Calculating high frequency betas with R EARL Conference 2014 September 2014 David Jessop Analyst Tel: +44 20 7567 9882 [email protected]Claire Jones Analyst Tel: +44 20 7568 1873 [email protected]This document has been prepared by UBS Limited ANALYST CERTIFICATION AND REQUIRED DISCLOSURES BEGIN ON SLIDE 39 UBS does and seeks to do business with companies covered in its research reports. As a result, investors should be aware that the firm may have a conflict of interest that could affect the objectivity of this report. Investors should consider this report as only a single factor in making their investment decision.
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Calculating high frequency betas with REARL Conference 2014
This document has been prepared by UBS LimitedANALYST CERTIFICATION AND REQUIRED DISCLOSURES BEGIN ON SLIDE 39UBS does and seeks to do business with companies covered in its research reports. As a result, investors should be aware that the firm may have a conflict of interest that could affect the objectivity of this report. Investors should consider this report as only a single factor in making their investment decision.
· We have access to a global database of high frequency tick data back to 2007.
· Our initial problem is how to use this to analyse changes in betas.
· There are a number of problems in doing this with high frequency data, which we will discuss below.
· Another difficulty comes when we move away from a single market to markets which are open at different times.
Introduction – What's the problem?
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Realised beta
· Divide the period (say one week) into N non overlapping intervals of size h, e.g. 5 minutes.
· Compute returns to the stock and to the index for each interval j=1,…,N.
· The approach is straightforward to implement: a simple regression without intercept of high frequency stock returns on high frequency market returns
j=1,…,N. rij is the return to stock i during the interval j. M indicates market returns.
Week t, N intervals in total
h h hh h=5 minutes hh
,ˆ1
2,
1,, /
N
jjM
N
jjMjiit rrr
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Realised beta: The intuition
· The denominator of realised beta is an estimator of the variance of market return over month t. The numerator is an estimator of the covariance of the returns to stock i with market returns over month t.
· Unlike GARCH volatility estimators, realised volatility is nonparametric and does not impose any assumptions on the dynamics of volatility.
N
jjM
N
jjMji
it
r
rr
1
2,
1,,
Realised covariance
Realised variance of market returns
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Dec 31 2007 Dec 01 2008 Dec 01 2009 Dec 01 2010 Dec 01 2011 Dec 03 2012 Nov 29 2013
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Tesco Beta
Realised beta for Tesco vs. FTSE 100
Source: UBS
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Dec 31 2007 Dec 01 2008 Dec 01 2009 Dec 01 2010 Dec 01 2011 Dec 03 2012 Nov 29 2013
0.4
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Tesco Beta
20 day beta20 day MA of daily betas
20 day beta vs 20 moving average
Source: UBS
The data
Section 2
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Data
· We have access to tick data from end 2007 globally.
· Sheer size of the database is huge (~ 1 PB)
· Our Universe:– Transaction prices for all historical constituents of FTSE 100, FTSE 350
– We use fixed intervals (e.g. of 5 minutes) and sample the latest transaction price before the end of the period
· Data cleaning (Bandorff-Nielsen et al., 2009): this is available in the "highfrequency" package- Eliminating corrected trades and entries with abnormal Sale Condition - Eliminating any trades outside the 8:00am – 4:30pm interval- Dealing with holidays
· Keeping track of any change of tickers, re-issued tickers
· Overnight interval (Dividend source: HSBC)
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More practical problems
· Dealing with the plethora of languages / systems High frequency data stored in a kdb database– Index constituent / other data stored in an Oracle database
– Mapping table extracted from original MSCI files using Python
– KEM algorithm originally in Matlab (Octave) language
– Our analysis done in R
– We use a number of packages, including reshape2, plyr, highfrequency, xts, timeDate, rhdf5, lubridate plus our internal packages to link between data sources (none of which are in SQL!).
· How to run analyses using data multiple times and at multiple frequencies?– Store locally (perhaps using HDF5) as too big for memory
– Run analysis in parallel
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Dealing with dates
· The hardest problem to overcome from a programming perspective was dates / times. We are storing and using data down to the millisecond, and often across markets in different time zones.
· Australia sometimes starts trading at 11pm UK time – so you have to be aware of which dates / time zones you ask for the data in.
· Dates that don’t have times often are retrieved from a database at midnight – BUT when you switch to summertime in London they get moved back a day!
