –  (or ‘Beta’) –the risk premium on the market portfolio

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 32 The Security Market Line

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 33 Security Market Line –The plot of a security’s risk premium (or sometimes security returns) against security beta Note that the slope of the security market line is the market premiumNote that the slope of the security market line is the market premium By CAPM theory, all securities must fall precisely on the SML (hence its name)By CAPM theory, all securities must fall precisely on the SML (hence its name)

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 34 Practical Example –Some simulated data was generated under the assumptions that: the market portfolio return has an expected value of 0.15, a volatility of 0.20, and index 0 = 50 the share z has a return of 0.12, a volatility of 0.25, and price 0 = 30 (no dividends)the market portfolio return has an expected value of 0.15, a volatility of 0.20, and index 0 = 50 the share z has a return of 0.12, a volatility of 0.25, and price 0 = 30 (no dividends) the correlation between the returns is 0.90; and the risk-free rate is 0.05the correlation between the returns is 0.90; and the risk-free rate is 0.05

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 35

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 36 Data Set Used In order to display the material clearly, only one year of data is generated, and is collected monthly, resulting in 13 sets of pricesIn order to display the material clearly, only one year of data is generated, and is collected monthly, resulting in 13 sets of prices In a real simulation, much more data must be collected in order to provide an adequate confidence interval for parameter estimatesIn a real simulation, much more data must be collected in order to provide an adequate confidence interval for parameter estimates

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 37 Transformation of Prices into Returns –The prices are transformed into monthly holding period returns (mhpr_Ind, and mhpr_Z) –The mhprs are transformed into annual rates, compounded annually –The annual rates compounded annually are transformed to annual rates compounded continuously

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 38 Table of Prices

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 39

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 40 Financial Calculators Everything could have been done using a modern standard-issue financial or scientific calculatorEverything could have been done using a modern standard-issue financial or scientific calculator –Remember, the correct rate to use is the annual rate compounded continuously, and that month-to-year conversions of standard deviation involve a square root of 12 –Take care to enter the market rate as the independent variable, x

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 41 Accuracy Issue –We assumed that the  ’s and  ’s are constants, but they are random variables too –In order to achieve adequate confidence, a large sample is needed –Small movements in price are masked by transaction prices The result is a compromise between currency and confidenceThe result is a compromise between currency and confidence

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 42 Model and Measured Values of Statistical Parameters

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 43 Comment The illustrated trajectory is typical for monthly data collected over a yearThe illustrated trajectory is typical for monthly data collected over a year Caution: avoid using small data sets to estimate CAPM parametersCaution: avoid using small data sets to estimate CAPM parameters

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 44 Regression Line The slope of the regression line of dependent stock against independent market returns is betaThe slope of the regression line of dependent stock against independent market returns is beta

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 45

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 46 Observation All securities, (not just efficient portfolios) plot onto the SML, if they are correctly priced according to the CAPMAll securities, (not just efficient portfolios) plot onto the SML, if they are correctly priced according to the CAPM

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 47 The Beta of a Portfolio When determining the risk of a portfolioWhen determining the risk of a portfolio –using standard deviation results in a formula that’s quite complex –using beta, the formula is linear

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 48 Computing Beta Here are some useful formulae for computing betaHere are some useful formulae for computing beta

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall Using the CAPM in Portfolio Selection Whether or not CAPM is a valid theory, indexing is attractive to investors becauseWhether or not CAPM is a valid theory, indexing is attractive to investors because –historically it has performed better than most actively managed portfolios –it costs less to implement that active management

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 50 A Paradox Resolved The last chapter posed a paradox with two securities co-existing, one having a lower standard deviation and higher return than the otherThe last chapter posed a paradox with two securities co-existing, one having a lower standard deviation and higher return than the other If we accept the CAPM as a valid theory, we have a resolutionIf we accept the CAPM as a valid theory, we have a resolution Both securities lie on the SML, and both securities lie below the CMLBoth securities lie on the SML, and both securities lie below the CML

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 51  risk and  risk A security has two kinds of risk: risk that may be diversified away, and risk that is associated with the marketA security has two kinds of risk: risk that may be diversified away, and risk that is associated with the market –The CAPM theory states that the lower return on the  riskier security implies that it has a lower level of market  risk, and this is the only relevant risk –The  riskier security contains relatively more (irrelevant) security-specific risk

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 52 A Brand Manager Most investors have the opportunity to eliminate most individual risk from their portfolio; but consider a product manager’s exposure to riskMost investors have the opportunity to eliminate most individual risk from their portfolio; but consider a product manager’s exposure to risk If a brand manager’s productsIf a brand manager’s products perform well, promotion, higher salary, and greater autonomy followperform well, promotion, higher salary, and greater autonomy follow perform badly, humiliation, unemployment and poverty followperform badly, humiliation, unemployment and poverty follow

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 53 A Brand Manager Now assume that a new product is available for inclusion in the brand, but given its  -risk and expected return, it falls below the sml, and hence is not in the investors’ interestsNow assume that a new product is available for inclusion in the brand, but given its  -risk and expected return, it falls below the sml, and hence is not in the investors’ interests –The manager discovers that the new product reduces his total risk, and acts in his own interests (rather than the investors’), and accepts the product (agency problem)

