THE ALUMINIUM PRICE FORECASTING BY REPLACING THE INITIAL CONDITION VALUE BY THE DIFFERENT STOCK EXCHANGES

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1 Acta Metallurgca Slovaca, Vol. 20, 2014, No. 1, p THE ALUMINIUM PRICE FORECASTING BY REPLACING THE INITIAL CONDITION VALUE BY THE DIFFERENT STOCK EXCHANGES Marcela Lascsáková 1) *, Peter Nagy 1) 1) Techncal Unversty of Košce, Faculty of Mechancal Engneerng, Košce, Slovaka Receved: Accepted: * Correspondng author: e-mal: marcela.lascsakova@tuke.sk, Tel.: , Department of Appled Mathematcs and Informatcs, Faculty of Mechancal Engneerng, Techncal Unversty of Košce, Letná 9, Košce, Slovaka Abstract In mathematcal models, for forecastng prces on commodty echanges dfferent mathematcal methods are used. In the paper the numercal model based on the eponental appromaton of commodty stock echanges was derved. The prce prognoses of alumnum on the London Metal Echange were determned as numercal soluton of the Cauchy ntal problem for the 1 st order ordnary dfferental equaton. To make the numercal model more accurate the dea of the modfcaton of the ntal condton value by the alumnum prce (stock echange) was realsed. The derved numercal model was verfed by determnng the nfluence of the length of the ntal condton drft on the accuracy of the obtaned prognoses. The types of the ntal condton drft durng dfferent movements of alumnum prces were studed. The most accurate prognoses were the most often obtaned by usng the longest ntal condton drft. In ths type of drft the ntal condton value was replaced by alumnum stock echange n the month n whch the absolute percentage error of the prognoss had at least selected value. The advantage of ths drft was manfested especally n the stable prce course and wthn larger changes n prces. If there was prce fluctuatng wthn the observng perod, n the net forecastng the most accurate was the drft the ntal condton value of whch had been replaced by the prce that was the nearest to the stock echange prce evoluton. Keywords: alumnum, eponental appromaton, numercal modellng, prce forecastng, commodty echange 1 Introducton Observng trends and forecastng movements of metal prces s stll a current problem. There are a lot of approaches to forecastng prce movements [1, 2]. Some of them are based on mathematcal models [2-19]. Forecastng prces on commodty echanges often uses the statstcal methods that need to process a large number of hstorcal market data [5-12]. The quantty of needed market data can sometmes be a dsadvantage. In such cases, other mathematcal methods are requred. In our prognostc model numercal methods were used. Ther advantage s that, n comparson wth statstcal models, many fewer market data are needed. Our numercal model for forecastng prces s based on the numercal soluton of the Cauchy ntal problem for the 1 st order ordnary dfferental equatons [13-19]. The alumnum prces presented on the London Metal Echange (LME) were worked on. We dealt wth the monthly averages of the daly closng alumnum prces "Cash Seller&Settlement

2 Acta Metallurgca Slovaca, Vol. 20, 2014, No. 1, p prce" n the perod from December 2002 to June The market data were obtaned from the offcal web page of the London Metal Echange [20]. The course of the alumnum prces on LME (n US $ per tonne) wthn the observng perod s presented n Fg. 1. As we can see n Fg. 1 the course of the alumnum prces wthn the consdered perod changes dramatcally. Fg. 1 Course of alumnum prces on LME n the years We started from the orgnal model calculatng the prognoses wthn s months followng the appromaton term after modfcaton of the ntal condton value by the obtaned monthly prce prognoses [17-19]. The orgnal model forecasts the alumnum prce relably wthn the stable prce course, when the prce does not changed rapdly. Wthn the rapd ncrease or decrease of stock echanges, but also n the case of changes n the prce course the forecastng fals. Snce the varablty wth rapd and sudden changes s typcal of the commodty prce course, we judged the possblty of makng the forecastng more accurate by usng the modfcaton of the ntal condton value by alumnum prce. We analysed the use of the ntal condton drft concernng dfferently chosen stock echanges for successful forecastng. 2 Mathematcal model The Cauchy ntal problem n the form y a, y (1) 1 y y ( 0 ) 0 a1 s consdered. The partcular soluton of the problem (1) s n the form y k e, where a10 k y0 e. The consdered eponental trend was chosen accordng to the test crteron of the tme seres trend sutablty. The values ln ( Y 1) ln ( Y ), for 0,1,..., 42 have appromately constant course. ( Y s the alumnum prce (stock echange) on LME n the month.) The prce prognoses are created by the followng steps: 1 st step: Appromaton of the values the values of the appromaton term are appromated a by the least squares method. The eponental functon n the form ~ y 1 a0e s used. When observng the nfluence of the appromaton term length on the prognoses accuracy, we found out that the prognoses obtaned by longer appromaton terms are more accurate [14, 18]. The appromaton of larger number of the values s not subject to abrupt changes and follows a longer prce course evoluton. Let us consder two dfferent varants:

3 Acta Metallurgca Slovaca, Vol. 20, 2014, No. 1, p Varant B: The values from the perod January June 2003 are appromated. The net appromaton terms are created by sequental etenson of ths perod by 3 months. Thus th the duraton of the appromaton terms s etended (the n appromaton term has 6 3( n 1) stock echanges) (Fg. 2). Fg. 2 Varant B (A appromaton term, P forecastng term) (as can be seen n [17-19]) Varant E: We appromate values wthn 12 months and each term s shfted by 1 month (Fg. 3). (The frst appromaton term s January December 2003.) Fg. 3 Varant E (A appromaton term, P forecastng term) (as can be seen [17-19]) 2 nd step: Formulatng the Cauchy ntal problem accordng to the acqured appromaton functon y ~, the Cauchy ntal problem (1) s wrtten n the form y a y, y ( ) Y, (2) 1 where s the last month of the appromaton term, Y s the alumnum prce on LME n the month. 3 rd step: Computng the prognoses the formulated Cauchy ntal problem (2) s solved by the numercal method based on the eponental appromaton of the soluton. A detaled soluton method s seen n [21]. The method uses the followng numercal formulae: 1 h, v vh y 1 y bh Qe e 1,

4 Acta Metallurgca Slovaca, Vol. 20, 2014, No. 1, p for 1, 2,3,..., where h 1 s the constant sze step. The unknown coeffcents are calculated by means of these formulae: v f (, y ) f (, y ), Q 2 f (, y ) f (, y ) (1 v) v e v, b f (, y ) f (, y ) v. If we consder the Cauchy ntal problem (2), the functon f (, y ) has the form f (, y ) a1 y and then f (, y ) a1 y ( ) a1 y, f (, y ) a y 1 ( ) a1 y. We calculate the prognoses wthn s months that follow the end of the appromaton term n ths way: The frst month prognoss s determned by solvng the Cauchy ntal problem n the form (2). The nterval, 1 of the length h 1 month s dvded nto n parts, where n s the number of tradng days on LME n the month 1. We get the sequence of the dvson ponts 0, j h. j n, for j 1,2,..., n, where n 1. For each pont of the subdvson of the nterval, the Cauchy ntal problem n the form (2) s solved by the chosen numercal method. In ths way we obtan the prognoses of the alumnum prces on sngle tradng days y j. By calculatng the arthmetc mean of the daly prognoses we obtan n the monthly prognoss of the alumnum prce n the month 1. So, y 1 y n. j 1 j The prognoses for the followng months shall be calculated after modfcaton of the ntal condton value. The ntal condton value n the month s, s 1,2, 3,4, 5 s replaced ether by the calculated monthly prognoss y s or by some alumnum stock echange (n case of hgher absolute percentage error of gven monthly prognoss y s ). The Cauchy ntal problem y a1 y, y( s ) y s, respectvely y a y, y ( ) 1 s Yp (where Y p s chosen alumnum stock echange) s used for calculatng daly prognoses and ther arthmetc mean serves to defne the monthly prce prognoss y s 1 for the month s 1. By comparng the calculated prognoss y s n the month s wth the real stock echange Y s, the absolute percentage error ps ys Ys Ys.100 % s determned. The prce prognoss y s n the month s s acceptable n practce, f p s 10 %. Otherwse, t s called the crtcal forecastng value of. To compare the accuracy of forecastng of all forecastng terms, the mean t absolute percentage error (MAPE) p p t s 1 s s determned, where, n our case, t 6. The modfcaton of the ntal condton value by the real alumnum stock echange prce s called the ntal condton drft. Let us name the selected mnmal absolute percentage error of the prognoss, causng the ntal condton drft, the lmtng value error. The month n whch the absolute percentage error of the prognoss has at least the lmtng value error wll be consdered as the lmtng month. We have chosen the lmtng value error of 7 %. Let us consder three dfferent types of the ntal condton drft: 1-month drft, drft before the lmtng month, drft to the lmtng month. 3 Results 3.1 The orgnal model the model wthout usng ntal condton drft In the orgnal model the prognoses n the months s, s 2,3, 4,5, 6 (where s the last month of the appromaton term) are calculated after modfcaton of the ntal condton value