Dates are a problem – especially in the UK time zone
· If we knew the stock and market return perfectly then all would be well. However we don't!
· There are a number of problems including:
1. noise (bid / offer prices and having discrete stock prices) which biases variance estimates up
2. trading frequency - some stocks don't trade that often, which makes calculating betas against non-traded indices difficult
3. and asynchronicity (stocks trade at different times) which biases covariances to zero (Epps, 1979).
Realised beta: the problemsWe don't work in a perfect market
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· In a signature plot we calculate the realised variance using a number of frequencies over a given period. We then plot the average realised variance against time. This tends to give (at least for liquid stocks) a chart such as that shown below for Microsoft
Realised beta: the problemsIllustration of the realised variance problem
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0 20 40 60Minutes
Std
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v (%
an
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Source: UBS
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Realised beta: the problemsFor realised covariance we have the opposite problem
Source: UBS
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· For Rio against the FTSE 100.
· We note that up to around 10 minutes the correlation seems to be understated.
Realised beta: the problemsIllustrating this for correlations …
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0 20 40 60Minutes
Co
rre
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Source: UBS
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· Hayashi and Yoshida (2005): all returns with overlap
· We take the KEM approach from Corsi et al (2013)
· Treat asynchronicity as a missing variables problem and the microstructure noise as measurement errors of the latent prices
· Suggests a state space model + missing values -> Kalman Filter + Expectation Maximisation (EM)
· See the paper for the technical details.
The KEM methodology
Source: UBS
What frequency to use?
Section 2
· We have a few questions to answer– What time period to use? 1 day, 5 days … ?
– What frequency of data? And how should we measure time?
– Should we use overnight data?
– Should we smooth the results? And how?
– Should we use different frequencies for the covariance and variances?
· There is also the problem of the objective function , i.e. how do we tell if a beta is a good beta?
Deciding on the frequency
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· The chart shows the average number of trades a day for the FTSE 350.
· One trade per minute would be 510 trades (and log (510) = 6.23
· Hence for the FTSE 100 we are probably OK using calendar time, but as we go into the FTSE 250 then perhaps another approach will be needed.
Liquidity, numbers of trades and time FTSE 100 vs FTSE 250
Source: UBSNote: Chart shows the mean number of trades per day for the constituents of the FTSE 350 as at 30 Sep 2013. The average is taken from 2nd January 2013 to 30th September 2013.
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· There are a number of approaches to picking the best period for estimating the variance. These include– Optimal sampling frequency (Bandi and Russell, 2008)
– TSRV (Zhang, Mykland and Ait-Sahalia, 2005)
– RK (Barndorff-Nielsen, Hansen, Lunde and Shephard, 2008)
– Signature plots (Andersen, Bollerslev, Diebold and Labys, 2000)
· We used the signature plot approach to investigate the frequency for estimating the market variances.
Signature plotsFinding the frequency for the variances
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· In a signature plot we calculate the realised variance using a number of frequencies over a given period. We then plot the average realised variance against time. This tends to give (at least for liquid stocks) a chart such as that shown below for Microsoft
Signature plotsRevisiting our signature plots
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Source: UBS
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· Repeating that exercise on the S&P 500 and SPY we obtain the chart below.
Signature plotsIndices are different
Source: UBSNote: Chart shows the average realised volatilities calculated daily from 2nd January 2013 to 31st March 2014.
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· We use three measures of how good our beta is (although we do not report all of them due to space):– The in sample pricing error
– The one day out of sample pricing error
– The five day out of sample pricing error
· If we have a beta estimate then we calculate the pricing error for stock i on day t as
· where s = t for the in sample pricing error and s = t – 1 for the out of sample one.
· Our measure of the goodness of an approach to estimating beta is the average of - a lower value of the average is better. We report the square root of this number.
How good is a beta? How to measure whether our beta is good
Results: FTSE 100
Section 3
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· As a benchmark we chose to use 6 month daily betas. Their loss functions are:
Min. 1st Qu. Median Mean 3rd Qu. Max.
6m beta 0.389 1.198 1.483 1.775 1.824 5.753
6m SW beta 0.364 1.208 1.508 1.790 1.832 6.007
All 1 0.828 1.271 1.503 1.841 1.898 5.718
All 0 0.301 1.498 1.921 2.186 2.268 6.141
And the same at a weekly frequency
Min. 1st Qu. Median Mean 3rd Qu. Max.
