An application of parametric quantile regression to extend the two-stage quantile regression for ratemaking

This paper deals with the use of parametric quantile regression for the calculation of a loaded premium, based on a quantile measure, corresponding to individual insurance risk. Heras et al. have recently introduced a ratemaking process based on a two-stage quantile regression model. In the first stage, a probability to have at least one claim is estimated by a GLM logit, whereas in the second stage several quantile regressions are necessary for the estimate of the severity component. The number of quantile regressions to be performed is equal to the number of risk classes selected for ratemaking. In the actuarial context, when a large number of risk classes are considered (e.g. in Motor Third Party Liability), such approach can imply an over-parameterization and time-consuming. To this aim, in the second stage, we suggest to apply a more parsimonious approach based on Parametric Quantile Regression as introduced by Frumento and Bottai and never used in the actuarial context. This more conservative approach allows you not to lose efficiency in the estimation of premiums compared to the traditional Quantile Regression

Riserva sinistri stocastica e misure di risk capital per assicurazioni non-life

ItIn generale, la solvibilità implica la disponibilità da parte di un impresa assicurativa di mezzi finanziari sufficienti a soddisfare gli impegni presenti e futuri nei confronti degli assicurati su un orizzonte temporale di riferimento, con un prestabilito livello di confidenza, corrispondente al livello di probabilità del percentile della distribuzione della variabile aleatoria importo complessivo del danno, assunto come target di sicurezza. L’impiego di metodologie stocastiche per la determinazione della riserva sinistri, pari alla stima degli impegni di risarcimento aleatori per sinistri avvenuti e non ancora liquidati alla chiusura dell’esercizio, fornisce, oltre alla stima puntuale di tale posta, intervalli di variazione della stessa secondo prefissati livelli di probabilità in modo coerente con l’impianto metodologico rappresentato nel Solvency II introducendo tuttavia rischi di modello, di parametro e di processo. In particolare, nel presente lavoro partendo dal contributo originale di Pentikainen e Rantala (1992), in cui è analizzato l’errore commesso dai metodi di stima della riserva sinistri attraverso l’uso di un modello simulativo di tipo Monte Carlo, è effettuata la stima dell’errore di modello riferito a modelli attuariali di stima della riserva sinistri stocastici del tipo descritti da Mack (1993 e 1999) e da England e Verral (2001 e 2002); l’analisi è inoltre estesa per dare evidenza dell’effetto dell’errore di stima sul capitale economico determinato secondo la logica di un approccio per quantili

Alcune considerazioni sulle basi tecniche delle assicurazioni Dread Disease

Collana: Rapporti Scientifici AMASESSeries: Scientific Reports AMASE

A health insurance pricing model based on prevalence rates: Application to critical illness insurance

The Italian health insurance market is currently undersized. The paucity of assured data and the discon- tinuous statistical surveys carried out by the National Institute of Statistics (ISTAT) represent one of the main obstacles to the insurance market development. The paper sets forth a parametric model to estimate technical basis for health insurance policies when data are limited and only aggregated information on mortality and morbidity is available. The probabilistic framework is based on a multiple state continu- ous and time inhomogeneous Markov model. We provide an estimate of transition intensities from the healthy state to the sickness state when only prevalence rates of sickness are available, according to an extension and modification of the methodology proposed in Olivieri (1996) for Long Term Care insur- ance. We assume that mortality intensity of both healthy and sick lives is modelled by two independent Gompertz–Makeham models

A quantitative analysis of disability surveys in five European countries

Università degli studi di Roma “La Sapienza

AN ACTUARIAL MODEL FOR LOSS GIVEN DEFAULT ESTIMATION VIA SEMI-MARKOV PROCESS

According to Basel II (Revised International Capital Framework), banks have to hold adequate capital to cover losses resulting from specific sources of risk, among them credit risk. Credit risk refers to the possibility of a loss occurring due to counterparty failure, against that, the bank has taken credit exposure. The minimum amount of capital that banks should hold against credit risk can be calculated using a standardized approach - based on assessments developed by external Rating Agencies and approved by Authority – or using a Internal Rating Based (IRB) approach - based on the use of an internal rating system. In this framework a strategic variable is the Loss Given Default (LGD), which quantifies the loss which is subject to a credit institute in case of counterparty default, usually referred to any exposure of bank’s client; LGD can also be interpreted as the complement to one of the recovery rate. The aim of this paper is to illustrate an actuarial model for LGD estimation in a IRB approach – i.e. based on a credit institute’s database – by means of a semi-markov approach to describe the recovery process. In fact, creditor’s recovery process can be represented as a multistate model with a finite number of states, where each state corresponds to a possible step of the recovery process. Unlike classical Markov models, the proposed methodology joins the multistate model with a semi- markov probabilistic structure, in which the transition probabilities among states take into account, not only the status previously occupied, as in Markov models, but also the permanence time in the last visited state. The semi-Markov models are an evolution of Markov processes as integrate the information on time spent in the most recent visited state. The semi-Markov model captures a large amount of information, so it seems more adequate for the problem to be treated. In fact, in financial sense, recovery rate is the present value of the cash flows associated with different steps of recovery process, until the settlement, for any unit of exposure. Therefore, to quantify the loss incurred by the creditor, a key variable is the time required to settle each recovery step. Once defined the probabilistic structure, the estimation of LGD needs to define cash flows – usually called rewards - associated with each system state, making a distinction between the rewards associated with permanence in a state and those associated with transitions among different states. When a set of rewards are defined, it is possible to determine LGD on t periods; this measure can be extended to the entire portfolio of outstanding loans, obtaining the Total Loss Given Default (TLGD). Finally it is proposed an application of the model to estimate semi-markov transition probabilities, LGD and TLGD for bankruptcy proceedings, characterized by 12 steps of recovery. Finally, for an entire portfolio of outstanding loans of a financial institution, semi-Markov approach has been compared with Markov approach.According to Basel II (Revised International Capital Framework), banks have to hold adequate capital to cover losses resulting from specific sources of risk, among them credit risk. Credit risk refers to the possibility of a loss occurring due to counterparty failure, against that, the bank has taken credit exposure. The minimum amount of capital that banks should hold against credit risk can be calculated using a standardized approach - based on assessments developed by external Rating Agencies and approved by Authority – or using a Internal Rating Based (IRB) approach - based on the use of an internal rating system. In this framework a strategic variable is the Loss Given Default (LGD), which quantifies the loss which is subject to a credit institute in case of counterparty default, usually referred to any exposure of bank’s client; LGD can also be interpreted as the complement to one of the recovery rate. The aim of this paper is to illustrate an actuarial model for LG

Elementi tecnico-attuariali per le assicurazioni sulla vita e contro i danni

E' fornita una descrizione delle fondamentali regole tecniche di gestione dei rischi nell'ambito delle assicurazioni vita e delle assicurazioni danni, tramite il richiamo delle fonti normative primarie e secondarie del settore assicurativo