519 Wahrscheinlichkeiten, angewandte Mathematik
Refine
Has Fulltext
- yes (14)
Is part of the Bibliography
- yes (14)
Year of publication
Document Type
- Doctoral Thesis (9)
- Journal article (5)
Keywords
- A-priori-Wissen (2)
- Audit sampling (2)
- Extremwertstatistik (2)
- Konfidenzintervall (2)
- ARFIMA-Modell (1)
- ARFIMA-Modelle (1)
- ARMA-Modell (1)
- Agentenbasierte Modellierung (1)
- Anpassungstest (1)
- Approximation (1)
Institute
ResearcherID
- C-2593-2016 (1)
EU-Project number / Contract (GA) number
- 304617 (2)
Measurements of the centrality and rapidity dependence of inclusive jet production in \(\sqrt{^SNN}\)=5.02 TeV proton–lead (p+Pb) collisions and the jet cross-section in \(\sqrt{s}\)=2.76 TeV proton–proton collisions are presented. These quantities are measured in datasets corresponding to an integrated luminosity of 27.8 nb\(^{−1}\) and 4.0 pb\(^{−1}\), respectively, recorded with the ATLAS detector at the Large Hadron Collider in 2013. The p+Pb collision centrality was characterised using the total transverse energy measured in the pseudorapidity interval −4.9<η<−3.2 in the direction of the lead beam. Results are presented for the double-differential per-collision yields as a function of jet rapidity and transverse momentum (\(p_T\)) for minimum-bias and centrality-selected p+Pb collisions, and are compared to the jet rate from the geometric expectation. The total jet yield in minimum-bias events is slightly enhanced above the expectation in a \(p_T\)-dependent manner but is consistent with the expectation within uncertainties. The ratios of jet spectra from different centrality selections show a strong modification of jet production at all \(p_T\) at forward rapidities and for large \(p_T\) at mid-rapidity, which manifests as a suppression of the jet yield in central events and an enhancement in peripheral events. These effects imply that the factorisation between hard and soft processes is violated at an unexpected level in proton–nucleus collisions. Furthermore, the modifications at forward rapidities are found to be a function of the total jet energy only, implying that the violations may have a simple dependence on the hard parton–parton kinematics.
Based on the work of Eisenberg and Noe [2001], Suzuki [2002], Elsinger [2009] and Fischer [2014], we consider a generalization of Merton's asset valuation approach where n firms are linked by cross-ownership of equities and liabilities. Each firm is assumed to have a single outstanding liability, whereas its assets consist of one system-exogenous asset, as well as system-endogenous assets comprising some fraction of other firms' equity and liability, respectively. Following Fischer [2014], one can obtain no-arbitrage prices of equity and the recovery claims of liabilities as solutions of a fixed point problem, and hence obtain no-arbitrage prices of the `firm value' of each firm, which is the value of the firm's liability plus the firm's equity.
In a first step, we consider the two-firm case where explicit formulae for the no-arbitrage prices of the firm values are available (cf. Suzuki [2002]). Since firm values are derivatives of exogenous asset values, the distribution of firm values at maturity can be determined from the distribution of exogenous asset values. The Merton model and most of its known extensions do not account for the cross-ownership structure of the assets owned by the firm. Therefore the assumption of lognormally distributed exogenous assets leads to lognormally distributed firm values in such models, as the values of the liability and the equity add up to the exogenous asset's value (which has lognormal distribution by assumption). Our work therefore starts from lognormally distributed exogenous assets and reveals how cross-ownership, when correctly accounted for in the valuation process, affects the distribution of the firm value, which is not lognormal anymore. In a simulation study we examine the impact of several parameters (amount of cross-ownership of debt and equity, ratio of liabilities to expected exogenous assets value) on the differences between the distribution of firm values obtained from our model and correspondingly matched lognormal distributions. It becomes clear that the assumption of lognormally distributed firm values may lead to both over- and underestimation of the “true" firm values (within the cross-ownership model) and consequently of bankruptcy risk, too.
