Optimization in portfolio using maximum downside deviation stochastic programming model

Transcription

1 Avalable onlne at Advances n Appled Scence Research, 2010, 1 (1): 1-8 Optmzaton n portfolo usng maxmum downsde devaton stochastc programmng model Khlpah Ibrahm, Anton Abdulbasah Kaml * and Adl Mustafa School of Dstance Educaton, Unverst Sans Malaysa, 11800, Penang, Malaysa ABSTRACT Portfolo optmzaton has been one of the mportant research felds n fnancal decson makng. The most mportant character wthn ths optmzaton problem s the uncertanty of the future returns. To handle such problems, we utlze probablstc methods alongsde wth optmzaton technques. We develop sngle stage and two stage stochastc programmng wth recourse wth the objectve s to mnmze the maxmum downsde sem devaton. We use the socalled Here-and-Now approach where the decson-maker makes decson now before observng the actual outcome for the stochastc parameter. We compare the optmal portfolos between the sngle stage and two stage models wth the ncorporaton of the devaton measure. The models are appled to the optmal selecton of stocks lsted n Bursa Malaysa and the return of the optmal portfolo s compared between the two stochastc models. The results show that the two stage model outperforms the sngle stage model n the optmal and n-sample analyss Keywords: Portfolo optmzaton, Maxmum Sem devaton Measure, Downsde rsk, Stochastc Lnear Programmng. INTRODUCTION Portfolo optmzaton has been one of the mportant research felds n fnancal decson makng. The most mportant character wthn ths optmzaton problem s the uncertanty of the future returns. To handle such problems, we utlze probablstc methods alongsde wth optmzaton technques. Stochastc programmng s our approach to deal wth uncertanty. Stochastc Programmng s a branch of mathematcal programmng where the parameters are random. The objectve of stochastc programmng s to fnd the optmum soluton to problems wth uncertan data. Ths approach can deal the management of portfolo rsk and the dentfcaton of optmal portfolo smultaneously. Stochastc programmng models explctly consder uncertanty n 1

2 some of the model parameters, and provde optmal decsons whch are hedged aganst such uncertanty In the determnstc framework, a typcal mathematcal programmng problem could be stated as mn f(x) x (1.1) s.t g (x) 0, = 1,...m, n n where x s from R or Z. Uncertanty, usually descrbed by a random element ξ ( ), where s a random outcome from a space Ω, leads to stuaton where nstead of just f(x) and g (x) one has to deal wth f(x, ξ()) and g (x, ξ()). Tradtonally, the probablty dstrbuton of ξ s assumed to known or can be estmated and s unaffected by the decson vector x. The problem becomes decson makng under uncertanty where decson vector x has to be chosen before the outcome from the dstrbuton of ξ( ) can be observed. Markowtz used the concept of rsk nto the problem and ntroduced mean-rsk approach that dentfes rsk wth the volatlty (varance) of the random objectve [5, 6]. Snce 1952, meanrsk optmzaton paradgm receved extensve development both theoretcally and computatonally. Konno and Yamazak proposed mean absolute devaton from the mean as the rsk measure to estmate the nonlnear varance-covarance of the stocks n the mean-varance model [4]. It transforms the portfolo selecton problem from a quadratc programmng nto a lnear programmng problem. At the same tme, the popularty of downsde rsk among nvestors s growng and mean-return-downsde rsk portfolo selecton models seem to oppress the famlar mean-varance approach. The reason for the success of the former models s that they separate return fluctuatons nto downsde rsk and upsde potental. Ths s especally relevant for asymmetrcal return dstrbutons, for whch mean-varance model punsh the upsde potental n the same fashon as the downsde rsk. Ths led Markowtz to propose downsde rsk measures such as (downsde) sem varance to replace varance as the rsk measure [6]. Consequently, one observes growng popularty of downsde rsk models for portfolo selecton [7]. Young [8] ntroduced another lnear programmng model whch maxmze the mnmum return or mnmze the maxmum loss (mnmax) over tme perods and appled to the stock ndces from eght countres, from January 1991 untl December The analyss showed that the model performs smlarly wth the classcal mean-varance model. In addton, Young argues that, when data s log-normally dstrbuted or skewed, the mnmax formulaton mght be more approprate method, compared to the classcal mean-varance formulaton, whch s optmal for normally dstrbuted data. Dantzg [5] and ndependently Beale [1] suggested an approach to stochastc programmng and termed as stochastc programmng wth recourse. Recourse s the ablty to take correctve acton after a random event has taken place. The man nnovaton s to amend the problem to allow the decson maker the opportunty to make correctve actons after a random event has taken place. In the frst stage a decson maker a here and now decson. In the second stage the decson maker sees a realzaton of the stochastc elements of the problem but he s allowed to make 2

