Upravljanje portfeljima

master's thesis

• Title (english): Portfolio management
• Author: Sara Bonassin
• Mentor: Nela Bosner (mentor)
• Committee member: Nela Bosner (predsjednik povjerenstva)
• Committee member: Eduard Marušić-Paloka (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-03-04, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

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.

Abstract (english)

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.

• Keywords: investicije u vrijednosnice; teorija portfelja; iterativne metode; programski jezik MATLAB
• Keywords (english): investments in securities; portfolio theory; iterative methods; MATLAB
• Language: croatian
• URN:NBN: urn:nbn:hr:217:915896
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-03-23 10:55:37