MATHEMATICAL ENGINEERING TECHNICAL REPORTS. Sequential Optimizing Investing Strategy with Neural Networks

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1 MATHEMATICAL ENGINEERING TECHNICAL REPORTS Sequental Optmzng Investng Strategy wth Neural Networks Ryo ADACHI and Akmch TAKEMURA METR February 2010 DEPARTMENT OF MATHEMATICAL INFORMATICS GRADUATE SCHOOL OF INFORMATION SCIENCE AND TECHNOLOGY THE UNIVERSITY OF TOKYO BUNKYO-KU, TOKYO , JAPAN WWW page:

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3 Sequental optmzng nvestng strategy wth neural networks Ryo Adach Akmch Takemura February 2010 Abstract In ths paper we propose an nvestng strategy based on neural network models combned wth deas from game-theoretc probablty of Shafer and Vovk. Our proposed strategy uses parameter values of a neural network wth the best performance untl the prevous round (tradng day) for decdng the nvestment n the current round. We compare performance of our proposed strategy wth varous strateges ncludng a strategy based on supervsed neural network models and show that our procedure s compettve wth other strateges. 1 Introducton A number of researches have been conducted on predcton of fnancal tme seres wth neural networks snce Rumelhart [10] developed back propagaton algorthm n 1986, whch s the most commonly used algorthm for supervsed neural network. Wth ths algorthm the network learns ts nternal structure by updatng the parameter values when we gve t tranng data contanng nputs and outputs. We can then use the network wth updated parameters to predct future events contanng nputs the network has never encountered. The algorthm s appled n many felds such as robotcs and mage processng and t shows a good performance n predcton of fnancal tme seres. Graduate School of Informaton Scence and Technology, Unversty of Tokyo, Hongo, Bunkyo-ku, Tokyo , JAPAN Insttute of Industral Scence, Unversty of Tokyo, Komaba, Meguro-ku, Tokyo , JAPAN 1

4 Relevant papers on the use of neural network to fnancal tme seres nclude [5], [6], [8] and [14]. In these papers authors are concerned wth the predcton of tme seres and they to not pay much attenton to actual nvestng strateges, although the predcton s obvously mportant n desgnng practcal nvestng strateges. A forecast of tomorrow s prce does not mmedately tell us how much to nvest today. In contrast to these works, n ths paper we drectly consder nvestng strateges for fnancal tme seres based on neural network models and deas from game-theoretc probablty of Shafer and Vovk (2001) [11]. In the game-theoretc probablty establshed by Shafer and Vovk, varous theorems of probablty theory, such as the strong law of large numbers and the central lmt theorem, are proved by consderaton of captal processes of bettng strateges n varous games such as the con-tossng game and the bounded forecastng game. In game-theoretc probablty a player Investor s regarded as playng aganst another player Market. In ths framework nvestng strateges of Investor play a promnent role. Predcton s then derved based on strong nvestng strateges (cf. defensve forecastng n [13]). Recently n [9] we proposed sequental optmzaton of parameter values of a smple nvestng strategy n mult-dmensonal bounded forecastng games and showed that the resultng strategy s easy to mplement and shows a good performance n comparson to well-known strateges such as the unversal portfolo [3] developed by Thomas Cover and hs collaborators. In ths paper we propose sequental optmzaton of parameter values of nvestng strateges based on neural networks. Neural network models gve a very flexble framework for desgnng nvestng strateges. Wth smulaton and wth some data from Tokyo Stock Exchange we show that the proposed strategy shows a good performance. The organzaton of ths paper s as follows. In Secton 2 we propose sequental optmzng strategy wth neural networks. In Secton 3 we present some alternatve strateges for the purpose of comparson. In Secton 3.1 we consder an nvestng strategy usng supervsed neural network wth back propagaton algorthm. The strategy s closely related to and reflects exstng researches on stock prce predcton wth neural networks. In Secton 3.2 we consder Markovan proportonal bettng strateges, whch are much smpler than the strateges based on neural networks. In Secton 4 we evaluate performances of these strateges by Monte Carlo smulaton. In Secton 5 we apply these strateges to stock prce data from Tokyo Stock Exchange. Fnally we gve some concludng remarks n Secton 6. 2

