1. Time-Varying Beta: Heterogeneous Autoregressive Beta Model Kunal Jain Spring 2010 Economics 201FS May 5, 2010

2. Background Beta (β) is a commonly defined measure of market risk, which essentially measures the volatility of returns on assets or securities and co-movements of the market portfolio. Demonstrated Through: Security Characteristic Line (SCL) Markets are efficient (α = 0) Effective Risk Free Rate = 0 R i,t = β*R m,t + ε i,t and Capital Asset Pricing Model (CAPM)- conventionally estimate betas through monthly returns over a 5-year time horizon (Banz, 1981). Static Beta unable to satisfactorily explain cross-section of average returns on stock. (Harvey (1989), Ferson and Harvey (1993), Jagannathan and Wang (1994)). Recent literature suggests that a static-beta model may produce incoherent results.

3. Data SPY –January 2, 2001 – January 3, 2009 KO, PEP, MSFT, JPM, BAC, JNJ, WMT, XOM –January 2, 2001 – January 3, 2009 In Sample Time Interval –January 2, 2001 – January 2, 2006 Out of Sample Time Interval (A) –January 3, 2006-January 2, 2008 Out of Sample Time Interval (B) –January 3, 2006-January 2, 2009

4. Optimal Sampling Frequency Hansen and Lunde (2006) define the market microstructure noise: u(t) = p(t) – p*(t) –p(t) is the observable log price in the market at time t –p*(t) is the latent real log price at time t. p(t + θ) – p(t) = [p*(t + θ) - p*(t)] + [u(t + θ) – u(t)] –θ is a real number increment –[p(t + θ) – p(t)] represents the change in price over a time interval –u(t) is i.i.d. and represents the microstructure noise applicable to the price change over the specified time interval. Volatility Signature Plot [Andersen, Bollerslev, Diebold and Labys (2000)] –This graphical approach displays how average realized variance corresponds to sampling frequency.

5. Volatility Signature Plots 10 minutes chosen as optimal sampling frequency for subsequent analysis.

6. Motivation Table 1: Standard Deviation of Beta (In Sample) CompanyStandard Deviation Coca Cola Company (KO) 0.3925 Pepsico, Inc. (PEP) 0.2745 Microsoft Corporation (MSFT) 0.3417 JPMorgan Chase & Co. (JPM) 0.4485 Bank of America Corporation (BAC) 0.3721 Johnson & Johnson (JNJ) 0.3822 Wal-mart Stores Inc. (WMT) 0.4783 Exxon Mobil Corporation (XOM) 0.2997 Given that beta was in fact constant over time, one would expect the standard deviation to be relatively zero omitting market microstructure noise.

7. Motivation- Cont.

8. AR(1)- F.O. Autocorrelations Table 2: AR(1)- First Order Autocorrelation of Beta Equity First Order Autocorrelati on Coca Cola Company (KO)0.5345 Pepsico, Inc. (PEP)0.2910 Microsoft Corporation (MSFT)0.0866 JPMorgan Chase & Co. (JPM)0.3296 Bank of America Corporation (BAC)0.3550 Johnson & Johnson (JNJ)0.4673 Wal-mart Stores Inc. (WMT)0.5701 Exxon Mobil Corporation (XOM)0.1531

9. HAR-Beta Model Realized Beta = Realized betas are realizations of the underlying ratio between the integrated stock and market return covariance and the integrated market variance –Integrated Variance = –Integrated Covariance = Underlying spot volatility is impossible to observe Discrete measures of variation can be used to numerically approximate integrated variance and integrated covariance. Realized Beta = –Cov(r i,t, r m,t ) is the covariance of realized returns on an asset i and returns on the market on sampling interval t – Var(r m,t ) is the variance of realized returns on asset i on the sampling interval t –N t is the number of units into which the sampling interval is partitioned into.

10. HAR-Beta Model Rβt = –t represent time –n represents the number of units Predict Daily Returns using: where the dependent variables correspond to lagged daily (t =1), weekly (t = 5) and monthly (t = 22) regressors. Rβ t+1,t = β 0 + α D* *Rβ t-1,t + α W *Rβ t-5,t + α M *Rβ t-22,t + ε t+1

11. HAR-Beta Regression Coefficients Table 3: HAR-Beta Regression Coefficients β0β0 Rβ t, t-1 Rβ t, t-5 Rβ t, t-22 Sum of Coefficients R2R2 KO0.0096**0.0289*0.2695*0.6367**0.94470.4784 PEP0.0051**0.0201*0.0880*0.6754**0.78860.0019 MSFT0.0107** 0.1023**- 0.0226- 0.3291*- 0.23870.0113 JPM0.0078*0.0210*0.4598**0.3476**0.83620.2547 BAC0.0011*0.0416*0.0864*0.7470**0.87610.2797 JNJ0.0066*- 0.0375*0.3764**0.5924**0.93790.4361 WMT0.0094*- 0.01510.4014**0.5613**0.95700.5391 XOM0.0034*0.0094- 0.0934*0.8429**0.76230.1106 The significance levels of the coefficients are denotes by the asterisk: * → p < 0.05, ** → p < 0.01

12. Benchmark Comparison (1) Constant Mean Return: –logarithmic returns at time t, R(t), are observed as a sum of all latent logarithmic returns leading up to time t-1, R(t - 1), divided by the number of observations n.

