Using the attached, research the variables that impact the pricing of options. Focus your energy on comparing the attributes of the two widely accepted models used for option pricing: Black-Scholes and Binomial Models. Your paper should be completed in Word and be no less than two pages in length following APA format.

Inzinerine Ekonomika-Engineering Economics, 2012, 23(2), 144-153 Development, Implementation and Evaluation of Multistage Investment Strategies Eimutis Valakevicius, Kristina Vaznelyte Kaunas University of Technology Studentu st. 50-219, LT-3031, Kaunas, Lithuania e-mail: eimutis. valakevicius@ktu. It, kristina. vaznelyte(^stud. ktu. It cross-* http://dx.doi.Org/10.5755/j01.ee.23.2.1542 Construction of optimal investment portfolio is very complicated task due to many diverse factors which might affect risk and return of the portfolio in the future. Values and impact level of unique factors on the portfolio are changing over time; therefore every investor should take into account the fact that there always will be a certain level of risk associated with any portfolio involving stocks. There is a number of ways to form a collection of most appropriate stocks and bonds for investment portfolio as well as to allocate weights of assets based on various criteria. All of these methods, dedicated f'or selection and allocation of assets, have their specific features and some disadvantages. In order to be able to conclude which of the asset selection methods have least disadvantages, four popular techniques were analysed and compared. These techniques were based on different variables: correlation coefficients between asset returns, maximisation of the utility function (diverse values of risk aversion coefficients were analysed), selection of assets with highest historical returns, and employment of modified priceto-earnings ratio. The article deals with multistage extension to the mean-variance and expected utility maximisation portfolio choice. Multistage investing consists of several essential stages, where each stage forms a basis for the next stage by providing useful input data, derived by stage-specific analysis. For construction of optimal portfolio the following stages are used: asset allocation, security selection, investment strategy development, construction of the model and its evaluation. After asset allocation was made and stocks for the portfolios had been selected, different theoretical asset allocation models (equal weight asset allocation, Markowitz model Capital Asset Pricing Model (CAPM), model where risk free asset is incorporated when constructing a portfolio) have been modified and adapted in order to become suitable for real market situation. Such prerequisites as normal distribution of stock returns were not satisfied by most Lithuanian companies ' stocks when different interims were investigated, therefore authors set a presupposition that distribution properties of the stocks can be disregarded when Markowitz and CAPM models were applied to real market. Some other changes for the prerequisites of models were made; otherwise these theoretical models could not have been applied to Lithuanian market. After all models had been applied in Lithuanian equity market, back testing was carried out and certain characteristics of outcomes of different investment strategies were compared. Results were judged against characteristics of popular stock exchange indices of Baltic States in order to obtain conclusions. Most models were developed for broad financiis markets (global markets). In the paper we analyse financial market of Lithuania. Since this market does not fit assumed conditions of general models, the models were slightly modified to apply for Lithuania market. The results of portfolio were compared with Baltic States index. It was concluded that the highest retum rate is achieved by constructing the investment portfolio with employing modified Capital Asset Pricing Model. The best technique for selecting stocks proved to be the maximisation of utility function when risk aversion coefficient A=3. In addition to this, after comparison of different asset selection methods, it was noted that the highest value of the Sharpe ratio was achieved by utilising the same technique. After investigation it was noted that investors should add a risk free asset into portfolio of stocks because it usually improves the results of most portfolios, irrespective of their contents. Constructing portfolios based on asset allocation according to indices analysed in the paper is not recommended because characteristics of indices were worse than the ones of constructed portfolios. Stocks of every company quoted in NASDAQ OMX Baltic (2011) Stock Exchange in Vilnius Official list for more than 10 years (2001 beginning-2010 end) were investigated. Stocks of 14 companies satisfied preset 10 year interim criterion. Keyword..