5 2 Sequental optmzng strategy wth neural networks Here we ntroduce the bounded forecastng game of Shafer and Vovk [11] n Secton 2.1 and network models we use n Secton 2.2. In Secton 2.3 we specfy the nvestng rato by an unsupervsed neural network and we propose sequental optmzaton of parameter values of the network. 2.1 Bounded forecastng game We present the bounded forecastng game formulated by Shafer and Vovk n 2001 [11]. In the bounded forecastng game, Investor s captal at the end of round n s wrtten as K n (n = 1, 2,...) and ntal captal K 0 s set to be 1. In each round Investor frst announces the amount of money M n he bets ( M n < K n 1 ) and then Market announces her move x n [ 1, 1]. x n represents the change of the prce of a unt fnancal asset n round n. The bounded forecastng game can be consdered as an extenson of the classcal con-tossng game snce the bounded forecastng game results n the classcal con-tossng game f x n { 1, 1}. Wth K n, M n and x n, Investor s captal after round n s wrtten as K n = K n 1 + M n x n. The protocol of the bounded forecastng game s wrtten as follows. Protocol: K 0 =1. FOR n = 1, 2,... : Investor announces M n R. Market announces x n [ 1, 1]. K n = K n 1 + M n x n END FOR We can rewrte Investor s captal as K n = K n 1 (1 + α n x n ), where α n = M n /K n 1 s the rato of Investor s nvestment M n to hs captal K n 1 after round n 1. We call α n the nvestng rato at round n. We restrct α n as 1 < α n < 1 n order to prevent Investor becomng bankrupt. Furthermore we can wrte K n as K n = K n 1 (1 + α n x n ) = = Π n (1 + α k x k ). Takng the logarthm of K n we have log K n = n log(1 + α k x k ). (1) 3

6 The behavor of Investor s captal n (1) depends on the choce of α k. Specfyng a functonal form of α k s regarded as an nvestng strategy. For example, settng α k ɛ to be a constant ɛ for all k s called the ɛ-strategy whch s presented n [11]. In ths paper we consder varous ways to determne α k n terms of past values x k 1, x k 2,..., of x and and seek better α k n tryng to maxmze the future captal K n, n > k. Let u k 1 = (x k 1,..., x k L ) denote past L values of x and let α k depend on u k 1 and a parameter ω: α k = f(u k 1, ω). Then ω k 1 = argmax k 1 log(1 + f(u t 1, ω)x t ) t=1 s the best parameter value untl the prevous round. In our sequental optmzng nvestng strategy, we use ω n 1 to determne the nvestment M n at round n: M n = K n 1 f(u n 1, ω n 1). For the functon f we employ neural network models for ther flexblty, whch we descrbe n the next secton. 2.2 Desgn of the network We construct a three-layered neural network shown n Fgure 1. The nput layer has L neurons and they just dstrbute the nput u j (j = 1,..., L) to every neuron n the hdden layer. Also the hdden layer has M neurons and we wrte the nput to each neurons as I 2 whch s a weghted sum of u j s. As seen from Fgure 1, I 2 s obtaned as I 2 = L j=1 ω 1,2 j u j, where ω 1,2 j s called the weght representng the synaptc connectvty between the jth neuron n the nput layer and the th neuron n the hdden layer. Then the output of the th neuron n the hdden layer s descrbed as y 2 = tanh(i 2 ). As for actvaton functon we employ hyperbolc tangent functon. In a smlar way, the nput to the neuron n the output layer, whch we wrte I 3, s obtaned as I 3 = M =1 4 ω 2,3 y 2,

7 Fgure 1: Three-layered network where ω 1 s the weght between the th neuron n the hdden layer and the neuron n the output layer. Fnally, we have y 3 = tanh(i 3 ), whch s the output of the network. In the followng argument we use y 3 as an nvestment strategy. Thus we can wrte where α k = y 3 = f(u k 1, ω), ω = (ω 1,2, ω 2,3 ) = ( (ω 1,2 j ) =1,...,M, j=1,...,l, (ω 2,3 ) =1,...,M ). Investor s captal s wrtten as K n = K n 1 (1 + f(u n 1, ω)x n ). We need to specfy the number of nputs L and the number of neurons M n the hdden layer. It s dffcult to specfy them n advance. We compare varous choces of L and M n Secton 4 and Secton 5. Also n u n 1 we can nclude any nput whch s avalable before the start of round n, such as movng averages of past prces, seasonal ndcators or past values of other economc tme seres data. We gve further dscusson on the choce of u n 1 n Secton 6. 5

