Copulas provide a powerful and flexible tool for modeling the dependence structure of random vectors, and they have many applications in finance, insurance, engineering, hydrology, and other fields. One well-known class of copulas in two dimensions is the Farlie–Gumbel–Morgenstern (FGM) copula, since its simple analytic shape enables closed-form solutions to many problems in applied probability. However, the classical definition of the high-dimensional FGM copula does not enable a straightforward understanding of the effect of the copula parameters on the dependence, nor a geometric understanding of their admissible range. We circumvent this issue by analyzing the FGM copula from a probabilistic approach based on multivariate Bernoulli distributions. This paper examines high-dimensional exchangeable FGM copulas, a subclass of FGM copulas. We show that the dependence parameters of exchangeable FGM copulas can be expressed as a convex hull of a finite number of extreme points. We also leverage the probabilistic interpretation to develop efficient sampling and estimating procedures and provide a simulation study. Throughout, we discover geometric interpretations of the copula parameters that assist one in decoding the dependence of high-dimensional exchangeable FGM copulas.

In this paper, we use copula-GARCH models applied to daily data from March 2010 to March 2018 to test the time-varying dependence of the Liv-ex 50, a secondary market fine wine index comprised of the ten most recent vintages of the five Bordeaux First Growths, with a portfolio composed of the six main stock markets (S&P 500, CAC 40, DAX 30, FTSE 100, and Hang Seng). Our results suggest that the Liv-ex 50 underperforms the six stock indexes, but provides diversification benefits in terms of volatility, asymmetry, and extreme events. (JEL Classifications: G110, G120, Q14)

This paper explores dependencies between operational risks and between operational risks and other risks such as market, credit and insurance risk. The paper starts by setting the regulatory context and then goes into practical aspects of operational risk dependencies. Next, methods of modelling operational risk dependencies are considered with a simulation study exploring the sensitivity of diversification benefits arising from dependency models. The following two sections consider how correlation assumptions may be set, highlighting some generic dependencies between operational risks and with non-operational risks to assist in the assessment of dependencies and correlation assumptions. Supplementary appendices provide further detail on generic dependencies as well as a case study of how business models can lead to operational risks interacting with other risks. Finally, the paper finishes with a literature review of operational risk dependency papers including correlation studies and benchmark reports.

A non-parametric test based on nested L-statistics and designed to compare the riskiness of portfolios was introduced by Brazauskas et al. (2007). Its asymptotic and small-sample properties were primarily explored for independent portfolios, though independence is not a required condition for the test to work. In this paper, we investigate how performance of the test changes when insurance portfolios are dependent. To achieve that goal, we perform a simulation study where we consider three different risk measures: conditional tail expectation, proportional hazards transform, and mean. Further, three portfolios are generated from exponential, Pareto, and lognormal distributions, and their interdependence is modelled with the three-dimensional t and Gaussian copulas. It is found that the presence of strong positive dependence (comonotonicity) makes the test very liberal for all the risk measures under consideration. For types of dependence that are more common in an insurance environment, the effect of dependence is less dramatic but the results are mixed, i.e., they depend on the chosen risk measure, sample size, and even on the test’s significance level. Finally, we illustrate how to incorporate such findings into sensitivity analysis of the decisions. The risks we analyse represent tornado damages in different regions of the United States from 1890 to 1999.

In this article, possible use of climate forecasts in rainfall index insurance of hay and forage production is considered in a geographical area (southeast United States) relatively heavily impacted by the El Nino Southern Oscillation (ENSO). Analysis of the stochastic properties of rainfall, yields, and the ENSO forecasts using the copula technique shows that the forecast impact depends on the proximity to the Gulf Coast where the impact of the ENSO is more pronounced and earlier in the year. Stochastic modeling shows that the use of skillful long-term climate forecasts by the insured producers creates intertemporal adverse selection that can be precluded by offering forecast conditional premiums. The impacts on the efficiency of the rainfall index insurance and results of sensitivity analysis with respect to model parameters are discussed.

Government farm support programs such as Loan Deficiency Payments (LDP) and Counter-Cyclical Payments (CCP) have payoff structures that effectively make them costless price insurance instruments. A combination of these payments with yield insurance may provide a viable alternative to revenue insurance. This paper finds that, contrary to expectations, the revenue product analyzed is uniformly superior to yield insurance under both current (2002) and proposed (2008) Farm Bill structures of government payments. Given minor adjustments, however, yield insurance combined with government payments can provide more effective risk management than revenue insurance in production areas with low yield–price correlation.

In this paper, we study a continuous-time bivariate risk process in which each individual line of business implements a dividend barrier strategy. The insurance portfolios of the two insurers are correlated as they are subject to common shocks that induce dependent claims. To analyse the expected discounted dividends until the joint ruin time of the bivariate process (i.e. exit from the positive quadrant), we propose a discrete-time counterpart of the model and apply a bivariate extension of the Dickson−Waters discretisation with the use of a bivariate Panjer-type recursion. Detailed numerical examples under different dependencies via common shocks, copulas and proportional reinsurance are discussed, and applications to optimal problems in reinsurance, capital allocation and dividends are given. It is also illustrated that the optimal pair of dividend barriers maximising the dividend function is dependent on the initial surplus levels. A modified type of dividend barrier strategy is proposed towards the end.