- stocks, asset selection, Markowitz model, asset allocation, asset pricing, Sharpe ratio, utility function, rate of return. Introduction Quantitative models can he found in modem investment theory. There are models that quantify the relationship between the expected returns among a set of assets and their relative risk levels, such as the Capital Asset Pricing Model (CAPM) (Sharpe, 1964; Jagannathan, & Wang, 1996) or the Arbitrage Pricing Theory (APT) (Ross, 1976). There are models used to price financial products, including stocks, bonds, options, et cetera. And of particular interest for this article, there are models used to guide asset allocation decisions, such as the Markowitz Mean-Variance criterion or the set of expected utilitytheoretic paradigms. -144- Eimutis Valakevicius, Kristina Vaznelyte. Development, Implementation and Evaluation... Analysis of historical stock returns bad shown that investing in stock market is tbe best way to protect money from inflation and that variability of short-term investments is directly proportional to probability of exceeding the rate of inflation during certain interim (Valakevicius, 2007). There are various methods to construct a portfolio (Ang & Chen, 2002), and each of these methods have specific flaws. Therefore, the best combination of metbods used in various stages of forming, forecasting and managing the investment portfolio should be devised so that investors could obtain maximum benefit from tbeir money. Metbods of forming tbe portfolio differ in tbeir complexity, time and financial resources needed to be established, reliability, etc. E.g., the models involving copulas initially were regarded successful (Junker, Szimayer & Wagner, 2006); later these models, according to Salmon (2009), were called one of the Wall Street crash reasons. Variables (stock prices, interest rates, currency rates) in financial markets often cannot be described properly by using normal distribution (Glasserman, Heidelberger & Sbahabuddin, 2002) and not always it is possible to find appropriate copula fiinction for financial variables (Malevergne & Sornette, 2003), sometimes the combination of several copula functions has to be created (Rachev et al, 2009). When misinterpreted and unprofessionally applied these fiinctions can cause a lot of losses. One of tbe time-consuming models is the model based on factors (Grinblatt & Titman, 1983). Valakevicius & Zolyte (2003) used this method in tbeir researcb and extracted 8 factors - consumer price index, export and import of goods, unemployment level, etc. Roll (1980) states that for this type of model 3 to 4 factors should be enough if arbitrage pricing theory is adapted. Mansor (2011) had identified connections between stock market development and GDP, ratio of market capitalisation and investment, and aggregate price level. In this article we will concentrate on tbe tnodels which do not need such high amount of external information about country's economy. Input information for all the stages of forming the portfolio requires only one external macro economical variable retum norm of riskless asset. All other input information is calculated using price data of tbe companies in tbe Lithuanian market. Figure 1 sbows tbe change of prices of Litbuanian companies stocks, selected for tbe -APG GRG IVL LDJ -PTR P2V RSU research. Bonds are regarded riskless assets. They can be issued by Litbuanian government, public limited companies and private limited companies (LVPK, 2008). Let 5 denote tbe price of a single stock at time /' and let i denote tbe corresponding return over period from (i'-i.',:], defined as: wbicb represents tbe percentage change in value of tbe stock. If inequality /, E ( R E ) ) , then asset A will be prioritised during asset selection process. Tbe last technique, selection of assets having normal distribution of tbe returns, could not be implemented due to tbe nature of tbe data. Altbougb in modem portfolio theory models (i.e. Markowitz and CAPM) it is implied tbat asset retums are normally distributed, in paper publisbed by Kitt and Kalda (2005) it was stated tbat normal distribution does not represent appropriately real market stocks. Having implemented tbe analysis of retums distribution of Litbuanian companies' stocks, it was discovered that there are only few cases of normal distribution and only at few periods investigated, tberefore there was not possible to form tbe portfolio with assets having normal distribution because there were no assets having such distribution for several periods. rates of retum E C R J , ECR,J E(Rn3 and covariance crfj, where i = 1, n and j = l,n. If to eacb of n assets we bave invested part Wjof tbe entire initial portfolio, tben 1 = I!"