In a second step, the bankruptcy risk of one firm within the system is analyzed in more detail in a further simulation study, revealing that the correct incorporation of cross-ownership in the valuation procedure is the more important, the tighter the cross-ownership structure between the two firms. Furthermore, depending on the considered type of cross-ownership (debt or equity), the assumption of lognormally distributed firm values is likely to result in an over- resp. underestimation of the actual probability of default. In a similar vein, we consider the Value-at-Risk (VaR) of a firm in the system, which we calculate as the negative α-quantile of the firm value at maturity minus the firm's risk neutral price in t=0, i.e. we consider the (1-α)100%-VaR of the change in firm value. If we let the cross-ownership fractions (i.e. the fraction that one firm holds of another firm's debt or equity) converge to 1 (which is the supremum of the possible values that cross-ownership fractions can take), we can prove that in a system of two firms, the lognormal model will over- resp. underestimate both univariate and bivariate probabilities of default under cross-ownership of debt only resp. cross-ownership of equity only. Furthermore, we provide a formula that allows us to check for an arbitrary scenario of cross-ownership and any non-negative distribution of exogenous assets whether the approximating lognormal model will over- or underestimate the related probability of default of a firm. In particular, any given non-negative distribution of exogenous asset values (non-degenerate in a certain sense) can be transformed into a new, “extreme" distribution of exogenous assets yielding such a low or high actual probability of default that the approximating lognormal model will over- and underestimate this risk, respectively.
After this analysis of the univariate distribution of firm values under cross-ownership in a system of two firms with bivariately lognormally distributed exogenous asset values, we consider the copula of these firm values as a distribution-free measure of the dependency between these firm values. Without cross-ownership, this copula would be the Gaussian copula. Under cross-ownership, we especially consider the behaviour of the copula of firm values in the lower left and upper right corner of the unit square, and depending on the type of cross-ownership and the considered corner, we either obtain error bounds as to how good the copula of firm values under cross-ownership can be approximated with the Gaussian copula, or we see that the copula of firm values can be written as the copula of two linear combinations of exogenous asset values (note that these linear combinations are not lognormally distributed). These insights serve as a basis for our analysis of the tail dependence coefficient of firm values under cross-ownership. Under cross-ownership of debt only, firm values remain upper tail independent, whereas they become perfectly lower tail dependent if the correlation between exogenous asset values exceeds a certain positive threshold, which does not depend on the exact level of cross-ownership. Under cross-ownership of equity only, the situation is reverse in that firm values always remain lower tail independent, but upper tail independence is preserved if and only if the right tail behaviour of both firms’ values is determined by the right tail behaviour of the firms’ own exogenous asset value instead of the respective other firm’s exogenous asset value.
Next, we return to systems of n≥2 firms and analyze sensitivities of no-arbitrage prices of equity and the recovery claims of liabilities with respect to the model parameters. In the literature, such sensitivities are provided with respect to exogenous asset values by Gouriéroux et al. [2012], and we extend the existing results by considering how these no-arbitrage prices depend on the cross-ownership fractions and the level of liabilities. For the former, we can show that all prices are non-decreasing in any cross-ownership fraction in the model, and by use of a version of the Implicit Function Theorem we can also determine exact derivatives. For the latter, we show that the recovery value of debt and the equity value of a firm are non-decreasing and non-increasing in the firm's nominal level of liabilities, respectively, but the firm value is in general not monotone in the firm's level of liabilities. Furthermore, no-arbitrage prices of equity and the recovery claims of liabilities of a firm are in general non-monotone in the nominal level of liabilities of other firms in the system. If we confine ourselves to one type of cross-ownership (i.e. debt or equity), we can derive more precise relationships. All the results can be transferred to risk-neutral prices before maturity.
