Inflation

For the Motor Liability portfolio, a combined inflation must be taken into account, whereby for material claims (as for Motor Casco) an average inflation of new car purchases as well as vehicle repairs can be assumed. For personal injury claims, the CPI excluding energy is rather [1]obvious, because this compensation consists of a mixture of loss of working ­capacity (VAV) and reimbursement of other costs. The VAV in particular will be adjusted in the medium term towards the realized CPI excl. energy (assuming that wages in the longer term will be corrected for the high inflation in 2022), and given that this represents a large part of the incurred loss, this gives a substantiation for assuming the CPI excluding energy for this purpose.

In order to arrive at a general inflation figure for this Line of Business (LoB), a weighting between material claim and personal injury must therefore ­be determined. We can deduce this from the statistics in various ways:

1. When an insurer has information about the historical relationship of incurred losses between injury and material claims, this historical relationship can be used. If data is available for this, it is wise to ­investigate whether there are trends in the frequency of claims for both personal injury and material claims, as well as for the average claim amounts.
   If possible: correct for these trends to obtain a best estimate for the weighting between injury and material claims for the year 2023.
2. When the data quality does not allow for a historical weighting in one's own portfolio, public data can be used, among other things, to make a historical weighting between injuries and properties. From the STAR accident registration, we know that for the whole of the Netherlands in 2022, approximately 90,000 accidents have occurred by December 1. Of these, 20,000 involved injuries [2]. If we assume ­that the relationship between material and personal injuries is approximately the same for one-vehicle and two-vehicle accidents, then 22.2% personal injury claims could be assumed in determining the weighting factors. In order to arrive at a relationship of incurred losses between injury and material claims, an approximation must also be found for the average amount of claims to which we can apply this frequency weighting. For this purpose, the available data of minor and major claims for Motor Liability could be looked at. Practical experience shows that major claims almost always have an injury component and minor claims almost never have it. If a personal injury claim does not exceed a few thousand euros [3], then this usually does not deviate significantly from material claims in terms of inflation development, because it often concerns reimbursement of healthcare costs or temporary support.

[6] STAR: Smart Traffic Accident Reporting (star-verkeersongevallen.nl)
[7] Choosing an adequate threshold can be done on the basis of a rule of thumb that personal injury claims with a significant VAV and/or domestic help component can never, as a rule, have an incurred loss of less than 50,000 euros (ie at least 25% loss of work capacity based on full-time minimum wage). (12 x 1,934 euros) for 5 years already leads to an incurred loss of 29,000 euros for the VAV component alone). It is also customary to use a higher threshold than 50,000 euros.

To determine the additional inflation, one option is to assume the long-term inflation trend to be equal to the ECB's aim of keeping inflation at 2.0% on average over the long term. However, it is better to use historical claims inflation analysis to find out whether Line of Business-specific inflation deviates significantly from this 2% and ­to replace this ECB target with it. By deducting the LoB-specific long-term inflation from the figures for 2022, we get an additional inflation figure per LoB for 2022, which can be used in the modeling of the technical provisions.

An example

For home insurance, long-term ­inflation can be derived from CBS figures on the output price index for construction costs for new ­homes (see table below). Over the period 2016-2021, average inflation has been 5%. This could be used as a starting point to compare the inflation figures for 2022 (and thus the determination for the additional inflation for home ­insurance policies).

Solvency II Technical Provisions

Premium provision
The expected claims ratio, the cost ratio, the reinsured claims ­, the expected cancellation and the settlement pattern of the incurred losses play a role in the premium provision.

It is important for the premium provision that the Combined Ratio (COR) is adequately adjusted to the inflation ­expectation. For example, the expected future claims ratio can be determined on the basis of a long-term average of the (annual) claims ratios, taking into account historical inflation and premium adjustments (general inflation correction method). However, at the end of 2022, two things must be taken into account:

1. The additional inflation over 2022, 2023 and 2024, which cumulatively increases the claims ratio;
2. Adjusted premium rates to correct for inflation, which lowers the claims ratio.

