Modeling and Optimization of Assets Portfolio with Consideration of Profits Reinvestment

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1 Global Joural of Pure ad Applied Mathematics. ISSN Volume 12, Number 3 (2016), pp Research Idia Publicatios Modelig ad Optimizatio of Assets Portfolio with Cosideratio of Profits Reivestmet Iria Leoidova Kashiria, Tatiaa Vasilieva Azarova, Yulia Valetiova Bodareko, Iria Naumova Shchepia Voroezh State Uiversity, Russia, , Voroezh, Uiversitetskaya sq. 1. Abstract The goal of the article: developmet of mathematical modelig apparatus of portfolio ivestmet processes, applied for the substatiatio of ivestmet decisios. The article examies the model of assets portfolio maagemet for a log-term ivestmet period, divided ito a certai umber of short-term periods, moreover, after each short-term period the portfolio profit is ot withdraw but reivested. The model is based o the maximizatio of the portfolio yield growth rate durig the whole log-term period. The deviatio of the average geometric portfolio yield growth rate for the eve growth rate is used as the assessmet of such portfolio risk. The article provides the results of the calculatio experimet, which demostrates both distictive features of the suggested approach ad i some cases similarity with the results obtaied with the applicatio of the classical Markowitz model. The article provides theoretical explaatio of the possible similarity of the modelig results. AMS subject classificatio: Keywords: Portfolio maagemet, portfolio yield, risk assessmet, capital growth rate. 1. Itroductio The problems of the formatio ad maagemet of securities portfolio take a importat place both i the theoretical literature ad scietific research ad i the system of real ecoomy (Davis et al., 2013), (Berkolaiko, Russma, 2004). Securities portfolio is a itegrated fiacial tool, purposefully developed i accordace with the defiite ivestmet strategy ad it is the totality of cotributios i the selected ivestmet objects (Watsham ad Parramore, 1996). The mai goal of the securities portfolio maagemet

2 2024 Iria Leoidova Kashiria, et al. is a itetioto obtai the highest possible yield with the lowest possible (or limited) risk level durig a certai period of time. Geeraly, this goal is achieved, through the portfolio diversificatio, i.e. redistributio of ivestor s fuds amog differet assets ad through the optimal choice of parts of the said assets i the portfolio (Barkalov et al., 2014). With the traditioal approach i the portfolio, as a rule, they purchase securities of well-kow compaies, which hold good productio ad fiacial idicators. Alogside with this, the portfolios structure is determied ot oly by the availability or absece of a security i them, but also by its share. Usually, yield ad risk of the portfolio, expected by the ed of the ivestmet period, are assessed o the basis of statistical data for previous periods of time of the same legth. I additio, the most commo approach is the oe suggested by H. Markowitz (Markowitz, 1952). Despite a large umber of studies, dedicated to the metioed topic, the problems of modelig ad optimizatio of fiacial tools portfolios are still urget. The developmet of moder market, globalizatio, ad availability of a broad statistical basis provide the emergece of ew methods of fiacial tools portfolio maagemet, based o ew coceptual approaches. The goal of this article is developmet of assets portfolio formatio model, based o the maximizatio of the portfolio yield growth rate durig the period of ivestmet, which takes ito accout the chages i the portfolio yield durig said period ad which has priciple differeces from H. Markowitz s model. 2. Classical approach to portfolio optimizatio The problem of the selectio of securities portfolio optimal structure was comprehesively studied for the first time by H. Markowitz i 1952 (Markowitz, 1952). The model of portfolio optimizatio suggested by him became the basis for the studies i the field of moder theory of ivestmet decisios makig. Accordig to Markowitz s model the portfolio yield is assessed as weighed by ivestmet shares average yield of each of the assets icluded i the portfolio, ad the risk is measured as average quadratic deviatio of the yield. With the set level of risk it is possible to maximize the portfolio yield: d p = x i m i max (1) σ p = v ij x i x j R max (2) x i = 1 (3) x i 0, i = 1, 2,..., (4) where x i is the share of ivestmet i i-th security; m i is the expected yield of the i-th security; d p is the portfolio yield; v ij is covariatio betwee i-th ad j-th securities;

