1. Run length and the Predictability of Stock Price Reversals Juan Yao Graham Partington Max Stevenson Finance Discipline, University of Sydney

2. 2 Structure of the paper Background and motivation Empirical design Data In-sample analysis Out-of-sample evaluation Conclusion

3. 3 Motivation of the study Evidence of predictability McQueen and Thorley (1991,1994) Low (high) returns follow runs of high (low) returns – probability that a run ends declines with the length; Maheu and McCurdy (2000) Markov-switching model – probability that a run ends depends on the length of the run in the markets; Ohn, Taylor and Pagan (2002) The turning point in a stock market cycle is not a purely random event.

4. 4 Motivation of the study The function form of the occurrence of such events is not known, and the baseline hazard can be given many parametric shapes Cox’s proportional hazard approach is a semi- parametric techniques – doesn’t need to specify the exact form of the distribution of event times Successful forecasting of price reversal in property market index by Partington and Stevenson (2001) The technique seems to also work on consumer sentiment index (a work is currently on going)

5. 5 Some definitions Events: reversal of price Event time versus calendar time Up-state and down-state: Up-state: positive runs, when P t – P t-1 >0 Down-state: negative runs, when P t - P t-1 <0 State transition Probability of transition Not predicting a price reversal, but the probability of a reversal

6. 6 Cox regression model We define the hazard of price state transition to be: where p ij (t,t+s) is the probability that the price in state i at time t will be in state j at time t+s.

7. 7 Cox Proportional Hazards Model The hazard function for each individual run will be: The log-likelihood:

8. 8 Cox Proportional Hazards Model The cumulative survival probability is defined as: S(t) = P(T>t) where T is time of the event S(t) can be calculated from: S(t) is the probability that the current run will persist beyond the time horizon t.

9. 9 Empirical design Two models are estimated: a transition from an up-state to a down-state and a transition from a down-state to an up- state Covariates: lagged price changes up to 12 lags for monthly data, 30 lags for daily data the number of state transitions in the previous period a dummy variable to distinguish a bull and bear market

10. 10 Identification of Bull and Bear Market Pagan and Sossounov (2003) criterion Identify the peaks and troughs over a window of eight months The minimum lengths of bull and bear states are four months. The complete cycle has minimum length of sixteen months The minimum four months for a bull or bear state can be disregarded if the stock price falls by 20% in a single month. This enables the accommodation of dramatic events such as October 1987.

11. 11 Forecast evaluation At aggregate level: Sum the estimated probability of survival to time t for each run in holdout sample to obtain the expected number of runs survive beyond t: At individual level: Brier score

12. 12 Brier score Assessing probabilistic forecasts: When the price has reversed a n = 1, and when it has not a n = 0. A lower Brier score implies better forecasting power

13. 13 Data Monthly All Ordinary Price Index Feb. 1971 – Dec. 2001 holdout sample: last five years Daily All Ordinary Price Index 31 st Dec. 1979 – 30 th Jan. 2002 holdout sample: last two years

14. 14 Table 1. Summary Statistics for Run Length. Run Type CountMin Length Max Length MeanStd. Dev. Skew.Kurt. Panel A: Daily Price Changes Positive (Up state) 1092 1 d14 d 2.511.8961.928 (0.074) 4.944 (0.148) Negative (Down state) 1093 1 d13 d 2.131.5372.031 (0.074) 5.420 (0.148) Panel B: Monthly Price Changes Positive (Up state) 65 1 m10 m 2.461.8462.299 (0.297) 6.208 (0.586) Negative (Down state) 65 1 m7 m 1.891.3011.833 (0.297) 3.508 (0.586)

15. 15 Table 2. Bull and Bear Market Identification. TroughPeakBear (months) Bull (months) 11/197101/1973N.A.14 09/197411/19802074 03/198209/19873266 02/198808/19895*18 12/199010/19911610 10/199201/19941215 01/199509/19971232 08/199806/2001**1134 Average Length15.432.9

16. 16 Figure 1. All Ordinary Price Index (monthly) Feb. 1971 – Dec. 2001.

17. 17 Figure 2. Return of All Ordinary Price Index (monthly) Mar. 1971 – Dec. 2001.

18. 18

19. 19

20. 20 Out-of-sample Monthly: 21 completed negative price runs and 21 complete positive runs. Daily:121 negative price runs and 121 positive price runs. The out-of-sample survival functions are estimated according to:

21. 21 Comparisons Two benchmarks Naïve forecast: setting the survival probability for time t equal to the proportion of runs survived within sample Random forecast: the probability of each independent state change is 0.5, the survival probability at t is (0.5) t

22. 22 Figure 5. Comparison of the number of actual and expected runs of varying lengths (positive runs, daily data).

23. 23 Figure 6. Comparison of the number of runs of varying lengths (negative runs, daily data).

24. 24 Figure 7. Brier Scores for monthly negative runs.

25. 25 Figure 8. Brier Scores for monthly positive runs.

26. 26 Figure 9. Brier Scores for daily negative runs.

27. 27 Figure 10. Brier Scores for daily positive runs.

28. 28 Figure 11. Probability forecasts of positive runs.

29. 29 Figure 12. Probability forecasts of negative runs.

30. 30 Conclusions Lagged price changes and the previous number of transitions are significant predictor variables. In an up-state, the lagged positive (negative) changes decreases (increases) the possibility of reversal; in a down-state, the lagged positive (negative) changes increases (decreases) the possibility of reversal. State of the market, bull or bear is not significant.

31. 31 Conclusions Predictive power exists at the aggregate level. For individual runs, the model forecasts less accurate than naïve and random-walk forecasts. The random-walk and naïve forecasts are almost identical.

32. 32 Conclusions Are the price changes random? -No, price reversals are related to information in previous prices, specifically, the signs and magnitude of lagged price changes as well as the previous volatility. Is the market efficient? - Probably, model forecasts poorly in out-of-sample, no profitable trading.. Or, it is a bad specification?

33. 33 Future Research Test for duration dependence (whether the hazard function is a constant, or the density is exponential) Examine the runs of the individual stocks (choice of stocks? Frequency of the data?) Any suggestions are welcome!