AIR Currents

Mar 29, 2022

Editor's Note: This is the final article in our four-part series about Verisk’s Next Generation Modeling (NGM) initiative. Part I discussed our new loss accumulation methodologies with spatial correlations. Part II discussed the implementation of our new loss accumulation methodologies for residential and small business lines and how we model secondary uncertainty loss distributions, coverage correlations, and single risk terms. We also discussed financial modeling for secondary perils, as applicable to residential and small business lines, using the next generation financial module. Part III discussed the propagation of uncertainty to commercial lines—loss accumulation and complex (re)insurance structures. In this article, we discuss how the next generation loss module calculates direct, facultative, and treaty contracts losses.

We define facultative and treaty per-risk, or simply risk reinsurance, as reinsurance contracts placed on location, excess policy layer, insurance contract, line-of-business, and selected geographical resolutions. There are a few distinct tiers, or as we call them in our product—target types—for placing and applying facultative and risk reinsurance. The transformation of insured gross loss as it passes through these tiers of reinsurance contracts is a good illustration of complexity in catastrophe modeling (Figure 1). One of the main achievements of Verisk’s Next Generation Models for risk reinsurance is that this multi-tiered transformation is now done with fully probabilistic methodologies for loss accumulation and application of market terms and conditions.

The key benefits Next Generation Models provide you are twofold: All accumulation of facultative loss from the location to the portfolio is done in a way that captures and propagates spatial distance and peril-specific dependencies; and facultative loss is now reported with its uncertainty parameters providing a fuller and more comprehensive view of risk for the facultative reinsurer.

Figure 1
Figure 1. Multi-tiered transformation of insurer loss. (Source: Verisk)

In this article we provide three generalized loss perspectives: the facultative reinsurer; the risk treaty reinsurer; and a net of contract loss of the insurer after the application of all tiers of facultative and treaty risk reinsurance. We also provide a worked-out example of the key benefits and opportunities for new workflows that the application of our new methodologies provides you.  

Reinsurance Loss Accumulation with Spatial Correlation in Verisk’s Next Generation Financial Module

A typical portfolio rollup starts with loss accumulation of insurance coverages to locations and sub-limits. Then location losses are accumulated to a policy and, in the final step of loss accumulation, policy losses are integrated with the total portfolio loss. In the next generation financial module, the loss aggregation algorithm accounts for dependencies between different insurance coverages and for spatial correlation between risks. This involves the computation of gross reinsurance loss after each set of financial conditions is applied, as described in Part I.

In the next generation financial module, risk aggregation is performed by combining probability distributions with differing loss sizes. Therefore, retentions and limits are applied directly onto these distributions. A typical loss distribution is shown in the right panel of Figure 2. To apply the retention, we first placed it on the loss axis (dashed red line, upper panel in Figure 2), then accumulated the probability mass below the retention onto the retention, and finally set the retention as the zero point of the new loss axis. Similarly, the application of a limit (dashed red line, lower panel in Figure 2) takes all the probability mass above the limit and accumulates it onto the limit.

Figure 2
Figure 2. An example of the application of a retention (upper panel) and limit (lower panel) to a loss distribution. (Please note: For ease of visualization, this figure depicts the loss probability as a continuous distribution. In reality, the software uses a discrete distribution, as illustrated in Figure 1.) (Source: Verisk)

Rafał WójcikRafał Wójcik, Ph.D.

Ivelin ZvezdovIvelin Zvezdov
Assistant Vice President

Edited by Sara Gambrill, CEEM, and Serge Gagarin, CEEM

Reinsurance loss estimation for a portfolio can be thought of as the mathematical function composition, that is, the application of one function to the result of another. In this case, functions represent the application of retentions and limits at different tiers of financial terms, such as layers, treaties, etc.