3 Modelig ad Optimizatio of Assets Portfolio 2025 σ p is portfolio risk; R max is a parameter, limitig the maximum risk; is a umber of securities, regarded as the objects of ivestmet. While formulatig the problem of the portfolio optimizatio Markowitz used theoretical ad probabilistic formalizatio of the cocepts yield ad risk, assumig, alogside with this, that securities yield coforms to the ormal law of distributio. Moder researchers i the sphere of portfolio optimizatio sigle out a umber of deficiecies, (Kasimov, 2005) i the Markowitz model. 1. Groudlessess of the assumptio of ormal distributio of yields. Statistical observatios show that distributio of assets yields is characterized by large probabilities of extreme deviatios from the average, tha it ca be typical for ormal distributio (i.e. it has so called thick tails ). 2. Covariatio betwee yields of two assets is ot a costat value ad it chages i time, which, as a result, leads to the wrog assessmet of the future risk of ivestmet portfolio (Beati, 2003). 3. I practical calculatios accordig to Markowitz model the future yield of securities is defied as a average arithmetical of a umber of their historical yields. The said forecast does t iclude the ifluece of macroecoomic (the level of GDP, iflatio, uemploymet, brach idexes, etc.) ad microecoomic (liquidity, profitability, compay s fiacial sustaiability) factors (Davis et al., 2012). 4. The asset s risk i the Markowitz model is assessed o the basis of the measure of variability of the relatively of average value of the yield, but the deviatio of the yield to a bigger side is ot a risk but a super yield of the asset. 5. The Markowitz s model does t take ito accout may importat factors ad limitatios (trasactio costs, limitatios o the assets shares). The attempts to develop ad improve the Markowitz model have geerated a large umber of scietific results i the sphere of modelig ad optimizatio of securities portfolio. Particularly, i a umber of papers the behavior of securities yield is described by the law of distributio, differet from the ormal oe (Taaka, Guo 1999), or by o-static methods (Kashiria et al., 2014). Some papers use other tha Markowitz s approaches to the assessmet of the yield ad risk (Fusai, Luciao, 2001), (Kashiria et al., 2008). Some researchers while calculatig the risk do t take ito accout the positive deviatios from the average yield (Sortio Price 1994). May papers take ito accout the additioal limitatios of the ivestor ad the exteral eviromet (Chag T. J. et al., 2000). I some papers authors use idecipherable logics i order to take ito accout the idefiiteess of the expected risk ad yield (Xia et al., 2000), (Liu, 1999). Nevertheless, util ow there is a umber of usolved problems, which decrease the quality of the portfolio optimizatio results, i particular:

4 2026 Iria Leoidova Kashiria, et al. 1. I classical Markowitz model the problem of portfolio maagemet is solved for oe fixed period of ivestmet. Such approach does t suit very much for logterm ivestmets, as it does t allow to take ito cosideratio the chages of the portfolio yield durig the said period. 2. Mechaisms of portfolio optimizatio i may scietific studies are the modificatios of the Markowitz s approach ad they practically do ot cotai ay scietific ovelty. 3. Models, suggested by some authors, ofte do ot cotai ay theoretical substatiatio of the practical advatages i compariso with the classical approach. That is why the developmet of the models, which do ot cotai the above metioed deficiecies, is urget. 3. Portfolio optimizatio model with cosideratio of profits reivestmet Let us cosider problem of the securities portfolio maagemet for a log-term period, which, i its tur, cosists of short-term periods of ivestmet. Alogside with this, it is assumed that profit, obtaied from the ivestmet i i-th short-term period is ot withdraw but reivested i (i + 1)-th period. Let us idicate the portfolio yield i i-th short-term period with d i, i = 1,...,, i.e. d i = c i c i 1, c i 1 where c i portfolio value i i period, i = 0,...,. The the portfolio yield durig periods equals to d = c c 0 c 0 = c 0 (1 + d i ) c 0 c 0 = (1 + d i ) 1. The value 1 + d i is called capital growth rate i i-th period (ralph Vice, 2007), ad value T = 1 + d = (1 + d i ) is a aggregate capital growth rate durig all periods (Yaovsky, Vladyki, 2009). The average arithmetic portfolio growth rate ca be calculated by this formula: T ca = 1 (1 + d i ) = d i, (5) ad average geometric growth rate equals to T c = (1 + d i ), (6)

