Chapter 8 Index Model, Index Model, Index Model

Index Model Investment Management MBA Subject.

1. Chapter 8 Index Model MARIBEL V. VILLAVERDE MBA 511 Investment Management

2. Chapter overview Advantages of Single-factor model Risk decomposition Systematic vs firm specific Single Index & its estimation Optimal Risky Portfolio in the index model Index model vs Markowitz procedure

3. Emergence to Single Index Model 1952 “harry markowitz ” published portfolio selection model that maximized a portfolio’s return for a given level of risk This model required the estimation of: Expected returns for each security Variances for each security A covariance matrix (calculated the covariance between each possible pair or securities w/in the portfolio based on historical data through a scenario analysis.

4. 8.1 A Single-Factor Security Market The Input List of Markowitz Model To perform the necessary calculations, you need the following data pieces for a portfolio with N assets. N estimates of returns N estimates of variances (N 2 - N)/2 estimates of co variances A 50 asset portfolio requires: (N 2 - N)/2 =2500-50 =2,450/2 =1,225 estimate of covariances =1,325 total estimates

5. A Single factor model Model ß i - response of an individual securities return to the common factor, m- measures systematic risk m - common macroeconomic factor(systematic risk) S & P500 often used as proxy e i r i =(E( ri )+ unacticipated surprise r i =(E( ri )+ ß i m ( ei ) R i -rate if returns Variances = systematic risk + firm specific risk σ i 2 = ß 2 i σ 2 m + σ 2 ( e i ) Covariances-product of betas x market risk Cov ( r i r j )= ß 1 ßj σ 2 M  

6. Single index models-returns Approach leads to an equation similar to the single factor, which uses the market index to proxy for the common factor. Regression equation R i (t) = α i + ß i R M (t) + e i (t) Expected return Beta-Relationship E(R i ) = ß i E (R M )

7. Correlation Product of correlations with the market index

8. Answer the following questions What is the mean index excess return of the portfolio? What is the covariance between A & B, B & C, C& D and A & C, A & D, B & D? 3. What is the covariance between the stock A & the index B, C, & D ? 4. Breakdown the variance of the 4 assets The standard deviation of market portfolio is 35%

9. variance of an equally weighted portfolio with risk coefficient ß p in the single-factor economy It summarize as diversification increases; the total variance of portfolio approaches the systematic variance, defined as the variance of the market factor multiplied by square of the portfolio sensitivity coefficient, ß 2 / p

10. Thank you