8 2.3 Sequental optmzng strategy wth neural networks In ths secton we propose a strategy whch we call Sequental Optmzng Strategy wth Neural Networks (SOSNN). We frst calculate ω = ω n 1 that maxmzes n 1 φ = log(1 + f(u k 1, ω)x k ). (2) Ths s the best parameter values untl the prevous round. If Investor uses α n = f(u n 1, ω n 1) as the nvestng rato, Investor s captal after round n s wrtten as K n = K n 1 (1 + f(u n 1, ω n 1)x n ). For maxmzaton of (2), we employ the gradent descent method. Wth ths method, the weght updatng algorthm of ω 2,3 wth the parameter β (called the learnng constant) s wrtten as where ω 2,3 = ω 2,3 + ω 2,3 = ω 2,3 + β φ ω 2,3, φ ω 2,3 = φ f = = n 1 n 1 f ω 2,3 = n 1 φ f f k I 3 k I 3 ω 2,3 x k 1 + f(u k 1, ω)x k (1 tanh 2 ( k I 3 )) k y 2 k δ k 1 y 2, and the left superscrpt k to I 3, y 2 ndexes the round. Thus we obtan ω 2,3 = β φ ω 2,3 = β n 1 k δ 1 k y 2. Smlarly, the weght updatng algorthm of ω 1,2 j s expressed as ω 1,2 j = ω 1,2 j + ω 1,2 j = ω 1,2 j 6 + β φ ω 1,2 j,

9 where φ ω 1,2 j Thus we obtan = = = n 1 n 1 n 1 φ k I 2 k I 2 ω 1,2 j = n 1 φ f f k I 3 k I 3 k y 2 k δ 1 ω 2,3 (1 tanh 2 ( k I 2 )) (u k 1 ) j k δ 2 (u k 1 ) j. n 1 k y 2 k I 2 ω 1,2 j = β φ ω 1,2 = β k δ 2 (u k 1 ) j. j Here we summarze the algorthm of SOSNN at round n. k I 2 ω 1,2 j 1. Gven the nput vector u k 1 = (x k 1,..., x k L ) (k = 1,..., n 1) and L the value of ω n 1, we frst evaluate k I 2 = ω 1,2 j (u k 1) j and then j=1 k y 2 = tanh( k I 2 ). Also we set the learnng constant β. 2. We calculate k I 3 = M =1 ω 2,3 k y 2 and then k y 3 = tanh( k I 3 ) wth k I 2 and k y 2 of the prevous step. Then we update weght ω wth the weght updatng formula ω 2,3 = ω 2,3 +β k δ k 1 y 2 and ω 1,2 j = ω 1,2 j +β k δ 2 (u k 1 ) j n 1 n Go back to step 1 replacng the weght ω n 1 wth updated values. After suffcent tmes of teraton, φ n (2) converges to a local maxmum wth respect to ω 1,2 j and ω 2,3 and we set ω 1,2 j = ω 1,2 j and ω 2,3 = ω 2,3, whch are elements of ωn 1. Then we evaluate Investor s captal after round n as K n = K n 1 (1 + f(u n 1, ωn 1)x n ). 3 Alternatve nvestment strateges Here we present some strateges that are desgned to be compared wth SOSNN. In Secton 3.1 we present a strategy wth back-propagatng neural network. The advantage of back-propagatng neural network s ts predctve ablty due to learnng as prevous researches show. In Secton 3.2 we show some sequental optmzng strateges that use rather smple functon for f than SOSNN does. 7

