CHAPTER EIGHT INDEX MODELS University of Economics, Ho Chi Minh City - UEH Slide Team • • • • Trần Thị Thanh Thủy Phạm Thanh Nhất Võ Thị TrúcXuân Huỳnh Thị Bé Tư Giảng viên hướng dẫn: TS Trần Thị Hải Lý 8.1 A SINGEL FACTOR SECURITY MARKET Drawbacks of Markowitz model The Input List of the Markowitz Model: - The estimates of expected security returns - The covariance matrix EX: n = 50 estimates of expected returns n = 50 estimates of variances (n2 - n)/2 = 1,225 estimates of covariances estimates   1,325 total Drawbacks of Markowitz model If n = 100 , we need 5,150 estimates n = 300, we need 4,5 million estimates  This work is very hard Errors in the assessment or estimation of correlation coefficients can lead to nonsensical results   EX:     Standard   Asset Deviation (%)   A 20 B 20 C 20       A     B C 1.00 0.90 0.90 0.90 1.00 0.00 0.90 0.00 1.00 Drawbacks of Markowitz model EX: - Suppose that you construct a portfolio with weights: -1.00; 1.00; 1.00, for assets A; B; C, respectively, and calculate the portfolio variance - You will find that the portfolio variance appears to be negative (200) This of course is not possible because portfolio variances cannot be negative Drawbacks of Markowitz model  This chapter we introduce index models that simplify estimation of the covariance matrix and greatly enhance the analysis of security risk premiums Normality of Returns and Systematic Risk - The rate of return on any security, i, into the sum of its expected plus unanticipated components: ri = E (ri) + ei + E (ri) : expected return + ei : unexpected return ei, has a mean of zero and a standard deviation of бii that measures the uncertainty about the security return Normality of Returns and Systematic Risk - Joint normally distributed: security returns are driven by one common variables - Multivariate normal distribution: When more than one variable drives normally distributed security returns Normality of Returns and Systematic Risk ri = E (ri) + m + ei + m: The macroeconomic factor, measures unanticipated macro surprises (mean =0, бm ) + ei : measures only the firm-specific surprise + m and ei are uncorrelated Normality of Returns and Systematic Risk - The variance of ri : бi2 = бm2 + б2 (ei ) - The covariance between any two securities i and j: Cov (ri, rj) = cov (m + ei, m + ej) = бm2 - We can capture this refinement by assigning each firm a sensitivity coefficient to macro conditions, βi 8.4 Portfolio Construction and the Single-Index Model The Optimal Risky Portfolio in the Single-Index Model The optimal risky portfolio turns out to be a combination of two component portfolios: (1) an active portfolio (A), comprised of the n analyzed securities (2) the market-index portfolio, called the passive portfolio (M) Assume that the active portfolio has a beta of 1: • • The optimal weight in the active portfolio: αA/σ (eA) The analogous ratio for the index portfolio: E(R M)/σ M (8.20) 8.4 Portfolio Construction and the Single-Index Model The Optimal Risky Portfolio in the Single-Index Model The correlation between the active and passive portfolios is greater when the beta of the active portfolio is higher This implies less diversification benefit from the passive portfolio and a lower position in it Correspondingly, the position in the active portfolio increases The precise modification for the position in the active portfolio is: (8.21) 8.4 Portfolio Construction and the Single-Index Model The Information Ratio (8.22)   • • Shape ratio of an active portfolio >Shape ratio of the market-index portfolio The Information Ratio: the extra return we can obtain from security analysis to maximize the overall Sharpe ratio, we must maximize the information ratio of the active portfolio 8.4 Portfolio Construction and the Single-Index Model The Information Ratio The weight in each security is: (8.23) the contribution of each security to the information ratio of the active portfolio is the square of its own information ratio, that is: (8.24) 8.4 Portfolio Construction and the Single-Index Model Summary of Optimization Procedure 8.4 Portfolio Construction and the Single-Index Model Summary of Optimization Procedure 8.4 Portfolio Construction and the Single-Index Model Summary of Optimization Procedure 8.4 Portfolio Construction and the Single-Index Model AN EXAMPLE Risk Premium Forecasts 8.4 Portfolio Construction and the Single-Index Model AN EXAMPLE The Optimal Risky Portfolio 8.4 Portfolio Construction and the Single-Index Model AN EXAMPLE The Optimal Risky Portfolio Figure 8.5 Efficient frontiers with the index model and full-covariance matrix 8.5 Practical Aspects of Portfolio Management with the Index Model Is the Index Model Inferior to the Full-Covariance Model? 8.5 Practical Aspects of Portfolio Management with the Index Model The Industry Version of the Index Model A portfolio manager who has neither special information about a security nor insight that is unavailable to the general public will take the security’s alpha value as zero, according to Equation 8.9: E(ri) = rf + ßi[E(rM) – rf] (8.25) There are several proprietary sources for such regression results, sometimes called “beta books.” Table 8.3 is a sample of a typical page from a beta book 8.5 Practical Aspects of Portfolio Management with the Index Model The Industry Version of the Index Model * r = a + brM + e (8.26) instead of: r - rf = α + ß.(rM - rf) + e (8.27) • To see the effect of this departure, we can rewrite Equation 8.27 as: r = rf + α + ß.rM – ß.rf + e = α + rf(1-ß) + ß.rM (8.28) 2 2 2 RSD Intel 6.27% tháng R = 0.369 Có nghĩa là: Xichma (e) = 6.27 = 39.31 R = 1- xichma (e)/ xichma , tính tổng độ lệch chuẩn intel sau: intel 8.5 Practical Aspects of Portfolio Management with the Index Model What was Intel’s index-model a per month during the period covered by the Table 8.3 regression if during this period the average monthly rate of return on T-bills was 2%? THANK YOU FOR WATCHING ... sensitivity coefficient to macro conditions, βi 8.2 THE SINGLE INDEX MODEL  •- The Regression Equation of the Single -Index Model The market index : M Excess return: RM = rM – rf Standard deviation:... Market index risk Cov(ri, rj) =   - Correlation = Product of correlations with the market index Corr(ri, rj) = = = Corr(ri, rM) x Corr(rj, rM) The Set of Estimates Needed for the Single -Index. .. security’s risk premium is due to the risk premium of the index Systematic risk premium: E(RM)   Nonmarket premium: Risk and Covariance in the Single -Index Model - Total risk = Systematic risk + Firm-specific