The European Project Solvency II is devoted to the appraisal of a Solvency Capital Requirement that should capture the overall risk profile of insurance companies. In this framework there is a growing need to develop so-called internal risk models to get accurate estimates of liabilities. In the context of non-life insurance, it is crucial to correctly assess risk from different sources, such as underwriting risk with particular reference to premium, reserving and catastrophe risks.In particular the underwriting cycle, providing additional volatility, can lead to considerable capital requirement. In fact in adverse development of financial positions, company’s management typically tends to raise premium rates or reinsurance, whereas otherwise it increases dividends or reduces premium rates.A correct analysis of this phenomena is also significant to understand the evolution of the reserving cycle, which is often correlated, with a lag period, with the underwriting cycle; in fact, it has been ascertained the tendency of insurers to over-estimate technical reserves during the hard part of the cycle when loss ratios are low and under-estimate these reserves in the opposite case.The aim of this paper is to correctly model the underwriting cycle for non-life insurance companies, also taking into account differences between LoB in the Italian market, especially for the heaviness of the loss distribution tail.The basic model is derived from Collective Risk Theory where the solvency ratio u(t) (i.e. risk reserve U(t+1) on risk premium P(0), see Daykin et al., 1994 for more details) at the end of the year t+1 is (not considering expenses and relative loadings): (1)where r is a function (constant for our purposes) of the rate of return j, the rate of portfolio growth g and the inflation rate i (supposed constants): , typically with values strictly lower than 1; x(t+1) is the ratio of present value of aggregate loss X(t+1) on risk premium; p(t+1) is the ratio of risk premium P(t+1) = on initial level risk premium P(0) = ; (t+1) is the safety loading.Starting from the idea of Daykin et al. 1994, in this paper a dynamic control policy is defined to specify the relationship between solvency ratio and premium rates (underwriting cycle). For this reason it is assumed the following dynamic equation for the safety loading: . (2)Equation (2) shows how, starting from a basic level 0, safety loading will be dynamically: increased, with a percentage of c1, if u(t) decreases under a floor level R1 or decreased, with a percentage of c2, if u(t) is higher than a roof level R2.Under the rough assumption that aggregate loss distribution doesn’t change in time, so that p(t+1) = p(t) = p(t-1) = …= 1 (not considering also time lag effects), we define a simplified version of (1) that assumes the form of a one dimensional piecewise linear map in the state variable u(t): (3)Firstly it is analyzed a deterministic version of this map, where x(t+1)=x is simply regarded as a parameter. In this case local and global analysis of (3) are performed, showing that the behaviour of the solvency ratio u(t) can switch between extremely simple and extremely complex dynamics as the main parameters of the model vary. These bifurcations, called “Border-collision bifurcations” (see Di Bernardo et al., 2008 for details), are related to the crossing of the trajectory of (3) into regions where the definition of the map changes.Stochastic assessments of (3) and parameters estimation for the Italian case conclude the work.

### An analysis of the underwriting cycle for non-life insurance companies

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*LAMANTIA F.*

##### 2009-01-01

#### Abstract

The European Project Solvency II is devoted to the appraisal of a Solvency Capital Requirement that should capture the overall risk profile of insurance companies. In this framework there is a growing need to develop so-called internal risk models to get accurate estimates of liabilities. In the context of non-life insurance, it is crucial to correctly assess risk from different sources, such as underwriting risk with particular reference to premium, reserving and catastrophe risks.In particular the underwriting cycle, providing additional volatility, can lead to considerable capital requirement. In fact in adverse development of financial positions, company’s management typically tends to raise premium rates or reinsurance, whereas otherwise it increases dividends or reduces premium rates.A correct analysis of this phenomena is also significant to understand the evolution of the reserving cycle, which is often correlated, with a lag period, with the underwriting cycle; in fact, it has been ascertained the tendency of insurers to over-estimate technical reserves during the hard part of the cycle when loss ratios are low and under-estimate these reserves in the opposite case.The aim of this paper is to correctly model the underwriting cycle for non-life insurance companies, also taking into account differences between LoB in the Italian market, especially for the heaviness of the loss distribution tail.The basic model is derived from Collective Risk Theory where the solvency ratio u(t) (i.e. risk reserve U(t+1) on risk premium P(0), see Daykin et al., 1994 for more details) at the end of the year t+1 is (not considering expenses and relative loadings): (1)where r is a function (constant for our purposes) of the rate of return j, the rate of portfolio growth g and the inflation rate i (supposed constants): , typically with values strictly lower than 1; x(t+1) is the ratio of present value of aggregate loss X(t+1) on risk premium; p(t+1) is the ratio of risk premium P(t+1) = on initial level risk premium P(0) = ; (t+1) is the safety loading.Starting from the idea of Daykin et al. 1994, in this paper a dynamic control policy is defined to specify the relationship between solvency ratio and premium rates (underwriting cycle). For this reason it is assumed the following dynamic equation for the safety loading: . (2)Equation (2) shows how, starting from a basic level 0, safety loading will be dynamically: increased, with a percentage of c1, if u(t) decreases under a floor level R1 or decreased, with a percentage of c2, if u(t) is higher than a roof level R2.Under the rough assumption that aggregate loss distribution doesn’t change in time, so that p(t+1) = p(t) = p(t-1) = …= 1 (not considering also time lag effects), we define a simplified version of (1) that assumes the form of a one dimensional piecewise linear map in the state variable u(t): (3)Firstly it is analyzed a deterministic version of this map, where x(t+1)=x is simply regarded as a parameter. In this case local and global analysis of (3) are performed, showing that the behaviour of the solvency ratio u(t) can switch between extremely simple and extremely complex dynamics as the main parameters of the model vary. These bifurcations, called “Border-collision bifurcations” (see Di Bernardo et al., 2008 for details), are related to the crossing of the trajectory of (3) into regions where the definition of the map changes.Stochastic assessments of (3) and parameters estimation for the Italian case conclude the work.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.