10 3.1 Optmzng strategy wth back propagaton In ths secton we consder a supervsed neural network and ts optmzaton by back propagaton. We call the strategy NNBP. It decdes the bettng rato by predctng actual up-and-downs of stock prces and can be regarded as ncorporatng exstng researches on stock prce predcton. Thus t s sutable as an alternatve to SOSNN. For supervsed network, we tran the network wth the data from a tranng perod, obtan the best value of the parameters for the tranng perod and then use t for the nvestng perod. These two perods are dstnct. For the tranng perod we need to specfy the desred output (target) T k of the network for each day k. We propose to specfy the target by the drecton of Market s current prce movement x k. Thus we set +1 x k > 0 T k = 0 x k = 0. 1 x k < 0 Note that ths T k s the best nvestng rato f Investor could use the current movement x k of Market for hs nvestment. Therefore t s natural to use T k as the target value for nvestng strateges. We keep on updatng ω k by cyclng through the nput-output pars of the days of the tranng perod and fnally obtan ω after suffcent tmes of teraton. Throughout the nvestng perod we use ω and Investor s captal after round n n the nvestng perod s expressed as K n = K n 1 (1 + f(u n 1, ω )x n ). Back propagaton s an algorthm whch updates weghts k ω 1,2 j and k ω 2,3 so that the error functon E k = 1 2 (T k k y 3 ) 2 decreases, where T k s the desred output of the network and k y 3 s the actual output of the network. The weght k ω 2,3 of day k s renewed to the weght of day k + 1 as k+1 ω 2,3 where k+1 ω 2,3 = k ω 2,3 + k ω 2,3 = k ω 2,3 = k ω 2,3 β k ɛ 1 k y 2, β E k k ω 2,3 = k ω 2,3 β E k k I 3 k I 3 k ω 2,3 k ɛ 1 = E k k I 3 = E k k y 3 k y 3 k I 3 = (T k k y 3 )(1 tanh 2 ( k I 3 )). 8

11 Also weght k ω 1,2 j s renewed as k+1 ω 1,2 j = k ω 1,2 j + k ω 1,2 j = k ω 1,2 j = k ω 1,2 j β k ɛ 2 (ũ k 1 ) j, β E k k ω 1,2 j = k ω 1,2 j β E k k I 3 k y 2 k I 3 k y 2 k I 2 where k ɛ 2 = E k k I 3 k y 2 = k ɛ 1 ω 2,3 k I 3 k y 2 k I 2 (1 tanh 2 ( k I 2 )). At the end of each step we calculate the tranng error defned as k I 2 k ω 1,2 j tranng error = 1 2m m (T k k y 3 ) 2 = 1 m m E k, (3) where m s the length of the tranng perod. We end the teraton when the the tranng error becomes smaller than the threshold µ, whch s set suffcently small. Here let us summarze the algorthm of NNBP n the tranng perod. 1. We set k = Gven the nput vector u k 1 = (x k 1,..., x k L ) and the value of ω k, L we frst evaluate k I 2 = k ω 1,2 j (u k 1) j and then k y 2 = tanh( k I 2 ). Also j=1 we set the learnng constant β. 3. We calculate k I 3 = M =1 k ω 2,3 k y 2 and then k y 3 = tanh( k I 3 ) wth k I 2 and k y 2 of the prevous step. Then we update weght ω k wth the weght updatng formula k+1 ω 2,3 = k ω 2,3 β k ɛ 1 k y 2 and k+1 ω 1,2 j = k ω 1,2 j β k ɛ 2 (u k 1 ) j. 4. Go back to step 2 settng k + 1 k and ω k+1 ω k whle 1 k m. When k = m we set k = 1 and ω 1 ω m+1 and contnue the algorthm untl the tranng error becomes less than µ. 3.2 Markovan proportonal bettng strateges In ths secton we present some sequental optmzng strateges that are rather smple compared to strateges wth neural network n Secton 2 and Secton 3.1. The strateges of ths secton are generalzatons of Markovan strategy n [12] for con-tossng games to bounded forecastng games. We 9

