Investment Analysis 107 August 2011


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Investment Analysis 107 August 2011

  1. 1. Investment Analysis and Portfolio Management August 2011 MCR
  2. 2. The Capital Asset Pricing Model (CAPM) <ul><li>The CAPM is a model of equilibrium in the market for securities. </li></ul><ul><li>Previous lectures have addressed the question of how investors should choose assets given the observed structure of returns. </li></ul><ul><li>Now the question is changed to: </li></ul><ul><ul><li>If investors follow these strategies, how will returns be determined in equilibrium? </li></ul></ul>
  3. 3. The Capital Asset Pricing Model (CAPM) <ul><li>The simplest and most fundamental model of equilibrium in the security market </li></ul><ul><ul><li>Builds on the Markowitz model of portfolio choice </li></ul></ul><ul><ul><li>Aggregates the choices of individual investors </li></ul></ul><ul><li>Trading ensures an equilibrium where returns adjust so that the demand and supply of assets are equal </li></ul><ul><li>Many modifications/extensions can be made </li></ul><ul><ul><li>But basic insights always extend </li></ul></ul>
  4. 4. Assumptions <ul><li>The CAPM is built on a set of assumptions </li></ul><ul><li>Individual investors </li></ul><ul><ul><li>Investors evaluate portfolios by the mean and variance of returns over a one period horizon </li></ul></ul><ul><ul><li>Preferences satisfy non-satiation </li></ul></ul><ul><ul><li>Investors are risk averse </li></ul></ul><ul><li>Trading conditions </li></ul><ul><ul><li>Assets are infinitely divisible </li></ul></ul><ul><ul><li>Borrowing and lending can be undertaken at the risk-free rate of return </li></ul></ul><ul><ul><li>There are no taxes or transactions costs </li></ul></ul>
  5. 5. Assumptions <ul><ul><li>The risk-free rate is the same for all </li></ul></ul><ul><ul><li>Information flows perfectly </li></ul></ul><ul><li>The set of investors </li></ul><ul><ul><li>All investors have the same time horizon </li></ul></ul><ul><ul><li>Investors have identical expectations </li></ul></ul>
  6. 6. Assumptions <ul><li>The first six assumptions are the Markowitz model </li></ul><ul><li>The seventh and eighth assumptions add a perfect capital market and perfect information </li></ul><ul><li>The final two assumptions make all investors identical except for their degree of risk aversion </li></ul>
  7. 7. Direct Implications <ul><li>All investors face the same efficient set of portfolios </li></ul>
  8. 8. Direct Implications <ul><li>All investors choose a location on the efficient frontier </li></ul><ul><li>The location depends on the degree of risk aversion </li></ul><ul><li>The chosen portfolio mixes the risk-free asset and portfolio M of risky assets </li></ul>
  9. 9. Separation Theorem <ul><li>The optimal combination of risky assets is determined without knowledge of preferences </li></ul><ul><ul><li>All choose portfolio M </li></ul></ul><ul><ul><li>This is the Separation Theorem </li></ul></ul><ul><li>M must be the market portfolio of risky assets </li></ul><ul><ul><li>All investors hold it to a greater or lesser extent </li></ul></ul><ul><ul><li>No other portfolio of risky assets is held </li></ul></ul><ul><ul><li>There is a question about the interpretation of this portfolio </li></ul></ul>
  10. 10. Equilibrium <ul><li>The only assets that need to be marketed are: </li></ul><ul><ul><li>The risk-free asset </li></ul></ul><ul><ul><li>A mutual fund representing the market portfolio </li></ul></ul><ul><ul><li>No other assets are required </li></ul></ul><ul><li>In equilibrium there can be no short sales of the risky assets </li></ul><ul><ul><li>All investors buy the same risky assets </li></ul></ul><ul><ul><li>No-one can be short since all would be short </li></ul></ul><ul><ul><li>If all are short the market is not in equilibrium </li></ul></ul>
  11. 11. Equilibrium <ul><li>Equilibrium occurs when the demand for assets matches the supply </li></ul><ul><ul><li>This also applies to the risk-free </li></ul></ul><ul><ul><li>Borrowing must equal lending </li></ul></ul><ul><li>This is achieved by the adjustment of asset prices </li></ul><ul><li>As prices change so do the returns on the assets </li></ul><ul><li>This process generates an equilibrium structure of returns </li></ul>
