1. Singular spectrum analysis for a forecasting of financial time series Financial Academy under the Government of the Russian Federation Moscow - 2009 Speaker: Kozlov Alexander A. Report

2. Content list: Introduction to nonlinear dynamics approach Overview of the main methods (including SSA) Financial time series analysis and forecasting: Schlumberger Limited Deutsche Bank Honda Motor Co., Ltd. Toyota Motor Corp. Starbucks BP plc. Conclusions

3. Time seriesTime series is a series of variable values taken in successive periods of time. Time series analysisTime series analysis is a part of nonlinear dynamics. Supposition:Supposition: market of shares is unstable and chaotic. Objective:Objective: analysis and forecasting of stock price time series with nonlinear dynamics methods In this report the following questions will be considered:In this report the following questions will be considered: Embedding dimension as “space” characteristic and its estimation К2-entropy and Lyapunov exponents as “time” characterics and their estimation SSA forecasting methodIntroduction

4. The idea of attractor reconstruction [11]:The idea of attractor reconstruction [11]: Satisfactory geometry picture of low-dimensional strange attractor can be obtained if instead of x-variables from dynamic system equations somebody use k-dimensional delay vectors: Takens theorem [2]:Takens theorem [2]: There is a transformation which can embed to on conditions that. It means that: - k – embedding dimension; -Overview __________________________________________________________________________________________________ [1] Packard N.H., Crutchfield J.P., Farmer J.D., Shaw R.S.,"Geometry from a time series", Phys.Rev.Lett. 45, p.712,1980. [2] F. Takens, "Dynamical Systems and Turbulence", Lect. Notes in Math, Berlin, Springer. №898, 1981, p. 336.

5. Grassberger- Procaccia method [3]:Grassberger- Procaccia method [3]: Limitation [4]:Limitation [4]: ______________________________________________________________________________________ [3] P. Grassberger, I. Procaccia, "Characterization of Strange Attractors",Phys.Rev.Lett.,50,346, 1983 [4] G.G. Malineckiy, A.B. Potapov, “Actual problems of nonlinear dynamics", М: URSS, 2002Overview Correlation integral on r >1 [4]:Correlation integral on r >1 [4]: 1.Find, having curves for each k, starting with k=1; 2.Starting with certain k-number stops growing and stabilizes; 3.This k-number is embedding dimension ; 4.Maximum value of is a so-called correlation dimension (or ) of the attractor.

6. K2-entropy [5].Having fixed r and investigating dependence С(r*,k) from k (k>>1), somebody can estimate K2-entropy [5]. K2 defines the time of predictability for the system in “volume” interpretation (growing of the volume in phase space which the system can occupy in the future) Lyapunov exponentsThe time of predictability also can be determined from Lyapunov exponents The maximal one is estimated in Wolf method [6]. ________________________________________________________________________________________________________ [5] Grassberger, I. Procaccia,"Estimation of the Kolmogorov Entropy from a Chaotic Signal",Phys.Rev.A,vol.28,4,1983,p.2591 [6] Wolf A., Swift J.B., Swinney H.L., "Determining Lyapunov exponents from time series", Physica D, 69 (1985), №3, p.285-317. Overview

7. SSA forecasting method [7]: 1) Construction of the delay matrix from time series and preliminary changes in it (centering and normalization) 2) Finding the components (M) and selection of the most important ones (r) This is equal to search of eigenvectors and и eigenvalues of the matrix. 3) Time series reconstruction with r main components and taking average on each diagonal. 4) Forecast constraction with «caterpillar» method: Equal to constraction of the new delay vector with one unknown coordinate. _____________________________________________________________________________________________________ [20] “The main components of time series: “caterpillar“ method”. Col.articles // ed. D.L. Danilov, А.А. Zhiglyavskiy – St.P.: St.P. University, 1997. - 308 p.Overview

8. Criteria for the selected companies:Criteria for the selected companies: - long time on the market of shares (NYSE) – more than 10 years; - publicity; - from different sectors; the following companiesThus the following companies were chosen: Schlumberger Limited Deutsche Bank Honda Motor Co., Ltd. Toyota Motor Corp. Starbucks BP plc. Forecasting parametersForecasting parameters: - delay number - M=20 - number of the main components – logarithmic profitDuring forecasting logarithmic profit is taken in to account: - positive in growth - negative in fall Analysis and forecasting

9. 1. Schlumberger Limited Period from 31.12.1981 to 31.12.2008 Time series consists of 6814 stock price values (on close). Analysis and forecasting

10. 1. Schlumberger Limited Analysis and forecasting

11. 2. Deutsche Bank Period from 18.11.1996 to 31.12.2008. Time series consists of 3033 stock price values (on close). Analysis and forecasting

12. 2. Deutsche Bank Analysis and forecasting

13. 3. Honda Motor Co., Ltd. Period from 11.08.1987 to 31.12.2008. Time series consists of 5390 stock price values (on close). Analysis and forecasting

14. 3. Honda Motor Co., Ltd. Analysis and forecasting

15. 4. Toyota Motor Corp. Period from 13.04.1993 to 31.12.2008. Time series consists of 3956 stock price values (on close). Analysis and forecasting

16. 4. Toyota Motor Corp. Analysis and forecasting

17. 5. Starbucks Period from 26.06.1992 to 31.12.2008. Time series consists of 4161 stock price values (on close). Analysis and forecasting

18. 5. Starbucks Analysis and forecasting

19. 6. BP plc. Period from 03.01.1977 to 31.12.2008. Time series consists of 8076 stock price values (on close). Analysis and forecasting

20. 6. BP plc. Analysis and forecasting

21. Final results of analysis are in the table:Компания Schlumberger Limited73,120,16560,15976,046,26 Deutsche Bank63,260,12080,11648,288,58 Honda Motor Co. Ltd.72,80,14950,14426,696,94 Toyota Motor Corp.62,640,11750,12228,518,18 Starbucks72,340,13890,13427,207,45 BP plc.62,450,14030,13697,127,31 Analysis and forecasting Компания 2007, % 2008, % Schlumberger Limited757575 Deutsche Bank8169 Honda Motor Co. Ltd.7558 Toyota Motor Corp.7581 Starbucks8657 BP plc.71 Percentage of coincidence between logarithmic profit signs of forecast and real time series

22. Nonlinear dynamics methods applied to stock price time series led to a “space” and “time” analysis of the trading system. Thus we determined number of the main components (=embedding dimension) and time of predictability (according to K2-entropy and Lyapunov exponents) for each company. Obtained results have both fundamental and applied sense for economics. Complex analysis permitted to make a forecast on the basis of SSA method (“caterpillar”). Forecasted values and logarithmic profit fits the real ones quite well. Thus SSA forecasting method can be a useful instrument in quantitative analysis of any risks connected with financial time series. Conclusions

23. Thank You for attention!