Chapter 9

1. Chapter 9 AN INTRODUCTION TO ASSET PRICING MODELS

2. Chapter 9 Questions • What are the assumptions of the capital asset pricing model? • What is a risk-free asset and what are its risk-return characteristics? • What is the covariance and correlation between the risk-free asset and a risky asset or portfolio of risky assets? • What is the expected return when we combine the risk-free assets and a portfolio of risky assets?

3. Chapter 9 Questions • What is the standard deviation when we combine the risk-free asset and a portfolio of risky assets? • When you combine the risk-free asset and a portfolio of risky assets on the Markowitz efficient frontier, what does the set of possible portfolios look like? • Given the initial set of portfolio possibilities with a risk-free asset, what happens when you add financial leverage (that is, borrow)?

4. Chapter 9 Questions • What is the market portfolio, what assets are included, and what are the relative weights? • What is the capital market line (CML)? • What do we mean by complete diversification? • How do we measure diversification for an individual portfolio? • What are systematic and unsystematic risk?

5. Chapter 9 Questions • Given the capital market line (CML), what is the separation theorem? • Given the CML, what is the relevant risk measure for an individual risky asset? • What is the security market line (SML) and how does it differ from the CML? • What is beta and why is it referred to as a standardized measure of systematic risk?

6. Chapter 9 Questions • How can we use the SML to determine the expected (required) rate of return for a risky asset? • Using the SML, what do we mean by an undervalued and overvalued security, and how do we determine whether an asset is undervalued or overvalued? • What is an asset’s characteristic line and how do we compute the characteristic line for an asset?

7. Chapter 9 Questions • What is the impact on the characteristic line when we compute it using different return intervals (such as weekly versus monthly) and when we employ different proxies (that is, benchmarks) for the market portfolio (for example, the S&P 500 versus a global stock index)?

8. Chapter 9 Questions • What is the basic conceptual difference between the CAPM and the several multifactor models currently available? • When dealing with multifactor models, what is the difference between macroeconomic and microeconomic models?

9. Capital Market Theory: An Overview • Capital market theory extends portfolio theory and seeks to develops a model for pricing all risky assets based on their relevant risks • Asset Pricing Models • Capital asset pricing model (CAPM) allows for the calculation of the required rate of return for any risky asset based on the security’s beta • Alternative asset pricing models allow for multiple factors in determining the required rate of return

10. Assumptions of Capital Market Theory • All investors are Markowitz efficient investors who invest on the efficient frontier. • Investors can borrow or lend any amount of money at the risk-free rate of return (RFR). • Investors have homogeneous expectations; that is, they estimate identical probability distributions for future rates of return. • All investors have the same one-period time horizon such as one-month, six months, or one year.

11. Assumptions of Capital Market Theory • All investments are infinitely divisible, which means that it is possible to buy or sell fractional shares of any asset or portfolio. • There are no taxes or transaction costs involved in buying or selling assets. • There is no inflation or any change in interest rates, or inflation is fully anticipated. • Capital markets are in equilibrium.

12. Making Assumptions • Some of these assumptions are clearly unrealistic • Relaxing many of these assumptions would have only minor influence on the model and would not change its main implications or conclusions. • The primary way to judge a theory is on how well it explains and helps predict behavior, not on its assumptions.

13. Capital Market Theory and a Risk-Free Asset Perhaps surprisingly, there are rather large implications for capital market theory when a risk-free asset exists. • What is a risk-free asset? • An asset with zero variance • Provides the risk-free rate of return (RFR) • It will be an “intercept” value on a portfolio graph between expected return and standard deviation. • Since it has zero variance, it will also have zero correlation with all other risky assets

14. Risk-Free Asset Covariance between two sets of returns is Because the returns for the risk free asset are certain, Thus Ri = E(Ri), and Ri - E(Ri) = 0 Consequently, the covariance of the risk-free asset with any risky asset or portfolio will always equal zero. Similarly the correlation between any risky asset and the risk-free asset would be zero.

