The fuzzy Bornhuetter–Ferguson method: an approach with fuzzy numbers

Abstract

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.

A Likelihood Approach to Bornhuetter–Ferguson Analysis

“…With the Bornhuetter et al (1972) method the chain ladder ultimates are adjusted using prior knowledge while the adjusted cash flow is proportional to the original chain ladder cash flow. Mack (2000) gave a credibility interpretation of the Bornhuetter-Ferguson method. The adjustment of the ultimates can be done in two ways. Either by correcting the levels of the ultimates or the relative levels of the ultimates.…”

“…Indeed, we provide an empirical illustration where this is the case. The levels approach is most common in the literature; see for instance (Mack 2000(Mack , 2006, Taylor (2000), Verrall (2004), Wüthrich and Merz (2008) and Heberle and Thomas (2016). The relative levels approach is more recent; see (Martínez-Miranda et al 2013.…”

“…A fundamental interpretation of the Bornhuetter-Ferguson method arises when combining chain ladder with credibility formulas. Credibility formulas have been investigated in reserving by, for instance, de Vylder (1982), Mack (2000). and more recently, Bühlmann and Moriconi (2015).…”

“…and more recently, Bühlmann and Moriconi (2015). We have been particularly influenced by Mack (2000), who gives a credibility formula showing that adjusting the ultimates with prior knowledge yields a partial adjustment of the reserves. He then continues to show that the iterations of the credibility formula leads to the Benktander (1976) approach.…”

On Unbalanced Data and Common Shock Models in Stochastic Loss Reserving

“…Outstanding claims reserves are typically some of the most critical components in the financial statement of a non-life insurer (Abdallah, Boucher and Cossette, 2015;Heberle and Thomas, 2016;Saluz and Gisler, 2014). When estimating reserves, the insurer often has to provide the central estimate as well as a risk margin to accommodate for the stochastic nature of outstanding claims.…”