By correcting the premium and claims basis for both effects, a pure estimate of the claims ratio is obtained. The first effect should be determined by weighting the additional inflation using the payment patterns. For multi-year contracts, the applied claims ratios can be adjusted by the same factor. As for the ­claims provision, it is assumed that core inflation will drop to the long-term average of 2% after 2024, which means that there will probably no longer be any significantly higher technical-claim inflation.

The same ­considerations apply to the expense ratio, but a different inflation ­correction is applied to the costs than is applied to the incurred losses, namely with the 5% expected increase in wage costs in 2023. For reinsurance, the inflation ­adjustments in the reinsurance ­premium and commission are incorporated in contractual percentages. For the reinsured ­incurred losses in the premium provision, the reinsured claims ­ratio ­continues to be determined on the basis of the gross claims ratio multiplied by the quota share percentage and on the basis of long-term Excess of Loss income (corrected for historically adjusted Quota Share percentages and deductibles) . This ratio relates the reinsured incurred losses to the gross incurred losses. The gross incurred losses have been adequately adjusted for the inflation-expectation, on the basis of the stated method, which means ­that the reinsured incurred losses have also ­been sufficiently adjusted automatically­.

The economic conditions not only lead to high inflation, but may also lead to higher cancellations in times of recession. In valuing the premium provision at year-end 2022, it may be too prudent to anticipate uncertainty surrounding ­economic ­developments and their impact on cancellations. This underlines the importance for non-life insurers to conduct an extensive sensitivity analysis of their assumptions regarding cancellation.

As for the claims provision, a temporarily higher inflation may have an impact on the run-out pattern of the premium provision. As with the claims provision, there seems to be no reason to adjust this pattern, but if it is done on the basis of expert judgement, then it is advisable to also adopt this in the premium provision.

After a number of years in which the Arcturus partner team consisted of two partners, the step was taken to expand the team to ensure the long-term continuity of Arcturus and to build a good foundation for future of the company.

Rachel Bonsel

Rachel has been working for Arcturus since 2013. She has made huge strides within the company in recent years. For example, she managed application development teams and is valued by customers for her substantive knowledge and commitment. She is described by colleagues as enthusiastic, eager to learn and solution-oriented. Thanks to her creativity and ideas about improving tooling and processes, she can make an important contribution to the company's strategy.

Michiel van Dellen

From a rich experience background at various insurers and regulator DNB, Michiel brings a lot of experience to the team. Michiel has been working at Arcturus as an actuarial manager since the beginning of 2019 and has made an important contribution to the services that we offer as a company. In his current role he has also been responsible for performing various second-line actuarial activities for our clients. Michiel can be described as analytical, communicative at different levels and solution-oriented.

Michiel van Dellen

From a rich experience background at various insurers and regulator DNB, Michiel brings a lot of experience to the team. Michiel has been working at Arcturus as an actuarial manager since the beginning of 2019 and has made an important contribution to the services that we offer as a company. In his current role he has also been responsible for performing various second-line actuarial activities for our clients. Michiel can be described as analytical, communicative at different levels and solution-oriented.

Melchior Mattens

Since the beginning of 2016, Melchior has made an important contribution to our organization in many ways. He is always looking for innovative ideas and insights that lead to more adequate risk management. With his knowledge and creativity, he won the Johan de Witt thesis prize in 2017 and was named Actuarial Talent of the Year in 2018. In recent years he has taken on more coordinating roles for clients. He is highly valued by customers for his customer-oriented approach.

Patrick Dannenburg

Patrick joined our team on 1 November. He brings substantive knowledge and years of experience in the non-life actuarial field to the team. He has worked at the regulator DNB, in consultancy and in various other important roles at insurers. In his role as a partner, he will have a responsibility in the execution of the actuarial role and also in acquisition and strategy development.

Patrick Dannenburg

Patrick joined our team on 1 November. He brings substantive knowledge and years of experience in the non-life actuarial field to the team. He has worked at the regulator DNB, in consultancy and in various other important roles at insurers. In his role as a partner, he will have a responsibility in the execution of the actuarial role and also in acquisition and strategy development.