15. Combining a Risk-Free Asset with a Portfolio Expected return is the weighted average of the two returns This is a linear relationship

16. Combining a Risk-Free Asset with a Portfolio Standard deviation: The expected variance for a two-asset portfolio is Substituting the risk-free asset for Security 1, and the risky asset for Security 2, this formula would become Since we know that the variance of the risk-free asset is zero and the correlation between the risk-free asset and any risky asset i is zero we can adjust the formula

17. Combining a Risk-Free Asset with a Portfolio Given the variance formula the standard deviation is Therefore, the standard deviation of a portfolio that combines the risk-free asset with risky assets is the linear proportion of the standard deviation of the risky asset portfolio.

18. Combining a Risk-Free Asset with a Portfolio Since both the expected return and the standard deviation of return for such a portfolio are linear combinations, a graph of possible portfolio returns and risks looks like a straight line between the two assets.

19. Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier D M C B A RFR

20. Risk-Return Possibilities with Leverage • To attain a higher expected return than is available at point M (in exchange for accepting higher risk) • Either invest along the efficient frontier beyond point M, such as point D • Or, add leverage to the portfolio by borrowing money at the risk-free rate and investing in the risky portfolio at point M

21. Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier CML Borrowing Lending M RFR

22. The Market Portfolio • Portfolio M lies at the point of tangency, so it has the highest portfolio possibility line • This line of tangency is called the Capital Market Line (CML) • Everybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML (the CML is a new efficient frontier) • Therefore this portfolio must include all risky assets (or else some assets would have no demand)

23. The Market Portfolio • Because the market is in equilibrium, all assets are included in this portfolio in proportion to their market value • Because 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

24. Systematic Risk • Only systematic risk remains in the market portfolio • Systematic risk is the variability in all risky assets caused by macroeconomic variables • Systematic risk can be measured by the standard deviation of returns of the market portfolio and can change over time

25. Factors Affecting Systematic Risk • Systematic risk factors are those macroeconomic variables that affect the valuation of all risky assets • Variability in growth of the money supply • Interest rate volatility • Variability in aggregate industrial production

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

27. Diversifying Away Unsystematic Risk • The purpose of diversification is to reduce the standard deviation of the total portfolio • As you add securities, you expect the average covariance for the portfolio to decline, but not to disappear since correlations are not perfectly negative. • About how many securities must you add to obtain a completely diversified portfolio? • About 90% of the benefit after 12-18 stocks • Maximum benefit needs between 30 and 40

28. The Portfolio Standard Deviation Standard Deviation of Return Unsystematic (diversifiable) Risk Total Risk Standard Deviation of the Market Portfolio (systematic risk) Systematic Risk Number of Stocks in the Portfolio

29. The CML and the Separation Theorem • The CML leads all investors to invest in the M portfolio (The Investment Decision) • Individual investors should differ in position on the CML depending on risk preferences (which leads to the Financing Decision) • Risk averse investors will lend part of the portfolio at the risk-free rate and invest the remainder in the market portfolio (points left of M) • Aggressive investors would borrow funds at the RFR and invest everything in the market portfolio (points to the right of M)

30. A Risk Measure for the CML If… the relevant risk in a portfolio is the average covariance with all other assets in the portfolio, and … the only relevant portfolio is the market portfolio (M), then it follows that … the covariance with the market portfolio is the relevant (systematic) risk of an asset.

31. A Risk Measure for the CML Because all individual risky assets are part of the M portfolio, an asset’s rate of return in relation to the return for the M portfolio may be described using the following linear model: Rit = ai +biRMt +e where: Rit = return for asset i during period t ai = constant term for asset i bi = slope coefficient for asset i RMt = return for the M portfolio during period t e =random error term

32. The Capital Asset Pricing Model • The existence of a risk-free asset resulted in deriving a capital market line (CML) that became the relevant frontier • An asset’s covariance with the market portfolio is the relevant risk measure • This can be used to determine an appropriate required rate of return on a risky asset - the capital asset pricing model (CAPM)

33. The Capital Asset Pricing Model • CAPM indicates what should be the expected or required rates of return on risky assets • This helps to value an asset by providing an appropriate discount rate to use in dividend valuation models • You can compare an expected rate of return to the required rate of return implied by CAPM - over/ under valued?

