Vrstilne statistike in tvegana vrednost : delo diplomskega seminarja

Candellari, Nikolaj (author), Kokol-Bukovšek, Damjana (mentor), Toman, Aleš (co-mentor)

V delu diplomskega seminarja smo rešili nalogo, s katero se srečujejo vse banke: kako določiti količino kapitala, ki ga imamo v rezervi in ki služi preprečevanju nesolventnosti, da bomo zavarovani pred morebitnimi nepričakovanimi izgubami. Določitev regulatornega kapitala, kot ga imenujemo, je odvisna od predpisane stopnje tveganja, ki jo določi regulator bančnega trga. Naša naloga je, da določimo delež vrednosti portfelja, ki ga moramo držati v rezervi. To je pomembna naloga, saj pri podcenitvi le-tega tvegamo nesolvetnost ali kazen regulatorja, pri njegovi precenitvi pa nastanejo neželeni oportunitetni stroški. V delu diplomskega seminarja smo potrebno višino regulatornega kapitala ocenjevali z vrstilnimi statistikami. Analitično in s pomočjo simulacije smo primerjali štiri različne cenilke. Za izhodišče smo vedno vzeli zadnjih 250 zaporednih dnevnih logaritemskih donosov portfelja. Cenilke smo izbrali preko teorije vrstilnih statistik. Za ocenjevanje višine regulatornega kapitala pri 99 % stopnji tveganja smo kot potencialne cenilke videli drugo vrstilno statistiko, tretjo vrstilno statistiko, njuno povprečje ter drugo vrstilno statistiko, kjer smo vzorec zmanjšali na zadnjih 200 dnevnih logaritemskih donosov. Po izvedeni simulaciji smo zaključili, da sta med štirimi cenilkami nepristranski povprečje med drugo in tretjo vrstilno statistiko na celem vzorcu in druga vrstilna statistika na zmanjšanem vzorcu, vendar ima slednja večji standardni odklon. Kot najboljšo cenilko zato razglasimo povprečje med drugo in tretjo vrstilno statistiko. Poleg tega smo ugotovili, da cenilke ocenjujejo podobno ne glede na porazdelitev logaritemskih donosov portfelja, kar je dobrodošlo spoznanje za uporabo opisanih cenilk v praksi.

• Language: Slovenian
• Keywords: tvegana vrednost, vrstilna statistika, regulatorni kapital
• Work type: Bachelor thesis/paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2021
• PID: 20.500.12556/RUL-125151
• UDC: 519.2
• COBISS.SI-ID: 58579971
• Publication date in RUL: 05.03.2021

Secondary language

• Language: English
• Title: Order statistics and value at risk

Abstract

In this diploma thesis, we tackled the task that all banks face. How to determine the amount of capital they must hold in reserve to prevent insolvency and provide insurance against possible unexpected losses. The amount of regulatory capital, as we call it, depends on the level of risk set by the banking market regulator. Our task was then to calculate the proportion of the portfolio value that we need to hold in reserve. This is an important task as underestimating regulatory capital increases the risk of insolvency or punishment by the regulator. On the other hand, overestimating the capital leads to undesirable opportunity costs. In this diploma thesis, we used order statistics to estimate regulatory capital. We compared four different estimators analytically and using simulations. All estimators were supported by the theory of order statistics and based on the last 250 daily logarithmic returns of the portfolio. To estimate the regulatory capital corresponding to the 99% risk level, possible estimators are the second and third order statistics, their average, and the second order statistic based on the reduced sample of the last 200 daily logarithmic returns. Using simulations, we concluded that the average between the second and third order statistics and the second order statistic on the reduced sample are unbiased estimators, and the latter has a larger standard deviation. Therefore, the best estimator is the average between the second and third order statistics. We also found that the performance of the estimators is independent of the logarithmic returns distribution, which further supports the use of these estimators in practice.

• Keywords: value at risk, order statistic, regulatory capital