13. Benchmark Comparison (2) The usage of a beta computed from monthly returns over a 5-year time period has been noted by numerous studies including Banz (1981). Table 4: Constant betas computed using monthly data over 5-year period (In-Sample) CompanyBeta Β KO 0.4989 Β PEP 0.4074 Β MSFT 1.1039 Β JPM 1.5781 Β BAC 0.4830 Β JNJ 0.2958 Β WMT 0.6255 Β XOM 0.5899

14. Root Mean Squared Error (RMSE) Mean square forecast error (RMSE) is defined as: MSE = –n is the number of predictions contained –R a,t is out of sample realized logarithmic return on Asset a at time t –Ř a,t is the predicted return on Asset a at the corresponding time t. RMSE = –The result is the root mean squared error (RMSE) which can be interpreted in annualized standard deviation units.

15. Results- In Sample January 2, 2001 – January 2, 2006 Average SD of beta at 10 minute level is 0.3737 (statistically significant) Positive Autocorrelations –Average First Order Autocorrelation is 0.3484 –Notable that MSFT and XOM display low F.O. autocorrelation (0.0866, 0.1521) Less predictable Beta Coefficients = sum equal to one – Persistency –Notable exception is MSFT, sum = 0.2387 –XOM sum = 0.7623 Congruency

16. Results – Out of Sample (A) Table 5: Root Mean Squared Error (RMSE)-HAR-Beta, Constant Beta, Constant Return, and Standard Deviation of Beta (Out of Sample-A) RMSE HAR- BetaRMSE Constant BetaConstant Returns Standard Deviation of Beta (Out Sample) KO0.17320.18670.49350.2628 PEP0.19630.20950.25110.2703 MSFT0.28310.32710.28330.3800 JPM0.43470.47631.14330.4410 BAC0.33400.34100.52890.3479 JNJ0.16500.16740.27160.2425 WMT0.34410.36520.22150.3614 XOM0.28920.31420.30760.3669 All units are expressed in Annualized Standard Deviation Units.

17. January 3, 2006 – January 3, 2008 Visual comparison between RMSE displays significant reductions –Average 21.94% reduction of HAR-Beta RMSE when compared to constant returns. –Average 6.62% reduction of HAR-Beta RMSE when compared to constant beta. OLS, non-robust regression –WMT has a 35.56% increase in RMSE compared to constant returns Surprising –SD of Beta (0.4783) – highest –F.O. Autocorrelation (0.5701) – highest –R 2 (0.5391) – highest Poon and Granger (2003) issues with sample outliers and volatility estimation. MSFT and XOM less predictable? –Constant returns, low reduction 0.07% and 5.98% respectively –BUT constant beta, RMSE reduction 13.45% and 7.96% respectively –R 2 values of WMT and XOM lowest, 0.0113 and 0.1106. Salient point, given low predictability, the RMSE of the HAR-Beta was still reduced. –When MSFT and XOM constant beta compared to constant return, there is a significant increase in RMSE. Results – Out of Sample (A)

18. Results- Out of Sample (B) January 3, 2006 – January 2, 2009 Table 6: Root Mean Squared Error (RMSE)-HAR-Beta, Constant Beta, Constant Return, and Standard Deviation of Beta (Out of Sample-B) RMSE HAR- Beta RMSE Constant Beta Constant Returns Standard Deviation of Beta (Out Sample) KO0.19530.24960.59940.2442 PEP0.19860.21950.35700.2575 MSFT0.22910.37960.17740.3613 JPM0.42620.63291.24930.4884 BAC0.47060.49240.63480.4208 JNJ0.21490.22300.57670.2207 WMT0.22930.31060.32740.3339 XOM0.21880.27190.41350.3476 All units are expressed in Annualized Standard Deviation Units.

19. RMSE reduced by average 19.67% when compared to constant beta RMSE reduced by average of 39.28% when compared to constant return (including MSFT) Results- Out of Sample (B)

20. Final Conclusions Results in line with recent literature –Alternative to constant beta used in CAPM Average SD over whole sample was.3663 when using optimal sampling frequency Positive Autocorrelations – suggesting predictability of betas HAR-Beta showed overall reduction of RMSE across equities in different industry sectors. Weakness- OLS –Despite limitation, usage of logarithmic returns coupled with optimal sampling frequency provides relative sanity check Importance of conventional change from the constant beta to a time-varying beta.