5 Modelig ad Optimizatio of Assets Portfolio 2027 Let us explai practical meaig of the average geometric capital growth rate. If portfolio yield i each short-term periods was equal to some costat value d cp, the i order to receive aggregate capital growth rate durig periods, same as the earlier itroduced value T = (1 + d i ), the the equatio would have to be met: (1 + d cp ) = (1 + d i ). Therefore, T c = 1 + d cp = (1 + d i ) (Shvedov, 1999). Thus, maximum gai of the capital durig periods will esure the portfolio, for which the value of the average geometric capital growth rate is maximal: T c = (1 + d i ) max, (7) Sustaiability of such portfolio is evidetly coected ot with dispersio of yields d i aroud their average value, but with the deviatio of the average capital growth (1 + d i ) from some costat growth rate (1 + d cp ) (Yaovsky, Vladyki, 2010). As for the portfolio with fixed growth rate there is the equatio T ca = T c (average arithmetic is equal to average geometric), the the risk coected with the istability of such portfolio ca be assumed equal to the differece betwee metioed values: R = T ca T c = d i (1 + d i ). (8) Due to the kow mathematics ratio betwee average arithmetic ad average geometric values, R 0 as a alterative variat of the idicator of portfolio risk it is possible to use the value equal to the differece betwee 1 ad ratio betwee average geometric ad average arithmetic values: R = 1 T c T ca = 1 (1 + d i ) (9) d i I this case 0 R<1. Now, let us ow cosider the objective of costructig the portfolio with optimal average geometric capital growth rate. Supposig for the iclusio i the portfolio s regard there is k differet fiacial istrumets. Let s idicate the share of i asset i the ivestor s portfolio with x i, i = 1, 2,...,k, ad the yield of i-asset i j-short-term period, i = 1, 2,...,k, j = 1,...,. with d ij. The the average arithmetic capital growth rate ca be calculated accordig to

6 2028 Iria Leoidova Kashiria, et al. the formula: T ca = 1 = 1 ( k ) ( (1 + d ij )x i = 1 k ) k x i + d ij x i ( ) k 1 + d ij x i = k d ij x i, (10) alogside with this, ormalizatio of shares is take ito accout: k x i = 1. The formula for the calculatio of the average geometric capital growth rate will acquire the view: T c = k ( (1 + d ij )x i ) = ( k ) k x i + d ij x i = ( ) k 1 + d ij x i (11) Thus, the objective of the portfolio formatio with maximum geometric capital growth rate durig short-term periods ca be put dow like this: T c = ( ) k 1 + d ij x i max, (12) R = 1 (1 + ) k d ij x i k R max, (13) d ij x i k x i = 1, (14) x i 0,i = 1, 2,...,k. (15) Here R max is a parameter, which limits the portfolio maximum risk level. Istead of the limitatio (13) it is also possible to use the alterative limitatio (13), based o the applicatio of the formula (8): R = k d ij x i ( ) k 1 + d ij x i R max (13a) But for practical calculatio the limitatio appeared to be more coveiet (13) as the risk i this case is a ulimited value, takig values from the segmet [0.1].

7 Modelig ad Optimizatio of Assets Portfolio 2029 Table 1: Portfolio, obtaied o the basis of Markowitz model. VSMPO Diksi Video GUM Magit Apple MTS NMLK Roseft Sberbak x1 x2 x3 x4 x5 x6 x7 x8 x9 x Table 2: Portfolio, obtaied o the basis of model (12)-(15). VSMPO Diksi Video GUM Magit Apple MTS NMLK Roseft Sberbak x1 x2 x3 x4 x5 x6 x7 x8 x9 x Results Despite the priciple differeces i the approaches to the portfolio formatio, the calculatio experimet showed that the portfolios formed o the basis of model (12)-(15), i which a log-term period of ivestmet is divided ito short-term periods, are close i their compositio to the portfolios, obtaied o the basis of the Markowitz model. Thus, Table 1 provides the structure of the portfolio, obtaied o the basis of the Markowitz model with the target fuctio, which expresses the average mothly portfolio yield. Alogside with this, the value of the portfolio risk, expressed by the average quadratic deviatio of its yield, was limited by the value 0.1. Table 2 provides the structure of the portfolio, obtaied o the basis of the model (12)- (15) by 12 short-term periods, each of which has the duratio of oe moth. Herewith, we used the value equal to 0.01i order to limit the portfolio risk, expressed by the ratio betwee average arithmetic ad average geometric growth rate by formula (13). It was experimetally oted that the portfolios, formed o the basis of the Markowitz model ad models (12)-(15), are close i their compositio, if the ratio of the maximum risks set for them is i the limits I order to more vividly uderstad the specificity of the model (12)-(15), let s examie the example of the formatio of the portfolio with two assets, for which we kow the growth rates i two short-term periods. Table 3 demostrates, that despite the fact that asset II growth rates differ (they are Table 3: Example 1: Iitial data for the portfolio of two assets. Moth Asset I value Assets II value Asset I growth rates Asset II growth rates Average arithmetic asset growth rate Average geometric asset growth rate