12 present these smple strateges for comparson wth SOSNN and observe how complexty n functon f ncreases or decreases Investor s captal processes n numercal examples n later sectons. Consder maxmzng the logarthm of Investor s captal n (1): log K n = n log(1 + α k x k ). We frst consder the followng smple strategy of [9] n whch we use α n = α n 1, where α n 1 = argmax Π n 1 (1 + αx k). In ths paper we denote ths strategy by MKV0. As a generalzaton of MKV0 consder usng dfferent nvestng ratos dependng on whether the prce went up or down on the prevous day. Let α k = α + k when x k 1 was postve and α k = α k when t was negatve. We denote ths strategy by MKV1. In the bettng on the nth day we use α n + = α n 1 + and αn = α n 1, where u k 1 = (x k 1 ) and (α n 1 +, αn 1 ) = argmax Π n 1 (1 + f(u k 1, α +, α )x k ), f(u k 1, α +, α ) = α + I {xk 1 0} + α I {xk 1 <0}. Here I { } denotes the ndcator functon of the event n { }. The captal process of MKV1 s wrtten n the form of (1) as log K n = n log(1 + f(u k 1, α + k 1, α k 1 )x k ). We can further generalze ths strategy consderng prce movements of past two days. Let u k 1 = (x k 1, x k 2 ) and let f(u k 1, α ++, α +, α +, α ) = α ++ I {xk 2 0, x k 1 0} + α + I {xk 2 0, x k 1 <0} + α + I {xk 2 <0, x k 1 0} + α I {xk 2 <,x k 1 <0}. We denote ths strategy by MKV2. We wll compare performances of the above Markovan proportonal bettng strateges wth strateges based on neural networks n the followng sectons. 10

13 4 Smulaton wth lnear models In ths secton we gve some smulaton results for strateges shown n Secton 2 and Secton 3. We use two lnear tme seres models to confrm the behavor of presented strateges. Lnear tme seres data are generated from the Box-Jenkns famly [2], autoregressve model of order 1 (AR(1)) and autoregressve movng average model of order 2 and 1 (ARMA(2,1)) havng the same parameter values as n [15]. AR(1) data are generated as and ARMA(2,1) data are generated as x n = 0.6x n 1 + ɛ n (4) x n = 0.6x n x n 2 + ɛ n 0.5ɛ n 1, (5) where we set ɛ n N(0, 1). After the seres s generated, we dvde each value by the maxmum absolute value to normalze the data to the admssble range [ 1, 1]. Here we dscuss some detals on calculaton of each strategy. Frst we set the ntal values of elements of ω as random numbers n [ 0.1, 0.1]. In SOSNN, we use the frst 20 values of x n as ntal values and the teraton process n gradent descent method s proceeded untl ω 1,2 j < 10 4 and ω 2,3 < 10 4 wth the upper bound of 10 4 steps. As for the learnng constant β, we use learnng-rate annealng schedules whch appear n Secton 3.13 of [7]. Wth annealng schedule called the search-then-converge schedule [4] we put β at the nth step of teraton as β(n) = β (n/τ), where β 0 and τ are constants and we set β 0 = 1.0 and τ = 5.0. In NNBP, we tran the network wth fve dfferent tranng sets of 300 observatons generated by (4) and (5). We contnue cyclng through the tranng set untl the tranng error becomes less than µ and we set µ = 10 2 wth the upper bound of steps. Also we check the ft of the network to the data by means of the tranng error for some dfferent values of β, L and M. In Markovan strateges, we agan use the frst 20 values of x n as ntal values. We also adjust the data so that the bettng s conducted on the same data regardless of L n SOSNN and NNBP or dfferent number of nputs among Markovan strateges. In Table 1 we summarze the results of SOSNN, NNBP, MKV0, MKV1 and MKV2 under AR(1) and ARMA(2,1). The values presented are averages 11

14 of results for fve dfferent smulaton runs of (4) and (5). As for SOSNN, we smulate ffty cases (combnatons of L = 1,..., 5 and M = 1,..., 10), but only report the cases of L = 1, 2, 3 and some choces of M because the purpose of the smulaton s to test whether L = 1 works better than other choces of L under AR(1) and L = 2 works better under ARMA(2,1). For NNBP we only report the result for one case snce fttng the parameters to the data s qute a dffcult task due to the characterstc of desred output (target). Also once we obtan the value of β, L and M wth tranng error less than the threshold µ = 10 2, we fnd that the network has successfully learned the nput-output relatonshp and we do not test other choces of the above parameters. (See Appendx for more detal.) We set β = 0.07, L = 12 and M = 30 n smulaton wth AR(1) model and β = 0.08, L = 15 and M = 40 n smulaton wth ARMA(2,1) model. Investor s captal process for each choce of L and M n SOSNN, NNBP and each Markovan strategy s shown n three rows, correspondng to rounds 100, 200, 300 of the bettng (wthout the ntal 20 rounds n SOSNN and Markovan strateges). The fourth row of each result for NNBP shows the tranng error after learnng n the tranng perod. The best value among the choces of L and M n SOSNN or among each Markovan strategy s wrtten n bold and marked wth an astersk and the second best value s also wrtten n bold and marked wth two astersks. Also calculaton results wrtten wth are cases n that smulaton dd not gve proper values for some reasons. Notce that SOSNN whose bettng rato f s specfed by a complex functon gves better performance than rather smple Markovan strateges both under AR(1) and ARMA(2,1). Also the result that NNBP gves captal processes whch are compettve wth other strateges shows that the network has successfully learned the nput-output relatonshp n the tranng perod. 5 Comparson of performances wth some stock prce data In ths secton we present numercal examples calculated wth the stock prce data of three Japanese companes SONY, Nomura Holdngs and NTT lsted on the frst secton of the Tokyo Stock Exchange for comparng bettng strateges presented n Secton 2 and Secton 3. The nvestng perod (wthout days used for ntal values) s 300 days from March 1st n 2007 to June 19th n 2008 for all strateges and the tranng perod n NNBP s 300 days from December 1st n 2005 to February 20th n We use a shorter tranng perod than those n prevous researches, because longer perods resulted n 12

