Urn-based models for dependent credit risks and their calibration through EM algorithm

In this contribution we analyze two models for the joint probability of defaults of dependent credit risks that are based on a generalisation of Polya urn scheme. In particular we focus our attention on the problems related to the maximum likelihood estimation of the parameters involved, and to this purpose we introduce an approach based on the use of the Expectation-Maximization algorithm. We show how to implement it in this context, and then we analyze the results obtained, comparing them with results obtained by other approaches.

Estimating the Value of the Wincat Coupons of the Winterthur Insurance Convertible Bond: A Study of the Model Risk

The three annual 2¼% interest coupons of the Winterthur Insurance convertible bond (face value Chf 4 700) will only be paid out if during their corresponding observation periods no major storm or hail storm on one single day damages at least 6 000 motor vehicles insured with Winterthur Insurance. Data for events, where storm or hail damaged more than 1 000 insured vehicles, are available for the last ten years. Using a constant-parameter model, the estimated discounted value of the three Wincat coupons together is Chf 263.29. A conservative evaluation, which accounts for the standard deviation of the estimate, gives a coupon value of Chf 238.25. However, fitting models which admit a trend or a change-point, leads to substantially higher knock-out probabilities of the coupons. The estimated discounted values of the coupons can drop below the above conservative value; a conservative evaluation as above leads to substantially lower values. Hence, already the model uncertainty is higher than the standard deviations of the used estimators. This shows the dominance of the model risk. Consistency, dispersion, robustness and sensitivity of the models are analysed by a simulation stud

Large deviations of UU-empirical measures in strong topologies and applications

ABSTRACT. – We prove large deviation principles (LDP) for m-fold products of empirical measures and forU-empirical measures, where the underlying i.i.d. random variables take values in a measurable (not necessarily Polish) space (S,S). The results can be formulated on suitable subsets of all probability measures on (Sm,S⊗m). We endow the spaces with topologies, which are stronger than the usual τ-topology and which make integration with respect to certain unbounded, Banach-space valued functions a continuous operation. A special feature is the non-convexity of the rate function for m 2. Improved versions of LDPs for Banach-space valued U- and V-statistics are obtained as a particular application. Some further applications concerning the Gibbs conditioning principle and a process level LDP are mentioned. 2002 Éditions scientifiques et médicales Elsevier SA

Forecasting Leading Death Causes in Australia using Extended CreditRisk++

Recently we developed a new framework in Hirz et al (2015) to model stochastic mortality using extended CreditRisk$^+$ methodology which is very different from traditional time series methods used for mortality modelling previously. In this framework, deaths are driven by common latent stochastic risk factors which may be interpreted as death causes like neoplasms, circulatory diseases or idiosyncratic components. These common factors introduce dependence between policyholders in annuity portfolios or between death events in population. This framework can be used to construct life tables based on mortality rate forecast. Moreover this framework allows stress testing and, therefore, offers insight into how certain health scenarios influence annuity payments of an insurer. Such scenarios may include improvement in health treatments or better medication. In this paper, using publicly available data for Australia, we estimate the model using Markov chain Monte Carlo method to identify leading death causes across all age groups including long term forecast for 2031 and 2051. On top of general reduced mortality, the proportion of deaths for certain certain causes has changed massively over the period 1987 to 2011. Our model forecasts suggest that if these trends persist, then the future gives a whole new picture of mortality for people aged above 40 years. Neoplasms will become the overall number-one death cause. Moreover, deaths due to mental and behavioural disorders are very likely to surge whilst deaths due to circulatory diseases will tend to decrease. This potential increase in deaths due to mental and behavioural disorders for older ages will have a massive impact on social systems as, typically, such patients need long-term geriatric care.Comment: arXiv admin note: text overlap with arXiv:1505.0475

Actuarial Applications and Estimation of Extended~CreditRisk+^+

We introduce an additive stochastic mortality model which allows joint modelling and forecasting of underlying death causes. Parameter families for mortality trends can be chosen freely. As model settings become high dimensional, Markov chain Monte Carlo (MCMC) is used for parameter estimation. We then link our proposed model to an extended version of the credit risk model CreditRisk$^+$. This allows exact risk aggregation via an efficient numerically stable Panjer recursion algorithm and provides numerous applications in credit, life insurance and annuity portfolios to derive P\&L distributions. Furthermore, the model allows exact (without Monte Carlo simulation error) calculation of risk measures and their sensitivities with respect to model parameters for P\&L distributions such as value-at-risk and expected shortfall. Numerous examples, including an application to partial internal models under Solvency II, using Austrian and Australian data are shown.Comment: 34 pages, 5 figure

Estimating the Value of the Wincat Coupons of the Winterthur Insurance Convertible Bond: A Study of the Model Risk

ISSN:0515-0361ISSN:1783-135

Exponential Approximations In Completely Regular Topological Spaces And Extensions Of Sanov's Theorem

. This paper is devoted to the well known transformations that preserve a large deviation principle (LDP), namely, the contraction principle with approximately continuous maps and the concepts of exponential equivalence and exponential approximations. We generalize these transformations to completely regular topological state spaces, give some examples and, as an illustration, reprove a generalization of Sanov's theorem, due to de Acosta [1]. Using partition-dependent couplings, we then extend this version of Sanov's theorem to triangular arrays and prove a LDP for the empirical measures of exchangeable sequences with a general measurable state space. 1. Introduction and statement of the results The question when a large deviation principle (LDP) for a family of laws can be deduced from the LDP for another family was studied for the first time in [3]. There the probability measures are defined on a metric space. The concepts nowadays called "exponential equivalence" and "exponential a..