Finally, following Gouriéroux et al. [2012] and as a kind of extension to the above sensitivity results, we consider how immediate changes in exogenous asset values of one or more firms at maturity affect the financial health of a system of n initially solvent firms. We start with some theoretical considerations on what we call the contagion effect, namely the change in the endogenous asset value of a firm caused by shocks on the exogenous assets of firms within the system. For the two-firm case, an explicit formula is available, making clear that in general (and in particular under cross-ownership of equity only), the effect of contagion can be positive as well as negative, i.e. it can both, mitigate and exacerbate the change in the exogenous asset value of a firm. On the other hand, we cannot generally say that a tighter cross-ownership structure leads to bigger absolute contagion effects. Under cross-ownership of debt only, firms cannot profit from positive shocks beyond the direct effect on exogenous assets, as the contagion effect is always non-positive. Next, we are concerned with spillover effects of negative shocks on a subset of firms to other firms in the system (experiencing non-negative shocks themselves), driving them into default due to large losses in their endogenous asset values. Extending the results of Glasserman and Young [2015], we provide a necessary condition for the shock to cause such an event. This also yields an upper bound for the probability of such an event. We further investigate how the stability of a system of firms exposed to multiple shocks depends on the model parameters in a simulation study. In doing so, we consider three network types (incomplete, core-periphery and ring network) with simultaneous shocks on some of the firms and wiping out a certain percentage of their exogenous assets. Then we analyze for all three types of cross-ownership (debt only, equity only, both debt and equity) how the shock intensity, the shock size, and network parameters as the number of links in the network and the proportion of a firm's debt or equity held within the system of firms influences several output parameters, comprising the total number of defaults and the relative loss in the sum of firm values, among others. Comparing our results to the studies of Nier et al. [2007], Gai and Kapadia [2010] and Elliott et al. [2014], we can only partly confirm their results with respect to the number of defaults. We conclude our work with a theoretical comparison of the complete network (where each firm holds a part of any other firm) and the ring network with respect to the number of defaults caused by a shock on a single firm, as it is done by Allen and Gale [2000]. In line with the literature, we find that under cross-ownership of debt only, complete networks are “robust yet fragile" [Gai and Kapadia, 2010] in that moderate shocks can be completely withstood or drive the firm directly hit by the shock in default, but as soon as the shock exceeds a certain size, all firms are simultaneously in default. In contrast to that, firms default one by one in the ring network, with the first “contagious default" (i.e. a default of a firm not directly hit by the shock) already occurs for smaller shock sizes than under the complete network.
The purpose of confidence and prediction intervals is to provide an interval estimation for an unknown distribution parameter or the future value of a phenomenon. In many applications, prior knowledge about the distribution parameter is available, but rarely made use of, unless in a Bayesian framework. This thesis provides exact frequentist confidence intervals of minimal volume exploiting prior information. The scheme is applied to distribution parameters of the binomial and the Poisson distribution. The Bayesian approach to obtain intervals on a distribution parameter in form of credibility intervals is considered, with particular emphasis on the binomial distribution. An application of interval estimation is found in auditing, where two-sided intervals of Stringer type are meant to contain the mean of a zero-inflated population. In the context of time series analysis, covariates are supposed to improve the prediction of future values. Exponential smoothing with covariates as an extension of the popular forecasting method exponential smoothing is considered in this thesis. A double-seasonality version of it is applied to forecast hourly electricity load under the use of meteorological covariates. Different kinds of prediction intervals for exponential smoothing with covariates are formulated.
Das Ziel der Arbeit ist eine Zusammenfassung über den Stand der Forschung über das Thema der fraktionalen Integration und Kointegration sowie Weiterentwicklungen der aktuellen Methoden im Hinblick darauf, dass sie robuster auf eine Reihe von empirischen Gegebenheiten anwendbar sind. Hierzu wurden insbesondere die Möglichkeiten von Strukturbrüchen in deterministischen Prozessanteilen vorgeschlagen sowie deren Auswirkungen auf Schätzeigenschaften analysiert. Mit diesem Wissen können Schätzstrategien entwickelt werden, die auch im empirischen Teil der Arbeit angewandt wurden.