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 54 The Portfolio Manager Remember (last chapter) we had not resolved the issue how to evaluate the performance of a portfolio manager, but given the CAPM a resolution is at handRemember (last chapter) we had not resolved the issue how to evaluate the performance of a portfolio manager, but given the CAPM a resolution is at hand If your portfolio is producing actual returns with a lower beta than the sml specifies (with statistical significance), then you should certainly not be firedIf your portfolio is producing actual returns with a lower beta than the sml specifies (with statistical significance), then you should certainly not be fired

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 55 The Portfolio Manager The further a well diversified portfolio consistently lies above (below) the sml, the better (worse) the fund manager’s performanceThe further a well diversified portfolio consistently lies above (below) the sml, the better (worse) the fund manager’s performance –There are several measures of this distance, but this topic is better left for another day

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 56 Alpha Fund and the Security Market Line

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 57 Alpha Fund and the Capital Market Line

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 58 How to Win Investment Games – –You may have been asked to take part in an investment game where you ‘given’ $100,000 to manage for a semester; winner takes all – –The overwhelming chances are that the winning student uses poor financial practices

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 59 How to Win Investment Games (Continued) – –The criteria of success for the game differs significantly from real-life investing, so your strategy for winning is likely to be different – –If you diversify away unsystematic risk--even if you have some kind of informational advantage over your competition--you are very unlikely to win the game – –To win, you need individual risk to separate you from the crowd – –Unlike a real investor you don’t have real downside-risk your upside-potential materializes only by being first

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall Valuation and Regulating Rates of Return Beta may be used to obtain the discount factor for a projectBeta may be used to obtain the discount factor for a project Assume a project is similar to the projects undertaken by another firm, ‘Betaful’Assume a project is similar to the projects undertaken by another firm, ‘Betaful’ Betaful is financed by 20% short-term debt, and 80% equity, and its  is 1.3 (assume debt is risk-free)Betaful is financed by 20% short-term debt, and 80% equity, and its  is 1.3 (assume debt is risk-free) Your optimal capital structure is 40% (risk- free) debt, and 60% equityYour optimal capital structure is 40% (risk- free) debt, and 60% equity

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 61 Valuation and Regulating Rates of Return Assume the market rate is 15%, and the risk- free rate is 5%Assume the market rate is 15%, and the risk- free rate is 5% Compute the beta of Betaful’s operationsCompute the beta of Betaful’s operations

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 62 Valuation and Regulating Rates of Return Beta of Betaful’s operations is equal to the beta of our new operationBeta of Betaful’s operations is equal to the beta of our new operation To find the required return on the new project, apply the CAPMTo find the required return on the new project, apply the CAPM

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 63 Valuation and Regulating Rates of Return Assume that your company is just a vehicle for the new project, then the beta of your unquoted equity isAssume that your company is just a vehicle for the new project, then the beta of your unquoted equity is

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 64 Valuation and Regulating Rates of Return Assume that your company has an expected dividend of $6 next year, and that it will grow annually at a rate of 4% forever, the value of a share isAssume that your company has an expected dividend of $6 next year, and that it will grow annually at a rate of 4% forever, the value of a share is

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 65 Valuation and Regulating Rates of Return Regulators use the CAPM to establish a ‘fair’ rate of return on invested capital in public utilities, given the level of riskRegulators use the CAPM to establish a ‘fair’ rate of return on invested capital in public utilities, given the level of risk

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall Modifications and Alternatives to the CAPM Starting in the 1970s researchers found that CAPM did not seem to fully explain the structure of expected returns on assets.Starting in the 1970s researchers found that CAPM did not seem to fully explain the structure of expected returns on assets. A consensus emerged that the original version of the CAPM needed to be modified.A consensus emerged that the original version of the CAPM needed to be modified.

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 67 Potential explanations for the apparent deviations from the CAPM: 1. 1.CAPM actually does hold, but the “market” portfolios used in the testing were incomplete and inadequate representations of the true market portfolio.

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 68 Potential explanations for the apparent deviations from the CAPM: 2. 2.The focus on market imperfections is not contemplated in the CAPM. Such as: – –borrowing costs and constraints – –short sales restrictions and costs – –different tax treatments for various assets – –The nontradability of some important assets such as human capital

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 69 Potential explanations for the apparent deviations from the CAPM: Add greater realism to the modeling assumptions, while maintaining the CAPM’s basic methodology. One such model is the multifactor Intertemporal Capital Asset Pricing Model (ICAPM)

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 70 Intertemporal Capital Asset Pricing Model (ICAPM) In ICAPM equilibrium risk premiums on securities in this dynamic model come from several dimension of risks – –reflected not only by their return sensitivities or beta on the market portfolio – –but also by their sensitivity to other systematic risks such as: changes in interest rates expected returns on assets changes in consumption good prices.

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 71 Alternative Theories Arbitrage Pricing Theory (APT) – –APT states that a relation similar to the Security Market Line can exist even if investors are not mean-variance optimizers.

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 72 Arbitrage Pricing Theory (APT) If there are enough different securities to “diversify away” all but market risk Then an expected-return-to-beta relation will exist as a consequence of there not being any arbitrage opportunities