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This paper shows how the well-known Bornhuetter–Ferguson claims-reserving method can be extended by applying fuzzy methods. The a priori information for the ultimate claims derives from market statistics, organisational data, etc. and might contain vagueness. Likewise, the parameters of the claims development pattern can be vague or are adapted, retrospectively, due to subjective judgement. With the help of fuzzy numbers we develop new predictors for the ultimate claims. Furthermore, we quantify the uncertainty of the ultimate claims for single and aggregated accident years.

This paper considers the algorithmic implementation of the heteroskedasticity and autocorrelation consistent (HAC) estimation problem for covariance matrices of parameter estimators. We introduce a new algorithm, mainly based on the fast Fourier transform, and show via computer simulation that our algorithm is up to 20 times faster than well-established alternative algorithms. The cumulative effect is substantial if the HAC estimation problem has to be solved repeatedly. Moreover, the bandwidth parameter has no impact on this performance. We provide a general description of the new algorithm as well as code for a reference implementation in R.

The prediction of the outstanding loss liabilities for a non-life run-off portfolio as well as the quantification of the prediction error is one of the most important actuarial tasks in non-life insurance. In this paper we consider this prediction problem in a multivariate context. More precisely, we derive the predictive distribution of the claims reserves simultaneously for several correlated run-off portfolios in the framework of the Chain-ladder claims reserving method for several correlated run-off portfolios.Zusammenfassung Die Prognose der noch ausstehenden Schadenaufwendungen und die Quantifizierung des damit verbundenen Prognosefehlers ist für NichtlebenVersicherer von großer Bedeutung. In der vorliegenden Arbeit wird die prädiktive Verteilung für die Schadenreserve mehrerer korrelierter Run-Off-Portfolios mittels Bootstrapping im Rahmen eines multivariaten Chain-Ladder-Reservierungsverfahrens berechnet. Die Betrachtung mehrerer korrelierter Run-Off-Portfolios ist vor allem dadurch motiviert, dass es in der Praxis der Schadenreservierung häufig notwendig ist, ein gegebenes Run-Off-Portfolio erst in mehrere (Sub-) Run-Off-Portfolios zu zerlegen, bevor die für eine sinnvolle Anwendung des verwendeten Schadenreservierungsverfahren notwendigen Homogenitätseigenschaften zumindest ansatzweise erfüllt sind.

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