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Un’estensione stocastica del modello "Fisher-Lange"

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  • Massimo De Felice
  • Franco Moriconi

Abstract

Tra i metodi di stima puntuale della riserva sinistri nell’assicurazione danni `e largamente utilizzato, in particolare dalle compagnie italiane, un approccio noto come “metodo di Fisher- Lange” (FL). L’FL `e un metodo “a costi medi”, nel senso che la valutazione degli impegni di rimborso sinistri `e scomposta in una stima del costo medio del singolo sinistro e in una stima del numero di sinistri da rimborsare. Una seconda caratteristica dell’FL `e che la stima del numero di sinistri da pagare `e basata su una case outstanding development technique, in cui lo sviluppo del numero di sinistri da pagare ha come variabile esplicativa dati da riserva di inventario (case reserves). L’FL `e un metodo essenzialmente deterministico; si propone qui un “modello FL stocastico” (SFL), cio`e una versione probabilistica del metodo FL. L’approccio stocastico consente di costruire l’intera distribuzione di probabilit`a dei costi di rimborso futuri, producendo quindi una adeguata misurazione del reserve risk e una quantificazione del corrispondente requisito patrimoniale, come richiesto ai modelli interni nell’ambito della Direttiva Quadro Solvency II. Una caratteristica saliente del modello SFL `e quella di fornire una estensione stocastica coerente dell’approccio deterministico, nel senso che gli stimatori per i valori attesi prodotti dall’SFL coincidono con gli stimatori empirici adottati nella procedura di stima puntuale, e sono dotati delle qualit`a statistiche necessarie per garantire l’affidabilit`a previsiva (non distorsione, varianza minima). Il modello proposto ha struttura trivariata, nel senso che lo sviluppo dei costi `e determinato da tre processi stocastici: il processo del costo medio, il processo del numero di sinistri “con seguito” e il processo del numero di sinistri pagati. L’informazione case outstanding agisce sui processi del numero, che hanno come variabile esplicativa il numero dei sinistri messi a riserva in ogni anno di sviluppo. La struttura dei processi ipotizzati consente di costruire per simulazione la distribuzione di probabilit`a di tutti gli impegni futuri, sia secondo l’approccio Liability-at-Maturity, sia secondo l’approccio Year-End Expectation. L’incertezza di stima dei parametri `e inclusa nelle distribuzioni effettuando le simulazioni con un approccio conditional parametric bootstrap. Vengono anche ricavate espressioni in forma chiusa per il mean square error of prediction (MSEP) dei costi di rimborso. Una approssimazione lineare di queste formule consente di riottenere come caso particolare, reinterpretando opportunamente le quantit`a coinvolte, le formule di MSEP ricavate in un modello di loss reserving recentemente proposto da Dahms. Parole chiave: stochastic loss reserving, average cost, case outstanding development technique, mean square error of prediction, conditional parametric bootstrap.

Suggested Citation

  • Massimo De Felice & Franco Moriconi, 2011. "Un’estensione stocastica del modello "Fisher-Lange"," Quaderni del Dipartimento di Economia, Finanza e Statistica 86/2011, Università di Perugia, Dipartimento Economia.
  • Handle: RePEc:pia:wpaper:86/2011
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    References listed on IDEAS

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    Keywords

    stochastic loss reserving; average cost; case outstanding development technique; mean square error of prediction; conditional parametric bootstrap;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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