SOLVING OF PORTFOLIO OPTIMIZATION PROBLEMS WITH MATHEMATICA

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1 SOLVING OF PORTFOLIO OPTIMIZATION PROBLEMS WITH MATHEMATICA Iria Bolshaova Belarusia State Uiversity. ABSTRACT: Optimizatio models play a icreasigly role i fiacial decisios. Portfolio imizatio problems are based o mea-variace models for returs ad for ris-eutral desity estimatio. The mathematical portfolio imizatio problems are the quadratic or liear parametrical programmig sometimes with iteger variables. This paper aalyzes the mathematical models ad imizatio techiques for some classes of portfolio imizatio problems by usig the computig system Mathematica. KEYWORDS: Marowitz, portfolio imizatio, absolute deviatio, portfolio diversificatio, efficiet frotier, Sharpe ratio, miima model.. INTRODUCTION Coceptio of a imal portfolio of assets was first time metioed by Louis Bacheliers i his doctoral thesis which was defeded i 9 i Paris. Ufortuately, this thesis eactly lie the theory of imizatio created by L. Katorovich ad T. Kupmas the Nobel Prize wiers i ecoomy were less commo amog fiacial maagers. They maaged to use primary sills of actuarial mathematics, elemetary cocepts of share fare value (price). The moder portfolio theory was firstly reviewed i the wor writte by Marowitz [5] ad Sharpe [7], who were awarded Nobel Prize i Ecoomics i 99. This theory is seems to be of high importace. If you mae a iquiry about "portfolio theory" ad "portfolio imizatio" usig the search egie Google.com you will be give about, 5 ml lis for the first oe ad about, ml lis for the secod oe. However, the demad for such techiques amog Belarusia baers was earlier small. Now the situatio has chaged: a small margi of iterest maes bas maage assets usig moder imizig methods. As a rule the direct methods to obtai the imal ris portfolio are complicated []. Therefore the moder tools for solvig portfolio imizatio problems are very sigificat, as far as allows to simplify ad to reduce calculatios. This paper cosiders usig computig system Mathematica [] for solvig some classes of portfolio imizatio problems.. THE STANDARD MARKOWITZ PORTFOLIO MODEL AND IT S APPROACHES Let's suppose that ivestor has the possibility to choose from the variety of differet fiacial assets lie securities, bods ad ivestmet proects. The mai poit is to defie ivestmet portfolio = (, K, ), where is proportio of the asset. The the budget costrait is = =,, =,. () It is valuable to say, that absolute weightigs of assets could be icluded i the Marowitz. For istace, by K we deote the ivestor's iitial capital. The the budget costrait () might be replaced for: K = K =,, =,, (.) where K is the price of asset. If all assets are ifiitely divisible replaced variables K =, K we get budget costrait ().

2 Marowitz portfolio model [6] assumes to use two criterios: portfolio epected retur ad portfolio volatility (measure of ris adusted). Importat to add that theory uses the historical parameter, volatility, as a proy for ris, while retur is a epectatio o the future. The retur ( ) R of the portfolio is the compoet-weighted epected the retur R of the costituet assets. The epected retur of a asset is a probability-weighted average of the retur i all scearios. Callig r the retur i sceario t, we may write the epected retur as p t the probability of sceario t ad t T ( R ) = t = r = E p t r t. TIt's assumed that all sceario t (historical) are equal probability i the future, the r = r T t t / (see eample ). = The fuctio of the epected retur of the portfolio is eeded to be maimized: p t = / T ad r () = E( R() ) = r ma =. () If we suppose that r K r the imal solutio of the problem (), () is = (,, K,), i.e. all capital should ivest i the most profitable asset (greedy solutio). Clearly, it is very risy. That is why ivestors add (upper boud costrait) u, =, to budget costraits. I this case greedy solutio has followig form = u, K, u, u,, K, =, where u = ad + u = ad stays imal. It is possible further to add costraits for diversificatio of riss. However, Marowitz proposed other approach. Some authors use fuzzy umbers to represet the future retur of assets that approimated as fuzzy umbers the epected retur ad ris are evaluated by iterval-valued meas []. Oe of the best-ow measures T of ris is stadard deviatio of epected returs. Let's σ i is covariace of the returs i ad, i.e. σ i = ( rit ri )( rt r ). T t= Marowitz derived the geeral formula for the stadard deviatio of the portfolio (ris of the portfolio) as follows: σ ( ) = E( R( ) r( ) ) = i = = i σ i mi. () The variace of all asset's returs is the epected value of the squared deviatios from the epected retur: = T σ p t ( r t E() r ). t = Remar that the covariace matri ( ) σ = σ i is positively semi-defiite ad cosequetly σ () ad σ () are cove fuctios. That is why stadard Marowitz portfolio model () () is bi-criteria imizatio problem with liear () ad cove quadratic () obective fuctios. I some occasios stadard deviatio could be substituted for -order target ris: / σ = E R r. [ ] () ( () ()) Let s apply Marowitz's model to the problem of the imizatio portfolio of blue chips, Hi-Tech corporatio's shares, real estate ad Treasure bods. The aual times series for the retur are give below for each asset betwee si years. Eample. Portfolio problem with four assets.

