2011 IEEE Symposium on Business, Engineering and Industrial Applications (ISBEIA), Langkawi, Malaysia Comparative Analysis of Geometric Brownian motion Model in Forecasting FBMHS and FBMKLCI Index in

Aslina Omar* and Maheran Mohd Jaffar** *, ** Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, Shah Alam, Selangor, Malaysia. *[email protected] **[email protected]

Abstract— On April 17, 1999, the Kuala Lumpur Stock The stock market in the past few years has become an Exchange, today known as Bursa Malaysia, launched a new index integral part of the global economy. Each of these market called Syariah Index (SI) to facilitate participation in the equity fluctuations affects personal and corporate financial lives, and investment in accordance with Islamic syariah’s principles. the economic health of a nation. The stock market has always Syariah-based equity is basically shares of the company meeting been one of the most popular investments because of its high the criteria of Islamic jurisprudence. Indices are used as a performance benchmark for portfolios such as mutual fund returns [4]. shares. The index is a device that allows investors to measure the On April 17, 1999, the Kuala Lumpur , performance of the group share of the market. This paper today known as Bursa Malaysia, launched a new index called forecasts the FTSE Bursa Malaysia Hijrah Shariah (FBMHS) Syariah Index (SI) to facilitate participation in the equity and FTSE Bursa Malaysia KLCI (FBMKLCI) index using the better model of Geometric Brownian motion in terms of volatility investment in accordance with Islamic syariah’s principles. models and number of data. This paper shows that forecasting Syariah-based equity is basically shares of the company using log volatility and 4 week daily data gives accurate meeting the criteria of Islamic jurisprudence. The Bursa forecasting. Malaysia SI is a weighted average of the index and its components was originally composed of the appointment of Keywords-indices; Bursa Malaysia; Geometric Brownian Shariah Advisory Council (SAC) of the Securities motion; stock market Commission of Malaysia. Investors who want to make investments based on Islamic principles will use SI as a I. INTRODUCTION benchmark to make better decisions on the information [5]. Investment is money or capital commitment for the The is a single number calculated from purchase of financial instruments or other assets to recover the a different stock prices. Indices are used as a performance benefits in the form of interest income, or appreciation of the benchmark for portfolios such as mutual fund shares. The instruments value. An investment involves the choices of index is a device that allows investors to measure the individual, or organizations such as pension funds, after some performance of the group share of the market. It can establish analysis or thoughts, or lends money to place the instruments a benchmark for portfolio includes active or passive control of which has a certain risk level and provides the possibility of certain markets. Being part of the index is a status symbol for making a profit during the period of time [1]. Examples of the constituents and trading of constituent shares obviously instrument, vehicle or property are hotels, commodities, stocks supports the share prices. The index also allows the creation and bonds, financial derivatives such as futures or options or of the investment products that give investors exposure to assets in a foreign currency [2]. stock markets or groups that can help in a market where there Meanwhile, the stock market where shares in companies are are barriers to investment. Some investment funds (index bought and sold, gives the company the option to access funds) manage their portfolios so that their performances in a capital and investors the opportunity to own shares in the stock market index or a sector of the stock market is stable [3]. company. Investors can only invest in shares through the Bursa Malaysia and Financial Times Stock Exchange stock exchange; an organized market in which shares are (FTSE) Group merged on 26 June 2006 to launch the FTSE bought and sold under the strict rules, regulations and Bursa Malaysia Index Series. FTSE Bursa Malaysia Index guidelines. The Malaysian stock exchange is called Bursa Series is designed to represent the company’s performance, Malaysia. Bursa Malaysia has more than 1,000 registered provides investors with comprehensive and complementary set companies that offer variety of investment options for local of indices. To date, the FTSE Bursa Malaysia Index Series and global investors. Companies are either listed on Bursa comprises 6 tradable indices and 7 benchmark indices as Malaysia Securities Main Market or the Ace market. described in Fig. 1: Approximately 88% of the current impacts of listing on Bursa Malaysia are syariah-compliant and represents two-third of the market capital of Malaysia [3].

