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Asset Pricing Models

Asset Pricing Models. A. What are Asset Pricing Models?. Explaining asset returns, r i In a well-functioning capital market, investors should be rewarded for bearing risks associated with investing in an asset What kind of risks (also known as risk factors, factors)? How do we quantify them?.

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Asset Pricing Models

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  1. Asset Pricing Models

  2. A. What are Asset Pricing Models? • Explaining asset returns, ri • In a well-functioning capital market, investors should be rewarded for bearing risks associated with investing in an asset • What kind of risks (also known as risk factors, factors)? • How do we quantify them?

  3. (Cont’d) • Questions: • In theory, which types of risk should be included in the asset pricing model? • Empirically, which types of risk have explanatory power? • Want to find a set of common risk factors that can explain most stock returns • Applications: both the “buy” and “sell” sides of finance

  4. Asset Pricing Models • General form: E(ri) = rf + f(F1, F2, F3,......., FN) • Expected return = risk-free rate and function of N risk factors • Risk-free rate + risk premium • The two most popular asset pricing models: • CAPM (one-factor model) • Fama-French (three-factor model) and variations of it

  5. CAPM • Sharpe (1964), Lintner (1965) • There is only one risk factor in this model • Systematic or market risk E(ri) = rf + i [E(rm) – rf] • i [E(rm) – rf] is the risk premium of asset i • i measures systematic risk of asset i relative to the market portfolio • i = Cov(ri, rm)/Var(rm)

  6. CAPM (Cont’d) • An extension of the Markowitz model • Make use of the result that all investor hold some combination of the risky-free asset and the optimal risky portfolio • In equilibrium, the optimal risky portfolio is the market portfolio, M • The CAL linking F and M has the highest reward-to-variability (or Sharpe) ratio • Next few slides – intuition behind the model, its origin…..etc.

  7. Capital Market Line (CML) • In the Markowitz model, why do all investors hold a certain percentage of the market portfolio, M? • The CAL that has M as its risky portfolio is call the CML E(r) CML M Efficient frontier (of risky assets) ● ● rf 

  8. CML • Utility maximizing portfolio for different investors E(r) CML Aggressive Investor ● M Moderate Investor Efficient frontier (of risky assets) ● ● rf Conservative Investor 

  9. CML • Risk and return tradeoff of efficient portfolios • The slope represents reward per unit of market risk (also known as the market price of risk) • Therefore, the expected return of an efficient portfolio on the CML (consisting of F and M) can be interpreted as: E(rc) = rf + market price of risk x quantity of risk in c

  10. Market Risk Premium • The model also has implications for the market risk premium (MRP), i.e., what it depends on • Recall that: • For example, if A = 4, E(rp) = 11%, rf = 5%, p = 14.2%, then y = 74.39% • In equilibrium, P is replaced by M

  11. Market Risk Premium • What is the average holding (among all investors, lenders and borrowers) of the risk-free asset, F? • What is the average holding of the market portfolio, M? That is, what is the average y in the market? • Using the last answer, can determine what the MRP, E(rM) – rf, is a function of

  12. Assumptions of the CAPM • Reason for these assumptions: To derive a simple, tractable model that is easy to use • 7 assumptions • Investors make decisions based on the mean and variance of returns • Investors are rational and risk averse • Investors subscribe to the Markowitz method of portfolio diversification

  13. Assumptions of the CAPM • Capital market is competitive (large number of buyers and sellers of assets), and frictionless (e.g., no taxes, no transaction costs) • All investors have the same investment horizon • There is a risk-free asset, and risk-free lending and borrowing • Investors have the same expectations about the mean and variance of all assets

  14. SML • The CML provides an exact relationship between the risk and return of efficient portfolios (consisting of F and M) • Given the investment universe, shows the efficient portfolios; used for picking the optimal combined portfolio for a client • The SML provides an exact relationship between the risk and return of individual assets • This is the CAPM, i.e., for pricing securities or portfolios • Sharpe shows that the relevant risk for an individual asset is systematic risk, i.e., risk that cannot be diversified away

  15. SML • The amount of systematic risk depends on how much the asset return co-varies with the market • To derive the CAPM, the textbook uses the following equilibrium condition: • COV(ri, rM) is the contribution of risk to the market portfolio, M, by asset i • Equilibrium condition in words: Ratio of “reward” to “contribution of risk to a diversified portfolio, M” should be the same across assets, including M

  16. SML • Hence, investors are not compensated for the total risk of the asset, just the systematic risk • Rearrange the equilibrium condition on the last slide to obtain the CAPM • Sell-side applications: • Hurdle rate for capital budgeting, cost of equity • Under- or over-pricing of securities

  17. Under-/Over-pricing E(r) Underpriced asset ● SML ● E(rM) rf ● Overpriced asset  1

  18. Beta of a Portfolio • A nice property of the CAPM • The portfolio beta is a weighted average of the individual stock betas where n is the number of stocks in the portfolio wi is the weight of stock i in the portfolio

  19. Testing the CAPM • Estimating beta for different securities, and see if indeed higher-beta securities have higher returns on average • In general, the CAPM fails the empirical tests • If the market portfolio is not on the MV Efficient Frontier – the CAPM does not hold • Buy-side applications of CAPM: portfolio performance • Given systematic risk in portfolio, was return higher or lower than that predicted by CAPM?

  20. Fama and French (1993) • Three-factor pricing model. Gaining recognition and acceptance in industry • Previous research has shown that beta is not the only factor that explains stock or portfolio returns • No theory behind the model, entirely driven by empirical evidence • Takes into account two additional factors: size and book-to-market (B/M)

  21. FF • Size: evidence of small firms generating higher returns (not captured by CAPM) • Book-to-market: evidence of value firms generating higher returns (not captured by CAPM) • Problems in model specification: • Book-to-market is a ratio • Size as measured by market capitalization is in dollars • Market risk premium, E(rM) – rf, is a return in % • FF create portfolios that mimic the size and the value factors

  22. The Lost Decade?

  23. FF • FF (1993) proposes this model: • Market risk premium • E(rM) - rf • SMB • Size premium: return on small-cap stocks minus return on large-cap stocks • HML • Value premium: return on high B/M stocks minus return on low B/M stocks

  24. FF • SMB and HML are called factor-mimicking portfolios • They can also be thought of as “zero-cost portfolio”: • HML: long $1 of the high B/M portfolio and short $1 of the low B/M portfolio • SMB: long $1 of the small-cap portfolio and short $1 of the large-cap portfolio • Kenneth French provides free data (annual, monthly, daily) for the three factors

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