I am having problem setting up part 2 in excel.
UNIVERSITY OF ILLINOIS AT CHICAGO Liautaud Graduate School of Business Department of Finance Professor Hsiu-lang Chen1 Practice Problem III Part I. Validity of Multi-Factor Models (Lecture 5) In this exercise we want to verify the statistics presented on the slide of Validity of Multi-Factor Models in showing the benefit of using multi-factor models in asset pricing. Download monthly returns on the UAL and GE from http://wrds.wharton.upenn.edu (user name: uicfin15), and Fama-French 3 factors from Kenneth French web site. Company Name Stock Symbol Date Range United Continental Holdings Inc General Electric Co UAL GE 1/1990-12/2014 1/1990-12/2014 Perform regressions in two steps in three time periods: Step1 Regression: Compute residual returns for each stock using zero-, one-, and three-factor models. Zero-Factor Model: ri,t = i,t One-Factor Model: ri,t - rf,t = ai + RMRF,iFRMRF,t + i,t Three-Factor Model: ri,t - rf,t = ai + RMRF,iFRMRF,t + SMB,iFSMB,t + HML,iFHML,t + i,t Step2 Regression: Regress GE residual returns on UAL residual returns GE,t = a + b UAL,t + e What R2 in Step2 regression would you expect to find? Fill out the following table: R2 in Step2 Regression Period 1 Period 2 1/1990~12/1999 1/2000~12/2014 Period 3 1/1990~12/2014 Zero-Factor One-Factor Three-Factor 1 Part of this practice problem is constructed based on the homework assignment by Professor Schaumburg at Kellogg School of Management where I was visiting during my sabbatical leave in Fall 2005. Part II. Application of Multi-Factor Models in Portfolio Management (Lecture 5) Suppose you are the manager of the fund of domestic equity funds. - You have no stock picking ability - You do have factor forecasting/timing ability Consider investing in six portfolios, formed by sorting all stocks according to their market capitalization and book-to-market ratio: S/L (small growth), S/M (small core), S/H (small value), B/L (large growth), B/M (large core), and B/H (large value). You believe the following four-factor model holds: ri,t - rf,t = ai + bRMRF,iFRMRF,t + bTS,iFTS,t + bYS,iFYS,t + bOI,iFOI,t + i,t - FRMRF is the excess return on the stock market index - FTS is the yield spread between 10-year T-bond and 1-year T-bill. - FYS is the yield spread between Moody's Baa and Aaa corporate bonds. - FOI is Oil inflation (percentage change in Crude Oil Prices: WTI). (The data for FTS, FYS, FOI is from http://research.stlouisfed.org) Following the discussion in class, use the data in the spreadsheet file \"fundret.xls\" to answer questions below. (1) Suppose that at the beginning of Year 2015 you want to minimize your total portfolio variance and target at least 12% per year in expected returns, but 1. You would not take oil-price risk. 2. You would like your portfolio to move one-to-one with the market. How do you construct such a portfolio? Instruction: Step 1: Estimate each underlying asset's factor loadings Step 2: Construct each underlying asset's expected (excess) returns based on the 4-factor model Step 3: Construct the covariance matrix of 4 factors Step 4: Construct underlying assets' return covariance matrix based on the 4-factor model Step 5: Set up the constraints Step 6: Set up the solver in Excel (2) Suppose that at the beginning of Year 2015 you create an optimal portfolio tracking Russell 3000 and earning at least 2% per year above the expected return on the Russell 3000. You want to minimize the variance of the tracking errors. What is your optimal portfolio? What is the factor sensitivity of your tracking errors to these risk factors? (3) Same as the (2). Additionally, you want to neutralize the tracking error risk exposures to both FRMRF and FOil. What is your new optimal portfolio? (4) Suppose that at the beginning of Year 2015 you create a \"2x Russell 3000 Index\" ETF, which is a portfolio delivering twice the returns of Russell 3000 Index. You want to minimize the variance of the tracking errors. What is your optimal portfolio? What is the factor sensitivity of your tracking errors? (In practice, leveraged ETFs might invest in equity index swaps, index futures, and options on securities.) Part III. Model Comparison in Out-of-Sample Test (Lecture 5) Compare the 4-factor model in Part II to the Fama and French 4-Factor model (RMRF, SMB, HML, and UMD) in explaining the six style portfolios in Part II. A model is said to be better if the model can have a smaller prediction error and/or a smaller volatility of prediction errors. Which model is better? Instruction: Step 1: Estimate factor loadings in each model every quarter based on returns over the prior 5-year period. (Factor loadings are updated every quarter.) Step 2: Compare the six style