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Title: Dual Beta Model

1
Dual Beta Model
• Ho Ken Jom, Li Wenru, Zhang Jian
• Department of Mathematics, NUS, 14 March 2011

2
The Reference Paper
• Does beta react to market conditions?
Estimates bull bear betas using a nonlinear
market model with an endogenous threshold
parameter
• -by George Woodward Heather M.Anderson,
Quantitative Finance, 25 March 2009

3
Outline of The Presentation
• The Model
• Strategy
• Back Testing

4
Definition of Bull and Bear market
Compare market index to a critical threshold value
How to differentiate market states
Trend-based scheme
5
Market indicator- transition variable R
• R 12 month moving average of logarithmic
returns
• smoother
• noise not useful

Figure 1 Return on the market index
Figure 2, The transition variable (R)
Jump in and out of market phases rapidly
Much smoother path
6
Models
Dual-beta market model
Market state indicator.
critical threshold value for each industry
Bear state Bull state
7
Models
Logistic Smooth Transition Market Model
When is large and negative
When is large and positive
8
Data
• 24 industry groupings within the Australian
Stock Exchange
• Observations are monthly
• Return series are calculated as the difference
of the logarithms of prices.

industry Duration Sample size
19 industries Dec. 1979 to Dec. 2001 265
solid fuels, oil and gas, and entrepreneurial investors Dec. 1979 to Oct. 1996 203
miscellaneous services Dec. 1979 to Aug. 1997 213
tourism and leisure Dec. 1990 to Dec. 2001 144
9
Data
10
EstimationLSTM model
LSTM models
• 15 industries significant at 5
• 6 industries significant at 10
• 11 negative
• 10 positive
• Smooth transition
• estimate LSTM

11
EstimationDBM model
• Parameter estimates are almost identical

12
EstimationDBM model
LSTM model
• DBM fits the data well(R2)
• Stocks spend more time in bull-market
• ESS is not affected

13
Summary
• Bull and bear betas are significantly
different for most industries
• Transition between states is abrupt, supporting a
dual-beta market modeling framework
• For many industries, stocks spend more time in
bull-market than bear-market states.
• The risk associated with bull states is not
always smaller than the risk in bear market
states.

14
Strategies for Dual Beta Market Model
• Terms used
• Model Calibration
• Theory behind the strategy
• Calculate the fair return of index
• Search mispriced spots and trade
• The shortage of the strategy

15
Terms
• Rm, Ri Market/Index return
• ßu,ßd Beta in up/down state
• a Alpha
• c Threshold (defined on Rm)
• Tu,Td Upper and lower thresholds (for trading)
• RMA Rolling moving average
• e the deviation from fair rate

16
Bull Bear
• Paper suggests dual beta model
• Differing betas for different market states
• Relatively sudden transition
• Suggests a market inefficiency
• Use a 12-month MA to determine state
• Market state only changes a few times (2-5 years)

17
Up and Down
• Attempts with a 12-month MA yielded losses
• Not suitable for forecasting, trading on
• Long-term/slow changes not useful
• Trials with daily market data (Rm) successful
• State can (and does) change from day to day
• Investor sentiment
• Smaller, but still significant changes

18
Model
• For a given c (Threshold value) partition data
into up/down sets
• Fit a dual beta (single alpha model)
• Choose the c, a, ßu,ßd that provide the best fit
(R2)
• Look for significant change in beta

19
Model Calibration
• Calibration for single beta model (SBM)
• Use the a of SBM to initiate a search for
parameters of dual beta single alpha model(DBSAM)
• The DBSAM should have a higher R-square as
expected
• Trade on pairs of high R-square

20
Data Process
• For a given set of estimated model parameters
• Calculate Rm, compare with c to obtain state
• Calculate Ri, compare with (a ßRm)
• Check where difference lies w.r.t thresholds

21
Fair Rate (Model Predicted rate)
• For a given set of estimated model parameters
• Calculate Rm, compare with c to obtain state
• Calculate the fair rate of index return
• if(RmltC) a ßd
Rm
• if(RmgtC) a ßu
Rm
• The deviation is given as
• e Ri-(a ßu Rm)

22
Thresholds
• Above Tu over-performing or cheap
• Below Tu under-performing or expensive
• (Short) Sell
• Between Td No significant difference (noise)
• Close out positions

23
Thresholds
• Higher thresholds are more conservative
• Buy only during bigger differences
• Close out positions more quickly
• Thresholds estimated by backtesting
• Conservative thresholds give less gains/losses
• Important to determine the best threshold

24
Strategy
• Impose a position limit 1, -1
• For p0 if(e gtTu), p 1 else p -1
• For p1 if(abs(e) ltTd), p 0, update PL else
if(e lt-Tu), p -1, update PL
• For p-1 if(abs(e) ltTd), p 0, update PL else
if(e gtTu), p 1, update PL

25
The Shortage of The Strategy
• Vulnerable to Systematical Risk (suppose p1 but
the market drops during holding period, or p-1,
but the market rise)
• Does not consider transaction cost
• Solution impose further restrictions
• -impose positive condition for each cycle of
• -when return is very high, dont short, and
vice versa do not hold for a long time for a
long position, etc

26
Back Testing
• Setup
• Stress Testing
• Future Works
• Conclusion

27
Setup
• Data Selection
• Using Spot rate of return instead of RMA
• Calibrating dual beta single alpha model (DBSAM)
• Parameterization of the strategy
• The state transition probability