5 Acta Metallurgca Slovaca, Vol. 20, 2014, No. 1, p by the obtaned monthly prce prognoses. 36 forecastng terms of the orgnal model n both varants B and E wthn perod from July 2003 to June 2006 are observed. From among all forecastng terms, 11 of them belong to varant B and 25 ones are part of varant E, whereby 9 forecastng terms are common for both varants. In the observng perod we ganed 216 orgnal monthly prce prognoses. Based on the prognoss accuracy analyss of the orgnal model, we classfed forecastng terms nto the followng classes: I. trouble free forecastng terms (18 terms) All absolute percentage errors of the monthly prognoses wthn 6-month forecastng term are 10 % ; a) The ntal condton drft does not occur (14 terms) ether all absolute percentage errors of the monthly prognoses are 7 % or the absolute percentage prognoss error n the last month of the forecastng term s from the nterval 7,10. b) The ntal condton drft occurs (4 terms) the absolute percentage monthly prognoses errors n some months wthn observed forecastng term are hgher than or equal to 7 % ; II. forecastng terms wth a small error (10 terms) The mean absolute percentage error of the forecastng term s less than 10 %, but the absolute percentage errors of some monthly prognoses are at least 10 % (there are the crtcal forecastng values n forecastng term); III. forecastng terms wth a bg error (8 terms) The mean absolute percentage error of the forecastng term s 10 %. From among 36 observed forecastng terms, a half of them are trouble free. Appromately n 3/4 of these terms, the forecastng s so accurate that the ntal condton drft does not occur. The ntal condton values are replaced only by calculated monthly prognoses; thereby the orgnal model has not changed. The second half of the orgnal terms conssts of the forecastng terms wth dfferent errors causng the ntal condton drft. Ths eplans why the forecastng results dffer from the orgnal model. The forecastng terms wth small and bg errors are almost equally met. By havng analysed the alumnum prce evoluton we found out that the trouble free forecastng terms are typcal of a moderate prce ncrease wth ts occasonal oscllaton. Wthn the forecastng terms wth a small error, the prce ncrease n the forecastng term s more rapd than the prce ncrease n an appromaton term, or there s a rapd decrease of the stock echanges wthn the forecastng term, even though at ts begnnng there s alumnum prce ncrease. In the forecastng terms wth a bg error we can observe a rapd alumnum prce decrease, or ncrease, mmedately after a perod of prce declne. 3.2 The nfluence of the ntal condton drft length on the accuracy of the forecast prces The forecastng terms n whch the ntal condton drft occurred (22 terms) were taken nto consderaton. We were nterested n whether dfferent ntal condton drft length affected the forecastng accuracy. Let us consder three types of the ntal condton drft wth regard to ther length: 1. 1-month drft, 2. drft before the lmtng month, 3. drft to the lmtng month. For each forecastng term we defned the type of drft wth the lowest MAPE. We also observed the ablty of ntal condton drft to elmnate the crtcal forecastng values. The followng

6 Acta Metallurgca Slovaca, Vol. 20, 2014, No. 1, p table shows the number of the forecastng terms n whch the forecastng by the determned types of the ntal condton drft was the most accurate. Table 1 The comparson of the success rate of the chosen types of the ntal condton drft for mprovng the prognoses accuracy of the orgnal model Type of forecastng term 1-month drft Type of drft drft before the lmtng month drft to the lmtng month I II III Wth regard to the ntal condton drft length, the most accurate forecastng results were obtaned by usng the drft to the lmtng month, n other words, the longest ntal condton drft. Usng t, we move really close to the real forecast prces. Ths type of drft, n comparson wth the other two drfts, had the lowest MAPE n 18 forecastng terms. In two of them we obtaned the same results by usng 1-month drft (the ntal condton drfts were the same). The 1-month drft was far less successful. It was the most advantageous type for 4 forecastng terms, 2 of them are already mentoned n the prevous drft type. The drft before the lmtng month was the most accurate only n 2 forecastng terms. Let us analyze the success rate of the determned types of the ntal condton drft wthn dfferent moves of the alumnum prce course: unstable moderate prce ncrease The unstable moderate ncrease of the stock echanges can be seen n the forecastng terms of the years 2003, 2004 and at the begnnng of the year Wthn these terms, the orgnal forecastng was mostly acceptable (the absolute percentage of prognoses errors were 10 % ), so the forecastng terms belong to the trouble free forecastng terms. In most of these terms, the absolute percentage errors were even 7 %, so the ntal condton drft dd not occur. In the trouble free forecastng terms, bgger prognoses errors are gven by the change of the ncrease rate of the forecast stock echanges n comparson wth the stock echanges wthn the appromatng term. The ntal condton drft occurred only n three dfferent forecastng terms. In each term, forecastng by usng dfferent ntal condton drft length was the most successful. Wth regard to the moderate course of the stock echanges, the ntal condton drft to the values approprate for the net stock echanges course was the most advantageous. steep prce decrease There was steep prce decrease from Aprl 2005 to June Wthn ths perod, the prce decrease was sgnfcant compared to the prevous ncrease. The forecastng terms wthn these months can be dvded nto two groups: a) perods of the prce ncrease wth the consecutve prce declne; b) perods of the prce declne wth the consecutve unstable ncrease of the stock echanges. In both groups of the forecastng terms, the stock echanges n appromaton terms are ncreasng, the appromaton functons have also an ncreasng course, and the prognoses

