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Investment Analysis and Portfolio Management Lecture 7 Gareth Myles The Capital Asset Pricing Model (CAPM) The CAPM is a model of equilibrium in the market for securities. Previous lectures have addressed the question of how investors should choose assets given the observed structure of returns. Now the question is changed to: If investors follow these strategies, how will returns be determined in equilibrium? The Capital Asset Pricing Model (CAPM) The simplest and most fundamental model of equilibrium in the security market Builds on the Markowitz model of portfolio choice Aggregates the choices of individual investors Trading ensures an equilibrium where returns adjust so that the demand and supply of assets are equal Many modifications/extensions can be made But basic insights always extend Assumptions The CAPM is built on a set of assumptions Individual investors Investors evaluate portfolios by the mean and variance of returns over a one period horizon Preferences satisfy non-satiation Investors are risk averse Trading conditions Assets are infinitely divisible Borrowing and lending can be undertaken at the riskfree rate of return There are no taxes or transactions costs Assumptions The risk-free rate is the same for all Information flows perfectly The set of investors All investors have the same time horizon Investors have identical expectations Assumptions The first six assumptions are the Markowitz model The seventh and eighth assumptions add a perfect capital market and perfect information The final two assumptions make all investors identical except for their degree of risk aversion Direct Implications All investors face the same efficient set of portfolios rp rf r MVP σ MVP σp Direct Implications All investors choose a location on the efficient frontier The location depends on the degree of risk aversion The chosen portfolio mixes the risk-free asset and portfolio M of risky assets Separation Theorem The optimal combination of risky assets is determined without knowledge of preferences All choose portfolio M This is the Separation Theorem M must be the market portfolio of risky assets All investors hold it to a greater or lesser extent No other portfolio of risky assets is held There is a question about the interpretation of this portfolio Equilibrium The only assets that need to be marketed are: The risk-free asset A mutual fund representing the market portfolio No other assets are required In equilibrium there can be no short sales of the risky assets All investors buy the same risky assets No-one can be short since all would be short If all are short the market is not in equilibrium Equilibrium Equilibrium occurs when the demand for assets matches the supply This also applies to the risk-free Borrowing must equal lending This is achieved by the adjustment of asset prices As prices change so do the returns on the assets This process generates an equilibrium structure of returns The Capital Market Line All efficient portfolios must lie on this line rM − r f Slope = Equation of the line rM σM rM − rf rp = rf + σM rp σ p rf σM σp Interpretation rf is the reward for "time" Patience is rewarded Investment delays consumption rM − rf is the reward for accepting "risk" σM The market price of risk Judged to be equilibrium reward Obtained by matching demand to supply Security Market Line Now consider the implications for individual assets Graph covariance against return The risk on the market portfolio is σM The covariance of the risk-free asset is zero The covariance of the market with the market is σ M2 Security Market Line Can mix M and the riskfree asset along the line If there was a portfolio above the line all investors would buy it No investor would hold one below The equation of the line is rM − r f ri = r f + σ iM 2 σ M rp rM M rf σ M2 σ iM Security Market Line σ iM = 2 σM Define The equation of the line becomes β iM [ ] ri = rf + rM − rf β iM This is the security market line (SML) Security Market Line There is a linear trade-off between risk measured by β iM and return ri In equilibrium all assets and portfolios must have risk-return combinations that lie on this line rp rf β iM Market Model and CAPM Market model uses β iI CAPM uses β iM β iI is derived from an assumption about the determination of returns β iM it is derived from a statistical model the index is chosen not specified by any underlying analysis is derived from an equilibrium theory Market Model and CAPM In addition: I is usually assumed to be the market index, but in principal could be any index M is always the market portfolio There is a difference between these But they are often used interchangeably The market index is taken as an approximation of the market portfolio Estimation of CAPM Use the regression equation [ ] ri − r f = α iM + β iM rM − r f + ε i Take the expected value [ ] [ E ri − r f = α iM + β iM E rM − r f The security market line implies It also shows α iM = 0 α iI = [1 − β iM ] r f ] CAPM and Pricing CAPM also implies the equilibrium asset prices The security market line is [ ] ri = r