3 \ t Average aual percetage r t 5 6 r = E( R ) Blue chips 8,, 5, 5,6,6,,68 Hi-Tech shares, 9,6 5,7,6 -,6-7,,98 Real estate Maret 8,' 8,96 8,5 9,6 8,5 7,9 8, Treasury Bods 8, 8,6 8, 9, 9, 8,95 8,67 Average aual percetage r t is specified P t+ P t r t =, P t where P t is asset price at istat time t. The retur ad covariace matries ca be easily fid i the Mathematica system by usig build-i fuctios Mea ad Covariace. The covariace matri is: σ = The first approach leads to the tas of miimizig the variace of the portfolio () retur give a lower boud o the epected portfolio retur r(), (.) i.e. uder all possible portfolios, cosider oly those which satisfy the costraits, i particular those which retur at least a epected retur of. The amog those portfolios determie the oe with the smallest retur variace. Problem (), (), (.) is quadratic imizatio problem with a positive semidefiite obective matri σ : ( ) σ = ,68 +,98 +8, +8,67, =,, =,. mi, This problem ca be solved by usig stadard quadratic programmig algorithms or i a very efficiet way by usig the computig system Mathematica ad it s build-i fuctio Miimize. Settig i the problem (), (), (.) for portfolio imizatio ad solvig it for guarateed retur =.7%, we get the imal portfolio ( =.95, =.7, =, =.) with ris σ ( ) = 5.959% (oe of the corer portfolio). The secod approach we cosider the tas of maimizig the mea of the portfolio retur ( ) give upper boud for the variace σ ( ) : r uder a σ (). (.) Problem (), (), (.) is a liear parametric programmig with a additioal cove quadratic costrait (.) ad parameter. This problem ca be also efficietly solved by usig the Mathematica system ad it s build-i fuctio Maimize. Settig i the problem (), (), (.) for portfolio imizatio ad solvig it for as eample =%, we get the imal portfolio ( =.89, =., =, =.7697) with retur r ( ) = 9.%.

4 A portfolio is efficiet (Pareto imal) if ad oly if o other feasible portfolio that improves at least oe of the two imizatio criteria without worseig the other. A efficiet portfolio is the portfolio of risy assets that gives the lowest variace of retur of all portfolios havig the same epected retur. Alteratively we may say that a efficiet portfolio has the highest epected retur of all portfolios havig the same variace. The efficiet frotier sur-plae ( r,σ ) is the image ( r( ) σ ( ) ), of all efficiet portfolios. Let s plot the efficiet frotier by usig the build-i fuctio ParametricPlot i Mathematica system: Fig.. The efficiet frotier s, % EFFICIENT FRONTIER r, % While choosig a efficiet portfolio we could apply for weightig obective fuctio approach. The third approach is based o usig the Carli theorem of coicidece Pareto-imal solutios i () () i imal solutios i the oe-criterio parametric imizatio with parameter : () ( )() ma Here the parameter ( ) r σ. () build-i fuctio Maimize i the system Mathematica. shows ivestor's ris. This problem ca be also easily solved by usig The lower = the less ris we apply for the model, ivestor is more coservative. Miimal ris is.88% with portfolio ( =.57, =, =.776, =.7687) ad retur 8.687% (aother corer portfolio). If = ivestor must accept ris i order to receive higher returs. Maimal ris is 6.57% with portfolio ( =, =, =, = ) ad retur.98%. This algorithm for parametric quadratic programmig solves the problem (), () for all i the iterval [ ;]. Startig from oe poit o the efficiet portfolio the algorithm computes a sequece of so called corer portfolios = (, K, ). These corer portfolios defie all efficiet portfolio are m cove combiatios of the two adacet corer portfolios: if ad are adacet corer portfolios with epected returs r ( ) ad r ( ), r ( ) r ( ) the for every r ( ) = λ r ( ) + ( λ) r( ) the efficiet portfolio is calculated as λ + λ, λ. = ( ) For istace, fid corer portfolios for Treasury bods ( ) with the portfolio retur 8.5;.9 by usig build-i fuctio Evaluate i the Mathematica system: [ ]