28
Data Selection
• Market variable SP 500(GSPC)
• Target indices scanned
• NYSE Composite (NYA)
• NASDAQ Composite ( IXIC)
• Vanguard Index Trust 500 Index (VFINX)
• PHLX OIL SERVICE SECTOR INDEX ( OSX)
• Bank of America (BAC)
• Period 01/01/200904/03/2011 (546 days), by
YahooEOD
• Add some plots to show the betas here

29
NYA and SP500, 1/1/20096/03/2011
30
R-Square for the First 294 Days
Target NYA IXIC VFINX OSX BAC
R2-SBM 0.978 0.923 0.999 0.717 0.481
R2-DBSAM 0.979 0.925 0.999 0.728 0.504
• Sector index and single stock has
• SP500 NYA

31
Using Spot Rate of return
• RMA data yields a bad fitting, and
even fails itself, so
• chose spot rate return of SP500 for

mLag 1 2 3 5 10
SBM R2 0.978 0.117 0.083 0.064 0.020
DBMSA R2 0.979 0.757 0.589 0.390 0.179
32
Calibration of DBSAM
• First conduct an SBM OLS on the whole 294 days
data
• Then partition the data to lower wing and upper
wing with different candidate threshold values
• Then for each partition, generate a-grid (see
below), search for a and the associate ßd ßu
such that the composite R-square is maximized
• In my program, s 0.01, i -25, ,25, but is
still expensive in computing

33
OSX SP500 SBM Fitting
• R2 0.69, Beta 1.46, ALPHA
0.175

34
OSXNYA DBSAM Fitting
• R2 0.76, Beta Lower 1.56, Beta Upper
1.44, ALPHA 0.415

35
Parameterization of The Strategy
US, set as 252
• ND number of days in
data, in our case this is 546
• NRD number of days for
regression NRD 546-NBT
• NBT number of days for
back-testing
• TH the threshold weight to
partition the market
• E the average modeling
error
• Tu the threshold to
trigger a position in scales of E
• Td threshold to close a
position in scales of E
• Limit position limit, set as
1, -1
• S the resolution for
search for alpha, fixed as 0.01
• FEE fees, set as zero for
the time-being

36
• Program uses daily holding period return
• Test shows log return performs better
• However, for the purpose of consistency, I keep
using daily holding return

37
State Transition Probability(1)
• Define modeling error at t, t0,1,..,N as Et.
Assume
• For all positive Es, sum over both sides
• Similarly, we have
• Once we know r, we can estimate the volatility
using

38
State Transition Probability(2)
• Now, suppose the initial error is Et which is
less than Td. After 1 time unit, the error
distribution is given as
• Then the probability that a long position
will be triggered is
• Eventually this will enable us to estimate
the holding period.

39
• The model is calibrated using the first NRD
days data, and back-tested against the left NBT
days data, assuming the model is stationary
• Search optimal Tu (trigger threshold)
• Search optimal Td (closing threshold)
• Search optimal TH (weighting threshold)
• Different NBT (number of back-testing days)

40
Optimal Tu
Tu 2 3 4 5 6 7 8
SBM PL -295.56 -392.94 -223.48 148.58 -47.18 -176.6 -341.26
DBSAM PL -264.78 -236.48 15.12 214.66 251.08 272.44 -13.26
• Setting NBT 126, Td
0.1, TH 0.2

41
Optimal Tu
42
Optimal Td
Td 1 0.75 0.5 0.25 0.1
SBM Return 4.02 4.65 4.24 4.74 3.71
DBSAM Return 10.33 14.73 9.52 9.06 5.37
• Setting NBT 126, Tu 5, TH
0.2

43
Optimal TH
• Setting NBT 126, Td
0.5, Tu 5

44
Different NBT
45
High Frequency Trading with Different NBT
NBT 60 90 126 180 252
SBM Return 24.85 1.81 4.02 7.93 19.75
DBSAM Return 24.00 5.61 10.33 10.76 24.14
• It can be observed that (i) DBSAM is better (ii)
return increases with NBT (Setting Td 1, Tu
2, TH 0.3)

46
• The issue of computing time
• -It takes 50s for Java to search the grid,
for 252 days, weekly update, need 42m on Dell
OptiFlex755
• Using C CUDA
• -Reduce to 7m (on Nvidia GTX580)
• Some results

47
Results Dynamical SBM vs Stationary SBM
48
Results Dynamic DBSAM vs Stationary DBSAM
49
Possible Reason for Under-Performance
• My own OLS algorithm for regression passing
through the origin (consistent with matlab) yield
slightly different parameters from AlgoQuant.

50
Stress Testing (stationary) against 20072009
Crisis
• Case I Regression before crisis and back testing
in crisis(1 Jan 20076 Mar 2009)
• Case II Both in crisis (3 Jan 20086 March 2010)
• Case III Regression in crisis and back testing
out crisis? But no such case!

Cases I II
SBM -3.75 7.08
DBSAM 0.077 32.74
51
Conclusion
• The simple strategy works well after optimization
for composite indices, and in most of the cases
the DBSAM outperforms SBM. But the model is
subject to Systematic risk .
• The Dynamical SBM outperforms Stationary SBM,
this is not true for DBSAM, possibly due to the
inconsistency OLS algorithm.

52
Future Works
• Refine the model
• -Resolving the inconsistency of OLS algorithm
• -Try daily model updating
• -Impose conditions to hedge the systematic
risk
• -Apply to sector index (R2 0.70.9)
• -Reduce the parameters of the model
• -Compute the Sharp ration/Sharp Omega if we
have time
• Try DUAL BETA DUAL ALPHA MODEL
• Try Pairs Trading with Dual Dynamical Beta Model

53
• Q A!