Impact of Spatial Correlation on Loss

The degree to which the spatial correlation between risks can impact reinsurance loss estimates is sensitive to the conditions of a particular portfolio, especially to the values of retentions and limits applied at different levels of loss aggregation. To illustrate this last statement in a more practical way, let us consider the reinsurance contract in Figure 3, which has two locations affected by an earthquake. Ground-up losses at these locations are correlated in space. The sum of losses from Location 1 and Location 2 is subject to a facultative contract and then the insurer’s net facultative loss goes to a per-risk XOL treaty. In what follows, we are going to focus on four different loss perspectives: layer gross loss, facultative underwriter loss, insurer net loss, and treaty underwriter loss.

Figure 3
Figure 3. An example of typical reinsurance contract. (Source: Verisk)

Portfolio rollup for the case of our contract can be conceptualized as a loss aggregation tree in Figure 4. Here transformation ɸ represents the application of a facultative contract on the layer X_1+X_2 and ψ represents the application of the XOL treaty on net. The bottom node of that tree computes the mixture distribution of losses entering the node. The top root node at which net loss enters the treaty is an apt example of function composition—the treaty is applied to the result of net loss computation.

Figure 4
Figure 4. Loss aggregation tree for the reinsurance contract. (Source: Verisk)

Three Loss Perspectives Explained

Figure 5. Animation of aggregate loss distributions for the reinsurance contract in Figure 4, where the correlation coefficient ρ increases from 0 to 1. Distribution of facultative underwriter loss, insurer net of facultative and treaty underwriter loss, respectively, is shown in the bottom panel. (Source: Verisk)

You can assume that losses at the two locations are represented by loss distributions in the upper panel of Figure 5. The correlation coefficient between these losses is ρ. Loss distributions characterizing facultative underwriter loss, insurer net loss, and treaty underwriter loss are shown in the bottom panel of Figure 5. The animation shows the loss distributions as ρ varies from 0 to 1. You can see that increasing correlation impacts the shape of the X_1+X_2 layer loss distribution, decreases mean facultative underwriter loss, and increases net and treaty underwriter loss. In all loss perspectives the standard deviation of loss increases with increasing correlation. This behavior is of course dependent on the shape of loss distributions and values of retentions and limits characterizing a particular portfolio.

Verisk’s methodology for loss aggregation preserves spatial correlation between losses during loss estimation in a statistically sound manner. The impact of spatial correlation on portfolio reinsurance loss depends on: the strength of the correlation between risks; the shape of the marginal distributions, which describe losses at locations; and the values of the retentions and limits describing reinsurance structures with different expected loss cost.

Generalized Workflows for Facultative, Treaty, and Net of Contract Losses: Key Benefits

The estimation of loss in each generalized loss perspective—facultative reinsurer; risk treaty reinsurer; and net of contract loss after application of all tiers of facultative and treaty risk reinsurance—is characterized first by fully probabilistic loss accumulation using the mixture method with correlations by the principles defined in the previous sections. Results are then produced by the fully probabilistic application of terms and conditions. Ultimately, loss results and parameters fully capture and contain all modeled uncertainty in this multi-tiered transformation process. Facultative contracts are typically the first tier of reinsurance, placed on the location and/or policy layer. Treaty contracts are the second tier of reinsurance and are typically applied after facultative treaties and involve a large number of locations. Net of contract losses are the final tier after all reinsurance terms have been applied. All reinsurance terms are applied in a fully probabilistic way. 

Figure 6
Figure 6. Accumulations and workflow for location facultative contracts. (Source: Verisk)

Risk treaties are typically two-tiered contracts with location recoveries and aggregate treaty terms. In our current modeling framework, we compute these deterministically with mean-based operations. The benefits from Next Generation Models come from the application of all new fully probabilistic methodologies. Again, we accumulate location treaty recoveries with all next generation principles of loss accumulation. These methodologies capture peril-specific insurance coverage and geographical distance dependencies. 

Figure 7
Figure 7. Accumulations and workflow for location treaty with aggregate terms. (Source: Verisk)

Treaty aggregate terms are applied with our new fully probabilistic methodology. This new methodology for the application of reinsurance aggregate terms is designed to encompass and propagate all modeled uncertainty, and thus produce the needed accuracy, fidelity, and veracity of loss outcomes. 

