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# Time-Varying Beta Model: HAR-Beta

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

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## Time-Varying Beta Model: HAR-Beta

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

2. Background • CAPM Model • Ra,t+1= Ba,t*Rm,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. βt+1 = β0 + αDβt + αWβt-5,t + αMβt-22,t + εt+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. 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)

7. Mean Squared Error’s

8. Mean Squared Error’s

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

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