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Capital Asset-Pricing Model (CAPM) and Arbitrage Pricing Theory (APT). Objectives Understand the capital market line (CML), which contains only efficient portfolios Explain the separation theorem

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Capital Asset-Pricing Model (CAPM) and Arbitrage Pricing Theory (APT)


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  1. Capital Asset-Pricing Model (CAPM) and Arbitrage Pricing Theory (APT) • Objectives • Understand the capital market line (CML), which contains only efficient portfolios • Explain the separation theorem • Understand why the security market line (SML) describes the CAPM equilibrium relationship between risk and expected rate of return • Explain the APT; compare and contrast the CAPM and APT Chapter 15: CAPM and APT

  2. Capital Asset Pricing Model (CAPM) • Investors demand a premium for bearing risk (the higher the riskiness of a security, the higher the expected return required to induce investors to buy (or to hold) it. • If investors are primarily concerned with portfolio risk rather than the risk of the individual securities in the portfolio, how should the riskiness of an individual stock be measured? • CAPM Chapter 15: CAPM and APT

  3. Capital Asset Pricing Model (CAPM) • CAPM – an important tool used to analyze the relationship between risk and rates of return • The primary conclusion of the CAPM is this: The relevant riskiness of an individual stock is its contribution to the riskiness of a well-diversified portfolio. Chapter 15: CAPM and APT

  4. Systematic and Non-systematic Risk • In discussing the CAPM, it is essential to stress the importance of distinguishing between 2 types of risk: i. Systematic risk (market risk) –the variability of a stock’s total return that is directly associated with overall movements in general economic (or stock market activity); and Chapter 15: CAPM and APT

  5. Systematic and Non-systematic Risk ii. Non-systematic risk – a specific component, representing that portion in the variability of a stock’s total return that is not related to the variability in general economic (market) activity. Chapter 15: CAPM and APT

  6. Total Risk = Systematic Risk + non-systematic risk = Market Risk + non-market risk = Non-diversifiable risk + diversifiable risk Virtually all securities have some systematic risk, whether bonds or stock, because systematic risk directly encompasses interest rate risk, market risk and inflation risk. Chapter 15: CAPM and APT

  7. Systematic and Non-systematic Risk • After the non-systematic risk is eliminated, what is left is the non-diversifiable portion (or market risk or systematic risk) • Market risk is inescapable because the risk of the overall market cannot be avoided (if stock market declines sharply, most stocks will be adversely affected) • Investors can construct a diversified portfolio and eliminate part of the total risk – non-systematic risk Chapter 15: CAPM and APT

  8. Systematic and Non-systematic Risk Portfolio Risk Diversifiable (non-systematic) risk Total risk Non-diversifiable (systematic) risk 1 30 40 10 20 Number of stocks in the portfolio Chapter 15: CAPM and APT

  9. Systematic and Non-systematic Risk • As more securities are added, the non-systematic risk becomes smaller and smaller, and the total risk for the portfolio approaches its systematic risk • No matter how much we diversify, we cannot eliminate systematic risk • Empirical evidence suggests that at the very least 30 stocks, and perhaps more, are needed for a well-diversified portfolio. Chapter 15: CAPM and APT

  10. The Concept of Beta () • A measure of the extent to which the returns on a given stock move with the stock market • Since the stock’s beta measures its contribution to the riskiness of a portfolio, beta is the theoretically correct measure of the stock’s riskiness Chapter 15: CAPM and APT

  11. The Concept of Beta () Return on stock i (%) Stock H, High Risk:  = 2 Stock A, Average Risk:  = 1 Stock L, Low Risk:  = 0.5 Return on the market (%) Risk of Individual Stocks Chapter 15: CAPM and APT

  12. Capital Asset Pricing Model (CAPM) • Equilibrium model that underlines all modern financial theory • Derived using principles of diversification with simplified assumption Chapter 15: CAPM and APT

  13. Capital Asset Pricing Model (CAPM) Assumptions • Individual investors are price takers • Single-period investment horizon • Investment are limited to traded financial asset • No taxes, and transaction costs • Information is costless and available to all investors • Investors are rational mean-variance optimizes • Homogeneous expectations Chapter 15: CAPM and APT

  14. Resulting Equilibrium Conditions • All investors will hold the same portfolio for risky assets – market portfolio • Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value • Risk premium on the market depends on the average risk aversion of all market participants • Risk premium on an individual security is a function of its covariance with the market Chapter 15: CAPM and APT

  15. Figure 5.1: Investment Opportunities in Risk-Return Space Markowitz Efficient Portfolios Individual assets contain both diversifiable and non-diversifiable risk and are not efficient investments. Efficient Frontier—these portfolios contain only undiversifiable risk Individual assets Chapter 15: CAPM and APT

  16. Borrowing and Lending at the Risk-Free Rate • If we add borrowing and lending at the risk-free rate to Figure 5.1 the investment opportunities can be extended • A riskless asset must have a variance of zero, by definition • The efficient frontier curve is now dominated by the capital market line Chapter 15: CAPM and APT

