Index Models Chapter 8

Advantages of the Single Index Model Reduces the number of inputs for diversification. Can be used to estimate covariance estimates for MPT-type analysis Easier for security analysts to specialize. Provides “easy-to-understand” expected return estimates if factor is the VW market excess return. CAPM based

Single Factor Model ri = E(Ri) + ßiF + e Where E(Ri) = return to the asset that is not due to risk (i.e., the risk-free rate component). ßi = index of a securities’ particular return to the factor F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns Assumption: a broad market index like the S&P500 is the common factor.

a Single Index Model (ri - rf) = i + ßi(rm - rf) + ei Risk Prem Market Risk Prem or Index Risk Prem a = the stock’s expected return if the market’s excess return is zero: i (rm - rf) = 0 ßi(rm - rf) = the component of return due to movements in the market index ei = firm specific component, not due to market movements

Risk Premium Format Let: Ri = (ri - rf) Rm = (rm - rf) Risk premium format Ri = i + ßi(Rm) + ei

Example: advantages of using factor model to estimate covariances

Study: Article on Modern Portfolio Theory and how well it really works See class web page, handouts, chapter 8, “Article on Modern Portfolio Theory and how well it really works.” Motivation How good of a job do single factor and multifactor models do at estimating covariances in a MPT setting? In other words, can we form a higher Sharpe ratio portfolio using factor model estimated covariances instead of a covariance matrix based off historical time-series of returns?

Study: Article on Modern Portfolio Theory and how well it really works Results The SR of the minimum variance portfolio is: 0.69 using a single factor model (with VW as the factor) 0.64 using historical return data 0.62 using a nine factor model Conclusions The single factor model does a better job than just using historical data. No improvement in the SR using a much larger number of factors.

Security Characteristic Line The slope = beta Excess Returns (i) SCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excess returns on market index . . . . . . . . . . . . . . . . . Ri =  i + ßiRm + ei

Example Excess GM Ret. Excess Mkt. Ret. Jan. Feb. . Dec Mean Std Dev 5.41 -3.44 . 2.43 -.60 4.97 7.24 .93 . 3.90 1.75 3.32

  Regression Results rGM - rf = + ß(rm - rf) ß Estimated coefficient Std error of estimate Variance of residuals = 12.601 Std dev of residuals = 3.550 R-SQR = 0.575 = (correlation coefficient)2 -2.590 (1.547) 1.1357 (0.309)

Components of Risk Market or systematic risk: risk related to the macro economic factor or market index. Unsystematic or firm specific risk: risk not related to the macro factor or market index. Total risk = Systematic + Unsystematic

The Components of Total Risk i2 = i2 m2 + 2(ei) The first term is market risk, and the second term is firm specific risk where; i2 = total variance (or total risk) i2 m2 = systematic variance 2(ei) = unsystematic variance

Simple: Just take the variance of the security characteristic line. I know what you are thinking! You are wondering how they derived the expression for total risk, right? Simple: Just take the variance of the security characteristic line.

Examining Percentage of Variance Total Risk = Systematic Risk + Unsystematic Risk Systematic Risk/Total Risk = 2 = R-SQR ßi2  m2 / 2 = 2 i2 m2 /[i2 m2 + 2(ei)] = 2 Lets do some examples!

Index Model and Diversification

Risk Reduction with Diversification St. Deviation Unique Risk s2(eP)=s2(e) / n bP2sM2 Market Risk Number of Securities

Oh no! Some more examples http://finance.yahoo.com/q/bc?t=6m&s=MSO&l=on&z=m&q=l&c=&c=%5EGSPC Her stock could be a real “steal” right now…hahaha

Industry Prediction of Beta Merrill Lynch Example Use returns not risk premiums a has a different interpretation a = a + rf (1-b) Forecasting beta as a function of past beta Forecasting beta as a function of firm size, growth, leverage etc.

Up next: the case for multi-factor models and a study Should the “correct” asset pricing model have multiple factors? Study on how well multifactor models predict the cross-section of future stock returns.

Example See external class web page, Excel handouts, chapter 8, “The use of macroeconomic variables to predict the S&P500”

Study: Economic Forces and the Stock Market See class web page, handouts, chapter 8, “Economic Forces and the Stock Market” Motivation Are betas from other factors (besides the market beta) priced in the cross-section of expected returns? Method Perform “two-pass” regressions. First, estimate factor loadings from time series regressions to obtain firm betas for multiple factors, then use cross-sectional regressions to see if the factors are priced.

Study: Economic Forces and the Stock Market Macroeconomic variables used in the study: MP = monthly growth in industrial production DEI = change in expected inflation UI = unexpected inflation UPR = risk premium (junk spread) UTS = term structure (slope of the yield curve)

Study: Economic Forces and the Stock Market

Study: Economic Forces and the Stock Market Conclusions: MP (industrial productions), UI (inflation), UPR (junk spread) appear to be priced. CAPM beta is not priced.

Study: Multifactor explanations of asset pricing anomalies See class web page, handouts, chapter 8, “Multifactor explanations of asset pricing anomalies ” Motivation Can a multifactor model, using returns from well diversified portfolios as the factors, explain most or all of the common return anomalies? Aside: size, E/P, C/P, B/M, lagged returns, and other firm-specific variables (which all appear to predict stock returns) are called anomalies because the CAPM cannot explain their ability to forecast future stock returns.

Study: Multifactor explanations of asset pricing anomalies The three-factor model Ri-rf = ai + bi(Rm-rf) + si(SMB) + hi(HML) + ei Where Ri = return to stock i (or portfolio) SMB = small minus big; the difference between the return on a portfolio of small stocks and the return on a portfolio of large stocks. HML= high minus low; the difference between the return on a portfolio of high book-to-market stocks and the return on a portfolio of low book-to-market stocks. bi, si, and hi = factor loadings See also on external web page: Excel Handouts, chapter 8, “Example performance calculations” for an example of estimating a 3-factor model alpha on IBM monthly stock returns. Handouts, chapter 4, “Factor models used in performance calculations.”

Study: Multifactor explanations of asset pricing anomalies Testing methodology: Create a time-series of returns to a given anomaly. Example: cash-to-price. Sort all firms each year into deciles on lagged annual c/p. Hold portfolios for one year, then rebalance. Excess return spread: Decile 1 (low c/p) = 0.43%/month Decile 10 (high c/p) = 1.20%/month Spread = 0.77% month. Can time series variations in the market return, SMB, and HML explain this spread? Estimate the time series model: R(cp10-cp1)-rf = ai + bi(Rm-rf) + si(SMB) + hi(HML) + ei Null: a=0. If we reject the null, then the 3-factor model can’t explain or “price” the anomaly.

Study: Multifactor explanations of asset pricing anomalies Conclusions The 3-factor model can explain most accounting ratio anomalies. The 3-factor model can’t explain momentum.