Hierarchical risk aggregation using aggregation trees and copulas : magistrsko delo

Ivanova, Lina (author), Stopar, Nik (mentor), Milošević, Bojana (comentor)

Several historical financial crises have repeatedly demonstrated the importance of accurately modeling dependence structures between different sources of risk. Financial institutions face significant challenges in managing and aggregating risks across portfolios while at the same time assessing possible losses and adhering to ever more demanding regulatory requirements. In particular, traditional risk aggregation techniques based on linear correlation and multivariate normal distribution have proven inadequate, as they fail to capture nonlinear, asymmetric, and tail-dependent behavior that becomes especially pronounced during periods of market stress. This thesis studies portfolio risk aggregation using copula-based methods, with a particular focus on hierarchical risk aggregation. Copulas provide a flexible framework for modeling dependence separately from marginal distributions, allowing for more realistic representations of joint extreme events. However, standard multivariate copulas become restrictive and difficult to interpret in high-dimensional settings. Hierarchical aggregation addresses this limitation by decomposing a high-dimensional portfolio into a tree structure, where dependence is modeled locally between low-dimensional sub-portfolios and aggregated step by step into a total portfolio. Methodologically, the thesis combines parametric modeling of the marginal distributions with copula-based dependence modeling and simulation techniques. Marginal distributions are estimated using heavy-tailed models, including the Student-$t$ and Lévy-stable distributions, while dependence is captured using a combination of different copulas. Estimation is carried out using the inference functions for margins (IFM) approach. The joint distribution of aggregated portfolio return is approximated via the sample reordering algorithm, enabling the computation of risk measures such as Value-at-Risk and Expected Shortfall at the portfolio level. The empirical analysis demonstrates that hierarchical copula-based aggregation provides a substantial improvement over the traditional approaches based on Gaussian distribution, particularly in capturing tail dependence and extreme losses. The results highlight the importance of modeling dependence in a flexible and structured way and show that hierarchical copulas offer a powerful and practical tool for risk aggregation in modern financial risk management.

• Language: English
• Keywords: copula, risk, hierarchical aggregation, aggregation trees, risk measures, sample reordering algorithm
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2026
• PID: 20.500.12556/RUL-184289
• UDC: 519.6
• COBISS.SI-ID: 283753219
• Publication date in RUL: 03.07.2026

Secondary language

• Language: Slovenian
• Title: Hierarhična agregacija tveganja z uporabo agregacijskih dreves in kopul

Abstract

Več zgodovinskih finančnih kriz je pokazalo, kako pomembno je natančno modeliranje odvisnosti med različnimi viri tveganja. Finančne institucije se soočajo z velikimi izzivi pri upravljanju in agregaciji tveganj v portfeljih, hkrati pa morajo ocenjevati morebitne izgube in izpolnjevati vse zahtevnejše regulativne zahteve. Tradicionalne tehnike agregacije tveganj, ki temeljijo na linearni korelaciji in večrazsežni normalni porazdelitvi, so se izkazale za neustrezne, saj ne zajamejo nelinearne, asimetrične in repno odvisne dinamike, ki postane posebej izrazita v obdobjih tržnih stresov. V magistrskem delu preučujemo agregacijo portfeljskega tveganja z metodami na osnovi kopul, s posebnim poudarkom na hierarhični agregaciji tveganj. Kopule zagotavljajo fleksibilen okvir za modeliranje odvisnosti ločeno od robnih porazdelitev, kar omogoča bolj realistične predstavitve skupnih ekstremnih dogodkov. Standardne večrazsežne kopule pa postanejo omejujoče in težko interpretabilne pri visokih dimenzijah. Hierarhična agregacija to omejitev odpravlja z razgradnjo visokorazsežnega portfelja v drevesno strukturo, kjer se odvisnost modelira lokalno med nizkodimenzionalnimi podportfelji in postopno agregira v skupni portfelj. Metodološko delo združuje parametrično modeliranje robnih porazdelitev in modeliranjem odvisnosti s kopulami ter simulacijskimi tehnikami. Robne porazdelitve so ocenjene z modeli s težkimi repi, vključno s Studentovo $t$-porazdelitvijo in Lévyjevo stabilno porazdelitvijo, odvisnost pa je zajeta z kombinacijo različnih kopul. Ocenjevanje poteka z metodo sklepanja po robnih funkcijah (IFM). Skupna porazdelitev agregiranega donosa portfelja je aproksimirana z algoritmom preurejanja vzorcev, kar omogoča izračun mer tveganja, kot sta VaR (ang. Value-At-Risk) in ES (ang. Expected Shortfall), na ravni portfelja. Empirična analiza kaže, da hierarhična agregacija na osnovi kopul zagotavlja bistveno izboljšanje v primerjavi s tradicionalnimi pristopi, ki temeljijo na Gaussovi porazdelitvi, še posebej pri zajemanju repne odvisnosti in ekstremnih izgub. Rezultati poudarjajo pomen fleksibilnega in strukturiranega modeliranja odvisnosti ter kažejo, da hierarhične kopule predstavljajo zmogljivo in praktično orodje za agregacijo tveganj v sodobnem finančnem upravljanju tveganj.

• Keywords: kopula, tveganje, hierarhično agregiranje, agregacijska drevesa, mere tveganja, algoritem preurejanja vzorcev