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# L Pch11

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• The unsystematic risk is diversified away because the variance of row three tends to 0 as N goes to infinity.
• ### L Pch11

1. 1. Investments Chapter 11: The Arbitrage Pricing Theory
2. 2. Factor Risk Models <ul><li>Relate the common movements of asset prices to a series of common risk factors. </li></ul><ul><li>Examined here: </li></ul><ul><li>SIM (Single Index Model) </li></ul><ul><li>MIM (Multiple Index Model) </li></ul>
3. 3. The Single Index Model <ul><li>Assumes two factors are responsible for a given asset’s rate of return: </li></ul><ul><li>1. Changes in a common risk factor. </li></ul><ul><li>2. Changes related to asset-specific events. </li></ul>
4. 4. The Single Index Model <ul><li>Linear Relationship: </li></ul><ul><ul><li>Where: </li></ul></ul><ul><ul><li>R i is the rate of return on asset i , </li></ul></ul><ul><ul><li>I is the percentage change in the common risk factor, </li></ul></ul><ul><ul><li>e i is asset-specific component, </li></ul></ul><ul><ul><li>β i measures the sensitivity of the i- th asset’s return to changes in the common risk factor. </li></ul></ul>
5. 5. The Single Index Model <ul><li>- Assumptions on e i : </li></ul><ul><li>E( e i ) = 0 </li></ul><ul><li>Independence of specific news, E( e i, e s ) = 0 </li></ul><ul><li>Independence of common factor, E((I-E(I) e i ) = 0 </li></ul><ul><li>- The model simplifies to: E( R i ) = a i +  i E( I ) </li></ul><ul><li>The risk is: </li></ul>
6. 6. SIM versus Mean-Variance Efficient Set <ul><li>- SIM simplifies calculations </li></ul><ul><li>Only betas need to be estimated; covariances are then estimated by definition: </li></ul><ul><li>- Beta estimates on the other hand are very sensitive, due to structural changes and the choice of “Index”. </li></ul>
7. 7. The Multiple Index Model <ul><li>Allows for several common factors to influence a given asset’s rate of return. </li></ul><ul><li>Relationship: </li></ul>
8. 8. Example with 3 factors <ul><ul><li>Inflation (Its market price, or risk premium) </li></ul></ul><ul><ul><li>GDP growth ( “ ) </li></ul></ul><ul><ul><li>The \$/€ spot exchange rate, S, ( “ ) </li></ul></ul><ul><li>Our model is: </li></ul>
9. 9. Example <ul><li>Suppose we have made the following estimates: </li></ul><ul><ul><li> I = -2.30 </li></ul></ul><ul><ul><li> GDP = 1.50 </li></ul></ul><ul><ul><li> S = 0.50. </li></ul></ul><ul><li>Finally, the firm was able to attract a “superstar” CEO and this unanticipated development contributes 1% to the return. </li></ul>
10. 10. Example <ul><li>We must decide what surprises took place in the systematic </li></ul><ul><li>factors. If it was the case that the inflation rate was expected </li></ul><ul><li>to be 3%, but in fact was 8% during the time period, then </li></ul><ul><li>F I = Surprise in the inflation rate </li></ul><ul><ul><li>= actual – expected </li></ul></ul><ul><ul><li>= 8% - 3% </li></ul></ul><ul><ul><li>= 5% </li></ul></ul>
11. 11. Example <ul><li>If it was the case that the rate of GDP growth was expected </li></ul><ul><li>to be 4%, but in fact was 1%, then </li></ul><ul><li>F GDP = Surprise in the rate of GDP growth </li></ul><ul><li>= actual – expected </li></ul><ul><li>= 1% - 4% </li></ul><ul><li>= -3% </li></ul>
12. 12. Example <ul><li>If it was the case that \$/€ spot exchange rate was expected to </li></ul><ul><li>increase by 10%, but in fact remained stable during the time </li></ul><ul><li>period, then </li></ul><ul><li>F S = Surprise in the exchange rate </li></ul><ul><li>= actual – expected </li></ul><ul><ul><li>= 0% - 10% </li></ul></ul><ul><ul><li>= -10% </li></ul></ul>
13. 13. Example <ul><li>Finally, if it was the case that the expected return on the </li></ul><ul><li>stock was 8%, then </li></ul>
14. 14. The Arbitrage Pricing Theory <ul><li>Arbitrage: </li></ul><ul><li>A strategy that makes a positive return without requiring an initial investment. </li></ul><ul><li>In other words: arbitrage opportunities exist when two items that are the same sell at different prices. </li></ul><ul><li>In efficient markets, profitable arbitrage opportunities will quickly disappear. </li></ul>
