Dual Beta Model

Ho Ken Jom, Li Wenru, Zhang JianDepartment of Mathematics, NUS, 14 March 2011

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

Outline of The Presentation

• The Model

• Strategy

• Back Testing

Trend-based scheme

Definition of “Bull” and “Bear” market

How to differentiate market states

Compare market index to a critical threshold value

Market indicator- transition variable R*

Jump in and out of market phases rapidly Much smoother path

Figure 1: Return on the market index Figure 2, The transition variable (R*)

R*= 12 month moving average of logarithmic returns• smoother• noise not “useful”

Models

Dual-beta market model:

Bear state: Bull state ：

: Market state indicator. critical threshold value for each industry

Logistic Smooth Transition Market Model

Models

When is large and negative When is large and positive

F is the logistic smooth transition function, : smoothness parameter

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

miscellaneousservices

Dec. 1979 to Aug. 1997 213

tourism and leisure Dec. 1990 to Dec. 2001 144

Data

Estimation—LSTM model

LSTM models ：

• “Smooth” transition• estimate LSTM

•15 industries: significant at 5%• 6 industries: significant at 10%•11: negative•10: positive

Estimation—DBM model

• Parameter estimates are almost identical

Estimation—DBM model

• DBM fits the data well(R2)

• Stocks spend more time in ‘bull’-market

•ESS is not affected

LSTM model

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.

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

Terms

• Rm, Ri: Market/Index return• βu,βd: Beta in “up”/”down” state• α: Alpha• c: Threshold (defined on Rm)• Tu,Td: Upper and lower thresholds (for trading)• RMA: Rolling moving average• ε: the deviation from fair rate

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)

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

Model

• For a given c (Threshold value) partition data into “up”/”down” sets– Fit a dual beta (single alpha model)– Choose the c, α, βu,βd that provide the best fit (R2)– Look for significant change in beta

Model Calibration

• Calibration for single beta model (SBM)• Use the α 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

Data Process

• For a given set of estimated model parameters– Calculate Rm, compare with c to obtain state– Calculate Ri, compare with (α + βRm)

• Check where difference lies w.r.t thresholds

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(Rm<C) α + βd Rm

if(Rm>C) α + βu Rm

– The deviation is given as

ε = Ri-(α+ βu Rm)

Thresholds

• Above +Tu: over-performing or “cheap”– Buy

• Below –Tu: under-performing or “expensive”– (Short) Sell

• Between ±Td: No significant difference (noise)– Close out positions

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

Strategy

• Impose a position limit +1, -1• For p=0: if(ε >Tu), p = 1; else p =-1• For p=1: if(abs(ε) <Td), p = 0, update P&L; else

if(ε <-Tu), p = -1, update P&L• For p=-1: if(abs(ε) <Td), p = 0, update P&L; else

if(ε >Tu), p = 1, update P&L

The Shortage of The Strategy

• Vulnerable to Systematical Risk (suppose p=1 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 trading

-when return is very high, don’t short, and vice versa; do not hold for a long time for a long position, etc…

Back Testing

• Setup • Stationary Trading• Dynamical Trading• Stress Testing• Future Works• Conclusion

Setup

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

Data Selection • Market variable: S&P 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: 01/01/2009~04/03/2011 (546 days), by YahooEOD Add some plots to show the betas here

NYA and S&P500, 1/1/2009~6/03/2011

NYA and S&P500

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

9/ 9/ 2008

12/ 18/ 2008

3/ 28/2009

7/ 6/ 2009

10/ 14/ 2009

1/ 22/2010

5/ 2/ 2010

8/ 10/2010

11/ 18/ 2010

2/ 26/2011

6/ 6/ 2011

Data

Adj

Quot

e

S&P500NYA

R-Square for the First 294 Days

*Sector index and single stock has low R-square thus trade

S&P500 & NYA

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

Using Spot Rate of return

RMA data yields a bad fitting, and even fails itself, so chose spot rate return of S&P500 for trading

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

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 α-grid (see below), search

for α and the associate βd & βu such that the composite R-square is maximized

• In my program, s = 0.01, i = -25, …,25, but is still expensive in computing

isSBM

isSBM

OSX ~ S&P500 SBM Fitting

R2 = 0.69, Beta = 1.46, ALPHA = 0.175

OSX~NYA DBSAM Fitting R2 = 0.76, Beta Lower = 1.56, Beta Upper = 1.44, ALPHA = 0.415

Parameterization of The Strategy

nTrade number of trading days in 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

Which Return to use?