U.S. Government work not protected by U.S. copyright 157 In this paper, methods that were used are stochastic calculus. In financial area, stochastic calculus is very important in mathematical modeling of financial processes. This is because of the underlying random nature of financial markets [10]. The role of Brownian motion in stochastic processes is similar to that of Normal random variables in elementary statistics. The concepts of random walk, and discrete counterpart of the (continuous time). Brownian motion is well known among economics experts since most macroeconomics time series behave in a similar trend (a Figure 1: FTSE Bursa Malaysia Index Series random walk is a special case of what is known as unit root Source: [3] process or processes). Stock market index can be classified in many ways. An extensive index is the basis of the overall stock market Even though there are many methods that are used in performance and reflects how investors feel about the forecasting stock market prices and indices, but most of them economy. Stock markets are the most frequently cited area of have their own constraints. In this research the method that the basic index of large companies listed on the nation’s will be used in forecasting the indices is geometric Brownian largest stock exchanges, such as the U.S. Dow Jones Industrial motion. Average and S & P 500 Index, the UK FTSE 100, French There are numbers of studies that used geometric Brownian CAC 40, DAX of German and Japan’s . Stock motion as a method in forecasting the stock market prices and indices have been developed in the last twenty years to indices. For example [8] and [9] who estimated the monthly become much more than economic indicators (market returns on the S&P 500 from January 1950 to July 1988 and barometer) and with growing developments in financial from a sample of individual stocks, he found persistence markets, the more technical functions of the index was among stock returns if stock prices followed the Brownian brought to the surface [3]. motion. Stock indices are used by investors and investment Reference [11] studied some foreign currency and found managers as a tool for assessing the performance of the stock that foreign currency followed either a fractional Brownian markets. Now the extensive applications, include the use of motion or a Pareto-Levy stable distribution. indices as benchmarks for comparison as a part of portfolio investors and the underlying financial products, such as Besides that, [12] measured the degrees of persistence of Exchange traded funds (ETFs) and derivatives. The Bursa the daily returns of several European stock market indices and Malaysia index is calculated using market capitalization they found the FTSE turns out to be an ultra-efficient market weighted methodology. It means the total market that exhibit abnormally fast mean-reversion, faster than capitalization of a company’s shares is registered in theoretically postulated by a Geometric Brownian motion. accordance with the current market price. Therefore larger Reference [13] studied Bombay stock exchange (BSE) companies give higher weightage compared to smaller index financial time series for fractal and multi-fractal companies [3]. behavior. He found that BSE index time series is mono-fractal There are several studies that examine the relationship and can be characterized by a fractional Brownian motion. between mutual funds and local stock market indices. One of Reference [14] used index series are from the Shanghai stock the earliest studies on Islamic Investment in Malaysia was market and Shenzhen A-shares and B-shares. They found that Khaled [6] who used parametric t statistic and the non- these stock markets move like geometric Brownian motion. parametric signed-rank test and also Jensen measure to The objective in this paper is to identify the better examine whether returns earned by investors who purchase geometric Brownian motion models in forecasting the shares in the Dow Jones Islamic Index and FTSE Global FBMKLCI and FBMHS indices in Bursa Malaysia. Islamic index are significantly different from those of the Dow Jones World Index and FTSE All-World Index, both in the II. THE MATHEMATICAL MODELS short-run and long-run. Mathematical theory of Brownian motion was implemented One of the method that has been used to forecast the stock in the context of far more than the movement of particles from market indices is Hurst exponent which was proposed by the liquid. Until now, the stock market researchers have faced Hurst [7]. It is used in hydrological studies that have been the same problem. While they can be mapped on the market performed for many areas of research [7]. The used of this center minute by minute basis, it is very difficult for them to exponent has also become more popular in the financial observe the purchase, the sell, and how demand and supply studies largely due to work of Peters [8] and [9] who estimated affect the price movement. This has attracted a lot of theories Hurst exponent for monthly returns on the S&P 500 from about how the behavior of investors can make prices move, January 1950 to July 1988. but there is no empirical evidence to support the important link between investors and the dynamic pricing decision [15].