  17. Borrowing and Lending at the Risk-Free Rate M = market portfolio Chapter 15: CAPM and APT

  18. The Capital Market Line (CML) • The formal equation for the capital market line is: RFR = Risk Free Rate E(rM) – RFR = market risk premium Chapter 15: CAPM and APT

  19. The Market Portfolio • Portfolio M is known as the market portfolio • Equilibrium portfolio containing all the assets in the world in the proportions they are supplied • Represents the single portfolio all rational investors want to own • Because it can be used to create the dominant CML • The portfolio expected return, E(Rp) is related positively to its standard deviation, p Chapter 15: CAPM and APT

  20. The Separation Theorem • All investors desiring Markowitz diversification will select Portfolio M • The next question is: • How should the investment in Portfolio M be financed? • Highly risk-averse investors will select a lending portfolio • Aggressive investors will select a leveraged (borrowing) portfolio Chapter 15: CAPM and APT

  21. The Separation Theorem • The decision to invest in portfolio M is separate from the decision as to whether the investor will be a borrower or a lender Chapter 15: CAPM and APT

  22. Assumptions Underlying Portfolio Theory • Four assumptions underlie all portfolio theories based on the efficient frontier • Rate of return is the most important investment outcome • Investor’s risk estimates are proportional to the standard deviation or variance they perceive • Investors are willing to base their decisions on only the expected return and variance (or standard deviation) of the expected return • For any risk class, investors desire a higher rate of return to a lower one Chapter 15: CAPM and APT

  23. Assumptions Underlying the CML, SML and CAPM • Any amount of money can be borrowed or lent at the risk-free rate of interest • All investors visualize the same expected return, risk and correlation for any specified asset (homogeneous expectations) • All investors have a one-period investment horizon • All investments are infinitely divisible • No taxes or transaction costs exist • No inflation or changes in interest rates exist • Capital markets are a static equilibrium (supply equals demand) • The market portfolio contains all assets in the proportions in which they exist Chapter 15: CAPM and APT

  24. Assumptions Underlying the CML, SML and CAPM • Assumptions are unrealistic • But provide a concrete foundation • Final test should be the theory’s predictive power, not the realism of its assumptions Chapter 15: CAPM and APT

  25. Rationale for the SML • Economic rationale based on the manner in which the assets in a diversified portfolio covary • Markowitz portfolio analysis • Find securities with low covariances with the market • Reduce portfolio risk • Since investors will prefer securities with low covariances, securities with high (low) covariances will fall (rise) in price Chapter 15: CAPM and APT

  26. Security Market Line • Equation of the SML Represents the risk-adjusted rate to be used when finding the present value of an asset (when considering systematic risk). Chapter 15: CAPM and APT

  27. Security Market Line In equilibrium every asset should be priced as a linear function of its covariance with the market. Chapter 15: CAPM and APT

  28. SML vs. CML • Individual securities will lie on the SML if they are correctly priced • But individual securities will lie below the CML • They have a high amount of diversifiable risk • Efficient portfolios will lie on the CML • The SML does not consider diversifiable risk • Can be eliminated via simple diversification Chapter 15: CAPM and APT

  29. Restating the SML • The beta regression slope coefficient is an index of an asset’s undiversifiable or systematic risk • An asset’s covariance of returns with the market is also a measure of undiversifiable risk • Two measures are mathematically equivalent Chapter 15: CAPM and APT

  30. Restating the SML • SML may be equivalently restated using the beta coefficient • The restated SML: Chapter 15: CAPM and APT

  31. Over- and Under-Priced Assets • Point U (Slide 27) is an underpriced asset • Has an abnormally high return for its systematic risk • Will experience high demand and a subsequent increase in price until return equates to U • Point O (Slide 27) is an overpriced asset • Has an abnormally low return for its systematic risk • Price will fall due to lack of demand • Assets on the SML are in equilibrium and will remain so until • Systematic risk changes, the risk-free rate changes, etc. Chapter 15: CAPM and APT

  32. Negative Correlation with the Market Portfolio • Point N (Slide 27) is a security with a negative covariance (beta) with the market • The equilibrium rate of return is below the risk-free rate • Gold mining stocks may have negative betas Chapter 15: CAPM and APT

  33. Discontinuity Between Expected Value Theory and Historical Data • Expected returns are determined by expected risk • Investors do not use only historical data to make investment decisions • Plans are made based on future expectations • If probability distributions of returns have changed over time, historical data will not impact future expectations Chapter 15: CAPM and APT

  34. Relaxing CML and SML Assumptions • Different interest rates for borrowing and lending • Realistically borrowers are charged a higher rate than lenders earn • Results in two tangency portfolios, Md and Mb The CML has a curved efficient section between Md and Mb. Different borrowing and lending rates result in two SMLs (with different intercepts). Chapter 15: CAPM and APT

  35. Relaxing CML and SML Assumptions • Transactions costs create friction • Transactions costs include taxes, commissions, search costs, fees, etc. • Can be modeled as a ‘band’ below the CML and SML • CML and lower edge of ‘band’ are 1-2 percentage points apart • Markets would never reach the theoretical equilibrium Chapter 15: CAPM and APT