15 Table 1: Log captal processes of SOSNN, NNBP, MKV0, MKV1, MKV2 under AR(1) and ARMA(2,1) AR(1) model SOSNN L\M NNBP MKV0 MKV1 MKV ( ) ARMA(2,1) model SOSNN L\M NNBP MKV0 MKV1 MKV ( ) poor fttng. The data for 300 days from December 1st n 2005 to February 20th n 2007 s used for nput normalzaton of x n to [ 1, 1], whch s conducted accordng to the method shown n [1] and the procedure s as follows. For the data of daly closng prces n the above perod, we frst obtan the maxmum value of absolute daly movements and then dvde daly movements n the nvestng perod by that maxmum value. In case x n 1.0 or x n 1.0 we put x n = 1.0 or x n = 1.0. Thus we obtan x n n [ 1, 1] and we use them for nputs of the neural network. We tred perods of varous lengths for normalzaton and decded to choose a relatvely short perod to avod Investor s captal processes stayng almost constant. Also n NNBP we used β = 0.07, L = 12 and M = 90 for SONY, β = 0.07, L = 15 and M = 100 for 13

16 Nomura and β = 0.03, L = 15 and M = 120 for NTT wth upper bound of 10 5 teraton steps. Other detals of calculaton are the same as n Secton 4. We report the results n Table 2. In Fgure 2 we show the movements of closng prces of each company durng the nvestng perod. In Fgures 3-5 we show the log captal processes of the results shown n Table 2 to compare the performance of each strategy. Fgure 3 s for SONY, Fgure 4 s for Nomura Holdngs and Fgure 5 s for NTT. For SOSNN we plotted the result of L and M that gave the best performance at n = 300 (the bottom row of the three rows) n Table Sony Nomura NTT 2 SOSNN NNBP MKV Closng Prce Log Captal Days Rounds Fgure 2: Closng prces Fgure 3: SONY 2 SOSNN NNBP 1 SOSNN NNBP MKV0 MKV Log Captal -1-2 Log Captal Rounds Rounds Fgure 4: Nomura Fgure 5: NTT As we see from above fgures, NNBP whch shows compettve performance for two lnear models n Secton 4 gves the worst result. Thus t s obvous that the network has faled to capture trend n the bettng perod even f t fts n the tranng perod. Also the results are favorable to SOSNN f we adopt approprate numbers for L and M. 14

17 Table 2: Log captal process of SOSNN, NNBP, MKV0, MKV1 and MKV2 for TSE stocks SONY SOSNN L\M NNBP MKV0 MKV1 MKV ( ) Nomura SOSNN L\M NNBP MKV0 MKV1 MKV ( ) NTT SOSNN L\M NNBP MKV0 MKV1 MKV ( ) 15

