1. Time-Varying Beta Model: HAR-Beta Kunal Jain Economics 201FS Duke University April 21, 2010

2. Background CAPM Model  R a,t+1 = B a,t *R m,t  Conventional CAPM model uses a constant beta computed from monthly returns over a 5-year time period. (Banz, Journal of Financial Economics, 1981) Harvey (1989), Ferson and Harvey (1991,1993), Jagannathan and Wang (1996) all question the notion of a constant beta element –Try different modeling strategies to estimate a time varying beta. HAR-Beta Model Calculate Realized Betas over a 1-day, 5-day, and 1-month time interval to build the conditional betas. t=1 corresponds to daily realized Beta, t=5 corresponds to weekly realized Beta, t=22 corresponds to monthly realized Beta. β t+1 = β 0 + α D β t + α W β t-5,t + α M β t-22,t + ε t+1

3. Motivation Motivation: Test the validity of the HAR-Beta model, using daily, weekly, and monthly realized Betas, to substantiate a time-varying Beta model to estimate daily returns. Method: –Find mean return from 5 year-daily data Compute differentials over a specified time interval to find MSE –Calculate Constant Betas from monthly 5-year data Simulate returns using constant Beta to find MSE over specified time interval –Calculate HAR-Beta Coefficients Model calculated Beta Coefficients over specified time interval to find predicted Betas. Simulate returns using time-varying HAR-Beta to find MSE over specified time interval

4. Data SPY –January 2, 2001 – January 3, 2009 KO, PEP, MSFT, BAC, JNJ, WMT, XOM, AMZN, JPM (9 equities) –January 2, 2001 – January 3, 2009 Calculated Time Interval –January 2, 2001-January 2, 2006 Simulated Time Interval –January 3, 2006 – January 2, 2008 Sampling Frequency- 10 minutes Units – Annualized Standard Deviation

5. EquityBeta (2001-2006) 5-year monthly returns (10-minute sampling) Coca Cola (KO)0.498943 Pepsi (PEP)0.407434 Microsoft (MSFT)1.103915 Bank of America (BAC).483028 Johnson & Johnson (JNJ).295775 Wal-mart (WMT).625543 Exxon Mobil (XOM).589867 Amazon (AMZN)2.632052 J.P. Morgan Chase (JPM)1.578181 Constant Beta

6. HAR-Beta (KO,SPY) Calculate HAR-Beta coefficients over calculated time interval (January 2, 2001-January 2, 2006) Calculate Beta predictions using calculated HAR-Beta coefficients over simulated time interval (January 3, 2006 – January 2, 2008) Use Beta predictions to calculate expected return and compare with actual return to find differentials. –MSE:.17317 (Annualized Standard Deviation Units) β0β0 0.0096 Β t-1 0.0289 Β t-5 0.2695 Β t-22 0.6367

7. Mean Squared Error’s 10-Minute Sampling Standard Deviation Units EquityMean ReturnConstant BetaHAR-Beta Coca Cola (KO) 0.49350.18670.1732 Pepsi (PEP) 0.25110.20950.1963 Microsoft (MSFT) 0.28330.32710.2831 Bank of America (BAC) 0.52890.34100.3340 Johnson & Johnson (JNJ) 0.27160.16740.1650 Wal-mart (WMT) 0.22150.36520.3441 Exxon Mobil (XOM) 0.30760.31420.2892 Amazon (AMZN) 1.01000.73070.6414 J.P. Morgan Chase (JPM) 1.14330.47630.4347

8. Mean Squared Error’s 5-Minute Sampling Standard Deviation Units EquityMean ReturnConstant BetaHAR-Beta Pepsi (PEP) 0.71560.18560.1809 Microsoft (MSFT) 0.64810.32840.3189 Bank of America (BAC) 0.49130.32970.3326 Johnson & Johnson (JNJ) 0.16310.16260.1625

9. Future Research Analysis with more equities Different Sampling Frequencies Test HAR-Beta estimates with weekly returns Specific Literature