Der Aufbau der Arbeit gestaltet sich so, dass nach der Einleitung und Problemstellung im zweiten Kapitel der Arbeit zunächst in die Zeitreihenanalyse eingeführt wird. Hierbei wird auch eine intuitive Motivation für die Betrachtung von Long-Memory-Prozessen gegeben. Diese gestaltet sich so, dass der klassischerweise als ganzzahlig angenommene Integrationsgrad eines Prozesses nun jede beliebige Zahl, also auch Brüche, annehmen kann. Diese Annahme führt wiederum dazu, dass hiermit sehr langfristige Abhängigkeiten von Zeitreihen effizient beschrieben werden können, da diese lediglich von einem einzigen Parameter abhängen.
Die Schätzung dieses nunmehr fraktionalen Integrationsgrads wird im dritten Kapitel ausführlich beschrieben und in mehreren Simulationsstudien ausgiebig analysiert. Hierzu werden neben parametrischen Schätzmethoden, die einer genauen Spezifizierung der Korrelationsstruktur von Zeitreihen bedürfen, auch semiparametrische Methoden angeführt, die in der Praxis robuster einsetzbar sind, da ihre Schätzgenauigkeit und Effizienz nicht von einer korrekten Klassifizierung von sog. Short-Memory-Komponenten beeinflusst werden. Die Analyse dieser Methode erfolgt in erster Linie im Hinblick auf eine empirische Anwendbarkeit und bietet auch als Ergebnis Empfehlungen für eine optimale Schätzstrategie.
Das vierte Kapitel beschäftigt sich in erster Linie mit Integrationstests wie z.B. Einheitswurzeltests und deren Anwendbarkeit bei Existenz von Long-Memory-Prozessbestandteilen. Darüber hinaus werden auch Schätz- und Testmethoden für das Vorliegen von deterministischen Trends thematisiert, die wiederum auch die Möglichkeit von Strukturbrüchen zulassen.
Eine multivariate Betrachtungsweise ermöglicht das fünfte Kapitel mit der Einführung der fraktionalen Kointegration. Auch liegt der Fokus der Arbeit darin, die empirische Anwendbarkeit zu verbessern, indem in Simulationsstudien Effekte von empirischen Gegebenheiten - wie Strukturbrüche - analysiert und optimale Schätzstrategien entwickelt werden.
Im sechsten Kapitel der Arbeit wird im Rahmen der ökonomischen Theorie der Markterwartungshypothese die Verzinsung deutscher im Zeitraum Oktober 1998 bis November 2011 untersucht. Diese Hypothese impliziert, dass zwischen den einzelnen Zinssätzen eine multivariate Beziehung in Form von Kointegrationsbeziehungen bestehen sollte, da die Zinssatzdifferenzen einer Liquiditätsprämie entsprechen. Von dieser wurde in bisherigen Studien angenommen, dass sie stationär ist, d.h. dass sie allenfalls eine Short-Memory-Eigenschaft aufweist, welche nur relativ kurzfristige Abhängigkeit impliziert. Von dieser Sichtweise löst sich die Arbeit, indem sie die Möglichkeit von fraktionalen Kointegrationsbeziehungen ermöglicht, die eine Aussage über die Persistenz der Liquiditätsprämie ermöglicht.
Im Rahmen dieser Analyse konnten eine Reihe interessanter Erkenntnisse gewonnen werden, wie z.B. dass das Ausmaß der Persistenz (d.h. die Trägheit der Anpassung auf ökonomische Schocks) mit ansteigender Laufzeitdifferenz sukzessive größer wird und auch nicht mehr durch klassisch angenommene Prozessstrukturen erklärt werden kann. Nichtsdestotrotz können die Ergebnisse der empirischen Analyse die Annahmen der Markterwartungshypothese nicht bestätigen, da insbesondere der Integrationsgrad für sehr lange Laufzeitdifferenzen so groß ausfällt, dass selbst eine relativ schwache fraktionale Kointegrationsbeziehung abgelehnt werden muss.