Figure 8
Figure 8. Fully probabilistic method for the application of annual aggregate treaty terms. (Source: Verisk)

We apply the aggregate treaty terms on the first stochastic event in the year. Then we exhaust them with a fully probabilistic technique, and as a result we are able to estimate the applicable set of terms for the second event in the same year. Lastly, we use the applicable set of terms on this second event and repeat the process on all consecutive events. This method allows us to produce a comprehensive view of variability and uncertainty for the treaty reinsurer, which we define in our results tables with the standard deviation of treaty loss and maximum quantile of treaty loss.

The same principles of fully probabilistic loss accumulation from location and single risk to portfolio define the benefits for you in higher accuracy and higher fidelity of pre-CAT net losses. This is achieved with new metrics of all modeled uncertainty. 

Figure 9
Figure 9. Accumulations and workflow for insurer pre-CAT net loss. (Source: Verisk)

This view is reflected in the reporting of standard deviation of insurer pre-CAT net loss, and maximum quantile of pre-CAT net loss. As with treaty and facultative loss perspectives these are the new loss metrics available now.

Business Value of Next Generation Methodologies: Example

To illustrate the business value of the application of our new methodologies to the modeling of facultative and risk treaty reinsurance, we developed a realistic scenario with worked-out numbers. The following example is constructed for a notional business unit with three tiers of reinsurance—facultative contracts on each location, per-risk excess of loss (XOL) treaty on each location, and business unit (XOL) treaty. We used a group of high-value commercial risks in a coastal county of Florida and computed losses using the historical event of Hurricane Andrew 1992. Such a complex structure with three tiers of reinsurance (Figure 10) can only be modeled and computed in fully probabilistic mode, which captures all uncertainty, with Next Generation Models.  

Figure 10
Figure 10. Three-tiered reinsurance structure for a notional business unit. (Source: Verisk)

To bring clarity to the workflows of our example and to further illustrate the benefits of our new methodologies, we estimate both the deterministic and fully probabilistic outcomes for this business unit. We estimate the losses for the insurer in three perspectives—net of facultative, net of risk treaty, and pre-CAT net i.e., net of business unit treaty. We also estimate the losses for the facultative and treaty reinsurers.

The business value of the new actuarial methodologies, represented in deterministic form in Table 1, is evident in the critical aspects of providing support for the modeling of complex multi-tiered reinsurance stacks and the reporting of all relevant loss results and perspectives for all parties involved in the distribution of risk. These new loss perspectives are facultative, risk recovery, and treaty losses—by location and by top aggregations—layer, contract, and geography.

For multi-tiered reinsurance structures covering the business unit as well as the whole book, Next Generation Models enable you to compute the net loss of each reinsurance contract and pass it to upper tiers of treaties. In the “Actuarial Methodology” column in Table 1, we present an example of the conceptual and deterministic form of these algorithms.

Table 1. Estimated losses (in USD millions) with deterministic approach.
Table 1

In Next Generation Models these actuarial algorithms are implemented with a fully probabilistic approach, which entails: loss accumulation with the mixture method with correlations; application of all reinsurance terms and conditions on probabilistic secondary uncertainty distributions; and reporting the uncertainty metrics of standard deviation and maximum quantile of loss for all results and perspectives (Table 2).

This example requires that we revisit the benefits of the fully probabilistic mode of actuarial computations. In the deterministic mode, as employed in Table 1, all actuarial operations of the application of terms and conditions are done on the single scenario of the expected value of loss, (Figure 11). In probabilistic mode, these operations are performed on all scenarios defined as discrete loss iterations of a statistical secondary uncertainty distribution. This method provides a full set of all possible loss scenarios, assigned with varying probabilities, and thus supports the propagation and accounting of modeled uncertainty.

