## Reporting delays

When performing a mortality analysis, it is my practice to disregard the most recent six months or so of experience data.  The reason is delays in the reporting and recording of  deaths, i.e. occurred-but-not-reported (OBNR) to use the terminology of Lawless (1994).  We use the term OBNR, rather than the more familiar term IBNR (incurred-but-not-reported); IBNR is associated with "cost-orientated" delay distributions of insurance claims (Jewell, 1989), whereas we are focused on just the delay itself.

But what evidence is there to support my six-month rule of thumb?  As part of ongoing research in Richards (2021), I received separate extracts of annuitant mortality experience in June and September 2020 from two insurers: one French, one British.  By matching up the annuities in successive extracts, I identified cases that were marked as in-force in the June extract, but which were marked as pre-June deaths in the September extract.  These cases were therefore late-reported deaths, i.e. they had occurred before the June extract but hadn't been reported to the insurer at the time of extract. Figure 1 shows the three possibilities for a pre-June death occurring on a given date:

Figure 1. Reporting delays for three deaths A, B and C occurring on the same date before the June extract.  Source: Richards (2021).

In Figure 1, Death A occurs and is reported before the June extract. Death B occurs before the June extract, but isn't reported until after the September extract; we therefore know nothing of Death B.  Death C occurs before the June extract, but is reported sometime between the June and September extracts.  We note that comparing these two extracts gives us incomplete information on the reporting delays:

1. Type A.  Without separate recording of notification dates, we know nothing of the delay between death and its notification; our delay times are therefore left-truncated.
2. Type B.  Not all pre-June deaths will have been reported by the time of the September extract; our delay times are therefore right-truncated.
3. Type C.  We don't know precisely when such pre-June deaths were notified, only that they had been notified by the time of the September extract.  We therefore know that each delay time is at least the difference between the date of death and the date of the June extract (this is the "Min delay" in Figure 1).  We also know that the delay is at most the difference between the date of death and the date of the September extract (this is the Max delay in Figure 1).  Our delay times are therefore interval-censored.

With these limitations in mind, Table 1 shows the data that we have.  We note that the FRA portfolio has a greater problem with late-reported deaths, as it has more Type C deaths and fewer Type A deaths; the ratio of late-reported to reported deaths is therefore far higher for the FRA portfolio.

Table 1. Reporting delays for annuitant mortality observed between successive data extracts in 2020. Source: Richards (2021).

FRAUK
Last date of death in first extract ($$u_1$$) 17 June 17 June
Type A deaths in 2020 1,171 2,045
Last date of death in second extract ($$u_2$$) 17 Sept. 15 Sept.
Type C deaths reported in $$(u_1, u_2]$$ 853 488
Ratio of Type C deaths to Type A deaths 0.729 0.239
Median minimum delay 54 days 39 days
Longest minimum delay 15.95 years 24.48 years

Figure 2 shows the distribution of minimum delays for Type C late-reported deaths for each portfolio:

Figure 2. Cumulative distribution of minimum reporting delays for Type C deaths reported in June-September 2020. Source: Richards (2021).

Figure 2 shows that the UK portfolio has an initially strong rate of reporting, but that this then tails off quickly with a substantial proportion of delays lasting at least a year.  In contrast, the French portfolio has an initially slower rate of reporting, but most deaths seem to be reported within half a year.  While the delays in the FRA portfolio look better in Figure 2, it must be remembered that the FRA portfolio has a far higher ratio of Type C deaths to Type A, and so has a bigger OBNR problem overall.

So what does this mean for my six-month rule of thumb for mortality analysis?  We cannot just look at the distribution of delays in Figure 2, as the two portfolios have very different ratios of Type C deaths to Type A deaths.  We must therefore look at the impact on the mortality estimate overall.  To do this, we can use the semi-parametric estimator for time-varying mortality, $$\hat\mu_y$$, defined in my previous blog.  We define the following ratio:

$R=\frac{\hat\mu_y {\rm using\ June\ extract}}{\hat\mu_y {\rm using\ September\ extract}}\qquad(1)$

$$R$$ will give an idea of the impact of reporting delays by showing the proportion of deaths known at the extract date in June.  In fact, $$R$$ will under-state the true impact, as the denominator relies on all pre-June deaths having been reported by the September extract, something we know not to be quite true because of the existence of Type B deaths.  Plots of $$R$$ for the two portfolios are shown in Figure 3:

Figure 3. Ratio of $$\hat\mu_y$$ using June extract to $$\hat\mu_y$$ using September extract $$(c=0.1)$$. Source: Richards (2021).

Figure 3 broadly confirms the six-month rule-of-thumb: late-reported deaths have minimal impact half a year or more prior to the extract date, as the ratio is close to 1.  After that, however, the impact of OBNR deaths becomes more dramatic, especially for the FRA portfolio.  The reason that this is important is twofold: first, mortality levels would be under-estimated if we included the most recent OBNR-affected period; and second, improvement rates would be over-stated as the OBNR effects were conflated with the time trend.

References

Jewell, W. S. (1989) Predicting IBNYR Events and Delays: I. Continuous Time, ASTIN Bulletin, 19(1), pages 25–55, doi:https://doi.org/10.2143/AST.19.1.2014914

Lawless, J. F. (1994) Adjustments for reporting delays and the prediction of occurred but not reported events, Canadian Journal of Statistics, 22(1), pages 15–31, doi:https://doi.org/10.2307/3315826.n1.

Richards, S. J. (2021) Mortality shocks and reporting delays in portfolio data, Longevitas working paper.

The time-based Nelson-Aalen estimate and the corresponding hazard estimate can be obtained from running a data audit on any uploaded Longevitas file.  Clicking on the resulting icon downloads an Excel spreadsheet that allows the user to vary the smoothing parameter, $$c$$.