3 further decsons to avod the constrants of the problem becomng nfeasble. In ths paper we develop sngle stage and two stage stochastc programmng wth recourse for portfolo selecton problem and the objectve s to mnmze the maxmum downsde devaton measure of portfolo returns from the expected return. We use the so-called Here-and-Now approach where the decson-maker makes decson now before observng the actual outcome for the stochastc parameter. The man objectve of ths study s to solve portfolo optmzaton problem usng two dfferent stochastc programmng models. We apply these models to the optmal selecton of stocks lsted n Bursa Malaysa and compare the optmal portfolos between the sngle stage and two stage models. The remander of the paper s organzed as follows. In the next secton we dscuss the maxmum downsde sem devaton measure and formulate the equvalent sngle stage stochastc lnear programmng model for portfolo selecton problem. Then we extend the sngle stage model to two stage stochastc programmng wth recourse model. Secton 3 devoted to the expermental analyss on real-lfe data from Bursa Malaysa. Fnally, some concludng remarks are gven n secton 4. MATERIALS AND METHODS Consder a set of securtes I = {: = 1,2,..., n} for an nvestment. At the begnnng of the holdng perod the nvestor wshes to apporton hs budget to these assets by decdng on a specfc T allocaton x = (x1,x2,...,xn ) such that x 0 (.e., short sales are not allowed) and x = 1 (budget constrant). At the end of a certan holdng perod the assets generate returns, ~ r =(r ~ T 1,r ~ 2,...,r ~ n ). At the begnnng of the holdng perod the returns are random. Suppose that r~ are represented by a fnte set of dscrete scenaros Ω = { : = 1,2,...,S }, whereby the returns under a partcular scenaro p >, probablty 0 R Ω take the values r r,r,...,r ) = ( 1 2 n T I wth assocated p = 1. The portfolo return under a partcular realzaton of r s = R( x, r ) and the expected portfolo return s R( x, r ) = p R( x, r ). Let M[ R( x be the mnmum of the portfolo return. The maxmum (downsde) sem devaton measure s defned as MM [ R( x = [ E[ R( x, r - Mn [R( x, r (2.1) MM [ R( x s a very pessmstc rsk measure related to the worst case analyss. It does not take nto account the dstrbuton of outcomes other than the worst one., let η = max [R( x, r ) - R( x, r 3

4 Subject to Then, we have Subject to η max [R( x ) - R( x, r for MM [ R( x = η (2.2) η max [R( x ) - R( x, r for 2.1 Sngle Stage Stochastc Lnear Programmng Portfolo Optmzaton Model wth MM devaton measure Portfolo optmzaton problem where (2.1) s mnmzed constranng the expected portfolo return at the end of nvestment perod can be formulated as a sngle stage stochastc lnear programmng model, S_MM below: Mnmze η (2.3) Subject to : R ( x, r ) α R( x, r ) - R( x, r ) η x = 1 I L x U I 2.2 Two Stage Stochastc Lnear Programmng Model wth recourse formulaton for S_MM We now ntroduce dynamc model where future changesecourse, to the ntal compostons are allowed. Assumng the nvestor can make correctve acton after the realzaton of random values by changng the composton of the optmal portfolo, we formulate the sngle perod stochastc lnear programmng model of S_MM as a two-stage stochastc programmng problem wth recourse. Consder the case when the nvestor s nterested n a frst stage decson x that hedges aganst the rsk of the second-stage acton. At the begnnng of the nvestment perod, the nvestor selects the ntal composton of the portfolo, x assumng there s a known dstrbuton of future returns. At the end of the plannng horzon, once a partcular scenaro of return s realzed, the nvestor rebalances the composton by ether purchasng or sellng the selected stocks. Let a set of second stage varables, y to represent the composton of stock after, rebalancng s done,.e., y, = x + P, or y, = x - Q, where P, and Q, are the quantty purchased and sold respectvely. The maxmum downsde devaton of portfolo returns from the expected return n terms of the second stage varables y can be formulated as follows:. MM [ R( y = max[ R( y Ξ ) R( y For every scenaro, let the auxlary varable η = max [ R( y, ) R( y (2.4) 4

5 Then, we have subject to subject to η max [R( y MM [ R( x = η η ) - R( y for max [R( y ) - R( y for (2.5) We formulate the two stage stochastc lnear programmng model, 2S_MM, for portfolo optmzaton problem that mnmzes second stage MM and constranng the expected portfolo return as follows: Subject to Mnmze η (2.6) I I x y = 1 = 1 R( x ) + R( y ) α L x U L y U R( y ) η 3. NUMERICAL ANALYSIS I I, We tested our models on ten common stocks selected at random from a set of stocks that were already lsted on the man board of Bursa Malaysa on December 1989 and stll n the lst on May The closng prces were obtaned from Investors Dgest. We use emprcal dstrbutons computed from past returns as equprobable scenaros. Observatons of returns over N overlappng perods of length t are consdered as the N possble outcomes (or scenaros) S of the future returns and a probablty of obtan 100 scenaros of the overlappng perods of length 1 month,.e. S 1 s assgned to each of them. For each stock, we N s N S. To evaluate the performance of the two models, we examned the portfolo returns resultng from applyng the two stochastc optmzaton models. We make comparson between S_MM and 2S_MM models by analyzng the optmal portfolo returns, n-sample portfolo returns and outof-sample portfolo returns over 60-month perod from to 06/1998 to 05/2004. At each month, we use the hstorcal data from the prevous 100 monthly observatons as scenaros and solve the resultng optmzaton models usng the mnmum monthly requred return α equals to one. 3.1 Comparson of Optmal Portfolo returns between S_MM and 2S_MM Fgures 1 presents the graphs of optmal portfolo returns resultng from solvng the two models; S_MM and 2S_MM (see appendx). The optmal portfolo returns of the two models exhbt the same pattern. There s a decreasng trend n the optmal returns n both models. However, n 5

