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The Actuary The magazine of the Institute & Faculty of Actuaries

Escaping the triangle

Development triangles may not be the most accurate method for calculating claims reserves; Yiannis Parizas shares a methodology that could provide better results



When calculating claims reserves, the market practice is to use development triangles, or modifications of this method. Recent research has suggested that non-triangle techniques are more accurate, especially for estimating reserver volatility. Estimation of Incurred But Not Reported (IBNR) for monitoring and pricing purposes often requires drilling down into data segments. Those estimations could significantly benefit from a non-triangle method. However, the market is reluctant to abandon triangle techniques, due to their satisfactory levels of accuracy and simplicity. This article examines a new triangle-free methodology for modelling IBNR claims, and explains how and where the method can be practically applied and its benefits.

Claim triangles and their limitations

Triangulation methods work by banding homogeneous groups of claims together into either accident, underwriting or reporting period cohorts, and then observing the development patterns. The analyst observes the historical patterns and uses them to predict future development. This provides an acceptable liability estimate for a fixed claims segment.

For the purposes of portfolio monitoring (consider a dashboard that enables dynamic data drilling) or exposure modelling, there is the risk that the segment’s average historical development is applied to a sub-segment with significantly different development patterns, presenting a different picture. Ideally, we seek a methodology that identifies sub-segments with significantly different developments and parametrises them. 

Modelling pure IBNR and Incurred But Not Enough Reported (IBNER) separately can increase accuracy. Table 1 shows how reserves emerge through time, the current market practice and where the proposed IBNR split happens. 

Table 1

Reporting delay distribution decoded

Reporting delay is the time difference between when the claim happened and when it was reported. Maximum possible reporting delay is the time difference between the valuation date and the period’s inception date. If a probability distribution is applied to reporting delay, its cumulative distribution function (CDF) could be viewed as the expected percentage of reported claims (in terms of count) within a cohort. One minus this value would indicate the proportion of unreported claims during the policy duration. This will be proportional to pure IBNR exposure.

Another way of looking at this is to break the policy duration into days. For every day of the policy, we need the expected exposure relating to unreported claims. This translates to a reporting delay that is greater than the maximum reporting delay (here, this is the date difference between the specific day and the valuation day). From the start of the policy period moving to the end, then, each day has a maximum reporting delay of one less. 

The sum of expected unreported claims percentages for each day, estimated by summing the cumulative probabilities for every consecutive day, would give the earned exposure adjusted for pure IBNR exposure in days. This would not be practical, as the calculation would need to run for each day in the period considered and for each policy/risk/cover.

From reporting delay to pure IBNR

A way to achieve the required function is to integrate the cumulative distribution of the reporting delay. Integrating will sum the CDF from a minimum reporting delay (constrained above 0) to the maximum reporting delay. The integral takes the period size from a discrete day to continuous. 

It would be possible and simpler to model the reporting delay with a lognormal distribution, but this could provide less predictive outcomes than a triangle. Instead we can fit a lognormal regression to the reporting delay. Regression modelling enables us to identify the significant factors that affect the reporting delay and parameterise them.

A further refinement would be to model reporting delay with a right-truncated regression model, with the point of truncation being maximum possible reporting delay. This will enable granular results that are less prone to manipulation by the analyst and are not impacted by portfolio mix changes. It will also reveal patterns that were otherwise invisible to the analyst on an accident year triangle. Before modelling them together, it is crucial that the analyst checks the different segments have similar log-reporting delay volatility through time. Segments with significantly different log-reporting delay volatilities are better modelled separately.

It would be ideal to apply the output of the current loss cost models to the exposure obtained and take advantage of existing work. A loss ratio proxy on the exposure-adjusted premium will provide an acceptable approximation. In the case of regressions in pricing, modelling with only reduced exposure is more accurate than loading for IBNR and avoids having to apply an optimisation programme.



Benefits of the new method

Unlike triangles, this method requires less judgment from the analyst, leading to more granular developments, through a truncated regression model. 

More accurate factors are applied, as the method is continuous. The method enables the application of earnings patterns. Due to its regression nature, the model is insensitive to portfolio mix changes, and becomes more accurate in changing environments. This method can take advantage of cost-loss models’ built-in pricing, making the reserve prediction more accurate. The expertise for modelling the reporting delay regression already exists in pricing teams. 

As such, the process could bring pricing and reserving departments closer, allowing them to share expertise and promote specialisation and economies of scale within the organisation. Computationally, this process is fast, as the CDF integral provides an analytical solution. In terms of reserve distribution, this method is conceptually simpler than bootstrapping or the Mack model. Different reinsurance structures can be applied on top of the frequency-severity simulations, while the equivalent on a triangle would be more time consuming. There are practical applications of this method in different aspects of non-life work, although reserving may not benefit at the moment, due to reporting being based on triangles.

Yiannis Parizas is a pricing contractor at NetSim Analytics

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