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Cilj ovog rada bio je prikazati primjenu neglatke optimizacije u teoriji portfelja. Najprije smo dali pregled teorije koja je vezana uz povrate portfelja, njihove distribucije, učinkovitost i rizičnost. Nakon toga smo prikazali najpoznatije mjere rizika koje se koriste u optimizaciji portfelja te sugerirali korištenje mjere CVaR jer je koherentna i može se koristiti u situacijama kada gubici nisu normalno distribuirani. U drugome poglavlju prikazali smo model optimizacije portfelja u kontekstu mjere rizika CVaR. Budući da izraz za CVaR sadrži funkciju \(x^+=\max (0,x) \), to je problem neglatke optimizacije. Rješavanju problema moguće je pristupiti na dva načina: transformirati zadaću u problem linearnog programiranja ili direktno riješiti zadaću pogodnom metodom neglatke optimizacije. U radu smo predstavili oba načina, uz poseban naglasak na direktnu primjenu diskretne gradijentne metode. Takav pristup daje učinkovitije rješenje, pogotovo u situacijama kada želimo pronaći optimalne portfelje, a promatramo veliki broj scenarija. The aim of this thesis was to show application of non-smooth optimization in portfolio theory. First we gave an overview on theory which is related to portfolio returns, distribution of returns, efficiency and riskiness. After that we introduced the most popular risk measures in portfolio optimization and suggested use of CVaR because it is coherent and it can be used even if distribution of returns is not normal. In the second chapter we described portfolio optimization model in the context of CVaR measure. Since expression for CVaR contains a function \(x^+=\max (0,x) \), it is a non-smooth optimization problem. To solve this problem, one could use two ways: transform the problem into a linear programming model or directly solve the problem using an appropriate non-smooth method. In this thesis we introduced both ways, with special emphasis on direct application of discrete gradient method. Such approach gives a more efficient solution, especially when we want to find optimal solution for portfolios with large number of scenarios.

Upravljanje portfeljima je organizacija financijske imovine ulagača radi smanjenja rizika i maksimiziranja povrata. U ovom radu uvodimo mjeru za ishode pojedinih investicija u vrijednosnice te mjeru za nesigurnost ostvarivanja povrata pomoću matematičkog očekivanja i varijance slučajnih varijabli koje modeliraju povrate. Moderna teorija portfelja nudi rezultate kojima računamo portfelj minimalne varijance te efikasne portfelje, preferirane na tržištima kapitala, jer za odabrani očekivani povrat, imaju najmanji mogući rizik. Da bi izbjegli direktno rješavanje sustava linearnih jednadžbi s kovarijacijskom matricom, koristimo iterativne metode. Ukratko su opisane Jacobijeva, Gauss-Seidelova te SOR metoda. Predstavljeni su najbitniji teoremi konvergencije tih metoda te algoritmi implementirani u programskom jeziku MATLAB. Model vrednovanja kapitalne imovine promatra cijelo tržište te promjene u povratu na portfelj uslijed fluktuacija na tržištu. Takvo kretanje je opisano pomoću beta faktora portfelja. Predstavlja sistemski rizik kojeg nije moguće diversificirati dok se nesistemski rizici mogu ukloniti i to ulaganjem u tržišni portfelj. Odnos izmedu povrata na dani portfelj i povrata na tržišni portfelj opisan je linearnom vezom koja se u praksi procjenjuje linearnom regresijom. Opisane su metode za rješavanje linearnog problema najmanjih kvadrata uz pripadne algoritme: rješavanje sustava normalnih jednadžbi te QR faktorizacija kao preferirana metoda koja djeluje na matricu sustava i koja je numerički stabilnija. Portfolio management is the managing of an individual’s financial assets in terms of minimum risk and maximum return. In this thesis we introduce a measure for the outcomes of single investments in securities and a measure for the uncertainty of returns by defining them in the terms of expectation and variance of random variables that describe returns. Modern portfolio theory allows us to calculate a minimum variance portfolio and effcient portfolios, favored on capital markets for minimizing the risk for a given expected return. To avoid any direct solving of systems of linear equations with a covariance matrix, we use iterative methods of which are described the Jacobi, Gauss-Seidel and SOR method. The main convergence theorems are presented along with the algorithms implemented in MATLAB. The Capital Asset Pricing Model takes in consideration the whole market and the variations in the return on a portfolio due to market fluctuations. These variations are described with the beta factor of a portfolio. This value represents the undiversifiable risk, while the diversifiable risk can be eliminated by investing in the market portfolio. The relation between the return on a portfolio and the return on the market portfolio is linear and it is estimated with linear regression. We describe the methods for solving the linear least squares problem and their algorithms: solving the system of normal equations and the QR factorisation preferred for working directly on the system matrix and its numerical stability.

Tourism is currently one of the most important sectors for the economic development of the Republic of Croatia. It mainly focuses on foreigners from within the EU. Because of the dynamic and very competitive tourist market, it is hard to forecast foreign tourism demand nowadays. It can vary over time among tourists of different nationalities. Stability of inbound tourist demand forms an important condition for the development of tourism and foreign currency income. Considering that tourism policy-makers must distribute available resources to different tourism markets for use in promotion, the purpose of this study is to analyze, by country of origin, the number of overnight stays by inbound foreign tourists in accommodation establishments as well as their average daily spending in Croatia and to construct an optimal mixture of tourists of different nationalities that will help tourism policy-makers to optimize or maximize tourism revenues at a certain level of risk. The main idea of this research is to apply financial portfolio theory to Croatian tourism demand and to construct an optimal mixture of foreign inbound tourists when there is an infinite number of possibilities. Several optimal mixes were calculated with different risk/return options to show on which foreign tourist markets Croatia must focus. For example, to achieve the mixture of foreign tourists that provides the highest level of tourist consumption expenditures, tourism authorities should increase their resources on the German, Slovenian, Italian and Austrian markets. The results of this research can easily be modified according to policy maker’s risk/return preferences.

