Same slope forecasting method.

Pasic, Mugdim ; Bijelonja, Izet ; Sunje, Aziz 等

Abstract: In this paper a quantitative forecasting method has been developed, which stands out by its simplicity, but at the same time it also shows very good results in terms of forecasting the price of stock in comparison to the other existing methods of the same complexity level. To demonstrate the efficiency of this forecasting method we tested it on forecasting the price of stock for five issuers of Sarajevo Stock Exchange.

Key words: Forecasting, Quantitative Forecasting Models, Stock Price Forecasting, Weighted Forecasting Factor.

1. INTRODUCTION

Forecasting is an activity necessary for success of any individual and organization. There are two general approaches to forecasting i.e. two types of forecasting methods: qualitative and quantitative. Quantitative methods are further divided into causal methods and time series methods, what is focus of this paper.

All time series forecasting methods use historical data on demand based on which they forecast an outcome of an event in the future (future demand) e.g.: demand for a product, stock price and so on. Basic premise of all time series methods is that demand can be separated into components such as an average level, trend, seasonal peaks, cycles and errors that must be taken into consideration if relevant results are to be achieved (Heizer & Render, 2004).

To the present day majority of time series methods has been developed (Poon & Granger, 2003). Basic models of these methods, characteristic by simplicity, do not take into consideration all elements of demand (Render et al., 2003).

There are certainly more advanced versions of these methods, which include all elements of demand, but these methods are more complex and more demanding from the aspect of data application and analysis (Chase et al., 2006).

The difference in terms of complexity of the basic model and advanced one, which takes into consideration all elements of demand, can be noticed on the example of exponential smoothing (Gardner, 2006).

Efficiency of a model certainly depends on the nature of demand, thus we cannot speak of general superiority of one model over the others (Poon & Granger, 2003).

It has been established that stock price forecasting in one of the most demanding forecasting areas (Granger, 1992).

ARCH (Autoregressive Conditional Heteroskedasticity) model describes the forecast variance in terms of current observables (Engle, 2004).

Forecasting of the large covariance matrices relevant in asset pricing, asset allocation, and financial risk management applications and formal links between realized volatility and the conditional covariance matrix are developed in (Andersen et al., 2003)

Sarajevo Stock Exchange (SSE) is not an exception. Apart from the level and random error behavior of stock prices of the issuers on Sarajevo Stock Exchange demonstrates some other demand components such as cycles, trends and seasonal peaks. The model developed in this paper is applied to SSE.

2. BASICS OF THE SAME SLOPE MODEL

This method is based on the idea that the growth (decline) of stock price (demand) in the future period will have the same slope (trend) as in the prior period.

This idea can be presented mathematically as:

[F.sub.t] = [D.sub.t-1] - ([D.sub.t-2] - [D.sub.t-1]) (1) where:

[F.sub.t]--Expected value of stock price for the period t,

[D.sub.t-1]--Stock price in period t-1,

[D.sub.t-2]--Stock price in period t-2.

Figure 1. depicts functioning of the method developed in this paper.

It can seen from equation (1) that in order to forecast demand for the period t this method takes demand [D.sub.t-1] as a starting point and continues with the same trend the demand has had between the periods t-1 and t-2.

Mathematical equation of this method can be presented as follows:

[F.sub.t] = [D.sub.t-1] - [beta]([D.sub.t-2] - [D.sub.t-1]) (2)

This equation enables forecasting in a way to allow for the trend in the period between t and t-1 to be different from the trend the demand has had in the period between t-1 and t-2 periods. Depending on the value of coefficient [beta], we can have some of following scenarios:

[beta] <1--Forecasts smaller change rate in the period t to t-1 than in the prior period.

[beta] =1--Forecasts the same change rate in the period t to t-1 than in the prior period. This scenario presents the initial idea described in equation (1).

[beta] >1--Forecasts bigger change rate in the period t to t-1 than in the prior period.

This method shows major errors when demand function goes through local minimum or maximum. Figure 1. shows situation when demand function goes through local minimum, where demand starts climbing up, while the method falls a step behind and continues to fall, but right in the next step the method reacts to changes properly. The coefficient [beta] can be optimized by minimizing the forecast errors.

[FIGURE 1 OMITTED]

3. FORECAST ERRORS

Forecast error is difference between the forecast value and actual demand. In quantitative methods and statistics these errors are called residuals. Every forecast contains errors. The aim is to develop a model that will minimize these errors, and thus to get a model that will best describe future events.

To determine reliability (preciseness) of a forecasting method several measurements have been developed:

* Cumulative sum of forecast errors:

CFE = [n.summation over (t=1)] ([D.sub.t]-[F.sub.t]) (3)

where ([D.sub.t]-[F.sub.t] is forecast error for the period t.

* MSE--mean square error or model variance:

MSE [n.summation over (t=1)][([D.sub.t] - [F.sub.t]).sup.2] (4)

where n stands for number of periods for which the model error is measured.

