AIR Currents

Jul 26, 2021

Editor's Note: This article is the third in a series of four articles that AIR is publishing every few months about our 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. In this article, we discuss the propagation of uncertainty to commercial lines—loss accumulation and complex (re)insurance structures. In the final article of the series, we will discuss the next generation loss module direct treaty and facultative contracts.

The commercial, corporate, and specialty market segments have distinct attributes, which bring complexity to many modeling tasks. The contracts in this segment are multi-location and often close together geospatially, which makes the modeling of dependencies in loss accumulation of critical importance. Such contracts are sublimited and layered with detailed structures for covering individual and dependent sub-peril losses. Commercial and excess lines contracts are typically multi-layered with nested reinsurance placed on individual risks.

In this article we will discuss how gross loss accumulation with spatial correlation works in our next generation financial module in terms of the algorithm used; the basic financial operations the module performs; the impact of spatial correlations on gross loss; and two different loss perspectives—the sum of losses after the sublimit and the sum of losses after the application of a layer. We will then discuss practical use cases of the next generation financial module that illustrate how it can produce an accurate and realistic view of your risk for commercial and corporate policies.

Gross Loss Accumulation with Spatial Correlation in AIR’s Next Generation Financial Module

In AIR’s next generation financial module, the loss aggregation algorithm accounts for dependencies among different insurance coverage types and for spatial correlation between risks; this involves the computation of gross loss after each set of financial conditions is applied at each hierarchical level of portfolio rollup.

The Algorithm

Mathematically, gross loss estimation is equivalent to solving for the transformation of an arbitrary sum of risks S=X1+X2…+XD . In this context, the word “arbitrary” stands for “positively dependent,” i.e., positively correlated. Risks attributed to different types of locations are represented by random variables X1, X2,… ,XD. We assume that our arbitrary sum S is enclosed within two bounds: independent sum S, where risks are assumed to be independent, and comonotonic sum S+, where risks are assumed to be maximally correlated. The distribution of the arbitrary sum S is computed using the mixture method, as discussed in Part I, which reads:

formula 1


The value of the weight w depends on the positive correlation between pairs of risks and/or their partial totals. During portfolio rollup, the sum of risks S is subject to the transformation ɸ(S), which represents the application of financial terms. The distribution of the transformed sum reads:

formula 2


Where s′ = ɸ(s). As we can see, the application of financial terms to the sum S is equivalent to the application of financial terms separately to the independent sum S and to the maximally correlated sum S+, without affecting the value of the weight w. This implies that the correlation between pairs of risks and/or their partial sums is constant under the application of financial terms ɸ. In other words, the correlation remains unchanged during both ground-up and gross loss estimation.

Basic Financial Operations

Insurance financial terms are features designed to modify the loss payments. The most common insurance structure is the deductible, the amount the policyholder has to bear before the insurer will pay the claim. Application of the deductible d on the sum of risks S yields:

formula 3


For higher value properties, the insurer is likely to cover a specified limit u of the entire rebuild cost. Limits take a similar form as deductibles—a piecewise linear function:

formula 4


If, after consecutive application of deductible and limit,

formula 5


the insurance payoff is less than the rebuild cost, the difference will again fall to policyholder.

In the next generation financial module, risk aggregation is performed by combining probability distributions with differing loss sizes. Therefore, deductibles and limits are applied directly on these distributions. A typical loss distribution is shown in the right panel of Figure 1. To apply the deductible, we first placed it on the loss axis (dashed red line, upper panel in Figure 1), then accumulated the probability mass below the deductible onto the deductible, and finally set the deductible as zero of the new loss axis. Similarly, the application of a limit (dashed red line, lower panel in Figure 1) takes all the probability mass above the limit and accumulates it onto the limit.

Rafał WójcikRafał Wójcik, Ph.D.
Director of Financial Risk Modeling

Ivelin ZvezdovIvelin Zvezdov
Assistant Vice President

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

Figure 1
Figure 1. An example of the application of a deductible (upper panel) and limit (lower panel) to a loss distribution. Gross 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 deductibles and limits at different tiers of financial terms, such as sublimits, layers, policies, etc. (Source: AIR)

Impact of Spatial Correlation on Gross Loss

The degree to which the spatial correlation between risks can impact gross loss estimates is sensitive to the conditions of a particular portfolio, especially to the values of deductibles and limits applied at different levels of loss aggregation. To make this last statement a bit more practical, let us consider the insurance contract in Figure 2. We have three locations affected by a hurricane event. Losses at these locations are correlated in space. The sum of losses from Location 1 and Location 2 is subject to a sublimit and then gross loss after sublimit and loss from Location 3 go to a layer.

