When losses are expected to occur beyond the last development period of the loss development array, in the link ratio paradigm a ratio to ultimate is used to estimate the remaining portion of losses outside the array. These ratios are often based on smoothing the tail development factors. Just like any other link ratio calculation, the ratio to ultimate is impacted by calendar period trend changes.
As we have shown in previous case studies, link ratios do not work well when there are changing calendar year trends - so why would we expect ratios to ultimate to fare any better?
What happens in the case of no calendar year trend?
SDF is a simulated dataset with a single development trend (-30% / year), no changes in accident level, and no calendar year trends (thus its name). The model for this dataset, using the true parameters, is shown below.
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The trends, when estimated from the simulated data, do not exactly match the generating parameters – the data were simulated with volatility – but they are consistent statistically (a confidence interval from the estimated parameters include the true parameters).
When completing the square, the expected ultimate is 103,560. In order to reach projections of zero, we extend the forecast from 17 development periods to 45 development periods. The expected ultimate, based on the true parameters, is 104,195.
The true ratio to ultimate is: 1.006.
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Whenever we have loss data, however, we do not have the luxury of knowing the true parameters, instead we have to estimate them from the data. The estimated parameters, optimized so all parameters are statistically significant from each other and zero, are shown below.
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The estimated ratio to ultimate is 1.008 and the ratio to ultimate versus accident year is shown below.
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The ratio to ultimate is centered around a different mean ratio to ultimate, but aside from this, the plots look identical.
There is no pattern in the ratio to ultimate, they are scattered randomly around the estimated ratio to ultimate.
Summary
In the case of no statistically significant calendar year trend, the estimated ratio to ultimate for individual accident years are expected to be symmetric around the total ratio to ultimate.
The development year trends are orthogonal to any accident year level changes, so this will hold even there are multiple accident year or development year trend changes.
What about a single calendar year trend?
Data are simulated for a simple model where a single trend is applied for each direction (development, accident, and calendar time). The corresponding estimated trends are shown below.
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The estimates of the ultimates are calculated when completing the square (17 development periods) and when the final development year is extended to 45 development periods. The ratio of these two estimates of the ultimate form the estimated ‘ratio to ultimate’. This is shown plotted against accident year below.
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Once again, when the trends are estimated, the estimated ratio to ultimates by accident year, are randomly scattered around the mean ratio to ultimate.
Summary
For the examples considered, the ratios to ultimates by accident year are randomly scattered around the ratio to ultimate for the total ratio to ultimate in the cases of no calendar year trend or a single calendar year trend.
This is consistent
Mini-summary
How do changing calendar year trends impact estimated ratios to ultimate?
For example, consider the following data with two calendar year trends and no changes in development year trend or accident year level.
For this example, the calendar year trend since 1990 changes from a [statistically] zero calendar year trend to a calendar year trend of 16.35%+_4%.
What happens to the ratio to ultimates?
We calculate the ratio of: the ultimate from completing the square (16 development periods) to the ultimate from projecting to run-off completion (44 development periods). The plot of this ratio versus accident year is shown below.
There is a clear positive trend in the ratio to ultimate. It would be completely inappropriate to assume that the total ratio to ultimate was applicable to each individual accident year.
Why?
The calendar year trend projects onto the development and accident year periods. This means that the early accident years are not as affected by the changing calendar year trend as the more recent accident years. The effect of this new positive inflationary trend is that the ratio to ultimate must increase as the later development losses are increased by the inflation.
Similarly results apply if the more recent calendar year trend decreases, but where the ratio to ultimate is lower more recently.
The essential component is, the P&C Actuary must know what calendar year trend assumptions apply in the past as well as the future.