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We analyze the fluctuation of the loss from default around its large portfolio limit in a class of reduced-form models of correlated firm-by-firm default timing. We prove a weak convergence result for the fluctuation process and use it for developing a conditionally Gaussian approximation to the loss distribution. Numerical results illustrate the accuracy and computational efficiency of the approximation.
In this paper, we obtain the finite-horizon and infinite-horizon ruin probability asymptotics for risk processes with claims of subexponential tails for non-stationary arrival processes that satisfy a large deviation principle. As a result, the arrival process can be dependent, non-stationary and non-renewal. We give three examples of non-stationary and non-renewal point processes: Hawkes process, Cox process with shot noise intensity and self-correcting point process. We also show some aggregate claims results for these three examples.
We solve a version of the optimal trade execution problem when the mid asset price follows a displaced diffusion. Optimal strategies in the adapted class under various risk criteria, namely value-at-risk, expected shortfall and a new criterion called "squared asset expectation" (SAE), related to a version of the cost variance measure, are derived and compared. It is well known that displaced diffusions (DD) exhibit dynamics which are in-between arithmetic Brownian motions (ABM) and geometric Brownian motions (GBM) depending of the choice of the shift parameter. Furthermore, DD allows for changes in the support of the mid asset price distribution, allowing one to include a minimum permitted value for the mid price, either positive or negative. We study the dependence of the optimal solution on the choice of the risk aversion criterion. Optimal solutions across criteria and asset dynamics are comparable although differences are not negligible for high levels of risk aversion and low market impact assets. This is illustrated with numerical examples.
The paper analyzes risk assessment for cash flows in continuous time using the notion of convex risk measures for processes. By combining a decomposition result for optional measures, and a dual representation of a convex risk measure for bounded \cd processes, we show that this framework provides a systematic approach to the both issues of model ambiguity, and uncertainty about the time value of money. We also establish a link between risk measures for processes and BSDEs.
In this paper, we consider the asset-liability management under the mean-variance criterion. The financial market consists of a risk-free bond and a stock whose price process is modeled by a geometric Brownian motion. The liability of the investor is uncontrollable and is modeled by another geometric Brownian motion. We consider a specific state-dependent risk aversion which depends on a power function of the liability. By solving a flow of FBSDEs with bivariate state process, we obtain the equilibrium strategy among all the open-loop controls for this time-inconsistent control problem. It shows that the equilibrium strategy is a feedback control of the liability.
This paper contains an overview of results for dynamic multivariate risk measures. We provide the main results of four different approaches. We will prove under which assumptions results within these approaches coincide, and how properties like primal and dual representation and time consistency in the different approaches compare to each other.
We consider the Fractionally Integrated Exponential Generalized Autoregressive Conditional Heteroskedasticity process, denoted by FIEGARCH(p,d,q), introduced by Bollerslev and Mikkelsen (1996). We present a simulated study regarding the estimation of the risk measure $VaR_p$ on FIEGARCH processes. We consider the distribution function of the portfolio log-returns (univariate case) and the multivariate distribution function of the risk-factor changes (multivariate case). We also compare the performance of the risk measures $VaR_p$, $ES_p$ and MaxLoss for a portfolio composed by stocks of four Brazilian companies.
Hedging in the presence of transaction costs leads to complex optimization problems. These problems typically lack closed-form solutions, and their implementation relies on numerical methods that provide hedging strategies for specific parameter values. In this paper we use a genetic programming algorithm to derive explicit formulas for near-optimal hedging strategies under nonlinear transaction costs. The strategies are valid over a large range of parameter values and require no information about the structure of the optimal hedging strategy.
We investigate under which conditions a single simulation of joint default times at a final time horizon can be decomposed into a set of simulations of joint defaults on subsequent adjacent sub-periods leading to that final horizon. Besides the theoretical interest, this is also a practical problem as part of the industry has been working under the misleading assumption that the two approaches are equivalent for practical purposes. As a reasonable trade-off between realistic stylized facts, practical demands, and mathematical tractability, we propose models leading to a Markovian multi-variate survival--indicator process, and we investigate two instances of static models for the vector of default times from the statistical literature that fall into this class. On the one hand, the "looping default" case is known to be equipped with this property, and we point out that it coincides with the classical "Freund distribution" in the bivariate case. On the other hand, if all sub-vectors of the survival indicator process are Markovian, this constitutes a new characterization of the Marshall--Olkin distribution, and hence of multi-variate lack-of-memory. A paramount property of the resulting model is stability of the type of multi-variate distribution with respect to elimination or insertion of a new marginal component with marginal distribution from the same family. The practical implications of this "nested margining" property are enormous. To implement this distribution we present an efficient and unbiased simulation algorithm based on the Lévy-frailty construction. We highlight different pitfalls in the simulation of dependent default times and examine, within a numerical case study, the effect of inadequate simulation practices.
The Financial Crisis of 2008 is a worldwide financial crisis causing a worldwide economic decline that is the most severe since the 1930s. According to the International Monetary Fund (IMF), the global financial crisis gave impact on USD 3.4 trillion losses from financial institutions around the world between 2007 and 2010. Does the crisis give impact on the returns of the U.S. movie Box Office? It will be answered by doing an analysis on the financial risk model based on Extreme Value Theory (EVT) and calculations of Value at Risk (VaR) and Expected Shortfall (ES). The values of VaR and ES from 2 periods, 1982 to 1995 and 1996 to 2010, are compared. Results show that the possibility of loss for an investment in the movie industry is relatively lower than the possibility of gain for both periods of time. The values of VaR and ES for the second period are higher than the first period. We are able to conclude that the 2008 financial crisis gave no significant effect on these measurement values in the second period. This result describes the high potential opportunity in the investment of the U.S. movie makers.