15. 15. Arbitrage Pricing Theory (True example) <ul><li>Assume you bet on UEFA match Sakhim-Newcastle (sept 20, 2004), with the following odds: </li></ul><ul><li>1 gives 8, X gives 3.9 and 2 gives 1.7 times your money. </li></ul><ul><li>The following strategy is a money machine! </li></ul><ul><li>Result: 1 X 2 </li></ul><ul><li>Odds: 8 3.9 1.7 </li></ul><ul><li>Invest 1000: 129 264.5 606.5 </li></ul><ul><li>Return: 1032 1031 1031 </li></ul><ul><li>  This is an arbitrage free strategy since it generates a </li></ul><ul><li>risk free profit of SEK 31, no matter the result! </li></ul>
16. 16. Arbitrage Pricing Theory (True Example) <ul><li>Formulate the following linear programming to find </li></ul><ul><li>the allocation of your capital: </li></ul><ul><li>Min k </li></ul><ul><li>Subject to: </li></ul><ul><li>8x <= k </li></ul><ul><li>3.9y <= k </li></ul><ul><li>1.7z <= k </li></ul><ul><li>x + y + z = k </li></ul><ul><li>x >= 0, y >= 0, z >= 0 </li></ul>
17. 17. The Arbitrage Pricing Theory <ul><li>The APT investigates the market equilibrium prices when all arbitrage opportunities are eliminated. </li></ul><ul><li>The APT implies a linear equilibrium relationship between expected return and the factor sensitivities (betas) </li></ul>
18. 18. The APT: Assumptions <ul><li>Perfect competitive capital market </li></ul><ul><li>All investors have homogeneous expectations, regarding mean, variance and covariance </li></ul><ul><li>More wealth is preferred to less ( but no need to know for risk attitudes ) </li></ul><ul><li>Large number of capital assets exist </li></ul><ul><li>Short sales are allowed </li></ul>
19. 19. The Arbitrage Pricing Theory <ul><li>The expected return on a security under the APT with a single factor is given (precisely as by SML) by: </li></ul><ul><li>The expected return on a security under the APT with multiple factors is given by: </li></ul>
20. 20. Relationship Between the Return on the Common Factor & Excess Return Excess return The return on the factor F If F = 0, then  i > 0
21. 21. Relationship Between the Return on the Common Factor & Excess Return Excess return The return on the factor F If we assume that there is no unsystematic risk, then  i = 0
22. 22. Arbitrage Portfolios and Factor Models <ul><li>Now let us consider what happens to portfolios of stocks when each of the stocks follows a one-factor (F) model . </li></ul><ul><li>We will create portfolios from a list of N stocks and will capture the systematic risk with a 1-factor model. </li></ul><ul><li>The i th stock in the list have returns: </li></ul>
23. 23. Arbitrage Portfolios and Factor Models <ul><li>We know that the portfolio return is the weighted average of the returns on the individual assets in the portfolio: </li></ul>
24. 24. Arbitrage Portfolios and Factor Models <ul><li>The return on any portfolio is determined by three sets of parameters: </li></ul>In a large portfolio, the third row of this equation disappears as the unsystematic risk is diversified away. <ul><ul><li>The weighed average of expected returns. </li></ul></ul><ul><ul><li>The weighted average of the betas times the factor. </li></ul></ul><ul><ul><li>The weighted average of the unsystematic risks. </li></ul></ul>
25. 25. Arbitrage Portfolios and Factor Models <ul><li>So the return on a diversified portfolio is determined by two sets of parameters: </li></ul><ul><ul><li>The weighed average of expected returns. </li></ul></ul><ul><ul><li>The weighted average of the betas times the factor F. </li></ul></ul>In a large portfolio, the only source of uncertainty is the portfolio’s sensitivity to the factor.
26. 26. Arbitrage Portfolios and Factor Models <ul><li>The return on a diversified portfolio is the sum of the expected return plus the sensitivity of the portfolio to the factor. </li></ul>
27. 27. APT vs. CAPM <ul><li>Both models are based on completely different sets of assumptions. </li></ul><ul><li>None the less, both models can predict the same risk-return relationship: when the asset returns obey the SIM the APT relationship is identical to the SML. </li></ul><ul><li>APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio. </li></ul><ul><li>APT can be extended to multifactor models. </li></ul>
28. 28. Empirical Tests of the APT <ul><li>Strong indications that risk factors other than the market portfolio affect expected returns. </li></ul><ul><li>Important to remember that tests of the APT are joint tests of the validity of the APT, the research methodology and the quality of the data. </li></ul><ul><li>Empirical methods are based less on theory and more on looking for some regularities in the historical record. </li></ul>