Figure 11
Figure 11. Statistical loss distribution for probabilistic application of (re)insurance terms and conditions. (Source: Verisk)

The business value of fully probabilistic mode in Next Generation Models comes from the higher fidelity of accuracy and more realistic view of variability in outcomes, (Table 2). In general, reinsurance attachments in deterministic mode tend to overestimate insurer retentions and underestimate reinsurer contract losses when compared to their probabilistic scenario application. On the other hand, the application of limits in deterministic mode in general tends to overestimate the losses of the reinsurer and underestimates the loss on the reinsured vs. the same application on discrete actuarial distribution.

Table 2. Estimated losses (in USD millions) with fully probabilistic methods.
Table 2

Thus, we establish that the application of our new methodologies in loss accumulation and in actuarial mathematics provides the most accurate and realistic view of modeled uncertainty.  These methodologies better capture the range of losses that can result from events impacting these types of contracts.

Facultative and Treaty Contracts Applied on Insured Sub-Peril: Opportunities for New Workflows

As insurers are more frequently turning to complex contracts that offer a wide array of terms, the demand for flexibility, conditionality, and customization in reinsurance contract design is growing. Designing contracts for optimization and reuse of coverage capacity in different loss frequency and magnitude scenarios is an increasingly attractive option, particularly in an environment in which the frequency and severity of significant loss-causing events—such as severe thunderstorms and floods—are climbing as global temperatures and sea levels rise.

Next Generation Models support flexible workflows and complex contract structures with minimal to no changes and interventions in our existing exposure coding standards by providing new location-to-layer-to-contract rules for peril coding and loss accumulation. Through our enhanced actuarial methodologies NGM models can support a flexible schema of the splitting, combining, and accumulating of coverage, peril, sub-peril, and multi-peril losses into facultative, excess layer, and risk treaty structures. Figure 12 provides an example of the workflow for a location treaty with a policy layer in NGM.

Figure 12
Figure 12. Workflow for a location treaty with a policy layer in NGM. (Source: Verisk)

With Next Generation Models you can place facultative and treaty reinsurance by all insured perils and sub-perils. To develop this technique, it was necessary to first examine the accumulation of insurance gross loss by sub-peril to policy level—the definition of contract gross and pre-CAT net loss. In this approach, no longer is contract gross loss, also defined as policy gross loss, simply equivalent to the accumulated layer gross loss; in Next Generation Models, contract gross loss is the accumulation of all layered gross loss and any and all non-layered sub-peril gross loss from the location level. 

The NGM definition of accumulation of sub-peril gross and pre-CAT net insurance loss creates opportunities for modeling and computing many new workflows. As Figure 13 demonstrates, we support location/sub-peril combinations of loss, which can be defined to participate or not participate in layered gross loss. In both cases layered gross and non-layered sub-peril loss at the location will always accumulate to policy pre-CAT net loss. Thus, we can define separate and unique reinsurance contracts by peril. 

Figure 13
Figure 13. Placing risk treaty on wind layer and facultative on non-layered flood/surge location loss. (Source: Verisk)

Figure 13, an example of this enhanced flexibility in reinsurance workflows, shows an aggregate risk treaty placed on accumulated layer loss from all wind damage. Separately and independently facultative contracts are placed on locations with high likelihoods of surge and flood damage. 

Next Generation Modeling: An Exclusive Opportunity to Model Advanced Structures in Insurance

As it was when we applied new methodologies to insurance gross loss modeling, our fundamental motivation for propagating these principles to facultative and risk treaty loss was to address market and client challenges and complexities with higher accuracy and higher fidelity. How do these methodologies do this? First, they account for geospatial and peril-specific dependencies in loss modeling and more accurately propagate uncertainty in such loss accumulation flows. Second, with our enhanced actuarial methodologies we can support multi-tiered and nested policy structures. Third, our next generation modeling framework allows our clients to model facultative and treaty contracts uniquely and individually for secondary sub-perils with the full fidelity of modern probabilistic methods. Next Generation Models represent the only available opportunity to model advanced structures in the insurance market segment.



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