=i CJJ. If condition Wj > 0 is added, then sbort selling is prohibited. Otberwise sbort selling is allowed and weights ojjcan obtain negative values. Tbe aim of creation of sucb portfolio is to find minimum risk portfolio for specific average rate of retum and tbis problem can be solved by using linear programming. We have to find minimum of -p=il.f=i ^j^jcrjj, i = 1, n and j = l , n , with conditions Sf=i tojE(R|) = E(Rp) and I]f=i tOj = 1 (Luenberger, 1998). If we need to solve tbe problem wbere sbort selling is probibited, then we have to add one more condition: tiJj > 0. In this case we have formed the problem of nonlinear programming wbicb can be solved using various mathematical software packages. In tbis case MATLAB from Math Works bad been employed. Capital Asset Pricing Model Acronym for tbis asset allocation model is CAPM. CAPM expands theoretical model of portfolio diversification, proposed by H. Markowitz. CAPM tbeory was developed by William Sharpe, John Lintner and Jan Mossin (Sharpe, 1964; Lintner, 1965; Mossin, 1966).When model is developed according to CAPM, investor bas opportunity to incorporate riskless asset to tbe investment portfolio. Tbe results of tbe researcb showed tbat sometimes tbis feature is very advantageous, especially during crisis, because it belps form the portfolio which has positive rate of retum and bas no risk. Tbe researeb showed tbat during crisis efficient frontier quite often consists only of tbe one portfolio wbicb has tbe only asset and that asset is riskless while portfolios formed according to Markowitz tbeory often bring losses during crisis. If we compare CAPM, compared to Markowitz models at tbe same fixed level of rate of retum, CAPM bas the feature of providing lower risk. Several of tbe major CAPM prerequisites which bave to be ignored so that tbis model could be applied in tbe market were: all tbe assets are infinitely divisible, equity market is in balanced condition, all investors have tbe same level of access to needed financial information and act rationally according to that financial knowledge. These conditions do not reflect real market situation (they -147- Inzinerine Ekonomika-Engineering Economics, 2012,23(2), 144-153 simplify real market behaviour too much), so for the sake of the research modified CAPM without these prerequisites was applied to Lithuanian market. The results of the modified CAPM showed that formal prerequisites do not have to be satisfied for construction of the portfolio which brings profit. Main mathematical formula of CAPM is called Capital Allocation Line (CAL): where fC^) is average rate of retum of complete effective portfolio, containing risky and riskless assets, Or. - risk of complete portfolio; Rf - retum norm of risk-free portfolio; ECR^.) - average rate of retum of market portfolio; cr^r. - risk of market portfolio; expression ERn-J - Rf denotes risk premium. In CAPM theory it is stated that expected retum of entire portfolio should exceed retum of riskless asset due to proportionality risk premium and beta coefficient. However, (Fleuret, 2003) does not agree with this statement and says that empirical examinations do not reaffirm this theory. According to (Hagen, 1993), if market portfolio is effective, then there exists direct proportionality between beta coefficient of any asset and expected retum ofthat asset. The results of Black, Jensen and Scholes test, conducted in 1972 with the stocks quoted in NYSE 19261965, have shown that one can apply CAPM successilly to lend out, but not to borrow. Roll (1977) did not agree with the conclusions about validity of CAPM stating that the theory is not testable because exact composition of the market portfolio is unknown. Shanken (1987) after analysis of CAPM theory stated that unambiguous inference regarding validity of CAPM is probably unattainable. Connor & Korajczyk (1993) also stated that CAPM is not ideal model. However, CAPM is acknowledged model, used by rating agencies and other organisations. Stock exchange indices Stock exchange indices are useful benchmarks for evaluating stock market in general and its sectors (e.g., energy, manufacture, health care, financial institutions, commodities, information technologies, etc. (Standard & Poor's, 2011), individual companies in different sectors, mutual funds, groups of companies whose stocks are most actively bought and sold in the market, and other entities. Therefore characteristics of two popular indices (OMX Vilnius and OMX Baltic Benchmark) quoted in NASDAQ OMX Baltic Stock exchange were compared with those of constmcted portfolios; results of such comparison will show quantitatively how much parameters of models and indiees differ. OMX Vilnius: according to NASDAQ OMX Baltic (2009), index OMX Vilnius (OMXV), established on 1999-12-31, represents the situation in Lithuanian stock market and is composed of the stocks of all the companies which are quoted in the exchange's Main and Secondary lists, except for those whose 90% or more of stocks are owned by one shareholder. OMX Baltic Benchmark: this index is abbreviated as OMXBB. According to NASDAQ OMX Baltic (2009), OMXBB, established on 1999-12-31 and updated biannually (in order to ensure optimal investment strategy with minimal costs), it is composed from stocks of the companies having highest capitalisation, highest liquidity and belonging to all the sectors in the Baltic market; amount of the stocks of the company in the index depends on each company's value of the stocks and the amount of the stocks in the market. This index should be helpful in creating personal effective investment portfolio inexpensively and is very useil for managers of investment products and for investors as a benchmark (NASDAQ OMX Baltic, 2009). Therefore, we used this index in this article as representative benchmark. Sharpe ratio Sharpe ratio is also referred to as Sharpe measure and Sharpe index. It is popular, simple and representative characteristic to evaluate investments. Higher value of Sharpe ratio implies higher quality of investment portfolio or an individual stock (Sharpe, 1964). Sharpe ratio describes what premium (E(RA) - Ri) the investor will receive for each addition unit of risk. This ratio is written in such expression: where portfolio, C asset. ) is average rate of retum of a stock or its risk; Rf - rate of retum of a riskless Beta coefficient This coefficient describes the level of sensitivity of an asset with number i asset to the entire market volatility (Shanken, 1992). According to Fleuriet (2003), beta coefficient represents systemic risk of the asset comparing asset's change of the price with financial market's fiuctuation. The coefficient has such mathematical expression: i=Tf> = ^ . where CJJC-. represents covariance between the rate of retum of the asset with a number i and market portfolio, cr^. - variance of rate of retum of the market portfolio. Asset which as beta coefficient is equal to 1, has a risk equal to the risk of the market. Numerical value lower than 1 means that asset is less risky than entire market. Beta coefficient of entire portfolio is denoted this way: Equality below represents connection between average retum of a single asset and its beta coefficient: -148- Eimutis Valakevicius, Kristina Vaznelyte. Development, Implementation and Evaluation... Practical implementation of models There were several stages of the research which helped to isolate certain methods, models or techniques and compare them properly. Before beginning of practical implementation of models in Lithuanian equity market, it was assumed that year has 252 trading days, because such number of days is considered a standard (Borodin et al., 2004). Therefore, when models were implemented with interims of 6 months, 126 days data were used, and 63 days data was applied for interims of 3 months. Length of the interim determines how often contents (which companies' stock will be included and what weigh each asset will have) of the investment portfolio will be updated. In addition to this, chosen interim length is essential factor which is used to regulate forecasts of results of the portfolios. Another stage is the implementation of technique of asset selection. All techniques, which were described earlier, were applied in Lithuanian equity market. Models which were differing only in