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

using daily holding return

State Transition Probability(1)

Define modeling error at t, t=0,1,..,N as Et. Assume :

For all positive E’s, sum over both sides

Similarly, we have

Once we know r, we can estimate the volatility using

tE

dBrEE tt 1

0, 0, 0,0,0,

)/()(t t ttt Et Et Et

ttttEtEt

t EErErdBErE

0, 0, 0,0,0,

)/()(t t ttt Et Et Et

ttttEtEt

t EErErdBErE

1 tt ErEdB

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.

),(~ 21 tt rENE

uTL dErEEP ),,( 2

Stationary Trading

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)• High Frequency Trading

Optimal Tu

*Setting: NBT = 126, Td = 0.1, TH = 0.2

Tu 2 3 4 5 6 7 8

SBM P&L -295.56 -392.94 -223.48 148.58 -47.18 -176.6 -341.26

DBSAM P&L -264.78 -236.48 15.12 214.66 251.08 272.44 -13.26

Optimal Tu

Return f or Di ff erent Tu

- 12

- 10

- 8

- 6

- 4

- 2

0

2

4

6

8

0 1 2 3 4 5 6 7 8 9

Tu

Retu

rn SBMDBSAM

Optimal Td

*Setting: NBT = 126, Tu = 5, TH = 0.2

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%

Optimal TH

*Setting: NBT = 126, Td = 0.5, Tu = 5

Return VS TH

- 15

- 10

- 5

0

5

10

15

0 0. 05 0. 1 0. 15 0. 2 0. 25 0. 3 0. 35 0. 4 0. 45 0. 5

TH

Retu

rn (

%)

Different NBT

Di ff erent Back Test i ng Days

0. 00%

10. 00%

20. 00%

30. 00%

40. 00%

50. 00%

60. 00%

0 50 100 150 200 250 300

NBT

Retu

rn DBSAMSBM

High Frequency Trading with Different NBT

*It can be observed that (i) DBSAM is better (ii) return increases with NBT (Setting: Td = 1, Tu = 2, TH = 0.3)

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%

Dynamic Trading

• 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

Results: Dynamical SBM vs Stationary SBM

Dynamcal SBM and Stat i onary SBM

0. 00%

5. 00%

10. 00%

15. 00%

20. 00%

25. 00%

30. 00%

35. 00%

0. 00% 5. 00% 10. 00% 15. 00% 20. 00% 25. 00% 30. 00%

Stat i onary

Dyna

mica

l

Dynami cal SBMStat i onary SBM

Results: Dynamic DBSAM vs Stationary DBSAM

Dynami cal DBSAM vs Stat i onary SBSAM

0. 00%

5. 00%

10. 00%

15. 00%

20. 00%

25. 00%

30. 00%

35. 00%

40. 00%

45. 00%

0. 00% 10. 00% 20. 00% 30. 00% 40. 00% 50. 00%

Stat i onary DBSAM

Dyn

DBSA

M

Dynami cal DBSAMStat i onary DBSAM

Possible Reason for Under-Performance

My own OLS algorithm for regression passing through the

origin (consistent with matlab) yield slightly different parameters from AlgoQuant.

Stress Testing (stationary): against 2007~2009 Crisis

• Case I: Regression before crisis and back testing in crisis(1 Jan 2007~6 Mar 2009)

• Case II: Both in crisis (3 Jan 2008~6 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%

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.

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: 0.7~0.9) -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

Q& A!