TimeVarying Beta Model HARBeta Kunal Jain Economics 201

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

Background • CAPM Model Ø Ra, t+1= Ba, t*Rm, t Ø 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. βt+1 = β 0 + αD βt + αW βt-5, t + αM βt-22, t + εt+1 • t=1 corresponds to daily realized Beta, t=5 corresponds to weekly realized Beta, t=22 corresponds to monthly realized Beta.

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

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

Constant Beta Equity Beta (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

HAR-Beta (KO, SPY) • Calculate HAR-Beta coefficients over calculated time interval (January 2, 2001 -January 2, 2006) β 0 0. 0096 Βt-1 0. 0289 Βt-5 0. 2695 Βt-22 0. 6367 • 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)

Mean Squared Error’s 10 -Minute Sampling Standard Deviation Units Equity Mean Return Constant Beta HAR-Beta Coca Cola (KO) 0. 4935 0. 1867 0. 1732 Pepsi (PEP) 0. 2511 0. 2095 0. 1963 Microsoft (MSFT) 0. 2833 0. 3271 0. 2831 Bank of America (BAC) 0. 5289 0. 3410 0. 3340 Johnson & Johnson (JNJ) 0. 2716 0. 1674 0. 1650 Wal-mart (WMT) 0. 2215 0. 3652 0. 3441 Exxon Mobil (XOM) 0. 3076 0. 3142 0. 2892 Amazon (AMZN) 1. 0100 0. 7307 0. 6414 J. P. Morgan Chase (JPM) 1. 1433 0. 4763 0. 4347

Mean Squared Error’s 5 -Minute Sampling Standard Deviation Units Equity Mean Return Constant Beta HAR-Beta Pepsi (PEP) 0. 7156 0. 1809 Microsoft (MSFT) 0. 6481 0. 3284 0. 3189 Bank of America (BAC) 0. 4913 0. 3297 0. 3326 Johnson & Johnson (JNJ) 0. 1631 0. 1626 0. 1625

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