CAPM limitation 11 How much data should we collect?

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Investment theory, class #3 Page 1 of 26 ® All rights reserved to Tal Mofkadi A course in investment theory Class #3 – Factors model Arbitrage Pricing Theory By: Tal Mofkadi
Investment theory, class #3 Page 2 of 26 ® All rights reserved to Tal Mofkadi 1 CAPM LIMITATION .................................................... 3 1.1 H OW MUCH DATA SHOULD WE COLLECT ? ............................................... 3 1.2 P OSSIBLE ERRORS IN CORRELATION COEFFICIENTS ............................... 4 2 THE INDEX (FACTOR) MODEL ................................ 5 2.1 S INGLE INDEX MODEL ................................................................................ 5 3 THE CAPM IN PRACTICE (SINGLE INDEX MODEL) ................................................................................. 9 3.1 T HE RISK FREE RATE OF RETURN .............................................................. 9 3.2 C ALCULATE Β E ........................................................................................... 9 3.3 T HE MARKET RISK PREMIUM .................................................................. 10 3.3.1 Implied risk premium history ............................................................. 11 3.3.2 Implied vs. realized risk premium ...................................................... 12 3.4 S MALL SIZE PREMIUM .............................................................................. 12 3.5 M ULTIFACTOR MODELS ........................................................................... 13 3.5.2 Theoretical Foundations of Multifactor Models ............................... 14 3.5.3 Where do we look for factors (ICAPM)? ........................................... 15 4 ARBITRAGE PRICING THEORY ............................ 16 4.1 A RBITRAGE DEFINITION .......................................................................... 16 4.2 T HE APT ................................................................................................... 16 4.2.1 Example 1: ......................................................................................... 17 4.2.2 Example 2 .......................................................................................... 18 4.2.3 Conclusion: ........................................................................................ 18 4.2.4 The APT and the CAPM .................................................................... 20 4.3 M ULTIFACTOR APT ................................................................................. 22 4.3.2 Conclusion ......................................................................................... 23 5 ALTERNATIVE PRICING MODEL ......................... 24 5.1 C ONSUMPTION CAPM - CCAPM ........................................................... 24 5.1.1 Example ............................................................................................. 25 5.1.2 Results ................................................................................................ 26 6 FURTHER TOPICS ..................................................... 26 6.1 T HE EQUITY PREMIUM PUZZLE ............................................................... 26
Investment theory, class #3 Page 3 of 26 1 CAPM limitation 1.1 How much data should we collect?
® All rights reserved to Tal Mofkadi
Suppose your security analysts can thoroughly analyze 50 stocks. This means that your input list will include the following: n = 50 estimates of expected returns n = 50 estimates of variances (n 2 – n)/2 = 1,225 estimates of covariances Total of 1,325 estimates This is a formidable task, particularly in light of the fact that a 50-security portfolio is relatively small. Doubling n to 100 will nearly quadruple the number of estimates to 5,150. If n = 3,000, roughly the number of NYSE stocks, we need more than 4.5 million estimates. And what if our portfolio is build from 10,000 assets?
Investment theory, class #3 Page 4 of 26 ® All rights reserved to Tal Mofkadi 1.2 Possible errors in correlation coefficients Another difficulty in applying the Markowitz model to portfolio optimization is that errors in the assessment or estimation of correlation coefficients can lead to nonsensical results. This can happen because some sets of correlation coefficients are mutually inconsistent, as the following example demonstrates: If we construct a portfolio with weights: -1; 1; 1, for assets A; B; C, respectively, and calculate the portfolio variance. We will find that the portfolio variance appears to be negative (-200)!!! This of course is not possible. Of course, true correlation coefficients are always consistent 1 . We would like to seek a model that is easier to implement. 1 The mathematical term for a correlation matrix that cannot generate negative portfolio variance is “positive definite.”

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