one factor inclusion/exclusion of riskless asset were realised. Markowitz and CAPM models were implemented and comparative analysis was carried out. Due to large amount of the output data all results will not be presented in this article. Several examples of output are presented in Figure 2 and Figure 3. Figure 2 shows that the rate of retum of the Markowitz model, when 14 different companies' stocks are incorporated into the portfolio in both cases (when the model is compared with OMXV and OMBB), is higher, when value of the risk is similar. ratio on portfolio contents. Portfolios having higher numbers are located higher on the efficient frontier, this mean that they have both higher risk and higher rate of retum. Lower part of Figure 3 depicts how values of Sharpe ratio depend on interim number. The latter dependency is much more representative, values of Sharpe ratio depend more on Wellness of economic condition of the market rather than on of the composition of the portfolio. Comparison of rates of retum (based on OMXV) 0.QQ4Q Comparison of rates of retum (based on OMXBB) Figure 3. Dependence of Sharpe ratio values(depicted on y axis) on portfolio contents (upper part of the figure) and on interim number (lower part of the image), when efficient frontier portfolios are formed using 14 different assets and updated biannually Figure 2. Comparison of rates of retums (y axis) through periods (x axis) among index (blue line), efficient frontier (red line) and Markowitz model (including 14 different assets) being tested (green line) Upper part of Figure 3 shows graphical representation of comparative analysis dependence of values of Sharpe Results of the research The portfolios which were formed using equal asset weights method generally have tendency to provide lower rate of retum and higher risk, compared to portfolios, formed according Markowitz and CAPM models. Therefore it is not recommended to constmct the portfolio using equal asset weights method. However, if portfolio -149- Inzinerine Ekonomika-Engineering Economics, 2012, 23(2), 144-153 using this method has already been constmcted, then adding riskless asset is recommended. Most substantial disadvantage of CAPM and Markowitz models is that formal prerequisites have to be satisfied before the theoretical models are applied into practice. However, both literature review and this empirical research has shown that real market acts very unprcdictably and aforementioned models have to be modified by disregarding various prerequisites and only then applied to market. Application of CAPM and Markowitz models bring most benefits when market situation is stable or improving. This is because of the fact that in these cases future is successilly predicted using past historical data. After investigation of the portfolios constmcted according to Markowitz model, using the same amount of different companies' stocks, with and without incorporation of riskless asset, it was concluded that portfolios containing riskless asset generally convey several better characteristics simultaneously such as higher value of Sharpe's ratio, smaller value of beta coefficient and similar or higher value of rate of retum. Therefore it can be generally recommended to add riskless asset into the portfolio. If the portfolios are constmcted only according to some indices, it is better to choose index OMV Vilnius over OMX Baltic Benchmark, because OMX Vilnius conveys higher average value of Sharpe's ratio. When stocks are selected using correlation coefficients as well, there is no tendency for the portfolio's characteristics to improve when more stocks are added into portfolio. When stocks are selected using their price-to-eamings coefficients, higher amount of stocks in the portfolio conveys better characteristics, compared to the portfolio constructed from lower amount of companies' stocks. During recession it is recommended to invest only in riskless assets because stocks' rates of retum tend to be negative during crisis. Beta coefficient is not always appropriate characteristic to evaluate investments impartially. One of the reasons - lowest (i.e. best) values of beta coefficients were conveyed by portfolios which had lowest rate of retums. These portfolios were constructed using price-toeamings coefficient and updated quarterly. Conclusions When the portfolios constracted using different duration (3 months, 6 months and one year) of interims (ceteris paribus) were compared, it was noted that highest