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CHAPTER. 11. An Alternative View of Risk and Return: The APT. Chapter Outline. 11.1 Factor Models: Announcements, Surprises, and Expected Returns 11.2 Risk: Systematic and Unsystematic 11.3 Systematic Risk and Betas 11.4 Portfolios and Factor Models 11.5 Betas and Expected Returns

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  1. CHAPTER 11 An Alternative View of Risk and Return: The APT

  2. Chapter Outline 11.1 Factor Models: Announcements, Surprises, and Expected Returns 11.2 Risk: Systematic and Unsystematic 11.3 Systematic Risk and Betas 11.4 Portfolios and Factor Models 11.5 Betas and Expected Returns 11.6 The Capital Asset Pricing Model and the Arbitrage Pricing Theory 11.7 Parametric Approaches to Asset Pricing 11.8 Summary and Conclusions

  3. Arbitrage Pricing Theory Arbitrage arises if an investor can construct a zero investment portfolio with a sure profit. • Since no investment is required, an investor can create large positions to secure large levels of profit. • In efficient markets, profitable arbitrage opportunities will quickly disappear.

  4. = + R R U where R is the expected part of the return U is the unexpected part of the return 11.1 Factor Models: Announcements, Surprises, and Expected Returns • The return on any security consists of two parts. • First the expected returns • Second is the unexpected or risky returns. • A way to write the return on a stock in the coming month is:

  5. 11.1 Factor Models: Announcements, Surprises, and Expected Returns • Any announcement can be broken down into two parts, the anticipated or expected part and the surprise or innovation: • Announcement = Expected part + Surprise. • The expected part of any announcement is part of the information the market uses to form the expectation, R of the return on the stock. • The surprise is the news that influences the unanticipated return on the stock, U.

  6. 11.2 Risk: Systematic and Unsystematic • A systematic risk is any risk that affects a large number of assets, each to a greater or lesser degree. • An unsystematic risk is a risk that specifically affects a single asset or small group of assets. • Unsystematic risk can be diversified away. • Examples of systematic risk include uncertainty about general economic conditions, such as GNP, interest rates or inflation. • On the other hand, announcements specific to a company, such as a gold mining company striking gold, are examples of unsystematic risk.

  7. = + R R U becomes = + +  R R m ε where m is the systematic risk ε is the unsystemat ic risk 11.2 Risk: Systematic and Unsystematic We can break down the risk, U, of holding a stock into two components: systematic risk and unsystematic risk:  Total risk; U Nonsystematic Risk;  Systematic Risk; m n

  8. Cov ( R R ) b = i , M i s 2 ( R ) M 11.3 Systematic Risk and Betas • The beta coefficient, b, tells us the response of the stock’s return to a systematic risk. • In the CAPM, b measured the responsiveness of a security’s return to a specific risk factor, the return on the market portfolio. • We shall now consider many types of systematic risk.

  9. = + + R R m ε = + + + + R R β F β F β F ε I I GDP GDP S S β is the inflation beta I β is the GDP beta GDP β is the spot exchange rate beta S ε is the unsystemat ic risk 11.3 Systematic Risk and Betas • For example, suppose we have identified three systematic risks on which we want to focus: • Inflation • GDP growth • The dollar-eurospot exchangerate, S($,€) • Our model is:

  10. = + + + + R R β F β F β F ε I I GDP GDP S S = ε 1 % = - ´ + ´ + ´ + R R 2 . 30 F 1 . 50 F 0 . 50 F 1 % I GDP S Systematic Risk and Betas: Example • Suppose we have made the following estimates: • bI = -2.30 • bGDP = 1.50 • bS = 0.50. • Finally, the firm was able to attract a “superstar” CEO and this unanticipated development contributes 1% to the return.

  11. = - ´ + ´ + ´ + R R 2 . 30 F 1 . 50 F 0 . 50 F 1 % I GDP S = - ´ + ´ + ´ + R R 2 . 30 5 % 1 . 50 F 0 . 50 F 1 % GDP S Systematic Risk and Betas: Example We must decide what surprises took place in the systematic factors. If it was the case that the inflation rate was expected to be by 3%, but in fact was 8% during the time period, then FI = Surprise in the inflation rate = actual – expected = 8% – 3% = 5%

  12. = - ´ + ´ + ´ + R R 2 . 30 5 % 1 . 50 F 0 . 50 F 1 % GDP S = - ´ + ´ - + ´ + R R 2 . 30 5 % 1 . 50 ( 3 %) 0 . 50 F 1 % S Systematic Risk and Betas: Example If it was the case that the rate of GDP growth was expected to be 4%, but in fact was 1%, then FGDP = Surprise in the rate of GDP growth = actual – expected = 1% – 4% = – 3%