6m beta 0.858 2.729 3.499 4.516 4.626 17.047
6m SW beta 0.866 2.758 3.496 4.640 4.677 19.987
All 1 0.855 2.870 3.554 4.634 4.734 17.117
All 0 1.159 3.519 4.410 5.635 6.148 20.565
FTSE 100 – BenchmarksCan we improve on daily results?
Source: UBS
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· A more informative way to compare two models is to look at the cumulative difference in the cumulative pricing error. For example, comparing the 1 day errors for the calculated beta and all betas = 1 for the FTSE 100 gives us
FTSE 100 – Benchmarks: 6m dailyWhen does the model do better?
Source: UBS. Chart shows the difference between the 1 day errors for the 6m beta and the beta = 1. A falling number implies the calculated beta is better.
Jul 01 2008 Jan 04 2010 Jul 01 2011 Jan 02 2013
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· The table shows the Median Loss for the FTSE 100 on a daily basis
Daily results
Minutes
Days
1 3 5 15 30
1 1.727 1.712 1.687 1.708 1.757
3 1.676 1.655 1.648 1.667 1.677
5 1.670 1.651 1.643 1.651 1.654
10 1.659 1.642 1.636 1.633 1.631
15 1.658 1.642 1.636 1.630 1.628
20 1.657 1.641 1.639 1.633 1.628
25 1.655 1.644 1.642 1.636 1.631
30 1.653 1.646 1.643 1.638 1.631
Source: UBS
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· The table shows the Median Loss for the FTSE 100 on a weekly basis
Weekly results
Minutes
Days
1 3 5 15 30
1 3.618 3.587 3.597 3.655 3.755
3 3.596 3.582 3.608 3.624 3.655
5 3.571 3.540 3.552 3.552 3.569
10 3.554 3.530 3.517 3.487 3.489
15 3.575 3.523 3.519 3.484 3.487
20 3.558 3.496 3.493 3.466 3.454
25 3.559 3.517 3.503 3.479 3.466
30 3.550 3.493 3.490 3.479 3.468
Source: UBS
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· How should we smooth the data given the betas are quite volatile? We could use an AR(1) model but an underlying assumption of this model is that the mean is constant (at least during the period over which the model is estimated). Hence we choose to use a simple exponentially weighted moving average.
Weekly results – smoothing
Source: UBS
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· The table shows the Median Loss for the FTSE 100 using the smoothed data
· Smoothing the results with an EWMA gives better results than the unsmoothed data for either shorter windows and / or higher frequency.
Weekly results – smoothing
Minutes
Days
1 3 5 15 30
1 3.599 3.536 3.526 3.517 3.507
3 3.581 3.534 3.528 3.514 3.511
5 3.579 3.540 3.531 3.514 3.496
10 3.552 3.517 3.521 3.504 3.493
15 3.550 3.513 3.517 3.486 3.483
20 3.545 3.494 3.499 3.491 3.491
25 3.541 3.499 3.499 3.494 3.491
30 3.557 3.509 3.524 3.519 3.516
Source: UBS
Results: MSCI Asia Materials
Section 4
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· As a benchmark we chose to use 6 month daily betas. Their loss functions are:
Min. 1st Qu. Median Mean 3rd Qu. Max.
6m beta 0.885 1.992 2.415 2.714 2.829 9.870
6m SW beta 0.900 2.017 2.437 2.743 2.873 9.694
All 1 0.978 2.121 2.515 2.820 2.895 9.814
All 0 1.238 2.290 3.021 3.260 3.441 10.616
And the same at a weekly frequency
Min. 1st Qu. Median Mean 3rd Qu. Max.
6m beta 1.276 3.171 4.198 4.594 4.943 11.41
6m SW beta 1.281 3.189 4.230 4.609 4.961 11.39
All 1 1.322 3.379 4.328 4.703 4.985 11.66
All 0 2.111 3.971 5.260 5.934 6.238 16.14
MSCI Asia – Benchmarks
Source: UBS
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· Our first results with this approach are shown below. These are similar to the 6 month numbers.
· Again we believe the interest is where the higher frequency beta is different from the daily one.