158 Table I. A Scale of Judgment of Forecast Accurancy (Lewis) A stochastic process St is said to follow a Geometric Brownian motion if it satisfies the following stochastic MAPE Judgment of Forecast differential equation. Accurancy ()+= σμ < 10% Highly accurate tt dBtdtSdS . (1) 11% to 20% Good forecast Where μ is the percentage of drift and σ is the percentage of volatility (Wilmott, 2000). This equation has an analytic 21% to 50% Reasonable forecast solution: >51% Inaccurate forecast ⎛ 1 ⎞ ⎜ − 2 ⎟ +σσμ ()()− XtXt (0 ) Source: [16] ()= ( ) ⎝ 2 ⎠ 0 eStS . (2) From Table I it indicate that the smaller the values of MAPE, For an arbitrary initial value S0 . This model is used in the forecasting model used is more accurate. forecasting the indices . III. RESULTS AND DISCUSSION Equation (2) were used to forecast the indices for 2 weeks where tS )( can be the indices. The drift , μ will be Daily closing indices for the FTSE Bursa Malaysia Hijrah Shariah Index and FTSE Bursa Malaysia KLCI Index were calculated using. obtained from Bloomberg. The data contained information on 1 M the indices from May 31, 2010 to October 11, 2010. After the μ = R δ ∑ i public holidays were abolished, there were 93 observations tM i=1 . (3) In this paper, the volatility, σ is calculated using two (June 2, 2010 to July 9, 2010) were used to generate initial forecast. The next 20 observations (July 12, 2010 to August 6, different equations in order to find the better model of 2010) were used as the estimated values of the trading geometric Brownian motion for forecasting the indices. The produced for this period. two different equations are as follows: By using equations (2) and (3), the FBMHS and FBMKLCI A. Simple volatility of the asset, σ . index were forecasted where 4 week daily data were used. It can be estimated by Table II and Table III shows the forecasted FBMHS and FBMKLCI index using (4) and (5). M σ = 1 ()− 2 ∑ i RR Table II. Forecasted FBMHS Index ()− 1 δtM = i 1 . (4) where M is the number of returns in the sample, δt is the Dates Actual Forecast 1a Forecast 2a ('00) ('00) ('00) timestep (time between sampling), Ri is the indices at time i 12/07/2010 93.84 93.38 93.46 and R is the average indices. 13/07/2010 94.17 93.51 93.59 B. Log Volatility 14/07/2010 94.86 93.64 93.72 15/07/2010 94.61 93.77 93.85 If δt is sufficiently small the mean return R term can be 16/07/2010 94.55 93.9 93.99 ignored. For small δt 19/07/2010 94.25 94.03 94.12 M σ = 1 ()()− ( )2 20/07/2010 94.5 94.16 94.25 ∑ log i log tStS i −1 ()− 1 δtM = i 1 . (5) 21/07/2010 94.65 94.29 94.38 () 22/07/2010 94.34 94.42 94.51 where tS i is the closing indices on that ti . 23/07/2010 95.09 94.55 94.65 The best model will forecast the indices closest to the actual 26/07/2010 95.12 94.69 94.78 values. The error can be calculated by using Mean Absolute Percentage Error (MAPE). 27/07/2010 95.25 94.82 94.91 28/07/2010 95.2 94.95 95.04 ε The formula of MAPE, MAPE is as follows: 29/07/2010 95.46 95.08 95.18 30/07/2010 95.45 95.21 95.31 M − 1 XX Ap ε = . (6) 02/08/2010 95.44 95.35 95.45 MAPE M ∑ X j=1 p 03/08/2010 95.75 95.48 95.58 04/08/2010 95.86 95.61 95.71 where X p is the prediction value of indices and X A is the actual value of indices. Lawrence et al. (2009) states the scale 05/08/2010 95.53 95.74 95.85 of jugdment of forecast accuracy regarding MAPE as in Table 06/08/2010 95.5 95.88 95.98 I. MAPE 0.44% 0.37% a.Forecast 1 used (4) as volatility, Forecast 2 used (5) as volatility.

159 Table III. Forecasted FBMKLCI Index

Dates Actual Forecast 1a Forecast 2a ('00) ('00) ('00) 12/07/2010 13.27 13.37 13.31

13/07/2010 13.33 13.39 13.33

14/07/2010 13.41 13.41 13.35 15/07/2010 13.34 13.42 13.37

16/07/2010 13.37 13.44 13.38 Figure 4: Actual FBMKLCI vs Forecast 1 19/07/2010 13.33 13.46 13.4 20/07/2010 13.38 13.48 13.42 21/07/2010 13.41 13.5 13.44

22/07/2010 13.36 13.52 13.46

23/07/2010 13.46 13.54 13.48 26/07/2010 13.52 13.55 13.5 27/07/2010 13.52 13.57 13.51 28/07/2010 13.55 13.59 13.53 Figure 5: Actual FBMKLCI vs Forecast 2 29/07/2010 13.58 13.61 13.55 In order to find the best Geometric Brownian motion for 30/07/2010 13.61 13.63 13.57 forecasting the indices, we used four type of different number 02/08/2010 13.64 13.65 13.59 of data which are for 1 week, 2 week, 3 week and 4 week. Based on the data obtain, the results is as follows: 03/08/2010 13.64 13.67 13.61 04/08/2010 13.63 13.69 13.63 Table IV and Table V show the forecasting of FMBHS index by using (4) and (5). It indicates that the MAPE for this 05/08/2010 13.62 13.7 13.65 two volatility models is highly accurate since the error is less 06/08/2010 13.6 13.72 13.67 than 10% [16]. In terms of number of data, 4 week daily data MAPE 0.51% 0.26% gives the lowest error compare to others which are 0.44% and Fig. 2 and Fig. 3 show the graphs of forecasted FBMHS 0.37% respectively. index versus the actual index. Table IV. Forecasted FBMHS Index using Simple Volatility

Number of Data Error 1 Week 2 Weeks 3 Weeks 4 Weeks

MAPE 1.20% 1.39% 0.46% 0.44%

Table V. Forecasted FBMHS Index using Log Volatility

Figure 2: Actual FBMHS vs Forecast 1 Number of Data Error 1 Week 2 Weeks 3 Weeks 4 Weeks MAPE 1.25% 1.30% 0.39% 0.37% It is followed by 3 week daily data; 0.46% and 0.39%, 1 week daily data ; 1.20% and 1.25% and the last is 2 week daily data; 1.39% and 1.30% respectively.