The management of operational risk in the banking industry has undergone significant changes over the last decade due to substantial changes in operational risk environment. Globalization, deregulation, the use of complex financial products and changes in information technology have resulted in exposure to new risks very different from market and credit risks. In response, Basel Committee for banking Supervision has developed a regulatory framework, referred to as Basel II, that introduced operational risk category and corresponding capital requirements. Over the past five years, major banks in most parts of the world have received accreditation under the Basel II Advanced Measurement Approach (AMA) by adopting the loss distribution approach (LDA) despite there being a number of unresolved methodological challenges in its implementation. Different approaches and methods are still under hot debate. In this paper, we review methods proposed in the literature for combining different data sources (internal data, external data and scenario analysis) which is one of the regulatory requirement for AMA.
Recent financial disasters emphasised the need to investigate the consequence associated with the tail co-movements among institutions; episodes of contagion are frequently observed and increase the probability of large losses affecting market participants' risk capital. Commonly used risk management tools fail to account for potential spillover effects among institutions because they provide individual risk assessment. We contribute to analyse the interdependence effects of extreme events providing an estimation tool for evaluating the Conditional Value-at-Risk (CoVaR) defined as the Value-at-Risk of an institution conditioned on another institution being under distress. In particular, our approach relies on Bayesian quantile regression framework. We propose a Markov chain Monte Carlo algorithm exploiting the Asymmetric Laplace distribution and its representation as a location-scale mixture of Normals. Moreover, since risk measures are usually evaluated on time series data and returns typically change over time, we extend the CoVaR model to account for the dynamics of the tail behaviour. Application on U.S. companies belonging to different sectors of the Standard and Poor's Composite Index (S&P500) is considered to evaluate the marginal contribution to the overall systemic risk of each individual institution
I studied the convergence of regional house prices to national prices in USA by analyzing time-series of house price indices of 9 Census Divisions. I found the evidence of the convergence in some parts of the country using asymmetric unit root tests. The fact that the evidence of the convergence is not present in large parts of the country raises an issue of execution and interpretation of results of Federal Reserve Bank's annual stress testing of the US banking system.
In the wake of the ongoing global financial crisis, interdependencies among banks have come into focus in trying to assess systemic risk. To date, such analysis has largely been based on numerical data. By contrast, this study attempts to gain further insight into bank interconnections by tapping into financial discussion. Co-mentions of bank names are turned into a network, which can be visualized and analyzed quantitatively, in order to illustrate characteristics of individual banks and the network as a whole. The approach allows for the study of temporal dynamics of the network, to highlight changing patterns of discussion that reflect real-world events, the current financial crisis in particular. For instance, it depicts how connections from distressed banks to other banks and supervisory authorities have emerged and faded over time, as well as how global shifts in network structure coincide with severe crisis episodes. The usage of textual data holds an additional advantage in the possibility of gaining a more qualitative understanding of an observed interrelation, through its context. We illustrate our approach using a case study on Finnish banks and financial institutions. The data set comprises 3.9M posts from online, financial and business-related discussion, during the years 2004 to 2012. Future research includes analyzing European news articles with a broader perspective, and a focus on improving semantic description of relations.
In this paper, we introduce an insurance ruin model with adaptive premium rate, thereafter refered to as restructuring/refraction, in which classical ruin and bankruptcy are distinguished. In this model, the premium rate is increased as soon as the wealth process falls into the red zone and is brought back to its regular level when the process recovers. The analysis is mainly focused on the time a refracted Lévy risk process spends in the red zone (analogous to the duration of the negative surplus). Building on results from Kyprianou and Loeffen (2010) and Loeffen et al. (2012), we identify the distribution of various functionals related to occupation times of refracted spectrally negative Lévy processes. For example, these results are used to compute the probability of bankruptcy and the probability of Parisian ruin in this model with restructuring.
This paper provides a unified framework, which allows, in particular, to study the structure of dynamic monetary risk measures and dynamic acceptability indices. The main mathematical tool, which we use here, and which allows us to significantly generalize existing results is the theory of $L^0$-modules. In the first part of the paper we develop the general theory and provide a robust representation of conditional assessment indices, and in the second part we apply this theory to dynamic acceptability indices acting on stochastic processes.
A technique from stochastic portfolio theory [Fernholz, 1998] is applied to analyse equity returns of Small, Mid and Large cap portfolios in an emerging market through periods of growth and regional crises, up to the onset of the global financial crisis. In particular, we factorize portfolios in the South African market in terms of distribution of capital, change of stock ranks in portfolios, and the effect due to dividends for the period Nov 1994 to May 2007. We discuss the results in the context of broader economic thinking to consider capital flows as risk factors, turning around more established approaches which use macroeconomic and socio-economic conditions to explain Foreign Direct Investment (into the economy) and Net Portfolio Investment (into equity and bond markets).
In this paper, we obtain a property of the expectation of the inverse of compound Wishart matrices which results from their orthogonal invariance. Using this property as well as results from random matrix theory (RMT), we derive the asymptotic effect of the noise induced by estimating the covariance matrix on computing the risk of the optimal portfolio. This in turn enables us to get an asymptotically unbiased estimator of the risk of the optimal portfolio not only for the case of independent observations but also in the case of correlated observations. This improvement provides a new approach to estimate the risk of a portfolio based on covariance matrices estimated from exponentially weighted moving averages of stock returns.