This paper analyses the impact of using different correlation assumptions between lines of business when estimating the risk-based capital reserve, the solvency capital requirement (SCR), under Solvency II regulations. A case study is presented and the SCR is calculated according to the standard model approach. Alternatively, the requirement is then calculated using an internal model based on a Monte Carlo simulation of the net underwriting result at a one-year horizon, with copulas being used to model the dependence between lines of business. To address the impact of these model assumptions on the SCR, we conduct a sensitivity analysis. We examine changes in the correlation matrix between lines of business and address the choice of copulas. Drawing on aggregate historical data from the Spanish non-life insurance market between 2000 and 2009, we conclude that modifications of the correlation and dependence assumptions have a significant impact on SCR estimation.

In this paper we examine two aspects of extreme events: their calculation and their communication. In relation to calculation, there are two types of extreme events that are considered.

The first is the extent to which extreme events in two or more variables occur together. This can be gauged by using measures of tail association. Higher levels of tail association are useful for highlighting the extent to which there are concentrations of risk. We investigate the range of approaches used to measure tail association and propose a pragmatic alternative, the coefficient of finite tail dependence.

The second type of extreme event arises from combinations of losses from a series of risks that together result in total losses exceeding a particular level. This is measured using ruin lines or, in higher dimensions, planes and hyperplanes. The probability of ruin and the economic cost of ruin are considered here. In this context, it is important to consider what the term “loss” actually means, and whether it is in relation to a current set of exposures or a potential strategy.

The communication of extreme events is discussed not just in terms of the numbers that can be used, but in terms of the graphical methods that can be used to aggregate information on a range of risk combinations. This involves communicating not just the level of risk but also the importance of the risk considered.

A prudent assessment of dependence is crucial in many stochastic models for insurance risks. Copulas have become popular to model such dependencies. However, estimation procedures for copulas often lead to large parameter uncertainty when observations are scarce. In this paper, we propose a Bayesian method which combines prior information (e.g. from regulators), observations and expert opinion in order to estimate copula parameters and determine the estimation uncertainty. The combination of different sources of information can significantly reduce the parameter uncertainty compared to the use of only one source. The model can also account for uncertainty in the marginal distributions. Furthermore, we describe the methodology for obtaining expert opinion and explain involved psychological effects and popular fallacies. We exemplify the approach in a case study.

This paper considers insurance claims that are available by cause of loss, or peril. Using this multi-peril information, we investigate multivariate frequency and severity models, emphasizing alternative dependency structures. Although dependency models may be used for many risk management strategies, we focus on ratemaking.

Motivation for this research comes from homeowners insurance and so, for the frequency portion, we consider binary response models. Specifically, we examine several multivariate binary regression models that have appeared in the biomedical literature, focusing on a dependence ratio model. For multivariate severity, we use Gaussian copulas to represent dependencies among gamma regressions.

We calibrate competing models based on a representative sample of over 400,000 records and validate them using a held-out sample of over 350,000 records. We find that methods that allow for cross-dependencies among perils provide important economic value in pricing.

This paper demonstrates actuarial applications of modern statistical methods that are applied to detailed, micro-level automobile insurance records. We consider 1993-2001 data consisting of policy and claims files from a major Singaporean insurance company. A hierarchical statistical model, developed in prior work (Frees and Valdez (2008)), is fit using the micro-level data. This model allows us to study the accident frequency, loss type and severity jointly and to incorporate individual characteristics such as age, gender and driving history that explain heterogeneity among policyholders.

Based on this hierarchical model, one can analyze the risk profile of either a single policy (micro-level) or a portfolio of business (macro-level). This paper investigates three types of actuarial applications. First, we demonstrate the calculation of the predictive mean of losses for individual risk rating. This allows the actuary to differentiate prices based on policyholder characteristics. The nonlinear effects of coverage modifications such as deductibles, policy limits and coinsurance are quantified. Moreover, our flexible structure allows us to “unbundle” contracts and price more primitive elements of the contract, such as coverage type. The second application concerns the predictive distribution of a portfolio of business. We demonstrate the calculation of various risk measures, including value at risk and conditional tail expectation, that are useful in determining economic capital for insurance companies. Third, we examine the effects of several reinsurance treaties. Specifically, we show the predictive loss distributions for both the insurer and reinsurer under quota share and excess-of-loss reinsurance agreements. In addition, we present an example of portfolio reinsurance, in which the combined effect of reinsurance agreements on the risk characteristics of ceding and reinsuring company are described.

The authors review various facts about copulas linking discrete distributions. They show how the possibility of ties that results from atoms in the probability distribution invalidates various familiar relations that lie at the root of copula theory in the continuous case. They highlight some of the dangers and limitations of an undiscriminating transposition of modeling and inference practices from the continuous setting into the discrete one.

In nonlife insurance, actuaries usually resort to random effects to take unexplained heterogeneity into account (in the spirit of the Bühlmann-Straub model). This paper aims to study the kind of dependence induced by the introduction of correlated latent variables in the annual numbers of claims reported by policyholders. The effect of reporting claims on the a posteriori distribution of the random effects will be made precise. This will be done by establishing some stochastic monotonicity property of the a posteriori distribution with respect to the claims history.