An Introduction to Asset Pricing Models

1. Chapter 9 An Introduction to Asset Pricing Models Innovative Financial Instruments Dr. A. DeMaskey

2. Capital Market Theory: An Overview • Capital market theory extends portfolio theory and develops a model for pricing all risky assets • Capital asset pricing model (CAPM) will allow you to determine the required rate of return for any risky asset

3. Assumptions of Capital Market Theory • All investors are Markowitz efficient investors who choose investments on the basis of expected return and risk. • Investors can borrow or lend any amount of money at the riskfree rate of return (RFR). • All 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.

4. 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; that is, we begin with all investments properly priced in line with their risk levels.

5. Assumptions of Capital Market Theory • Some of these assumptions are unrealistic • Relaxing many of these assumptions would have only minor influence on the model and would not change its main implications or conclusions. • Judge a theory on how well it explains and helps predict behavior, not on its assumptions.

6. Riskfree Asset • Provides the risk-free rate of return (RFR) • An asset with zero variance and standard deviation • Zero correlation with all other risky assets • Covariance between two sets of returns is • Will lie on the vertical axis of a portfolio graph

7. Combining a Riskfree Asset with a Risky Portfolio • Expected return: • The expected variance for a two-asset portfolio: • Because the variance of the riskfree asset is zero and the correlation between the riskfree asset and any risky asset i is zero, this simplifies to:

8. Combining a Risk-Free Asset with a Risky Portfolio • Given the variance formula: • The standard deviation is: • Therefore, the standard deviation of a portfolio that combines the riskfree asset with risky assets is the linear proportion of the standard deviation of the risky asset portfolio.

9. 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 riskfree rate and investing in the risky portfolio at point M

10. The Market Portfolio • Because portfolio M lies at the point of tangency, it has the highest portfolio possibility line • Everybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML • Therefore, this portfolio must include ALL RISKY ASSETS • Since the market is in equilibrium, all assets are included in this portfolio in proportion to their market value. • Since 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.

11. 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

12. Factors Affecting Systematic Risk • Variability in growth of money supply • Interest rate volatility • Variability in

13. 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

14. Diversification and the Elimination of Unsystematic Risk • The purpose of diversification is to reduce the standard deviation of the total portfolio • This assumes that imperfect correlations exist among securities • As you add securities, you expect the average covariance for the portfolio to decline • How many securities must you add to obtain a completely diversified portfolio? • Observe what happens as you increase the sample size of the portfolio by adding securities that have some positive correlation

15. The CML and the Separation Theorem • The CML leads all investors to invest in the M portfolio • Individual investors should differ in position on the CML depending on risk preferences • How an investor gets to a point on the CML is based on financing decisions • Risk averse investors will lend part of the portfolio at the riskfree rate and invest the remainder in the market portfolio

16. The CML and the Separation Theorem • Investors preferring more risk might borrow funds at the RFR and invest everything in the market portfolio • The decision of both investors is to invest in portfolio M along the CML • The decision to borrow or lend to obtain a point on the CML is a separate decision based on risk preferences • Tobin refers to this separation of the investment decision from the financing decision as the separation theorem

17. A Risk Measure for the CML • Covariance with the M portfolio is the systematic risk of an asset • The Markowitz portfolio model considers the average covariance with all other assets in the portfolio • The only relevant portfolio is the M portfolio • Together, this means the only important consideration is the asset’s covariance with the market portfolio

18. A Risk Measure for the CML Since all individual risky assets are part of the M portfolio, an asset’s rate of return in relation to the return of the M portfolio may be described using the following linear model: 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

19. Variance of Returns for a Risky Asset Note: Var(biRMi) is variance related to market return Var(e) is the residual return not related to the market portfolio

20. The Capital Asset Pricing Model: Expected Return and Risk • The existence of a riskfree 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 expected rate of return on a risky asset - the capital asset pricing model (CAPM)

21. The Capital Asset Pricing Model: Expected Return and Risk • 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 • The estimated rate of return can also be compared to the required rate of return implied by CAPM to determine whether a risky asset is over- or undervalued

22. The Security Market Line (SML) • The relevant risk measure for an individual risky asset is its covariance with the market portfolio (Covi,m) • 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:

23. The Security Market Line (SML) The equation for the risk-return line is given as: We then define as beta

24. Determining the Expected Rate of Return for a Risky Asset • The expected rate of return of a risky 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)

25. Determining the Expected Rate of Return for a Risky Asset • In equilibrium, all assets and all portfolios of assets should plot on the SML • Any security with an estimated return that plots above the SML is underpriced • Any security with an estimated return that plots below the SML is overpriced • To earn better risk-adjusted rates of return than the average investor, a superior investor must derive value estimates for assets that are consistently superior to the consensus market evaluation

26. 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 return for the securities provide price and dividend outlooks