* Standard deviation is result of square root of variance:

[sigma] = [square root of MSE] = [square root of [n.summation (t=1)][([D.sub.t] - [F.sub.t]).sup.2]/n (5)

Error measurement will be used to determine which of the forecasting methods is the most efficient for forecasting in the given case.

4. RESULTS

Forecasting method developed in this work has been tested on forecasting of stock prices of Sarajevo Stock Exchange issuers. Forecasts are performed through some other established methods in order to demonstrate the efficiency of the method developed. Models of other methods used are of approximately the same level of complexity as the method developed in this work.

Since the stocks of the selected issuers have not been traded every day, the forecasts are made for average monthly stock price. The forecast was made for all months from September 1, 2004 to March 30, 2006 for five (of ten) issuers.

Error measures for certain methods and selected issuers are presented in Table 1. Signs given in the Table 1 stand for:

Stock companies:

* D1--BH Telecom,

* D2--Bosnalijek,

* D3--Energoinvest,

* D4--Hidrogradnja,

* D5--Sarajevo-Osiguranje.

Methods used:

* M1--Moving average method for n = 3

* M2--Weighted moving average for n = 3,

* M3--Exponential smoothing for [alpha] = 0,9,

* M4--Same Slope Method developed for [beta] = 1,

* M5--Same Slope Method developed for [beta] = 0,5.

It can be seen from Table 1. that the model developed in this paper shows the best performances in general. As it was explained in the previous paragraph, forecast errors can be minimized by choosing the proper value of [beta], as it is shown Table 1 comparing results using [beta]=0,5 and [beta]=1. Coefficient [beta] is sometimes called weighted forecasting factor, since it gives a weight to certain value of gradient of demand in the prior period.

5. CONCLUSION

The method developed is very simple and it reacts quickly on all changes in demand and it uses very little historical data (two prior demands).

Results of the research have shown that this method is more efficient than the other methods of the same complexity used for forecasting stock prices for the five issuers.

The method can be made even more efficient and it can keep its simplicity without making any detailed analyses of historical data. One of the approaches could be to determine the value of coefficient [beta], using the tools of operational studies. For the goal function, which should be minimized and which depends on coefficient [beta], one should take the sum of forecast errors made by this method in previous periods.

The future research should be focused on testing developed model on other cases in different fields.

6. REFERENCES

Andersen, T. G.; Bollerslev, T.; Diebold, F. X. & Labys, P. (2003). Modeling and Forecasting Realized Volatility. Econometrica, Vol. 71, No. 2, (March, 2003) pp. 579-625, ISSN 0012-9682

Chase B. R.; Jacobs F. R. & Aquilano N. J. (2006). Operations Management, McGraw-Hill/Irwin, ISBN 0-07-111552-8, New York, USA

Engle, R. (2004). Risk and Volatility: Econometric Models and Financial Practice. The American Economic Review, Vol. 94, No. 3, (June, 2004) pp. 405-420, ISSN 0002-8282

Gardner, E. (2006)., Exponential smoothing: The State of the Art--Part II. International Journal of Forecasting, Vol. 22, No. 4, (October-December, 2006) pp. 637-666, ISSN 0169-2070

Granger, C. (1992). Forecasting Stock-Market Prices: Lessons for forecasters. International Journal of Forecasting, Vol. 8, No. 1, (June, 1992), pp. 3-13, ISSN 0169-2070

Heizer, J. & Render, B. (2004). Operations Management, Prentice Hall, ISBN 0-13-120974-4, New Jersey, USA

Poon, H. & Granger, C. (2003). Forecasting Volatility in Financial Markets: A Review. Journal of Economic Literature, Vol. 41, No. 2, (June, 2003) pp. 478-539, ISSN 0022-0515

Render, B.; Stair, R. & Hanna, M. (2003). Quantitative Analysis for Management, Prentice Hall, ISBN 0-13-049543-3, New Jersey, USA

Table 1. Error measures of forecasting methods.

 Error
 measures M1 M2 M3 M4 M5

D1 CFE 1,26 0,87 0,74 -0,11 0,25
 MSE 10,92 6,36 4,42 3,93 2,92
 [sigma] 3,30 2,52 2,10 1,98 1,71
D2 CFE 1,46 1,06 0,79 0,18 0,47
 MSE 7,91 5,01 3,56 4,47 3,23
 [sigma] 2,81 2,24 1,89 2,12 1,80
D3 CFE 0,32 0,20 0,19 0,00 0,09
 MSE 3,28 1,97 1,37 1,68 1,19
 [sigma] 1,81 1,40 1,17 1,30 1,09
D4 CFE 0,40 0,30 0,17 0,05 0,12
 MSE 1,91 1,16 0,83 0,87 0,64
 [sigma] 1,38 1,08 0,91 0,93 0,80
D5 CFE 0,95 0,67 036 0,06 0,25
 MSE 5,72 3,44 2,43 2,75 1,93
 [sigma] 2,39 1,85 1,56 1,66 1,39