Figure 2
Figure 2. An example of typical insurance contract. (Source: AIR)

Portfolio rollup for our contract can be represented as the loss aggregation tree shown in Figure 3. The transformation ɸ represents the application of a sublimit and ψ represents the application of a layer. Each node of this tree computes the mixture distribution of losses entering that node, as discussed in Part I. Note that the top root node, at which the sum of losses is passed to the layer, is an apt example of function composition, i.e., ψ is applied to the linear combination of ɸ and free location.

Figure 3
Figure 3. Gross loss aggregation tree for the insurance contract. (Source: AIR)

Two Loss Perspectives Explained

We are now going to focus on two different loss perspectives: the sum of losses after the sublimit and the sum of losses after the application of a layer.

Figure 4. Animation of aggregate loss distributions for the insurance contract in Figure 2, where the correlation coefficient ρ increases from 0 to 1. Distribution of gross loss after the sublimit and after the application of a layer, respectively, is shown in red frames. (Source: AIR)

You can assume that losses at the three locations (X1, X2, and X3) are represented by loss distributions in the upper panel of Figure 4. The correlation coefficient between the three pairs of locations is ρ. Two loss distributions characterizing loss perspectives of interest are in red frames. The animation shows the loss distributions as ρ varies from 0 to 1. We can see that the effect of correlation is negligible, i.e., the change in the shape of the two distributions is small. This is due to the particular values of the deductibles and limits, which characterize the sublimit and the layer respectively. If the values of the limits and deductibles are set as in the Figure 5 animation, the effect of increasing spatial correlation is more pronounced: The shape of loss distributions in red boxes changes. This is due to the fact that truncation of the aggregate loss distributions at the sublimit and at the layer in Figure 5 is performed on different loss intervals than those in Figure 4. As a consequence, shape transition increases the mean and standard deviation of the total gross loss.

Figure 5. The same insurance contract as in the Figure 4 animation. The values of deductibles and limits, which parameterize the sublimit and layer, are higher than those in Figure 4. (Source: AIR)

AIR’s methodology for loss aggregation preserves spatial correlation between losses during gross loss estimation in a statistically sound manner. The impact of spatial correlation on portfolio gross loss depends on (i) the strength of the correlation between risks; (ii) the shape of the marginal distributions, which describe losses at locations; and (iii) the values of the deductibles and limits characterizing a particular portfolio.

Loss Modeling and Risk Transformation for Sub-Perils

The importance of modeling losses from secondary perils, or sub-perils, is growing in the industry as insurance policy coverage for them is becoming more comprehensive and more explicit. To accurately reflect market conditions, modelers are tasked with creating policy layers and sublimits for each individually covered sub-peril and for the combined peril loss of all risks in the contract. The critical enhancement provided by AIR’s Next Generation Models is the propagation of secondary uncertainty distributions of loss for each individual modeled sub-peril to the new financial module for the application of any permutation of contract and peril coverage. This new capability allows you to place terms and conditions specifically by sub-peril, thus enabling you to compute all components of modeled loss (Figure 6).

Figure 6
Figure 6. Support for secondary uncertainty distributions by all modeled sub-perils in AIR’s NGM U.S. hurricane model. (Source: AIR)

We have introduced this enhancement on a global scale—for all AIR tropical cyclone and earthquake models and for our U.S. severe thunderstorm model.

The specific structure of a storm surge sublimit as part of a policy excess layer is a good example to illustrate this enhancement. For our example we maintain loss accumulation for storm surge separately, as we already have support for secondary uncertainty distributions from the U.S. hurricane model. First, we apply the surge sublimit and compute the surge gross loss after the policy sublimit.  Second, we prorate the combined peril loss distribution to reflect the capping and reduction of surge losses. Third, the combined and prorated distribution of wind and surge flows into the policy excess layer (Figure 7).

Figure 7
Figure 7. Application of storm surge sublimit using AIR’s NGM U.S. hurricane model. (Source: AIR)

Lastly, we apply the excess layer on the prorated and combined loss distribution. The application of the layer in turn leads to the capping and reduction of the overall contract gross loss. To reflect this outcome on the individual losses for surge and wind, we prorate the surge and wind loss distribution to arrive at the final result for all three components (Figure 8). To enhance transparency and improve the usability of our metrics, we report combined and individual sub-peril losses with unique records in the year-event-loss tables (YELTs). The explicit support for sub-perils is an important enhancement that our Next Generation Models bring to the market.

Figure 8
Figure 8. Producing final contract gross loss results in AIR’s NGM U.S. hurricane model. (Source: AIR)

In addition to explicitly accounting for sub-perils, another key improvement the Next Generation Models provide is the vertical tiering and nesting of the insurance components of the commercial contract. Yet another is the nesting of individual per-risk reinsurance contracts within the insurance loss workflow. In both of these cases, the modeler’s task is to group risks by class and by geography and to sublimit them such that the modeling process and loss outcomes meet both the underwriting requirements of the insurers as well as the demands of the insured. In our Next Generation Models we provide two more tiers of policy sublimits to enhance flexibility and capabilities to transform risk accumulations by geography, by insurance coverage, or by sub-peril to realistically render the intricacies of large commercial and corporate insurance contracts (Figure 9).