  13. = - ´ + ´ - + ´ + R R 2 . 30 5 % 1 . 50 ( 3 %) 0 . 50 F 1 % S = - ´ + ´ - + ´ - + R R 2 . 30 5 % 1 . 50 ( 3 %) 0 . 50 ( 10 %) 1 % Systematic Risk and Betas: Example If it was the case that dollar-euro spot exchange rate, S($,€), was expected to increase by 10%, but in fact remained stable during the time period, then FS = Surprise in the exchange rate = actual – expected = 0% – 10% = – 10%

  14. = - ´ + ´ - + ´ + R R 2 . 30 5 % 1 . 50 ( 3 %) 0 . 50 F 1 % S = R 8 % Systematic Risk and Betas: Example Finally, if it was the case that the expected return on the stock was 8%, then

  15. = + + R R β F ε i i i i 11.4 Portfolios and Factor Models • Now let us consider what happens to portfolios of stocks when each of the stocks follows a one-factor model. • We will create portfolios from a list of N stocks and will capture the systematic risk with a 1-factor model. • The ith stock in the list have returns:

  16. e i Relationship Between the Return on the Common Factor & Excess Return Excess return If we assume that there is no unsystematic risk, then ei = 0 The return on the factor F

  17. Relationship Between the Return on the Common Factor & Excess Return Excess return If we assume that there is no unsystematic risk, then ei = 0 The return on the factor F

  18. = β 1 . 5 A = β 0 . 50 C Relationship Between the Return on the Common Factor & Excess Return Excess return Different securities will have different betas The return on the factor F

  19. L L = + + + + + R X R X R X R X R 1 1 2 2 P i i N N = + + R R β F ε i i i i = + + + + + + R X ( R β F ε ) X ( R β F ε ) 1 2 P 1 1 1 2 2 2 L + + + X ( R β F ε ) N N N N = + + + + + + R X R X β F X ε X R X β F X ε 1 2 P 1 1 1 1 1 2 2 2 2 2 L + + + X R X β F X ε N N N N N N Portfolios and Diversification • We know that the portfolio return is the weighted average of the returns on the individual assets in the portfolio:

  20. The weighed average of expected returns. • The weighted average of the betas times the factor. • The weighted average of the unsystematic risks. L = + + + R X R X R X R 1 2 N P 1 2 N L + + + + ( X β X β X β ) F 1 1 2 2 N N L + + + + X ε X ε X ε 1 1 2 2 N N Portfolios and Diversification The return on any portfolio is determined by three sets of parameters: In a large portfolio, the third row of this equation disappears as the unsystematic risk is diversified away.

  21. L = + + + R X R X R X R 1 2 N P 1 2 N L + + + + ( X β X β X β ) F 1 1 2 2 N N Portfolios and Diversification So the return on a diversified portfolio is determined by two sets of parameters: • The weighed average of expected returns. • The weighted average of the betas times the factor F. In a large portfolio, the only source of uncertainty is the portfolio’s sensitivity to the factor.

  22. L L = + + + + + R X R X R ( X β X β ) F 1 N P 1 N 1 1 N N β R P P Recall that and L L = + + = + + R X R X R β X β X β 1 P N P 1 1 N N 1 N = + R R β F P P P 11.5 Betas and Expected Returns The return on a diversified portfolio is the sum of the expected return plus the sensitivity of the portfolio to the factor.

  23. = + R R β F P P P Relationship Between b & Expected Return • If shareholders are ignoring unsystematic risk, only the systematic risk of a stock can be related to its expected return.

  24. R F = + - R R β ( R R ) P F F Relationship Between b & Expected Return SML Expected return D A B C b

  25. 11.6 The Capital Asset Pricing Model and the Arbitrage Pricing Theory • APT applies to well diversified portfolios and not necessarily to individual stocks. • With APT it is possible for some individual stocks to be mispriced - not lie on the SML. • APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio. • APT can be extended to multifactor models.

  26. 11.7 Empirical Approachesto Asset Pricing • Both the CAPM and APT are risk-based models. There are alternatives. • Empirical methods are based less on theory and more on looking for some regularities in the historical record. • Be aware that correlation does not imply causality. • Related to empirical methods is the practice of classifying portfolios by style e.g. • Value portfolio • Growth portfolio

  27. = + + + + R R β F β F β F ε I I GDP GDP S S 11.8 Summary and Conclusions • The APT assumes that stock returns are generated according to factor models such as: • As securities are added to the portfolio, the unsystematic risks of the individual securities offset each other. A fully diversified portfolio has no unsystematic risk. • The CAPM can be viewed as a special case of the APT. • Empirical models try to capture the relations between returns and stock attributes that can be measured directly from the data without appeal to theory.

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