34. The Security Market Line (SML) • The relevant risk measure for an individual risky asset is its covariance with the market portfolio (Covi,m) • This is shown as the risk measure • The return for the market portfolio should be consistent with its own risk, which is the covariance of the market with itself - or its variance

35. The Security Market Line (SML) We then define as beta

36. Graph of SML SML Negative Beta RFR

37. Determining the Expected Return • The expected rate of return of a risk asset is determined by the RFR plus a risk premium for the individual asset • The risk premium is determined by the systematic risk of the asset (beta) and the prevailing market risk premium (RM-RFR)

38. Determining the Expected Return • In equilibrium, all assets and all portfolios of assets should plot on the SML • The SML gives the market “going rate of return” or what you should earn as a return for a security • Any security with an expected return that plots above the SML is underpriced • Any security with an expected return that plots below the SML is overpriced

39. Identifying Undervalued and Overvalued Assets • Compare 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 • Independent estimates of expected return for the securities provide price and dividend outlooks

40. Calculating Beta: 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: Ri,t = the rate of return for asset i during period t RM,t = the rate of return for the market portfolio M during t

41. Issues in Beta Estimation • The Impact of the Time Interval • Number of observations and time interval used in regression vary • Value Line Investment Services (VL) uses weekly rates of return over five years • Merrill Lynch, Pierce, Fenner & Smith (ML) uses monthly return over five years • There is no “correct” interval for analysis • Weak relationship between VL & ML betas due to difference in intervals used • Interval effect impacts smaller firms more

42. Issues in Beta Estimation • The Effect of the Market Proxy • A measure of the market portfolio is needed • S&P 500 Composite Index is most often used • Includes a large proportion of the total market value of U.S. stocks • Value weighted series • Weaknesses of Using S&P 500as the Market Proxy • Includes only U.S. stocks • The theoretical market portfolio should include all types of assets from all around the world

43. Multifactor Models of Risk and Return • CAPM is criticized because of the difficulties in selecting a proxy for the market portfolio as a benchmark • An initial alternative pricing theory with fewer assumptions was developed: • Arbitrage Pricing Theory

44. Multifactor Pricing Models • Arbitrage Pricing Theory • A practical problem with implementation is that neither the identity nor the exact number of risk factors are part of the theory, so the risk factor specification is ad hoc • An alternative approach, similar to the APT is to directly specify the risk factors to model (the F’s in the slide that follows)

45. Multifactor Pricing Models = return on asset i during a specified time period = expected return for asset i = reaction in asset j’s returns to movements in a common factor = a common factor influences the returns on all assets = a unique effect on asset i’s return that, by assumption, is completely diversifiable in large portfolios and has a mean of zero Rjt Ei bjk

46. Multifactor Pricing Models Theory does not suggest a particular set of factors to price; two general approaches have been employed. • Macroeconomic-Based Risk Factor Models • Microeconomic-Based Risk Factor Models

47. Macroeconomic-Based Risk Factor Models • Chen, Roll, and Ross Developed a return model based on: • Return on a value-weighted NYSE index • Monthly growth in U.S. industrial production • Change in inflation • Difference between actual and expected inflation • Unanticipated change in the bond credit spread • Unanticipated term structure shift

48. Macroeconomic-Based Risk Factor Models Chen, Roll, and Ross General Findings: • Significance of various risk factors changes over time • The stock market index factor is never significant

49. Macroeconomic-Based Risk Factor Models • Burmeister, Roll, And Ross Identify five risk exposures: • Confidence risk • Time horizon risk • Inflation risk • Business cycle risk • Market timing risk

50. Microeconomic-Based Risk Factor Models Fama and French • Characteristic-based approach explaining returns based on: • Excess return on a stock market portfolio • SMB (Small minus big) • The return on a small-cap portfolio less the return on a large-cap portfolio • HML (High minus low) • The return of a high book-to-market value portfolio less the return on a low book-to-market value portfolio