f + rM − r f βiM But pi (1) − pi ( 0) ri = pi ( 0 ) where pi(0) is the value of the asset at time 0 and pi(1) is the value at time 1 CAPM and Pricing So the security market line gives p i (1) − p i ( 0) = r f + β iM rM − r f p i ( 0) [ ] This can be rearranged to find p i (1) p i ( 0) = 1 + r f + β iM rM − r f The price today is related to the expected value at the end of the holding period [ ] CAPM and Project Appraisal Consider an investment project It requires an investment of p(0) today It provides a payment of p(1) in a year Should the project be undertaken? The answer is yes if the present discounted value (PDV) of the project is positive CAPM and Project Appraisal If both p(0) and p(1) are certain then the riskfree interest rate is used to discount The PDV is p(1) PDV = − p( 0 ) + 1+ rf The decision is to accept project if p(1) p( 0) < 1+ rf CAPM and Project Appraisal Now assume p(1) is uncertain Cannot simply discount at risk-free rate if investors are risk averse For example using p (1) PDV = − p( 0) + 1+ rf will over-value the project With risk aversion the project is worth less than its expected return U ( p (1)) > EU ( p (1)) CAPM and Project Appraisal One method to obtain the correct value is to adjust the rate of discount to reflect risk But by how much? The CAPM pricing rule gives the answer The correct PDV of the project is p(1) PDV = − p(0) + 1 + r f + β p [rM − r f ] [...]... risk measured by β iM and return ri In equilibrium all assets and portfolios must have risk-return combinations that lie on this line rp rf β iM Market Model and CAPM Market model uses β iI CAPM uses β iM β iI is derived from an assumption about the determination of returns β iM it is derived from a statistical model the index is chosen not specified by any underlying analysis is derived... value at the end of the holding period [ ] CAPM and Project Appraisal Consider an investment project It requires an investment of p(0) today It provides a payment of p(1) in a year Should the project be undertaken? The answer is yes if the present discounted value (PDV) of the project is positive CAPM and Project Appraisal If both p(0) and p(1) are certain then the riskfree interest... analysis is derived from an equilibrium theory Market Model and CAPM In addition: I is usually assumed to be the market index, but in principal could be any index M is always the market portfolio There is a difference between these But they are often used interchangeably The market index is taken as an approximation of the market portfolio Estimation of CAPM Use the regression equation... security market line implies It also shows α iM = 0 α iI = [1 − β iM ] r f ] CAPM and Pricing CAPM also implies the equilibrium asset prices The security market line is [ ] ri = r f + rM − r f βiM But pi (1) − pi ( 0) ri = pi ( 0 ) where pi(0) is the value of the asset at time 0 and pi(1) is the value at time 1 CAPM and Pricing So the security market line gives p i (1) − p i ( 0) = r f + β iM... reward for "time" Patience is rewarded Investment delays consumption rM − rf is the reward for accepting "risk" σM The market price of risk Judged to be equilibrium reward Obtained by matching demand to supply Security Market Line Now consider the implications for individual assets Graph covariance against return The risk on the market portfolio is σM The covariance of the risk-free...Equilibrium Equilibrium occurs when the demand for assets matches the supply This also applies to the risk-free Borrowing must equal lending This is achieved by the adjustment of asset prices As prices change so do the returns on the assets This process generates an equilibrium structure of returns The Capital Market Line All efficient portfolios must lie on this line rM − r f Slope... The risk on the market portfolio is σM The covariance of the risk-free asset is zero The covariance of the market with the market is σ M2 Security Market Line Can mix M and the riskfree asset along the line If there was a portfolio above the line all investors would buy it No investor would hold one below The equation of the line is rM − r f ri = r f + σ iM 2 σ M rp rM M rf σ M2... decision is to accept project if p(1) p( 0) < 1+ rf CAPM and Project Appraisal Now assume p(1) is uncertain Cannot simply discount at risk-free rate if investors are risk averse For example using p (1) PDV = − p( 0) + 1+ rf will over-value the project With risk aversion the project is worth less than its expected return U ( p (1)) > EU ( p (1)) CAPM and Project Appraisal One method to obtain ... between risk measured by β iM and return ri In equilibrium all assets and portfolios must have risk-return combinations that lie on this line rp rf β iM Market Model and CAPM Market model uses β... market portfolio of risky assets All investors hold it to a greater or lesser extent No other portfolio of risky assets is held There is a question about the interpretation of this portfolio. .. the risk-free asset and portfolio M of risky assets Separation Theorem The optimal combination of risky assets is determined without knowledge of preferences All choose portfolio M This