8 2030 Iria Leoidova Kashiria, et al. Table 4: Portfolio structure depedig o the set value of maximum risk. Model (8)-(11) The Markowitz model Rmax x1 x2 Rmax x1 x2 > > , , equal to correspodigly 3 ad 2) i the first ad secod short-term periods, the average geometric growth rate of the said asset differs slightly from its average arithmetic growth rate, which meas low risk valuefor model (12)-(15), coected with ivestmet i this asset. But asset I has permaet growth rate, i.e. its risk assessmet i model (12)-(15) is zero. Alogside with this, asset II seems to be more attractive for ivestmet by model (12)-(15), as its average geometric growth rate is cosiderably higher tha that of the first asset (2.45 agaist 2.0). Experimetal calculatio accordig to model (12)-(15), preseted i Table 4, shows that asset 1 will be icluded i portfolio oly with rigid eough limitatio of the portfolio risk, ad it will be impossible to form the portfolio oly with the first asset with zero limitatio of the risk. For compariso the same table also presets the calculatios for portfolio formatio o the basis of the Markowitz model (o the basis of the same iitial data). It is possible to ote that i Markowitz model the risk of the first asset is also zero (as its yield is costat), but it is also impossible to form the portfolio of oly the first asset with zero limitatio of risk. Despite the geeral similarity of the results, received with the help of models (12)- (15) ad Markowitz s model it is easy to give the example i which model (12)-(15) provides a more ratioal result. Accordig to the iitial data, preseted i table 5, asset I looks more attractive for ivestmet accordig to the Markowitz model as its average yield is higher tha that of asset II. For the model (12)-(15), o the cotrary,asset IIis more attractive, as its average geometric growth rate is higher, tha that of asset I. Alogside with this, the results of the model (12)-(15) seem more substatiated, as by the results of the two periods the value of asset I dropped by 15%, while the value of the asset II, o the cotrary, icreased by 15%. I fairess, it should be oted that the dispersio of the asset I yield i this example is higher tha that of asset II ad that is why with the correspodig limitatio for maximum portfolio risk the Markowitz model would also give the preferece to asset II. 5. Results discussio Some theoretical explaatio of the similarity of the results of model (12)-(15) ad that of Markowitz ca be see o the example of the portfolio, built for two shortterm periods, for which average geometric capital growth rate,calculated by formula (2), equals T cð³ = (1 + d 1 )(1 + d 2 ). Let us idicate average arithmetic value of yields

9 Modelig ad Optimizatio of Assets Portfolio 2031 Table 5: Example 2: Compariso of iitial data for the portfolio of two assets. Moth Asset I value Asset II value Asset I growth rates Assets II growth rates Asset I yield Asset II yield II Average yield Average geometric growth rate durig two periods with d, i.e. d = d 1 + d 2. The the average quadratic deviatio of 2 the yield durig metioed periods will be the value: ( 1 σ = d 1 d ) 1 + d ( d 2 d ) 1 + d 2 (d1 ) 2 d 2 2 = = d 1 d (16) Assumig for certaity that d 1 d 2, the the yields ca be preseted as: d 1 = d +σ, d 2 = d σ. Thus, we ca write dow, that T c = (1 + d 1 )(1 + d 2 ) = (1 + d + σ)(1 + d σ) = (1 + d) 2 σ 2 (17) From formula (17) it is evidet, that average geometric capital growth rate will icrease with the growth of the average portfolio yield ad it will decrease with the growth of its dispersio, which correspods to the mai priciples of the Markowitz model. Now, let s calculate portfolio risk magitude by formula (9): Hece: 1 R max R = 1 (1 + d) 2 σ d R max (18) (1 + d) 2 σ 2 (1 + d) 2 σ 2 (1 + d) 2 R2 max + 2R max. With regard of o-egativity of the values (1 + d) ad σ we fially receive: σ (1 + d) Rmax 2 + 2R max (19) Thus, limitatio (18) coects average quadratic deviatio of the portfolio σ (risk accordig to Markowitz) with average arithmetic capital growth rate (1 + d), meaig,

