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Brainstorms: Modeling Payout Patterns

by David Sommer and Stephen Philbrick

For reserving actuaries, a long-standing challenge seems to be more insistent these days. There is an increasing level of interest in developing a range around our estimates of future liabilities, rather than just a best estimate. And while using the range of results from multiple methods provides an answer to this request, it doesn’t really address the real question—"How confident are we that future liabilities won’t exceed a given level?" Further, with the increasing awareness of DFA, variability in the payment pattern has become an issue of interest as well. We would like over the next few articles to discuss some of the challenges we’ve found in developing algorithms to address these issues. Our approach has been to abandon the search for the theoretically perfect approach and find some practical alternatives that, while addressing some issues adequately, have shortcomings that are acceptable. As a result, the focus will be towards simulation approaches, rather than analytic solutions.

For this discussion, let’s take the case of simulating a payment pattern. One problem when modeling payout patterns is ensuring that the resulting ultimate paid amounts match the incurred amounts. When simulating losses, it is common to simulate the ultimate losses for a particular year in one step. If we then randomly simulate the dollars paid in subsequent calendar periods in the obvious way, we have to deal with the fact that the simulated paid losses won’t necessarily add up to the original incurred pick. Solutions such as rescaling are not particularly appealing. We wanted to find a method that would address this in a "cleaner" way.

There is another problem when simulating payout amounts—does it make sense to simulate successive evaluation points (columns) independently? The reason that this question is important is that the answer affects how variability is measured. If we are looking at a column of losses and notice a particularly small value, we will generally have one of the following two thoughts: 1) losses are going to be more than usual next period to play catch-up, or 2) losses are going to be less than usual because this is a good year. While uncertain of the sign, we generally feel there is some correlation between columns. Given the ambiguity of our thoughts, the prospect of measuring this, much less modeling it, is daunting.

The approach that we suggest is appealing for two reasons. First, it avoids this problem, second, it has some intuitive basis.

In paying claims, adjusters not only look at how much they paid previously, but also at how much they have outstanding. As such, rather than analyzing columns of the triangle as percentage growth of cumulative payments over previous payments, we restate the triangle as incremental payments as a percentage of outstanding loss. Testing shows that this approach leads to columns that are more independent.

In defining outstanding loss, we prefer to include IBNR. While the claims people don’t know the amount of the IBNR, they are affected by incoming claims. But more importantly:

We now can calculate moments for each column and estimate the parameters of whichever distribution we choose to use. We can then simulate the percentage to be paid during each period and apply it to the estimated ultimate losses to determine the variation in future payments due to timing risk.

In summary, modeling payments, not as a function of paid-to-date, or exposures, but instead as a ratio of remaining outstanding:

Any thoughts?