  36. Relaxing CML and SML Assumptions • Different Tax Rates for Capital Gains • Most countries have a lower capital gains tax rate vs. ordinary income tax rate • Every investor would have a slightly different CML and SML in terms of after-tax returns • Static equilibrium could not emerge Chapter 15: CAPM and APT

  37. Relaxing CML and SML Assumptions • Indivisibilities • All assets are not infinitely divisible • Changes SML to a dotted line with each dot representing opportunity available with indivisible assets • Conclusion about assumptions • Even though model is not derivable under realistic assumptions it still rationalizes complex behavior • Offers suggestions about directions prices should move Chapter 15: CAPM and APT

  38. Criticisms and Tests of SMLS • Some criticize SML’s simplicity • Systematic risk is the sole determinant of expected returns and asset prices • Critics suggest adding more explanatory variables to SML Chapter 15: CAPM and APT

  39. Liquidity of Investments • Transactions costs represent liquidity costs • Amihud & Mendelson (1986) suggest that an illiquidity premium be added to SML • Should increase at a decreasing rate as liquidation costs increase Chapter 15: CAPM and APT

  40. Econometric Analysis of Empirical Data • SML is a positive linear relationship with betas explaining expected returns • Can be testing performing two-stage regression analysis • A first-pass regression is run to determine the betas for the N firms in the sample • The information obtained in this regression is used as inputs in a second-pass regression • The second-pass regression is: Chapter 15: CAPM and APT

  41. Econometric Problems with Characteristic Line • The characteristic line has been subject to statistical testing • Blume (1975) found that betas tend to regress toward +1 • Said to suffer from intertemporal instability or to be sample dependent • Francis (1979) found that betas, standard deviations and correlations with the market portfolio were all sample dependent • Fabozzi and Francis (1978-1979) found that this instability does not negate the value of the models Chapter 15: CAPM and APT

  42. Econometric Problems with SML • Theoretical SML argues that the asset’s beta will determine expected returns • Empirical studies report other variables have explanatory power, including: • The firm’s size as measured by total market value of equity • Book-value-to-market equity • Earnings-price ratio • Roll (1977) suggests that researchers did not use an efficient market portfolio Chapter 15: CAPM and APT

  43. Econometric Problems with SML • Basu (1977, 1983) and Reinganum (1981) find that excess returns on equity are due, in part, to earnings-price ratios • Banz (1981), Reinganum (1981) and Keim (1983) find that smaller firms tend to have larger averaged returns than predicted by SML • Fama and French (1992, 1993) among others suggest that book-value-to-market equity contributes to a firm’s returns Chapter 15: CAPM and APT

  44. Econometric Problems with SML • Most of the research suffered from the Errors-In-Variables (EIV) problem • Occurs because the true beta coefficients are unobservable • Researchers use estimates as proxies for the true betas • Hand, Kothari and Wasley (1993) and Kim (1993) show the EIV problem leads to an underestimation of beta as an explanatory variable and an overestimation of the other variables’ importance Chapter 15: CAPM and APT

  45. Econometric Problems with SML • Litzenberger and Ramaswamy (1979), Shanken (1992) and Kim (1995) suggest altering the two-pass regression • Kim (1995, 1997) suggests that using individual assets in empirical tests (vs. portfolios of assets) minimizes the EIV problem • Kim’s data fit the ‘pure’ SML better than it fit the SML including additional explanatory variables Chapter 15: CAPM and APT

  46. Arbitrage Pricing Theory • Arbitrage Pricing Theory (APT) was developed by Stephen Ross • One-factor APT assumes that assets’ returns are determined by a single systematic risk factor (F) • Factor betas measure how sensitive an asset’s return is to the risk factor F • Very similar to characteristic line except F cannot be the market portfolio Chapter 15: CAPM and APT

  47. Arbitrage Pricing Theory • An APT risk factor could be: • Gross domestic product • A market interest rate • The rate of inflation • Any other random variable that impacts security prices • The expected value of the risk factor (E(F)) = 0 • Because fluctuations around the mean always sum to zero Chapter 15: CAPM and APT

  48. Arbitrage Pricing Theory • We focus on betas and ignore unsystematic issues—residual variances and total variances • Because unsystematic risk can be easily diversified • Assets with identical betas should have identical rates of return, because they are equally risky • Otherwise, arbitrage would be possible • These assets should also have identical intercepts Chapter 15: CAPM and APT

  49. Arbitrage Pricing Theory Line • APT Line for a single risk factor is: • The arbitrage pricing line is similar to the SML • The difference is that their common risk factors must differ • Only the SML can have the market portfolio as its common risk factor Chapter 15: CAPM and APT

  50. E(ri) = Expected Return U Underpriced asset Slope =  = risk premium RFR O Overpriced asset 0 Risk class of assets O and U Factor beta Arbitrage Pricing Theory Line U and O violate the law of one price—they are in the same risk class but have different expected rates of return. Chapter 15: CAPM and APT