10 2032 Iria Leoidova Kashiria, et al. that risk revealed accordig to the Markowitz model, should ot grow faster, tha the average capital growth rate, multiplied by the costat, calculated depedig o the set value of the maximum risk R max. Although it should be oted oce agai that this vivid illustratio is give for the case with two short-term periods. 6. Coclusio The article examies the assets portfolio maagemet model, based o the maximizatio of the average geometric capital growth rate durig the ivestmet period. The suggested model of portfolio optimizatio is alterative to the traditioal approach of H. Markowitz (Markowitz, 1990). The model differs by takig ito accout the chages i portfolio yield ad risk durig the period of ivestmet ad by a differet approach to the assessmet of the yield ad risk of securities. By the results of the calculatio experimet we made the coclusio that model (12)-(15), i geeral, gives results similar to the results of the Markowitz model but there are examples of the iitial data, for which results of model (12)-(15) seem to be more preferable. The directio of further research ca be the cosideratio of certai cases of portfolio optimizatio with set limitatios (cosideratio of limitatios by the share of securities i the portfolio, cosideratio of trasactio costs etc.). Refereces [1] Barkalov, S.A., Ereshko, F.I., Kaaeva N.A. (2014). Diversificatio models aalysis. Sistemy Upravleia i Iformatsioye Tekhologii., V. 55, #1.1. P [2] Beati, S. (2003), The optimal portfolio problem with coheret risk measure costraits. EuropeaJouralofOperatioalResearch. #150. P [3] Berkolaiko, Z., Russma, I.B. (2004), O some methods of assets portfolio formatio ad maagemet. Part 1. Ekoomicheskaya auka sovremeoi Rossii. #1. P [4] Berkolaiko, Z., Russma, I.B. (2004), O some methods of assets portfolio formatio ad maagemet. Part 2. Ekoomicheskaya auka sovremeoi Rossii. #1. P [5] Chag T.J., Meade N., Beasley J.E., SharaihaY.M. (2000), Heuristics for cardiality costraied portfolio optimizatio, Computers & Operatioal Research. #27. P [6] Davis, V. V., Satki, S., Fetisov Vł., (2013), Optimal portfolio of securities i the coditios of globalizatio: approaches ad models. Sovremeaya ekoomika: problem i resheiya. # 8 (44). P [7] Davis V. V., Satki, S., Razisky Yu. (2012), Problems of real ivestmet o the basis of portfolio decisios. Sovremeaya ekoomika: problem i resheiya. #11 (35). P

11 Modelig ad Optimizatio of Assets Portfolio 2033 [8] Fusai G., Luciao E. (2001), Dyamic value at risk uder optimal ad suboptimal portfolio policies. Europea Joural of Operatioal Research. #135. P [9] Simov Yu. F. (2005), Itroductio i the theory of optimal portfolio of securities. 224 p. [10] Kashiria, I. L., Berkolaiko, M. Z., Ivaova K. G. (2008), Applicatio of Russma s D grades for assets portfolio maagemet. Vestik VGU, Series: Sistemy Aalizi Iformatsioye Tekhologii, #1P [11] Kashiria, I. L. (2014), Securities portfolio maagemet o the basis of the progostic methods of reachig boudary coditios i dual calculatio eviromet. Ekoomika I Meedzhmet System Upravleia. V.1. #1. P [12] Liu L. (1999), Approximate portfolio aalysis. Europea Joural of Operatioal Research. #119. P [13] Markowitz H. (1952), Portfolio Selectio. Joural of Fiace. #7. P [14] Markowitz H. (1990), Mea Variace Aalysis i Portfolio Choice ad Capital Markets. Basil. Blackwell. [15] Ralph Vice (2007), The hadbook of portfolio mathematics: formulas for optimal allocatio & leverage. Hoboke, New Jersey: Wiley & Sos, Ic. 422 p. [16] Shvedov, A. (1999), Theory of efficiet portfolios of securities. M., Published by GU VShE. 144 p. [17] Sortio F., Price L. (1994), Performace Measuremet i a Dowside Risk Framework, Joural of Ivestig. Vol. 3, # 3. P [18] Taaka H., Guo P. (1999), Portfolio selectio based o upper ad lower expoetial possibility distributios. Europea Joural of Operatioal Research. #114. P [19] Terry J. Watsham, Keith Parramore (1996), Quatitative Methods i Fiace, Cegage Learig EMEA. 395 p. [20] XiaY., Liu B., Wag S., Lai K.K. (2000), A model for portfolio selectio with order of expected returs. Computers & Operatioal Research. #27. P [21] Yaovsky, L. P., Vladyki S. N. (2009), SARM theory ad portfolio aalysis takig ito accout the ivestmet horizo ad optimal rates of capital growth. Vestik of the Russia State Uiversity of Trade ad Ecoomics (RGTEU), #7. P [22] Yaovsky, L. P., Vladyki S. N. (2009), Portfolio selectio takig ito accout ivestmet horizo. Fiasy i Kredit #29 (365). P [23] Yaovsky, L. P., Vladyki S. N. (2010), O the coectio betwee the selectio of portfolio refiacig strategy ad the ivestor s risk avoidace level. Sovremeaya ekoomika: problem i resheiya. # 1. P

Subject CT1 Financial Mathematics Core Technical Syllabus

Subject CT1 Financial Mathematics Core Technical Syllabus Subject CT1 Fiacial Mathematics Core Techical Syllabus for the 2018 exams 1 Jue 2017 Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig