7.1 A SINGLE-FACTOR SECURITY MARKET

 Input list (portfolio selection) ◦ N estimates of expected returns ◦ N estimates of variance ◦ n(n-1)/2 estimates of covariance  Errors in estimation of correlation coefficients  A model to simplifies the way describing the sources of security risk

 Decomposing uncertainty into the system- wide versus firm-specific sources ◦ Common economic factors  Business cycles, interest rates, technological changes, cost of labor and raw materials  Affect the fortunes of many firms ◦ Firm specific events  Assume one macroeconomic indicator moves the security market as a whole, all remaining uncertainty in stock returns is firm specific

 Reduces the number of inputs for diversification  Easier for security analysts to specialize Advantages of the Single Index Model

Decompose rate has mean of 0, SD=  Security return, If normal distribution and correlated across securities ◦ joint normally distributed ◦ driven by one or more common variables ◦ Multivariate normal distribution  Single factor security ◦ Only one variable rives the joint normally distributed return Expected unexpected

Holding-period return on security i =impact of unanticipated macro events on the security ’ s return, SD= = impact of unexpected firm specific event, SD=, have zero expected values, uncorrelated

 Variance of r arises from two uncorrelated sources  m generates correlation across securities  Covariance between any two securities i and j is

 m, unanticipated components of macro factor  ß i, responsiveness of security i to macro-events ◦ Different firms have different sensitivities to macroeconomic events  Single-Factor model

Single-Index Model Continued  Risk and covariance: ◦ Total risk = Systematic risk + Firm-specific risk: ◦ covariance Systematic risk

7.2 A SINGLE-INDEX MODEL

 Single-Index model ◦ Assumption: a broad market index like the S&P500 is a valid proxy for the common macroeconomic factor, as the common or systematic factor  Regression equation( regress Ri on RM)  M: market index, excess return SD=, serurity’s excess return

 Holding-period excess return on the stock Due to movements in overall market Residential, Due to firm specific factors Security’s expected excess return when market excess return is 0

Let: R i = (r i - r f ) R m = (r m - r f ) Risk premium format R i =  i + ß i (R m ) + e i

Single-Index Model  Regression Equation:  Expected return-beta relationship: Nonmarket premium Systematic risk premium: market risk premium multiplied by sensitivity

Single-Index Model  Risk and covariance: ◦ Total risk = Systematic risk + Firm-specific risk: ◦ Covariance = product of betas x market index risk: ◦ Correlation = product of correlations with the market index

 Input for single-index model ◦ n estimates of expected returns ◦n◦n ◦ n estimate of firm-specific variance ◦ 1 estimate of market risk premium ◦ 1 estimate of variance of macroeconomic factor  Index model abstraction is crucial ◦ for specialization of effort in security analysis ◦ provide a simple way to compute covariance

 Suppose choose an equally weighted portfolio of n securities, the excess return on each security is  The excess return on the portfolio of stocks is  To show: when n increases, nonmarket factors becomes smaller (diversified away), market risk remain

 Systematic risk component of the portfolio variance:  Nonsystematic component is attributable to firm-specific components  is average of the firm-specific variances. When n gets large, gets negligible.

7.3 ESTIMATING THE SINGLE-INDEX MODEL

 Using montly data for six stocks (IT/RETAIL/ENERGY), S&P 500, and T-bill from 2001 to 2006 (60 observations) to estimate the regression equation  Regress risk premiums for individual stocks against the risk premiums for the S&P 500  Slope is the beta for the individual stock  HP as an example

 Relationship between the excess returns on HP and the S&P 500 (regression equation)  SCL (security characteristic line) ◦ Regression estimates describe a straight line with,   is the sensitivity of HP to the market, slope of the regression line  intercept, representing the average firm-specific excess return when the market ’ s excess return is zero.  residual, vertical distance of each point from the regression line

Figure 8.2 Excess Returns on HP and S&P 500 April 2001 – March 2006 Annualized SD OF S&P=13.58% Annualized SD OF HP=38.17% Greater than average sensitivity to the index, beta>1

Scatter Diagram of HP, the S&P 500, and the Security Characteristic Line (SCL) for HP

Table 8.1 Excel Output: Regression Statistics for the SCL of Hewlett-Packard

◦ Variation in the S&P 500 excess return explains about 52% of the variation in the HP series. ◦ Correlation: ◦ SSR: sum of squares of the regression (0.3752) is the portion of the variance of the dependent variable (HP) that is explained by the independent variable (S&P) ◦ SSE: variance of the unexplained portion, independent of the market index SSR df=k SSE df=n-k-1 SST df=n-1

