COLLECTIVE RISK MODELS WITH DEPENDENCE UNCERTAINTY

2017 ◽  
Vol 47 (2) ◽  
pp. 361-389 ◽  
Author(s):  
Haiyan Liu ◽  
Ruodu Wang
Keyword(s):  
Value At Risk ◽  
Random Variable ◽  
Actuarial Science ◽  
Risk Models ◽  
Worst Case ◽  
Collective Risk ◽  
Approximation Errors ◽  
Claim Frequency ◽  
Aggregate Loss ◽  
The Individual

AbstractWe bring the recently developed framework of dependence uncertainty into collective risk models, one of the most classic models in actuarial science. We study the worst-case values of the Value-at-Risk (VaR) and the Expected Shortfall (ES) of the aggregate loss in collective risk models, under two settings of dependence uncertainty: (i) the counting random variable (claim frequency) and the individual losses (claim sizes) are independent, and the dependence of the individual losses is unknown; (ii) the dependence of the counting random variable and the individual losses is unknown. Analytical results for the worst-case values of ES are obtained. For the loss from a large portfolio of insurance policies, an asymptotic equivalence of VaR and ES is established. Our results can be used to provide approximations for VaR and ES in collective risk models with unknown dependence. Approximation errors are obtained in both cases.

scholarly journals An Appropriate Way to Switch from the Individual Risk Model to the Collective One

1993 ◽  
Vol 23 (1) ◽  
pp. 23-54 ◽  
Author(s):  
S. Kuon ◽  
M. Radtke ◽  
A. Reich
Keyword(s):  
Risk Model ◽  
Sufficient Conditions ◽  
Approximation Error ◽  
Individual Risk ◽  
Risk Models ◽  
Approximation Quality ◽  
Collective Risk Model ◽  
Collective Risk ◽  
The Individual ◽  
Necessary And Sufficient

AbstractFor some time now, the convenient and fast calculability of collective risk models using the Panjer-algorithm has been a well-known fact, and indeed practitioners almost always make use of collective risk models in their daily numerical computations. In doing so, a standard link has been preferred for relating such calculations to the underlying heterogeneous risk portfolio and to the approximation of the aggregate claims distribution function in the individual risk model. In this procedure until now, the approximation quality of the collective risk model upon which such calculations are based is unknown.It is proved that the approximation error which arises does not converge to zero if the risk portfolio in question continues to grow. Therefore, necessary and sufficient conditions are derived in order to obtain well-adjusted collective risk models which supply convergent approximations. Moreover, it is shown how in practical situations the previous natural link between the individual and the collective risk model can easily be modified to improve its calculation accuracy. A numerical example elucidates this.

scholarly journals Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures

2007 ◽  
Vol 37 (1) ◽  
pp. 93-112 ◽  
Author(s):  
Jun Cai ◽  
Ken Seng Tan
Keyword(s):  
Value At Risk ◽  
Negative Binomial ◽  
Risk Model ◽  
Sufficient Conditions ◽  
Individual Risk ◽  
Risk Models ◽  
Optimization Criteria ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Stop Loss

We propose practical solutions for the determination of optimal retentions in a stop-loss reinsurance. We develop two new optimization criteria for deriving the optimal retentions by, respectively, minimizing the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risks of an insurer. We establish necessary and sufficient conditions for the existence of the optimal retentions for two risk models: individual risk model and collective risk model. The resulting optimal solution of our optimization criterion has several important characteristics: (i) the optimal retention has a very simple analytic form; (ii) the optimal retention depends only on the assumed loss distribution and the reinsurer’s safety loading factor; (iii) the CTE criterion is more applicable than the VaR criterion in the sense that the optimal condition for the former is less restrictive than the latter; (iv) if optimal solutions exist, then both VaR- and CTE-based optimization criteria yield the same optimal retentions. In terms of applications, we extend the results to the individual risk models with dependent risks and use multivariate phase type distribution, multivariate Pareto distribution and multivariate Bernoulli distribution to illustrate the effect of dependence on optimal retentions. We also use the compound Poisson distribution and the compound negative binomial distribution to illustrate the optimal retentions in a collective risk model.

The individual risk model: a compound distribution

1989 ◽  
Vol 116 (1) ◽  
pp. 101-107 ◽  
Author(s):  
R. J. Verrall
Keyword(s):  
Risk Model ◽  
Individual Risk ◽  
Risk Models ◽  
Collective Models ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Compound Distribution ◽  
Compound Binomial ◽  
Mean And Variance ◽  
The Individual

The two approaches to modelling aggregate claims—the individual and the collective models—have been regarded as arising by considering a portfolio of policies in different ways. The individual risk model (IRM) is derived by considering the claims on individual policies and summing over all policies in the portfolio, while the collective risk model (CRM) is derived from the portfolio as a whole. This is sometimes held to be the main difference between the IRM and the CRM. In fact the IRM can be derived in exactly the same way as the CRM and can be regarded as a compound binomial distribution. This makes a unified treatment of risk models possible, simplifies the calculation of the mean and variance of the IRM, and facilitates the calculation of higher moments.