Priprema projekta vjetroelektrane obuhvaća mnogo međusobno povezanih koraka, a jedan od najvažnijih je svakako pravilna procjena vjetropotencijala. Procjena vjetropotencijala tipično se temelji na mjerenju karakteristika vjetra korištenjem mjernog stupa, te modeliranjem unutar nekog softverskog/matematičkog modela. U situacijama kada na lokaciji potencijalne vjetroelektrane postoji nekoliko mjernih stupova i mjernih uređaja, često se postavlja pitanje kako optimalno iskoristiti te podatke. U ovom radu prikazana je metoda korištenja takvih podataka kroz primjenu teorije portfelja, etablirane u ekonomskoj teoriji, ali i u mnogim drugim znanstvenim disciplinama. Metoda je vrlo fleksibilna u smislu ulaznih podataka i softverskih/matematičkih modela, a rezultati dobiveni unutar rada pokazuju da je njezinom primjenom moguće povećati točnost i smanjiti nesigurnost procjene vjetropotencijala. Prikazana metoda otvara širok prostor za daljnja istraživanja i poboljšanja, a sve u svrhu boljih rezultata procjene vjetropotencijala i u konačnici kvalitetnije pripremljenog projekta vjetroelektrane. Development of a wind farm project includes a lot of interconnected steps, with one of the most important ones being a proper energy yield assesment. Wind energy yield assesment is typically based on wind measurements on a measurement mast that are later modeled in one of the software/mathematical models. In cases where there are multiple wind measurements on the potential wind farm site, a question arises on how to optimally use all the available data. This paper shows a method of using such data through the application of the portfolio theory, a well established theory in economics, but also in other scientific disciplines. The method shown is very flexible in terms of input data and software/mathematical models, and the results of its application show that it is possible to increase accuracy and reduce uncertainty of energy yield assesment. The method opens up a wide space for further research and improvements, all with the objective of achieving better results of energy yield assesment and finally, better prepared wind project. The objective of the research was to improve wind potential assessment by using portfolio theory in cases when there are more wind measurements on a prospective wind farm site, as well as to increase the efficiency of using the avaliable wind measurement data. The basic hypothesis was that by using more than one measurement it is possible to increase the accuracy and reduce uncertainty of wind potential assessment, compared to using only one measurement for the entire location. Also, by using available measurement from nearby measurement masts, it is possible to assess the potential on a reference position. The paper describes the method of using portfolio theory for wind potential assessment, that can be used to upgrade the standard method. The method can be used when there are multiple measurements on site, it is flexible enough to be used with new measurement technologies (i.e. SODAR, LIDAR), as well as new wind flow models (i.e. CFD models). The method is realized through three key components: i) standard wind potential assessment procedure, ii) application of portfolio theory and iii) analysis and interpretation of results. The fist component represents the standard method of wind potential assessment that is usually done for any planned wind farm project. It consists of preparation and processing of measurement data, preparation and processing of orography and roughness data, and application of mathematical model for wind potential assessment – in this paper EMD's WindPro with integrated WAsP. In standard method, this procedure is done once, using mostly one prepared set of measuremet data (the data set can be a combination of several measurement sets, but in the end it is only one input data set). This paper offers a different apporach – each measurement set is treated separately and independently of the remaining measurements. In other words, if there are multiple measurement devices on a measurement mast on different heights, easch measurement is treated separately. Consequently, there are multiple results of wind potential assessment, each based on its own data set. For each calculated assessment, uncertainty of the assessment is calculated separately, as described in detail in the paper. Calculated wind potential and uncertainty represent a pair of input data that will be used for applying portfolio theory. Correlation of measurements, required for using portfolio theory, is also calculated. The second component of the descirbed method is based on applying the laws of portfolio theory to calculated input data with given shares (in this paper in steps of 10%). The paper first shows combinations of two measurements in different heights and positions, and later describes the application to multiple data sets. The results show that it is possible to reach better accuracy and lower uncertainty by using the proposed method. The third component of the method is analysis and interpretation of results. The results must be critically observed and related to reality, so that proper conclusions can be drawn. In this paper there are several analyses done: the influence of uncertainty, reference measurement in shorter period, lack of reference measurement, connection of wind speed (m/s) and production (kWh) and using portfolios with minimal error and uncertainty. The latter shows very promising results for further research. The reults show that when we use shorter period measurements for calculating optimal shares (the shares where both error and uncertainty are equal to zero), we can transfer those shares to longer term meaurements and get very good results. In this case it is recommended to have at least one measurement on a proposed wind generator site, even in a shorter period, as the quality of results increases dramatically. The analyis has also shown that it is possible to apply the method for vertical extrapolation, instead of logarithmic law. Also, some „common truths“ are questioned here. For example, this paper showed that the measurement closer to the reference does not need to be more accurate than a measurement further away. In this case, measurement four kilometers away was more accurate than the one two kilometers away. This paper only opens up the topic of applying portfolio theory in wind potential assessment. There is a wide space for further application and research, that would enhance this method, but also the standard method of wind potential assessment.