portfolio returns implied by each model to the actual returns over 3 months following the update of factor loadings. The implied return by a model is the pre-estimated factor loadings times the factor value. Step 3: Calculate prediction errors each month for each style portfolio. The prediction error by a model is the actual return minus the return implied by the model. Step 4: At the end of 2014, calculate the average and the standard deviation of monthly prediction errors by a model for each style portfolio. Step 5: Fill out the following table Style Portfolio SL SM SH BL BM BH Prediction Errors Part II 4-Factor FF 4-Factor Difference AVG STD AVG STD AVG STD AVG STD AVG STD AVG STD Part IV. Mutual Funds (Lecture 6) 1. Style AnalysisQuadratic Programming This exercise is to give you an idea of the mechanics behind Style Analysis presented in the lecture. Sharpe (1992) argues that most of the returns generated by active managers can be explained by the asset classes they invest in, the so-called Style Benchmarks. The analysis is carried out by \"regressing\" the monthly fund returns on a set of appropriately chosen stock indices (the Style Benchmarks) which span the manager's asset universe. However, since the manager typically cannot short, the regression coefficients are constrained to be non-negative and must add up to one. The coefficients interpreted as the fund portfolio weights are used to identify the fund's investment styles. You'll be using the data in fundret.xls which contains monthly returns of two popular equity funds as well as returns on Russell Style Benchmarks since 1979. The type of constrained regression we have in mind is ~ ~ ~ ~ R ab R b R b R 1 f 2 Russell 1000G 3 Russell 1000V Fund ~ ~ ~ b R b R 5 Russell 2000V 4 Russell 2000G 5 S .T . b 1 b ( Style Indication ) 0 j j j j 1 Instruction: Step 1. Calculate the residual for each month as a function of the coefficients {b1, b2, ..., b5}. Step 2. Calculate the variance of the residuals. Step 3. Have Solver search for the set of 5 coefficients which minimize the residual variance in step 2, subject to the constraint that the coefficients {b1, b2, ..., b5} be nonnegative and add up to one. Step 4. Finally, calculate the R2, i.e. the fraction of the total return variance explained by your model (with the coefficients you solved for in step 3). After learning Quadratic Programming, you perform style analysis and performance analysis on both Fidelity Magellan Fund and Growth Fund of America (Class A shares), which is managed by Capital Research and Management Company. a. Create the style history graph of Magellan Fund and Growth Fund of America and compare their investment style evolution. Given both are diversified funds, you may use 3-year or 5-year period to estimate their investment styles. The fund style needs to be updated at least annually. b. Create the cumulative style-adjusted returns graph of Magellan Fund and Growth Fund of America and compare their performance based on your t-statistics. Do both perform better than S&P500 Index? c. Analyze if changes in Magellan Fund managers affect its style and performance. 2. Fama and French (2010) examine the performance during 1984-2006 of actively managed US mutual funds that invest primarily in US equities. After reading their article, which is available at the blackboard, and commentary, which is available at the following web site, on luck versus skill in mutual fund performance, what do you conclude if a better performing fund is skilled or just lucky? http://www.dimensional.com/famafrench/essays/luck-versus-skill-in-mutual-fundperformance.aspx Part V. Factor Models and Markowitz (Lecture 5) Suppose you are a fund manager restricted to investing in 3 stocks, A, B and C. The risk free earns rf=3%. Your analyst tells you that a 2-factor model with uncorrelated factors accurately describes returns. Note that f1 and f2 are factor surprises and the intercepts hence are the respective expected returns (e.g. E(rA)=0.36) : rA = 0.36 + 2 f1 + 4 f2 + A rB = 0.225 + 3 f1 + 2 f2 + B rC = 0.12 + 1 f1 + 1 f2 + C with standard deviations given by f1=0.1 , f2=0.1 , A=0.15 , B=0.28 , C=0.05. 1. What is the covariance matrix between the three asset returns? [Hint: we are assuming that the factors are uncorrelated] 2. What is the three R2s that your analyst got when he regressed the returns of A, B and C respectively on the factor realizations? 3. What is the optimal risky portfolio of A, B and C? Its Sharpe ratio? Its factor loadings on f1 and f2? [Hint: Markowitz portfolio optimization again!] 4. Suppose that your clients do not want any exposure to Factor 2 risk. How does your optimal portfolio look now? PART V. BKM Problems (Lecture 6) Chapter 24, problems 8-11