Table VI and Table VII show the forecasting of FMBKLCI index by using simple volatility and log volatility. Similar to FBMHS index, the MAPE for this two volatility models is Figure 3: Actual FBMHS vs Forecast 2 highly accurate since the error is less than 10%. In terms of the number of data, 4 week daily data gives the lowest error Fig. 4 and Fig. 5 show the graphs of forecasted FBMKLCI compare to others. index versus actual index.

160 is managed by the Research Management Institute, Universiti Table VI. Forecasted FBMKLCI Index using Simple Volatility Teknologi MARA (600-RMI/FRGS 5/31/Fst (26/2008). Number of Data REFERENCES Error 1 Week 2 Weeks 3 Weeks 4 Weeks [1] A. O’. Sullivan, and S. M. Sheffrin, Economics: Principles in action ,New Jersey: Pearson Prentice Hall, 2003, pp. 271. MAPE 0.96% 1.12% 0.51% 0.34% [2] B. Graham, and D. Dodd, Security Analysis, McGraw-Hill Book Company, 1951.

[3] Bursa Malaysia (2010). Retrieved from www.bursamalaysia.com Table VII. Forecasted FBMKLCI Index using Simple Volatility [4] R.J. Kuo, L.C. Lee, and C.F. Lee, “Integration of Artificial NN and Fuzzy Delphi for Stock Market forecasting,” IEEE International Number of Data Error Conference on System, Man, and Cybernetics, Vol.2, pp. 1073-1078, 1996. 1 Week 2 Weeks 3 Weeks 4 Weeks [5] M. Sadeghi, “Financial Performance of Shariah-Compliant Investment: MAPE 0.67% 1.42% 0.34% 0.26% Evidence from Malaysia Stock Market,” International Research Journal of Finance and Economics, pp. 20, 2008. From the results, it shows that the two type of volatility [6] A.H. Khaled, Islamic Investment: Evidence from Dow Jones and FTSE used in this paper is highly accurate since its MAPE is less than Indices, 2004. 10%. So it becomes easier for researcher who do not have [7] H. Hurst, “Long-term storage capacity of reservoirs,” Transactions of insufficient data to forecast the indices by using 1 week, 2 the American Society of Civil Engineers, Vol.1, pp. 519-543, 1951. week, 3 week or 4 week daily data. But it is strongly [8] E. Peters, Chaos and Order in the Capital Markets: A new view of reccomended to used 4 week daily data since it is the most cycles, prices, and market volatility, John Wiley & Sons: New York, 1991. accurate and closest to the actual values. [9] E. Peter, Fractal Market Analysis: Applying Chaos Theory to Investment and Economics, John Wiley & Sons: New York, 1994. IV. CONCLUSION [10] P. Willmott, Quantitative Finance, Chichester: John Wiley & Son, Ltd, 2000. Among the two volatility models in forecasting the [11] M. Corazza, and A.G. Malliaris, “Multifractality in Foreign Currency FBMHS and FBMKLCI index, log volatility gives better Markets,” Multinational Finance Journal, Vol.6, pp. 387-401, 2002. results compared to simple volatility in terms of MAPE that [12] J. Lipka, & C. Los, Persistence characteristics of European stock produces the lowest error. This study also shows that indexes, Working paper (Kent State University, Kent, OH), 2002. forecasting the two indices is most accurate with the 4 week [13] A. Razdan, “Scaling in the Bombay stock exchange index,” daily data. Hence with these forecast values the investor will PRAMANA- Journal of Physics, Vol.58, pp. 537-544, 2002. be able to make quick decision in investing in Bursa Malaysia. [14] C.A. Los, and Y. Bing, “Persistence characteristics of the Chinese stock markets,” International Review of Financial Analysis, Vol.17, pp. 64- 82, 2008. ACKNOWLEDGMENT [15] W.N. Goetzmann, “Stock Markets, Behavior and the Limits of History,” National Bureau of Economics Research, 2000. This research is funded by the Fundamental Research Grant [16] K.D. Lawrence, R.K. Kllimberg and S.M. Lawrence, Fundamental of Scheme (FRGS), Ministry of Higher Education Malaysia, that forecasting Using Excel, Industrial Press Inc.,America, 2009.

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