retums are conveyed by portfolios updated annually. The second best option is half of the year. If investor wants to get the highest premium for every additional unit of risk obtained, in this case it is recommended to form the investment portfolio using maximisation of utility function method when value of risk aversion coefficient A=3. After comparative analysis of OMX Baltic Benchmark index and portfolios, formed using Markowitz model, which have similar risk, it was determined that portfolios, constmcted using Markowitz model, having even low amount of different assets, have better values of various characteristics (rate of retum, Sharpe's ratio and beta coefficient) than those of the index. Aforementioned statement is not valid when very short interims (i.e. 3 months) are chosen; such duration is too short to form the portfolio having better characteristics than OMX Baltic Benchmark. Comparative analysis has shown that portfolios, constructed according to CAPM and Markowitz models, and located on the effective frontier, generally have better characteristics (higher rate or retum, higher value of Sharpe's ratio and lower value of beta coefficient) when they are formed using more different companies' assets, compared to portfolios formed similarly, but using less different assets. Also, CAPM yielded better results compared to Markowitz model. Optimal portfolio, which showed better characteristics than any other portfolio, was constructed out of 14 companies' stocks using capital asset pricing model and updated annually, had 0.0013/0.0008=1,625 times higher rate of retum than OMX Vilnius, 0.0013/0.0007=1,86 times higher than OMX Baltic Benchmark retum rate and 0.0027/0.0013=2,08 times lower than ideal portfolio (efficient frontier portfolio which gives maximum rate of retum at corresponding interim) rate of retum. Sharpe's ratio of optimal portfolio is 0.1399/0.0976=2,43 times higher than the one of OMX Vilnius and 0.1341/0.0951=1,41 times higher than OMX Baltic Benchmark Sharpe's ratio. Values of beta coefficients of optimal portfolio are also better than the ones of OMX Vilnius and OMX Baltic Benchmark. References Ang, A., & Chen, J. (2002). Asymmetric Correlations of Equity Portfolios. Journal of Financial Economics, 63(3), 443494. http://dx.doi.org/10.1016/S0304-405X(02)00068-5 Auditum. Akcijos rinkos kainos ir grynojo peino koeficientas. Retrieved from: http://www.auditum.lt/index.php/ finansiniu-rodikliu-skaiciuokles/20-investiciniai-rodikliai/70-akcijos-rinkos-kainos-ir-grynojo-pelno-koeficientas Basu, S. (1977). Investment Performance of Common Stocks in Relation to Their Price-Eamings Ratios: A Test of the Efficient Market Hypothesis. Journal of Finance, 32, 663-82. http://dx.doi.org/10.2307/2326304 Bodie, Z., Kane, A., & Marcus, A. J. (2004). Essentials of Investments. McGraw-Hill, 5th ed. Borodin, A., El-Yaniv, R., & Gogan, V. (2004). Can We Leam to Beat the Best Stock. Joumal of Artificial Intelligence Research. 21(1), 579-594. -150- Eimutis Valakevicius, Kristina Vaznelyte. Development, Implementation and Evaluation... Campbell, R. A., Koedijk, C. G., & Kofman, P. (2002). Increased Correlation in Bear Markets. Financial Analysts Journal, 58(1), 87-94. http://dx.doi.org/10.2469/faj.v58.nl.2512 Connor, G. A., & Korajczyk, R. (1993). Test for the Number of Factors in an Approximate Factor Model. Journal of Finance, 48, 1263-1292. http://dx.doi.org/10.2307/2329038 Fleuriet, M. (2003). Finance, a Fine Art. John Wiley & Sons Ltd. Glasserman, P., Heidelberger, P., & Shahabuddin, P. (2002). Portfolio Value-at-risk With Heavy-Tailed Risk Factors. Mathematical Finance, 12(3), 239-269. http://dx.doi.0rg/lO.l 111/1467-9965.00141 Grinblatt, M., & Titman, S. (1983). Factor Pricing in a Finite Economy. Journal of Financial Economics, 12, 495-507. http://dx.doi.org/10.1016/0304-405X(83)90046-6 Hagen, R. A. (1993). Modern Investment Theory. Prentice-Hall Intemational, Inc., 3rded. Jagannathan, R., & Wang, Z. (1996). The Conditional CAPM and the Cross-Section of Expected Retums. Journal of Finance,5\\, 3-53. http://dx.doi.org/10.2307/2329301 Junker, M., Szimayer, A., & Wagner, N. (2006). Nonlinear Term Stmeture Dependence: Copula Functions, Empirics and Risk Implications. Journal of Banking & Finance, 30(4), 1171-1199. http://dx.doi.Org/10.1016/j.jbankfin.2005.05.014 Kitt, R., & Kalda, J. ( 2005). Leptokurtic Portfolio Theory. European Physical Journal B, 50 (1-2). Longin, F. M., & Solnik, B. (2001). Extreme Correlation of Intemational