  12. 12. The Capital Market Line <ul><li>All efficient portfolios must lie on this line </li></ul><ul><li>Slope = </li></ul><ul><li>Equation of the line </li></ul>
  13. 13. Interpretation <ul><li>r f is the reward for &quot;time&quot; </li></ul><ul><ul><li>Patience is rewarded </li></ul></ul><ul><ul><li>Investment delays consumption </li></ul></ul><ul><li>is the reward for accepting &quot;risk&quot; </li></ul><ul><ul><li>The market price of risk </li></ul></ul><ul><ul><li>Judged to be equilibrium reward </li></ul></ul><ul><ul><li>Obtained by matching demand to supply </li></ul></ul>
  14. 14. Security Market Line <ul><li>Now consider the implications for individual assets </li></ul><ul><li>Graph covariance against return </li></ul><ul><ul><li>The risk on the market portfolio is </li></ul></ul><ul><ul><li>The covariance of the risk-free asset is zero </li></ul></ul><ul><ul><li>The covariance of the market with the market is </li></ul></ul>
  15. 15. Security Market Line <ul><li>Can mix M and the risk-free asset along the line </li></ul><ul><ul><li>If there was a portfolio above the line all investors would buy it </li></ul></ul><ul><ul><li>No investor would hold one below </li></ul></ul><ul><li>The equation of the line is </li></ul>M
  16. 16. Security Market Line <ul><li>D efine </li></ul><ul><li>The equation of the line becomes </li></ul><ul><li>This is the security market line (SML) </li></ul>
  17. 17. Security Market Line <ul><li>There is a linear trade-off between risk measured by and return </li></ul><ul><li>In equilibrium all assets and portfolios must have risk-return combinations that lie on this line </li></ul>
  18. 18. Market Model and CAPM <ul><li>Market model uses </li></ul><ul><li>CAPM uses </li></ul><ul><li>is derived from an assumption about the determination of returns </li></ul><ul><ul><li>it is derived from a statistical model </li></ul></ul><ul><ul><li>the index is chosen not specified by any underlying analysis </li></ul></ul><ul><li>is derived from an equilibrium theory </li></ul>
  19. 19. Market Model and CAPM <ul><li>In addition: </li></ul><ul><ul><li>I is usually assumed to be the market index, but in principal could be any index </li></ul></ul><ul><ul><li>M is always the market portfolio </li></ul></ul><ul><li>There is a difference between these </li></ul><ul><li>But they are often used interchangeably </li></ul><ul><li>The market index is taken as an approximation of the market portfolio </li></ul>
  20. 20. Estimation of CAPM <ul><li>Use the regression equation </li></ul><ul><li>Take the expected value </li></ul><ul><li>The security market line implies </li></ul><ul><li>It also shows </li></ul>
  21. 21. CAPM and Pricing <ul><li>CAPM also implies the equilibrium asset prices </li></ul><ul><li>The security market line is </li></ul><ul><li>But </li></ul><ul><li>where p i (0) is the value of the asset at time 0 and p i (1) is the value at time 1 </li></ul>
  22. 22. CAPM and Pricing <ul><li>So the security market line gives </li></ul><ul><li>This can be rearranged to find </li></ul><ul><li>The price today is related to the expected value at the end of the holding period </li></ul>
  23. 23. CAPM and Project Appraisal <ul><li>Consider an investment project </li></ul><ul><li>It requires an investment of p (0) today </li></ul><ul><li>It provides a payment of p (1) in a year </li></ul><ul><li>Should the project be undertaken? </li></ul><ul><li>The answer is yes if the present discounted value ( PDV ) of the project is positive </li></ul>
  24. 24. CAPM and Project Appraisal <ul><li>If both p (0) and p (1) are certain then the risk-free interest rate is used to discount </li></ul><ul><li>The PDV is </li></ul><ul><li>The decision is to accept project if </li></ul>
  25. 25. CAPM and Project Appraisal <ul><li>Now assume p (1) is uncertain </li></ul><ul><li>Cannot simply discount at risk-free rate if investors are risk averse </li></ul><ul><li>For example using </li></ul><ul><li>will over-value the project </li></ul><ul><li>With risk aversion the project is worth less than its expected return </li></ul>
  26. 26. CAPM and Project Appraisal <ul><li>One method to obtain the correct value is to adjust the rate of discount to reflect risk </li></ul><ul><li>But by how much? </li></ul><ul><li>The CAPM pricing rule gives the answer </li></ul><ul><li>The correct PDV of the project is </li></ul>