18 6 Concludng remarks We proposed nvestng strateges based on neural networks whch drectly consder Investor s captal process and are easy to mplement n practcal applcatons. We also presented numercal examples for smulated and actual stock prce data to show advantages of our method. In ths paper we only adopted normalzed values of past Market s movements for the nput u n 1 whle we can use any data avalable before the start of round n as a part of the nput as we mentoned n Secton 2.2. Let us summarze other possbltes consdered n exstng researches on fnancal predcton wth neural networks. The smplest choce s to use raw data wthout any normalzaton as n [6], n whch they analyze tme seres of Athens Stock ndex to predct future daly ndex. In [5] they adopt prce of FAZ-ndex (one of the German equvalents of the Amercan Dow-Jones- Index), movng averages for 5, 10 and 90 days, bond market ndex, order ndex, US-Dollar and 10 successve FAZ-ndex prces as nputs to predct the weekly closng prce of the FAZ-ndex. Also n [8] they use 12 techncal ndcators to predct the S&P 500 stock ndex one month n the future. From these researches we see that for longer predcton terms (such as monthly or yearly), longer movng averages or seasonal ndexes become more effectve. Thus those long term ndcators may not have much effect n daly prce predcton whch we presented n ths paper. On the other hand, adoptng data whch seems to have a strong correlaton wth closng prces of Tokyo Stock Exchange such as closng prces of New York Stock Exchange of the prevous day may ncrease Investor s captal processes presented n ths paper. Snce there are numercal dffcultes n optmzng neural networks, t s better to use small number of effectve nputs for a good performance. Another mportant generalzaton of the method of ths paper s to consder portfolo optmzaton. We can easly extend the method n ths paper to the bettng on multple assets. Let the output layer of the network have P neurons as shown n Fgure 7 and the output of each neuron s expressed as yh 3, h = 1,..., P. Then we obtan a vector y3 = (y1, 3..., yp 3 ) of outputs. The number of neurons P refers to the number of dfferent stocks Investor nvests. Investor s captal after round n s wrtten as K n = K n 1 (1 + P f h (u n 1, ω h )x n,h ), h=1 where ω h = ( (ω 1,2 j ) =1,...,M, j=1,...,l, (ω 2,3 h ) =1,...,M). 16

19 Fgure 6: Three-layered network for portfolo cases Thus also n portfolo cases we see that our method s easy to mplement and we can evaluate Investor s captal process n practcal applcatons. Appendx Here we dscuss tranng error n the tranng perod of NNBP. In ths paper we set the threshold µ for endng the teraton to 10 2, whle the value commonly adopted n many prevous researches s smaller, for nstance, µ = We gve some detals on our choce of µ. Let us examne the case of Nomura Holdngs n Secton 5. In Fgure 7 we show the tranng error after each step of teraton n the tranng perod calculated wth (3). Whle the plotted curve has a typcal shape as those of prevous researches, t s unlkely that the tranng error becomes less than Also n Fgure 8 we plot E k = 1 2 (T k k y 3 ) 2 for each k calculated wth parameter values ω after learnng. We observe that the network fals to ft for some ponts (actually 9 days out of 300 days) but perfectly fts for all other days. It can be nterpreted that the network gnores some outlers and adjust to capture the trend of the whole data. 17

20 Tranng Error Tranng Error Epochs Perods Fgure 7: tranng error (epochs) Fgure 8: tranng error (perods) References [1] E. M. Azoff. Neural Network Tme Seres Forecastng of Fnancal Markets. Wley, Chchester, [2] G. P. E. Box and G. M. Jenkns. Tme Seres: Analyss Forecastng and Control. Holden-Day, San Francsco, [3] T. M. Cover. Unversal portfolos. Mathematcal Fnance, 1, No.1, 1 29, [4] C. Darken, J. Chang and J. Moody. Learnng rate schedules for faster stochastc gradent search. IEEE Second Workshop on Neural Networks for Sgnal Processng, 3 12, [5] B. Fresleben. Stock market predcton wth backpropagaton networks. Industral and Engneerng Applcatons of Artfcal Intellgence and Expert System 5th Internatonal Conference, , [6] M. Hanas, P. Curts and J. Thalassnos. Predcton wth neural networks: The Athens stock exchange prce ndcator. European Journal of Economcs, Fnance and Admnstratve Scences, 9, 21 27, [7] S. S. Haykn. Neural Networks and Learnng Machnes. 3rd ed., Prentce Hall, New York, [8] N. L. D. Khoa, K. Sakakbara and I. Nshkawa. Stock prce forecastng usng back propagaton neural networks wth tme and proft based adjusted weght factors. SICE-ICASE Internatonal Jont Conference, ,

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