Equity Markets. Journal of Finance, 56(2), 649676. http://dx.doi.0rg/lO.l 111/0022-1082.00340 Luenberger, D. G. (1998). Investment Science. Oxford University Press. Lietuvos Respublikos centrinis bankas. (2011). Govemment Securities Issuer, Ministry of Finance Records. Retrieved from: http://www.lbank.lt/wp/ Lietuvos vertybiniu popieriu komisija. svietimas/kur-investuoti/13162 Obligacijos, 2008. Retrieved from: http://www.vpk.lt/lt/investuotoju- Lintner, J. (1965). The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. Review of Economics and Statistics. Al, 13-37. http://dx.doi.org/10.2307/1924119 Malevergne, Y., & Somette, D. (2003). Testing the Gaussian Copula Hypothesis for Financial Assets Dependences. Quantitative Finance, 3(4), 231-250. http://dx.doi.Org/10.1088/1469-7688/3/4/301 Mansor, H. I. (2011). Stock Market Development and Macroeconomic Performance in Thailand. Inzinerine EkonomikaEngineering Economics, 22(3), 230-240. http://dx.doi.Org/10.5755/j0I.ee.22.3.513 Markowitz, H. M. (1952). Portfolio Selection. Journal of Finance. 7(1), 77-91. http://dx.doi.org/10.2307/2975974 Mossin, J. (1966). Equilibrium in a Capital Asset Market. Econometrica. 34, 768-783. http://dx.doi.org/10.2307/1910098 NASDAQ OMX Baltic (2011). Akciju ir indeksu istoriniai duomenys. Retrieved fi-om: http://www.nasdaqomxbaltic.com. Rachev, S. T., Stein, M., Sun, W. (2009/ Copula Concepts in Financial Markets. Retrieved fi-om: http://statistik.ets.kit.edu/download/Copula Concepts in Financial_Markets. Roll, R.(1977). A critique of the Asset Pricing Theory's Tests. Journal of Financial Economics 4, 129-176. http://dx.doi.org/10.1016/0304-405X(77)90009-5 Roll, R. (1980). An Empirical Investigation of the Arbitrage Theory. Joumal of Finance, 35, 1073-1103. http://dx.doi.org/10.2307/2327087 Ross, S. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory. 13, 341-60. http://dx.doi.org/10.1016/0022-0531(76)90046-6 Salmon, F. (2009). Recipe for Disaster: The Formula That Killed Wall Street. Retrieved from: http://www.wired.com/techbiz/it/magazine/17-03/wp quant?currentPage=all. Shanken, J. (1987). Multivariate Proxies and Asset Pricing Relations: Living with the Roll critique. Journal of Financial Economics, 18(1), 91-110. http://dx.doi.org/10.1016/0304-405X(87)90062-6 Shanken, J. (1992). On the Estimation of Beta-Pricing Models. Review of Financial Studies. 5, 1-33. http://dx.doi.0rg/lO.lO93/rfs/5.l.l Sharpe, W. (1964). Capital Asset Prices: a Theory of Market Equilibrium under Conditions of Risk. Journal of Finance, 33,885-901. Sharpe, W. F. (1964). The Sharpe Ratio. Portfolio Management, 21, 49-58. http://dx.doi.org/10.3905/jpm.1994.409501 Standard & Poor's (2011). GICS (Global Industry Classificafion Standard). Retrieved from: www.standardandpoors.com/indices/gics/en/us Valakevicius, E., & Zolyte, R. (2003). Lietuvos firmii akciju portfelio statistinis modelis ir jo tyrimas. Inzinerine Ekonomika-Engineering Economics{4), 7-12. -151 - Inzinerine Ekonomika-Engineering Economics, 2012, 23(2), 144-153 Valakevicius, E. (2007). Investiciju mokslas. Kaunas: Technologija. Valakevicius, E. (2008). Investavimas finansu rinkose. Kaunas: Technologija. Eimutis Valakevicius, Kdstina Vaznelyte Daugiapakopiq investavimo strategijq kOrimas, igyvendinimas ir vertinimas Santrauka Investieiniq portfeliu vidutin gr^zos nonna ir dzika priklauso nuo ivaidii veiksnii}, pavyzdziui, bendros ekonomins ir fmansins pasaulio, salies ar regiono padties, verslo sakos, kudai pdklauso kompanija bkls, kompanijos pelningumo ir kitii rodikliii (politiniii, socialiniii, ekologiniij faktodij). Visii veiksniij ir jn itakos, daromos akciju kainoms, tiksliai nustatyti neimanoma. Todl kudami, pritaikomi praktiskai ir vertinami bei tobulinami ivairs modeliai, skirti atdnkti ir paskirstyti aktyvams investieiniuose portfeliuose. Du svarbus dalykai, apie kuduos reikia pagalvoti pdes investuojant i vertybinius popierius - tai investavimo trukm ir tinkamii aktyvii pasidnkimas. Investieinius portfelius galima sudaryti is dzikingu ir nedzikingu vertybiniu popiedij. Akcijij pirkjai tampa bendroviii nuosavybs dalies turtojais. siuo darbu siekta patikdnti, kaip akcijn atdnkimo ir svodij portfeliuose optimizavimo bdai gali bOti pritaikyti Lietuvos vertybiniu popiedij dnkoje ( taip pat i portfelius buvo itraukiami nedzikingi aktyvai ir tidamas ju poveikis portfelio eharakteristikoms). Tyrimo tikslas: sukurti daugiapakopes investiciniii portfeliu formavimo strategijas, jas praktiskai pdtaikyti