6 fgure 1, t can be seen that the optmal portfolo returns from 2S_MM are hgher than the optmal portfolo returns from S_MM n all testng perods. Ths shows that an nvestor can make a better decson regardng the selecton of stocks n a portfolo when he takes nto consderaton both makng decson facng the uncertanty and the ablty of makng correcton actons when the uncertan returns are realzed compared to consders only makng decson facng the uncertanty alone Optmal Portfolo Return : S_MM and 2S_MM Portfolo Return n S_MM 2S_MM Jun-99 Dec-99 Jun-00 Dec-00 Jun-01 Dec-01 Jun-02 Dec-02 Jun-03 Dec-03 Tme Perod Fgure 1: Comparson of Optmal portfolo Returns S_MM and 2S_MM models 3.2 Comparson of Average In-Sample Portfolo returns between S_MM and 2S_MM We use average realzed returns to comparson In-Sample portfolo returns between S_MM model and 2S_MM model and the results are presented n Fgure 2. There s an ncreasng trend n the months from December 1999 untl Aprl 2000, then decreasng trend untl June Startng from June 2001 untl May 2004, both averages show an ncreasng trend. Average In-Sample Portfolo Return : S_MM and 2S_MM 1.15 S_MM 2S_MM Average Portfolo Return Jun-99 Dec-99 Jun-00 Dec-00 Jun-01 Dec-01 Jun-02 Dec-02 Jun-03 Dec-03 Tme Perod Fgure 2: Comparson of Average In-Sample Portfolo Return between S_MM and 2S_MM models 6

7 The average n-sample portfolo returns of 2S_MM are hgher than the average n-sample portfolo returns n all testng perods. 3.3 Comparson of Out-Of-Sample Portfolo returns between S_MM and 2S_MM models The comparson of out-of-sample portfolo returns between S_MM and 2S_MM s also done usng the average return. The results of Out-Of-Sample analyss are presented n Fgure 3. Throughout the testng perods, the average returns from the two models show smlar patterns. There s an ncreasng trend n the months from December 1999 untl December 2000, then decreasng trend untl June Startng from June 2001, both averages show an ncreasng trend. The average out-of-sample of the two-stage model, 2S_MM s hgher than those of sngle stage model, S_MM. Certanly, the models have been appled drectly to the orgnal hstorcal data treated as future returns scenaros thus loosng the trend nformaton. Possble applcaton of some forecastng procedures pror to the portfolo optmzaton models, we consder, seems to be an nterestng drecton for future research. For references on scenaros generaton see [2] Average Out-of-Sample Portfolo Returns: S_MM and 2S_MM Average Portfolo Return S_MM 2S_MM 0.98 Jun-99 Dec-99 Jun-00 Dec-00 Jun-01 Dec-01 Jun-02 Dec-02 Jun-03 Dec-03 Tme Perod Fgure 3: Comparson of Out-Of-Sample Analyss between sngle stage S_MM and two stage 2S_MM models CONCLUSION In ths paper, a portfolo selecton of stocks wth maxmum downsde sem devaton measure s modeled as a sngle stage and a two stage stochastc programmng models. Sngle stage model ncorporates uncertanty n the model and n the two stage model the uncertanty s ncorporated n the models and at the same consders rebalancng the portfolo composton at the end of nvestment perod. The comparson of the optmal portfolo returns, the n-sample portfolo returns and the out-of-sample portfolo returns shows that the performance of the two stage model s better than that of the sngle stage model. Here, we use hstorcal data as scenaros of future returns. In our future research we wll generate scenaros of future asset returns usng approprate scenaro generaton method before applyng to our developed models. 7

8 Acknowledgement The work funded by the FRGS (Fundamental Research Grant Scheme) of Mnstry for Hgher Educaton of Malaysa, Grant 203/PJJAUH/ Unverst Sans Malaysa. REFERENCES [1] Beale, E.M.L, Journal of the Royal Statstcal Socety. Seres B 17, 1955, [2] Carno, D.R., Myers, D.H. and Zemba, W.T, Operatons Research. 46, 1998, [3] Dantzg, G.B, Management Scence. 1, 1955, A16. [4] Konno, H. and Yamazak, H, Management Scence. 7, 1991, [5] Markowtz, H.M, Journal of Fnance. 8, 1952, [6] Markowtz,H.M, Portfolo Selecton:Effcent Dversfcaton of Investment, John Wley & Sons, New York, [7] Sortno, F.A. and Forsey, H.J, On the Use and Msuse of Downsde Rsk, Journal of Portfolo Management. Wnter, 1996, [8] Young, M.R, Management Scence. 44, 1998,