5 Fig.. The corer portfolios for Treasury bods Corer portfolios for other assets ca be fid by the same way. There are three corer portfolios: for returs =8.687%, =8.8% ad =.7%. Solvig the portfolio imizatio problem for retur =8.8%, get the imal portfolio ( =.757, =.5, =, =.99) with ris σ ( ) =.89% (the last corer portfolio). The efficiet portfolio is calculated as = λ λ + λ +, where ( =.57, =, =.776, =.7687), ( =.757, =.5, =, =.99), ( =.95, =.7, =, =.) ad λ + λ + λ =, λ.. MODEL WITH RISK-FREE ASSET Ris-free asset hypothetically correspods to be short-term govermet securities. Coditioally it is assumed that the variatio of the govermet securities retur r is equal zero. Cosiderig the followig Tobi model [9] for portfolio = (,, K, ) with ris free asset : + =,, =,. (.) = r (, ) = r + r() = r + r ma =. (.) (, ) = σ + σ + σ = σ = σ σ () σ p p p p p p p p = Obviously, the epected rates of retur o all risy assets are ot less asset, i.e. r r.. (.) If we tae some defiite efficiet portfolio, we could figure all portfolios with ris free assets o CML (Capital Maret Lie): ( R ) ( ) E r m r E C = r + σ C, σ m where r m is retur of the maret portfolio (depedig o the maret ide ad its ris is σ m ). It is iterestig to ote, if someoe has the possibility to choose ot oly betwee the give ris portfolio ad ris-free assets but also to choose a structure of the ris portfolio the there eists the uique imal solutio ( =.57, =-.59, =.6598, =.6897), ot depeded o ivestor s ris (solvig by the Mathematica system). 5

6 . SHARPE MODEL WITH FRACTIONAL CRITERIA The mai cotet of this model is replacemet of the bi-criterio model (), (), () for the oe-criterio model with budget costrait () ad liear-fractioal obective fuctio [8]: ma = r i = = i σ i. I [] describes a direct method to obtai the imal risy portfolio by costructig a cove quadratic programmig problem equivalet to Sharpe-ratio. I that form, this problem is ot easy to solve. But the Mathematica system easily does it by usig oly oe build-i fuctio Maimise. The uique imal portfolio is ( =.57, =, =.776, =.7687) with ris.88% ad retur 8.687% (the corer portfolio with miimal ris). 5. CONCLUSIONS The epected retur ad the ris measured by the variace are the two mai characteristics of a imal portfolio. The imal portfolio is desirable (the target portfolio). The real portfolio of assets ca ot be doe by huma ituitio aloe ad some other characteristics []: closeess to the target portfolio; eposure to differet ecoomic sectors close to that of the target portfolio; a small umber of ames; a small umber of trasactios; high liquidity; low trasactio costs. The mathematical problem ca be formulated i may ways but the pricipal problems ca be summarized as follows: bicriterial cove quadratic imizatio with simple budget costraits; liear imizatio with simple polymatroidal budget ad ris diversificatio costraits; cove quadratic or liear bicreterial imizatio with iteger (mied iteger variables) []. All models are easily ad visually solved by usig the Mathematica system. That allows to see the imal variat of capital ivestmets amog valid rage of solutios. 6. REFERENCES. Bertsimas D., Darell C, Soucy R. Portfolio costructio through mied-iteger programmig at Gratham, Moyo, Va Otterloo ad Compay. INTERFACES V.9, l, p Bolshaova I., Kovalev M. Fuzzy umbers i fiacial aalyses // The problems of forecast ad state regulatio of social ad ecoomic developmet: Publicatio of V iteratioal scietific coferece. Mis.. С Bolshaova I., Kovalev M., Girlich E. Portfolio imizatio problems: a survey. Preprit Nr 6. Otto-vo- Guerice Uiversity Magdeburg. 9. P Corueols G., Tutucu R. Optimizatio methods i fiace. Cambridge Uiversity Press. 7. p. 5. Marovitz H. Portfolio Selectio // Joural of Fiace. 95. V.. 7. р Marowitz H. Mea-variace aalysis i portfolio choice ad capital marets. Blacwell Publishers, Oford p. 7. Sharpe W. Simplified model for portfolio aalysis. Maagemet Sciece. 96. V. 9,, p Sharpe W. The Sharpe ratio. Portfolio Maagemet. 99. p Tobi J. Liquidity preferece as behavior towards ris. The Review of Ecoomic Studies V. 5, p Wolfram S. Mathematica. A system for doig mathematics by computer. -th ed. Addiso-Wesley Publishig Compay. 99. c. 6

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