7 Acta Metallurgca Slovaca, Vol. 20, 2014, No. 1, p calculated by the orgnal model are ncreasng too, but they are not suffcent to accommodate to a steep declne of the stock echanges. The longer s the declne perod, the hgher s the absolute percentage of prognoss error. In the perods when the prce declne ncludes a smaller part of the forecastng term, forecastng s trouble free even wthout the ntal condton drft. The perods n whch larger part ncludes the prce declne belong to the forecastng terms wth a small or bg error. In all these terms, the drft to the lmtng month was the most accurate. By usng the longest drft, the ntal condton value was the nearest to the declne prces. Thus, the followng crtcal values were elmnated, and the forecastng became much more accurate. As the ncrease of the stock echanges occurred n the net perod, the forecastng was successful and no coercons were needed. The greatest forecastng mprovement was obtaned n the forecastng term wth a bg error Aprl 2005 September The mean absolute percentage error n ths term was decreased by usng the ntal condton drft to the lmtng month from 12, 55 % to 4, 96 % (varant B) and from 12, 63 % to 4, 94 % (varant E). The number of crtcal values n both varants was reduced to one crtcal value aganst fve crtcal values wthn the orgnal forecastng. unstable moderate prce ncrease after prce declne followed by rapd ncrease The end of the year 2005 and the frst half of the year 2006 appear as the most problematc. In ths perod there are 5 forecastng terms wth a small error and 6 forecastng terms wth a bg error. The problems n forecastng are caused by the steep ncrease of the stock echanges after ther mportant declne. The appromaton terms wth the prce declne belong to the mentoned forecastng terms. The hgher s the number of the declne prces n appromaton term, the slower s ncrease of the appromaton functon. Its course could be even decreasng, as shown n Table 2. As these appromaton functons serve for the prce prognoses wthn the rapd ncrease perods, the orgnal forecastng beng hghly naccurate. More mportant changes n the course of appromaton functons can be seen n the varant E n whch, n comparson wth the varant B, fewer values are appromated. Thus the course of the appromaton functons of the varant E s more affected by the prce declne. Table 2 Summary of appromaton functons for ncreasng forecastng terms after alumnum prce declne n 2005 Forecastng term Varant B Varant E June November 2005 ~ 0,0103 y 1414,3 e July December 2005 ~ 0,0122 y 1340 e ~ y 0, ,2 e August January 2006 ~ 0,0026 y 1696,1 e September February 2006 ~ 0,0007 y 1792,2 e October March 2006 ~ 0,0110 y 1357,3 e ~ 0,0016 y 1918,2 e November Aprl 2006 ~ 0,00008 y 1848,3 e December May 2006 ~ 0,0033 y 1692,6 e January June 2006 ~ 0,0115 y 1348,5 e ~ y 0, ,4e Slower ncreasng prognoses obtan lower values than qucker ncreasng stock echanges; ths s why the forecastng accuracy s decreasng wth tme (crtcal values appled to the months at the end of the forecastng terms). The forecastng was the most accurate when usng the longest drft. By applyng t, we put prognoses the nearest to the steep ncreasng stock echanges. Only wthn one perod the 1-month drft was the most successful. At the moderate course of the stock echanges n ths forecastng term, the forecastng accuracy was affected