SSR df=k SSE df=n-k-1 SST df=n-1

◦ MSR=SSR/k=0.3752/1= ◦ MSE=SSE/n-k-1=0.3410/58= ◦ Standard error of the regression is square root of MSE, (firm-specific risk) ◦ Estimate of monthly variance of the dependent variable (HP) =0.7162/59=0.012 ◦ Annualized SD of dependent variable SSR df=k SSE df=n-k-1 SST df=n-1

 Estimate of Alpha ◦ Alpha=0.86%, t-statistic=0.8719<2, not reject null, too low to reject the hypothesis that the true value of alpha is 0 ◦ HP’s return net of the impact of market movements Explanatory Power of SCL for HP Nonmarket component of HP’s return actual return the return attributable to market movements

 Estimate of Beta ◦ Beta=2.0348, t-statistic=7.9888>2, reject null,  Firm specific risk ◦ Monthly SD of HP’s residual is 7.67%, or 26.6% annually (firm- specific risk) ◦ SD of systematic risk Explanatory Power of SCL for HP

 Six stocks: ◦ HP,DELL; ◦ TARGET, WALMART; ◦ BP,SHELL.

Excess Returns on Portfolio Assets

 Tremendous firm-specific risk (see excel)  For any pairs of securities, get the estimates of the risk parameters of the six securities and S&P500  Correlations of residuals ◦ for same-sector stocks are higher; ◦ cross-industry correlations are far smaller  Covariance matrix

Alpha and Security Analysis  Index model creates a framework that separates the two quite different sources of return variation, easier to ensure consistency across analysts ◦ Macroeconomic analysis is used to estimate the risk premium and risk of the market index ◦ Statistical analysis is used to estimate the beta coefficients of all securities and their residual variances, σ 2 ( e i ) ◦ establish expected return of the security absent any contribution from security analysis, the market-driven expected return is conditional on information common to all securities ◦ Security-specific expected return forecasts are derived from various security-valuation models, the alpha value distills the incremental risk premium attributable to private information developed from security analysis

Alpha and Security Analysis  The alpha helps determine whether security is a good or bad buy ◦ Risk premium on a security not subject to security analysis would be, any expected return beyond this benchmark risk premium (alpha) would be due to some non- market factors uncovered by security analysis ◦ Security with positive alpha is providing a premium over and above the premium it derives from its tendency to track the market index, should be over-weighted in portfolio

 To include the indexed portfolio as an asset of the portfolio to avoid inadequate diversification ◦ Beta=1, no firm-specific risk, alpha=0, no non- market factors in its return ◦ (n+1)th security ◦ The portfolio: n actively researched firms and a passive market index portfolio

Single-Index Model Input List  Risk premium on the S&P 500 portfolio  Estimate of the SD of the S&P 500 portfolio  n sets of estimates of ◦ Beta coefficient ◦ Stock residual variances ◦ Alpha values  Generate n+1 expected return, covariance matrix

Optimal Risky Portfolio of the Single-Index Model  Maximize the Sharpe ratio to get portfolio weights ◦ Expected return, SD, and Sharpe ratio:

 Basic trade-off of the model ◦ For diversification, holding the market index ◦ Security analysis gives chance to uncover nonzero alpha securities and take differential position ◦ Cost: unnecessary firm-specific risk ◦ The optimal risky portfolio trade off the search for alpha against departure from efficient diversification

Optimal Risky Portfolio of the Single- Index Model Continued  Combination of: ◦ Active portfolio denoted by A, comprised of the n analyzed securities ◦ Market-index portfolio, the (n+1)th asset which we call the passive portfolio and denote by M  Assume beta for A is 1. optimal weight of active portfolio

Optimal Risky Portfolio of the Single- Index Model Continued  Combination of: ◦ Modification of active portfolio position: ◦ When

The Information Ratio  The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy):  Information ratio: ratio of alpha to its residual SD, measures the extra return we can obtain from security analysis compared to the firm-specific risk we incur when we over-or-underweight securities relative to the passive market index.  Maximize Sharpe ratio, to maximize information ratio of A

 Maximize information ratio, get weight of each security in A  The total position in the active portfolio adds up to  The weight of each security in the optimal portfolio (M+A) is

 The positive contribution of a security to the portfolio is made by its addition to the nonmarket risk premium (alpha)  The negative impact is to increase the portfolio variance through firm-specific risk (residual variance)

 After security analysis, index-model estimates of security and market index parameters, to form the optimal risky portfolio ◦ Initial position of each security in A ◦ Scale Alpha: ◦ Residual variance of A: Initial position in A ◦ Beta of A Adjust ◦ Optimal risky portfolio weight ◦ Risk premium and variance of the optimal risky portfolio ◦

Figure 8.5 Efficient Frontiers with the Index Model and Full-Covariance Matrix

Table 8.2 Comparison of Portfolios from the Single-Index and Full-Covariance Models