ASYMPTOTIC BEHAVIOR OF EXTREMAL EVENTS FOR AGGREGATE DEPENDENT RANDOM VARIABLES

2013 ◽  
Vol 27 (4) ◽  
pp. 507-531
Author(s):  
Die Chen ◽  
Tiantian Mao ◽  
Taizhong Hu
Keyword(s):  
Value At Risk ◽  
Homogeneous Function ◽  
Individual Risk ◽  
Dependence Structure ◽  
Large Individual ◽  
Tail Behavior ◽  
Large Aggregate ◽  
Dependent Risks ◽  
Aggregate Loss ◽  
The Individual

Consider a portfolio of n identically distributed risks X1, …, Xn with dependence structure modelled by an Archimedean survival copula. It is known that the probability of a large aggregate loss of $\sum\nolimits_{i=1}^{n} X_{i}$ is in proportion to the probability of a large individual loss of X1. The proportionality factor depends on the dependence strength and the tail behavior of the individual risk. In this paper, we establish analogous results for an aggregate loss of the form g(X1, …, Xn) under the more general model in which the Xi's have different but tail-equivalent distributions and the copula remains unchanged, where g is a homogeneous function of order 1. Properties of these factors are studied, and asymptotic Value-at-Risk behaviors of functions of dependent risks are also given. The main results generalize those in Wüthrich [16], Alink, Löwe, and Wüthrich [2], Barbe, Fougères, and Genest [4], and Embrechts, Nešlehová, and Wüthrich [9].

scholarly journals PERHITUNGAN UKURAN RISIKO UNTUK MODEL KERUGIAN AGREGAT

2020 ◽  
Vol 14 (1) ◽  
pp. 21
Author(s):  
Nadya Pratiwi ◽  
Aprida Siska Lestia ◽  
Nur Salam
Keyword(s):  
Monte Carlo ◽  
Standard Deviation ◽  
Monte Carlo Method ◽  
Value At Risk ◽  
Risk Measure ◽  
Random Variable ◽  
Insurance Companies ◽  
Loss Model ◽  
Aggregate Loss ◽  
Conditional Tail Expectation

In the case of nonlife insurance, insurance companies are very potential to get losses if claims submitted by customers (policyholders) exceeds the reserves of budgeted claims. It is the risk that have to managed properly by insurance companies . One possible disadvantage is the aggregate loss model. The aggregate loss model is a random variable that states the total of all losses incurred in an insurance policy block. This kind of loss can be modeled using a collective risk approach where the number of claims is a discrete random variable and the size of claim is a continuous random variable. The purpose of this study is to determine risk measure of standard deviation premium principle, value at risk (VaR), and conditional tail expectation (CTE) of the aggregate loss model. Standard deviation premium principle risk measure of aggregate loss model is determined analytically by substituted it expected value and varians. Meanwhile, VaR risk measure is determined using numerical method by Monte Carlo method, then the quantile value and it confidence interval for the actual value will estimate. In the CTE calculation, based on the loss data obtained in the Monte Carlo method, the CTE value is estimated by calculating the average loss that exceeds the VaR value. If the data size is large enough, the CTE value estimation will converge to the actual value.Keywords: Aggregate Loss Model, Standard Deviation Premium Principle, Value at Risk (VaR), Conditional Tail Expectation (CTE).

scholarly journals Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures

2007 ◽  
Vol 37 (01) ◽  
pp. 93-112 ◽  
Author(s):  
Jun Cai ◽  
Ken Seng Tan
Keyword(s):  
Value At Risk ◽  
Negative Binomial ◽  
Risk Model ◽  
Sufficient Conditions ◽  
Individual Risk ◽  
Risk Models ◽  
Optimization Criteria ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Stop Loss

We propose practical solutions for the determination of optimal retentions in a stop-loss reinsurance. We develop two new optimization criteria for deriving the optimal retentions by, respectively, minimizing the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risks of an insurer. We establish necessary and sufficient conditions for the existence of the optimal retentions for two risk models: individual risk model and collective risk model. The resulting optimal solution of our optimization criterion has several important characteristics: (i) the optimal retention has a very simple analytic form; (ii) the optimal retention depends only on the assumed loss distribution and the reinsurer’s safety loading factor; (iii) the CTE criterion is more applicable than the VaR criterion in the sense that the optimal condition for the former is less restrictive than the latter; (iv) if optimal solutions exist, then both VaR- and CTE-based optimization criteria yield the same optimal retentions. In terms of applications, we extend the results to the individual risk models with dependent risks and use multivariate phase type distribution, multivariate Pareto distribution and multivariate Bernoulli distribution to illustrate the effect of dependence on optimal retentions. We also use the compound Poisson distribution and the compound negative binomial distribution to illustrate the optimal retentions in a collective risk model.

scholarly journals Heuristic Optimization of Overloading Due to Electric Vehicles in a Low Voltage Grid

Energies ◽  
2020 ◽  
Vol 13 (22) ◽  
pp. 6069
Author(s):  
Sajjad Haider ◽  
Peter Schegner
Keyword(s):  
Electric Vehicles ◽  
Low Voltage ◽  
Optimization Methods ◽  
Monte Carlo Analysis ◽  
Heuristic Optimization ◽  
Optimization Approach ◽  
Worst Case ◽  
Voltage Drops ◽  
The Individual ◽  
Nodal Voltage

It is important to understand the effect of increasing electric vehicles (EV) penetrations on the existing electricity transmission infrastructure and to find ways to mitigate it. While, the easiest solution is to opt for equipment upgrades, the potential for reducing overloading, in terms of voltage drops, and line loading by way of optimization of the locations at which EVs can charge, is significant. To investigate this, a heuristic optimization approach is proposed to optimize EV charging locations within one feeder, while minimizing nodal voltage drops, cable loading and overall cable losses. The optimization approach is compared to typical unoptimized results of a monte-carlo analysis. The results show a reduction in peak line loading in a typical benchmark 0.4 kV by up to 10%. Further results show an increase in voltage available at different nodes by up to 7 V in the worst case and 1.5 V on average. Optimization for a reduction in transmission losses shows insignificant savings for subsequent simulation. These optimization methods may allow for the introduction of spatial pricing across multiple nodes within a low voltage network, to allow for an electricity price for EVs independent of temporal pricing models already in place, to reflect the individual impact of EVs charging at different nodes across the network.

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