Figure 9
Figure 9. Accumulations and risk transformation with two tiers of sublimits in NGM. (Source: AIR)

Providing Support for Aggregate Policies

Aggregate policies in their many forms are a traditional market instrument; they can cover insurance losses from a single location as well as accumulated losses from groups of many locations in large commercial contracts. In both cases the aggregate limit protects the insurer from the accumulation of claims and losses from multiple catastrophe events in the contract year.  By the same token, the aggregate deductible protects the insured from the accumulation of retained losses in the case of multiple events and claims in the same contracted year. In our Next Generation Models we provide the ability to structure policies with annual aggregate terms at all three insurance tiers supported in Touchstone®: location, sublimit, and layer.

The actuarial settlement of the aggregate limit, at any of these three tiers, can be achieved in three intuitive steps. Figure 10 illustrates the overall logic of the new algorithm, which was discussed in the first section: (1). The full aggregate limit is applied on the first catastrophe event on the contract year; (2) the practitioner will estimate the remaining and applicable limit, which has not yet been exhausted from the first event; (3) this applicable limit itself will be applied to the occurrence of a second event in the year.

Figure 10
Figure 10. Actuarial procedure for application of annual aggregate limit in NGM. (Source: AIR)

We can transform the three steps described above (i.e., support for sub-perils, nested contract terms, and aggregate policies) from standard actuarial procedures to a fully probabilistic algorithm. We can then apply these algorithms to commercial policies, to apply aggregate structures on sublimits and layers. In this way, we can ensure that all actuarial operations are applied to fully probabilistic loss distributions. Aggregate policies in our Next Generation Models are structured and can be placed on both single-peril modeled loss analyses, as well as on multi-model loss analyses across all of their independent events. The enhanced financial module can handle all of this complexity and all you need to do is simply ensure that all of the appropriate peril codes are set on each location, contract, and layer level for each covered peril.

As Figure 11 shows, the aggregate policy structure is applied on the first stochastic event of the year; then the aggregate policy is eroded and transitioned to the second stochastic event of the year. This procedure continues until all events in the stochastic and contracted year are covered by the annual aggregate terms. The same principles are utilized in the case of aggregate policies for a single peril, as well as in the case of aggregate policies for multiple models and multi-perils.

Figure 11
Figure 11. Chronological transition of an annual aggregate layer. (Source: AIR)

Creating Nested, Per-Risk Reinsurance Workflows

A typical example of complexity and nesting within insurance workflows is the per-risk reinsurance treaty on location with a policy layer. With our Next Generation Models we provide the ability to place a location treaty, such as quota share, surplus share, or excess per risk, on the location level and complete the policy with a layer and sublimit structures in the same contract. The enhancement and innovation of our Next Generation Models in this case enables you to derive the actuarial net-of-treaty loss distribution in a fully probabilistic form—by event and by location (Figure 12). 

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

This methodology is based on the principles of comonotonic subtraction. Once these net-of-location reinsurance distributions are computed, we accumulate them across multiple risks. This aggregated distribution represents the net loss of a location treaty for the entire commercial contract. This accumulated distribution is then propagated to the policy layer.

The Business Benefits of Next Generation Financial Modeling

The implementation of this new methodology for annual aggregate deductibles and limits, and for any type of aggregate policy, enables you to build more complex and realistic contract structures than ever before. Given that the aggregate deductible and limit can be placed on both the sublimit level and on the policy layer level, this new design provides flexibility for creating multi-tiered and nested policies to address market demand for the accurate rendering of insurance and reinsurance terms, conditions, and clauses.

The business—and modeling—benefits that come from having a realistic representation of all terms and conditions of actual commercial policies and from the implementation of a modern methodology of loss accumulation that propagates all modeled dependencies and uncertainty is clear. Leveraging our Next Generation Model algorithms to produce an accurate and realistic view of your risk for commercial and corporate policies offers you a wide range of benefits.

In our next and final article in this series, we will discuss next generation financial modeling for the direct treaty and facultative market segments in addition to how you can compute the secondary uncertainty parameters of the standard deviation of loss and maximum quantile of loss for the reinsurance treaty and report these in the treaty-event-loss tables. Adding these metrics to your results will improve the understanding of uncertainty and the variability of all treaty, and net-of-treaty losses.


Hürlimann, W. (2002). On risk and price: stochastic orderings and measures. Transactions 27th International Congress of Actuaries, Cancun.

Mitchell-Wallace, K. Jones, M., Hillier, J. K., Foote, M. (2017) Natural catastrophe risk management and modelling: A practitioner’s guide. Wiley Blackwell.



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