27. Calculating Systematic Risk: 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 e = the random error term

28. 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 • Weak relationship between VL & ML betas due to difference in intervals used • There is no “correct” interval for analysis • Interval effect impacts smaller firms more

29. The Effect of the Market Proxy • Choice of market proxy is crucial • Proper measure must include all risky assets • Standard & Poor’s 500 Composite Index is most often used • Large proportion of the total market value of U.S. stocks • Value weighted series • Weaknesses

30. Arbitrage Pricing Theory (APT) • CAPM is criticized because of the difficulties in selecting a proxy for the market portfolio as a benchmark • An alternative pricing theory with fewer assumptions was developed: • Arbitrage Pricing Theory

31. Assumptions of Arbitrage Pricing Theory (APT) • Capital markets are perfectly competitive • Investors always prefer more wealth to less wealth with certainty • The stochastic process generating asset returns can be presented as K factor model

32. Assumptions of CAPMThat Were Not Required by APT • APT does not assume: • A market portfolio that contains all risky assets, and is mean-variance efficient • Normally distributed security returns • Quadratic utility function

33. Arbitrage Pricing Theory (APT) For i = 1 to N where: Ri = return on asset i during a specified time period Ei = expected return for asset i bik = reaction in asset i’s returns to movements in a common factor dk = a common factor with a zero mean that influences the returns on all assets ei = a unique effect on asset i’s return that, by assumption, is completely diversifiable in large portfolios and has a mean of zero N = number of assets

34. Arbitrage Pricing Theory (APT) • Multiple factors, dk, expected to have an impact on all assets: • Inflation • Growth in GNP • Major political upheavals • Changes in interest rates • And many more…. • Contrast with CAPM’s insistence that only beta is relevant

35. Arbitrage Pricing Theory (APT) • Bik determine how each asset reacts to this common factor • Each asset may be affected by growth in GNP, but the effects will differ • In applying the theory, the factors are not identified • Similar to the CAPM in that the unique effects (ei) are independent and will be diversified away in a large portfolio

36. Arbitrage Pricing Theory (APT) • APT assumes that, in equilibrium, the return on a zero-investment, zero-systematic-risk portfolio, is zero when the unique effects are diversified away • The expected return on any asset i (Ei) can be expressed as:

37. Arbitrage Pricing Theory (APT) Where: l0 = the expected return on an asset with zero systematic risk where l0 = E0 l1 = the risk premium related to each of the common factors, with i = 1 to k bi = pricing relationship between the risk premium and asset i

38. Example of Two Stocks and a Two-Factor Model l1 = changes in the rate of inflation. The risk premium related to this factor is 1% for every 1% change in the rate (l1 = 0.1) l2 = percent growth in real GNP. The average risk premium related to this factor is 2% for every 1% change in the rate (l2 = 0.02) l3 = the rate of return on a zero-systematic-risk asset (zero beta: boj = 0) is 3% (l3 = 0.03)

39. Example of Two Stocks and a Two-Factor Model bx1 = the response of asset X to changes in the rate of inflation is 0.50 (bx1 = 0.50) by1 = the response of asset Y to changes in the rate of inflation is 2.00 (by1 = 2.00) bx2 = the response of asset X to changes in the growth rate of real GNP is 1.50 (bx2 = 1.50) by2 = the response of asset Y to changes in the growth rate of real GNP is 1.75 (by2 = 1.75)

40. Example of Two Stocks and a Two-Factor Model = .03 + (.01)bi1 + (.02)bi2 Ex = .03 + (.01)(0.50) + (.02)(1.50) = .065 = 6.5% Ey = .03 + (.01)(2.00) + (.02)(1.75) = .085 = 8.5%

41. Empirical Tests of the APT • Studies by Roll and Ross and by Chen support APT by explaining different rates of return with some better results than CAPM • Reinganum’s study did not explain small-firm results • Dhrymes and Shanken question the usefulness of APT because it was not possible to identify the factors

42. Summary • When you combine the riskfree asset with any risky asset on the Markowitz efficient frontier, you derive a set of straight-line portfolio possibilities • The dominant line is tangent to the efficient frontier • Referred to as the capital market line (CML) • All investors should target points along this line depending on their risk preferences

43. Summary • All investors want to invest in the risky portfolio, so this market portfolio must contain all risky assets • The investment decision and financing decision can be separated • Everyone wants to invest in the market portfolio • Investors finance based on risk preferences

44. Summary • The relevant risk measure for an individual risky asset is its systematic risk or covariance with the market portfolio • Once you have determined this Beta measure and a security market line, you can determine the required return on a security based on its systematic risk

45. Summary • Assuming security markets are not always completely efficient, you can identify undervalued and overvalued securities by comparing your estimate of the rate of return on an investment to its required rate of return • The Arbitrage Pricing Theory (APT) model makes simpler assumptions, and is more intuitive, but test results are mixed at this point

46. www.valueline.com www.barra.com www.stanford.edu/~wfsharpe.com www.wsharpe.com The InternetInvestments Online