8 Acta Metallurgca Slovaca, Vol. 20, 2014, No. 1, p by the stock echange wth the ntal condton drft. The advantage of 1-month drft conssts n monthly shftng of the ntal condton value to approprate stock echange for the net prce evoluton. 4 Dscusson From among 36 observed forecastng terms, the ntal condton drft was notced n 22 terms. The orgnal forecastng fals wthn the perods wth consderable changes of the prce course. Wthn the observed perod from 2003 to 2006, t was namely the perod of the steep prce declne n 2005 followed by a rapd prce ncrease. Wthn the orgnal forecastng there were 8 forecastng terms wth the mean absolute percentage error 10 % (Fg. 4, Fg. 5). Observng these terms after ntal condton drft, only n two terms and just usng 1-month drft, the mean absolute percentage errors of prognoses were 10 %. However, f we consder the most successful type of drft n each term, all forecastng terms had the mean absolute percentage error 10 %. When there were crtcal values n the forecastng term, ther number was always reduced. Fg. 4 The mean absolute percentage errors of the chosen types of the ntal condton drft wthn the forecastng terms varant B Fg. 5 The mean absolute percentage errors of the chosen types of the ntal condton drft wthn the forecastng terms varant E

9 Acta Metallurgca Slovaca, Vol. 20, 2014, No. 1, p The most accurate prognoses were the most often obtaned by the longest ntal condton drft (Table 1, Fg. 4 and Fg. 5). 5 Conclusons The orgnal model forecasts the alumnum prce relably wthn the stable prce course, but wthn the rapd ncrease or decrease of stock echanges or n the case of changes n the prce course the forecastng fals. Wth regard to the chosen results, the strategy of the ntal condton drft sgnfcantly contrbutes to prognoses accuracy, and t s sutable way of the orgnal forecastng mprovement. The most accurate prognoses were the most often obtaned by the longest ntal condton drft. By usng t, calculated prognoses were moved closed to the real stock echanges. The ntal condton drft to the lmtng month allowed calculatng the most accurate prognoses, especally n the stable ncrease up to the rapd ncrease, or durng the longer-range declne n prce. When the prce fluctuatons appeared n the observed perod, the longest drft was not always the most accurate. The most accurate was the drft replacng the ntal condton value by stock echange that was the nearest to the net prce course. References [1] J. Dědč: Stock Echange and Commodty Echange, Prospektrum, Praha, 1992 (n Czech) [2] Z. Zmeškal: Fnancal Models, Ekopress, Praha, 2004 (n Czech) [3] H. Feng: Appled Economcs Letters, Vol. 18, 2011, No. 17, p [4] Z. Rahamneh, M. Reyalat, A. Sheta, S. Aljahdal: Forecastng stock echanges usng soft computng technques, In: Internatonal Conference on Computer Systems and Applcatons, AICCSA, Hammamet, 2010 [5] M. Bessec, O. Bouabdallah: Studes n Nonlnear Dynamcs and Econometrcs, Vol. 9, 2005, No. 2, Artcle 6 [6] J. Chajdak, E. Rublíková, M. Gudába: Statstcal Methods n Practce, STATIS, Bratslava, 1994, (n Slovak) [7] Z. Ismal, F. Jamaluddn: Asan Journal of Mathematcs and Statstcs, Vol. 1, 2008, No. 1, p [8] Z. Ismal, A. Yahya, A. Shabr: Amercan Journal of Appled Scences, Vol. 6, 2009, No. 8, p [9] E. Rublíková: Tme Seres Analyss, Iura Edton, Bratslava 2007, (n Slovak) [10] J. Seger, R. Hndls: Statstcal Methods n the Market Economy, Vctora Publshng, Praha 1995, (n Czech) [11] F. Sernald: Energy Economcs, Vol. 33, 2011, No. 6, p [12] M. Varga: Ekonome a Management, Vol. 11, 2008, No. 3, p [13] M. Lascsáková: Manufacturng Engneerng, Vol. 6, 2007, No. 4, p , (n Slovak) [14] M. Lascsáková: Acta Montanstca Slovaca, Vol. 12, 2007, No. 4, p , (n Slovak) [15] M. Lascsáková: Studes of the Unversty of Žlna, Vol. 23, 2009, No. 1, p [16] M. Lascsáková: Acta Metallurgca Slovaca, Vol. 14, 2008, No. 3, p [17] M. Lascsáková: Transfer Inovácí, Vol. 16, 2010, p , (n Slovak) [18] M. Lascsáková: Manufacturng and Industral Engneerng, Vol. 11, 2012, No. 1, p [19] M. Lascsáková: Transport and Logstcs, Vol. 12, 2012, No. 23, p. 1-8

10 Acta Metallurgca Slovaca, Vol. 20, 2014, No. 1, p [20] [ ] [21] V. Penjak, M. Lascsáková: Studes of Unversty n Žlna, Vol. 1, 2001, p Acknowledgement The research for ths artcle was supported by Slovak VEGA Grant 1/0130/12.