Calculating the covariance matrix using the KEM method
Oct 14 2011 Apr 06 2012 Oct 05 2012 Apr 05 2013 Oct 04 2013 Mar 31 2014
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China Blue Chemical
Source: UBS
An application
Section 5
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· We simulated the returns to three 12 month momentum portfolios in the FTSE 100, and then looked to hedge them using a 6m daily beta, a smoothed 5 day / 5 minute beta, and a 20 day / 30 minute beta.
· The simulated return to the portfolio is below.
Hedging momentum
Source: UBS.
Dec 24 2008 Dec 04 2009 Dec 03 2010 Dec 02 2011 Dec 07 2012 Nov 29 2013
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6m hedge 5/5 hedge
20/30 hedge
Average -0.092 -0.050 -0.043
Average abs 0.154 0.169 0.138
Standard dev 0.170 0.224 0.178
Hedging a book to price portfolioShort term betas do a better job of hedging
· The table shows the details of the out of sample beta of the hedged portfolios.
Source: UBS
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References· Andersen, T.G., Bollerslev, T., Diebold, F.X , Wu, J. (2006): Realized beta. Persistence and predictability, in: T. Fomby
and D. Terrell (eds.) Advances in Econometrics: Econometric Analysis of Economic and Financial Time Series, 1-39.
· Andersen, T.G., Bollerslev, T., Diebold, F.X , Wu, J. (2005): A Framework for Exploring the Macroeconomic Determinants of Systematic Risk, American Economic Review 92, 399-404.
· Ang, A., Chen, J. (2007): CAPM Over the Long Run: 1926-2001, Journal of Empirical Finance 14, 1-40.
· Ang, A., Kristensen, D. (2012): Testing Conditional Factor Models, Journal of Financial Economics, forthcoming.
· Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N. (2009): Realised kernels in practice: Trades and quotes, Econometrics Journal 12, 1–33.
· Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N. (2011): Multivariate realised kernels: Consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading, Journal of Econometrics 162, 149–169.
· Bollerslev, T., Tauchen, G., Zhou, H. (2009): Expected Stock Returns and Variance Risk Premia, Review of Financial Studies 22, 4463-4492.
· Bollerslev, T., Zhang, B.Y.B. (2003): Measuring and modelling systematic risk in factor pricing models using high-frequency data, Journal of Empirical Finance 10, 533-558.
· Corradi, V., Distaso, W., Fernandes, M. (2011): Conditional alphas and realized betas, Working paper.
· Corsi, F., Peluso, S. and Audrino, F., (2013): Missing in Asynchronicity: A Kalman-EM Approach for Multivariate Realized Covariance Estimation. Available at SSRN: http://ssrn.com/abstract=2000996
· de Pooter, M., Martens, M. & van Dijk, D. (2008): Predicting the daily covariance matrix for S&P 100 stocks using intraday data —but which frequency to use? Econometric Reviews 27, 199–229.
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References· Drechsler, I. & Yaron, A. (2011): What’s vol got to do with it. Review Financial Studies 24, 1-45.
· Epps, T.W. (1979) Comovements in Stock Prices in the Very Short Run, Journal of the American Statistical Association, 74, 291–298.
· Fleming, J., Kirby, C., Ostdiek, B. (2003): The economic values of volatility timing using ‘realized’ volatility, Journal of Financial Economics 67, 473–509.
· Ghysels, E. (1998): On Stable Factor Structures in the Pricing of Risk: Do Time-Varying Betas Help or Hurt? Journal of Finance 53, 549-573.
· Gilbert, T., Palacios, M., Wang, X. (2011): Macroeconomic Announcements and Firm-Level Risk Characteristics, Working paper.
· Griffin, J.E., Oomen R.C.A. (2011): Covariance measurement in the presence of non-synchronous trading and market microstructure noise, Journal of Econometrics 160, 58-68.
· Llewellyn and Nagel (2006): The Conditional CAPM does not Explain Asset-Pricing Anomalies, Journal of Financial Economics 82, 289-314.
· Merton, R.C. (1973): An intertemporal capital asset pricing model, Econometrica 41, 867-887.
· Patton, A.J., Verardo, M. (2012): Does beta move with news? Firm-specific information flows and learning about profitability, Review of Financial Studies, Forthcoming.
· Petkova, R., Zhang, L . (2005): Is Value Riskier Than Growth? Journal of Financial Economics 78, 187–202.
· Scholes, M., Williams, J. (1977): Estimating Betas from Non-synchronous Data. Journal of Financial Economics 5, 309-328.
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