ir nustatyti, kud strategija yra tinkamiausia Lietuvos akciju dnkai. Uzdaviniai: (1) nustatyti optimalii pedodo trukm, kuda^ pasidnkus bendru atveju btu gaunami gedausias charaktedstiku reiksmes tudntys investieiniai portfeliai; (2) po tyrimo padaryti isvad^ apie gedausi^ akeiju atdnkimo portfeliams modeli atsizvelgiant i skirtingus prioritetus; (3) nustatyti, kuds is aktyvu paskirstymo portfelyje bOdii yra gedausias; (4) atlikus is keliu pakopu sudarytu modeliu ir populiadu akeiju birzos indeksu lyginam^^ analiz padaryti isvadas apie sumodeliuotu portfeliu kokyb. Tyrimo objektai - akcijos, egzistuojancios Lietuvos vertybiniu popiedii dnkoje, nedzikingoji grcizos norma ir Baltijos saliu akeiju indeksai. Siame tyrime buvo istirti tokie akeiju atdnkimo bOdai: maziausiai koreliuojanciu su kitomis akeiju suradimas, didziausi^ naudingumo fnkcijos reiksm tudniu akeiju atdnkimas, kai rizikos vengimo koeficientas A=3, ir kai A=10, didziausi^ grizi turinciu akeiju atdnkimas, ir didziausias P/E koeciento reiksmes tudnciu akeiju atdnkimas. Atrenkant akcijas pagal koreliaeijos koefieientus remiamasi faktu, kad teodskai investieiniai portfeliai, sudaryti is neigiamai koreliuotu akeiju, yra geresni nei is teigiamai koreliuotu. Taip yra todl, kad akeijos, kudu g^zu normos kinta pdesingomis kryptimis, sumazina portfelio dziki. Atdnkus tu imoniu akcijas, kudu koreliaeiju koeficientu vidurkiai yra maziausi, buvo patikdnta, ar sis akeiju atrinkimo metodas yra naudingesnis praktikoje nei kti metodai. Aktyvus atrenkant pagal naudingumo funkeijos reiksmes yra atsizvelgiama i individualaus investuotojo dzikos vengimo koeciento reiksm A. Atrenkant akeijas pagal didziausiu akeiju gr^zu normu metod^ yra neatsizvelgiama i rizik^. Sis metodas - atskiras naudingumo funkeijos mtodo atvejis. Kitas bodas tinkamu imoniu akeijoms padnkti yra P/E koeciento (plg. angl. pdceto-eamings ratio), nusakancio akcijos dnkos kainos ir grynojo peino santyki, suradimas. Sis koeeientas nusako tam tikros imons akeiju paklaus^ rinkoje - juo parodoma, kiek litu investuotojas sutiktu mokti uz vien^ \\\\X\\ imons peino. Investieiniam portfeliui sudaryti buvo pasidnkti tokie modeliai; Markoviciaus modelis, kai tam tikrai gr^zai reikia rasti maziausi^ galimi^ dziki^ nansiniu aktyvu ikainojimo modelis, portfeliu sudaryinas i investicini portfeli pde dzikingu aktyvu aibs pdjungiant nerizikingi aktyvi ir portfelio sudarymas is vienodus svodus tudnciu aktyvu. Pagdndiniai Markoviciaus ir CAPM modeliu trkumai - pdelaidu, neatitinkanciu tikrovs, darymas. Todl sie modeliai buvo modikuojami atmetant realybs neatitinkancias pdelaidas ir tada pdtaikomi Lietuvos vertybiniu popiedu rinkoje. Tyrimui paimtos NASDAQ OMX Baltic" Vilniaus birzos Ocialiajame sirase desimt arba daugiau metu (2001-01-01 - 2010-12-31) egzistuojanciu imoniu akcijos. Is dabar kotiruojamu 18 imoniu akeiju si^ sq^lygi tenkina 14 imoniu. Buvo sumodeliuoti ir testuoti portfeliai, kudu sudtis atnaujinama taikant tds skirtingos trukms pedodus - metu, puss metu ir tnJH mnesiu. Vykdant skaiciavimus buvo laikoma, kad metuose yra 252 prekybos akeijomis dienos. Atitinkamai, pasidnkus pusmecio trukms pedodi skaiciavimai buvo atliekami su 126 dienu imoniu akeiju duomenu tTiatdca, o pasidnkus ketvircio metu gio pedod% - su 63 dienu. Nuo pasidnktos pedodo trukms priklauso, kaip daznai bus atnaujinama investicinio portfelio sudtis ir kokios trukms pedodui bus sudaromos prognozs apie bsimi portfelio gr^z^. Sukurtu ir rinkoje pdtaikytu daugiapakopiu modeliu charaktedstikos buvo palygintos ne tik vienos su kitomis, bet ir su OMXV ir OMXBB indeksais. Buvo padarytos isvados, kudos galtu tapti pagdndu tolesniems tyrimams. Nustatyta, jog portfeliai, sudaryti is aktyvu su vienodais svodais, bendruoju atveju tud mazesn gri^z^ ir didesn dziki negu analogiski portfeliai, sudaryti pagal Markoviciaus modeli arba CAPM. Todl nereiktu sudaryti portfeliu paskirstant aktyvus portfelyje vienodais svodais. Jei vis dlto toks portfelis sudaromas, norint gauti kuo didesn gr^z^ reiktu itraukti i portfeli nedzikingi aktyv^. Ant efektyviojo portfeliu krasto esanciu portfeliu, sudarytu is didesnio kiekio aktyvu, (palyginus tuo paciu metodu sudarytais portfeliais is mazesnio skaiciaus aktyvii), eharaktedstikos bendruoju atveju yra geresns (didesn gr^zos norma, Sarpo rodiklio ir mazesn beta koeeiento reiksm). Palyginus is to paties skirtingu akeiju kiekio sudarytus portfelius pagal Markoviciaus modeli su analogiskais portfeliais, sudarytais itraukus nedzikingi aktyv