Melchior Mattens MSc AAG
melchior.mattens@arcturus.nl 

Figurer 1: development of the number of cybercrimes.
Data 2020 until august. Source: data.politie.nl

Figuur 2 Number of offenses per 100 inhabitants by type of cybercrime.
* 2018 is interpolated because of missing data CBS.

Figure 3: Percentage share of insurable cybercrime offenses.

Melchior Mattens MSc AAG 
melchior.mattens@arcturus.nl 

Melchior Mattens MSc AAG
melchior.mattens@arcturus.nl 

SARS-CoV-2, also known as Covid-19 and Coronavirus, has dominated daily life in the Netherlands and far beyond for some time. The cabinet is assisted in policy choices by the RIVM. In order to fulfill this role as effectively as possible, RIVM collects large amounts of data from hospitals and the GGD in order to gain insight into the spread rate of the corona virus, but also into the development in hospital admissions and mortality. Relevant cross-sections of these data are made available to the public by the RIVM on a daily basis. This includes the number of hospital admissions and deaths by event day, as well as cross sections of patient characteristics.

In this article, we will discuss the type of data that RIVM publishes about the Netherlands and how it should be interpreted. Could actuarial practice be relevant to epidemiology? There appear to be a surprising number of topics within the epidemiological situation which look like this practice: IBNR, time series and forecasting models are just a few examples. In this article, we look at the best available actuarial and econometric models for making a mortality prediction.

Incurred But Not Reported (IBNR)

As mentioned, RIVM publishes the number of admissions and deaths by event day on a daily basis, and adjusts the number of reports from previous event days. Healthcare institutions usually have a reporting delay when someone with a positive Covid-19 test has been admitted or has died. This creates a reporting pattern in the number of admissions and deaths: only after about seven days do we know approximately how many people were actually admitted and died at a specific day. The number of deaths reported in the media daily is the sum of all newly reported deaths since the outbreak in the Netherlands and not just from the previous day.

The actuarial profession is very familiar with the problem that it is not yet exactly clear how many people have been admitted or died. Ideally, one has insight into the actual number of patients admitted and dying today: then trends become visible more quickly and more clearly.

Because we can follow the development for each event date through time, so-called triangular overviews are created of the number of patient admissions and deaths by event day and reporting day. These are shown in the figures below. The problem is that for the most recent days there can still be a lot of development of delayed reports on the number of admissions and deaths. However, the actuarial profession knows what to do with such overviews. A Chain Ladder model can be applied to estimate the total number of admissions and deaths per day. This applies the average run-off pattern from previous days to the numbers already reported at this time.

Figures 1 and 2: triangular overviews of admissions and fatalities in the Netherlands. The estimate of the final number is shown with the red arrows.

What is striking about the tables above is that for the last two days there is enormous uncertainty about the actual admissions and deaths. As with many (non-life) insurance policies, an alternative estimation method is needed for this. For example, by looking at the development in the total number of admissions / deaths (ultimate). Using statistical software, the following time series model (ARIMA) can be fitted that provides a projection for the number of hospital admissions and the 95% prediction interval surrounding it.

Figure 3: ARIMA model for daily number of hospital admissions in the Netherlands. Forecast up to 75 days ahead. Prediction interval: 75% and 90% probability.

Figure 3 shows that until around April 1st , an exponential relationship was observable in the number of daily hospital admissions including IBNR. Figure 4 zooms in on the period up to and including 13-4-2020. The dots indicate the number of admissions including IBNR and two curves have been added trying to fit a possible time relationship in the data. The red curve is the exponential best fit on the data and the green curve is the density function of a normal distribution. We see that the green curve fits much better and the decrease in the number of admissions has clearly started.

To predict the number of deaths this method is, however, too simplistic. After all, this is based on all kinds of risk factors, such as age and any comorbidities.

Mortality development and modeling

In addition to hospital admissions and deaths, RIVM publishes daily overviews of the registered patient group by age, gender and underlying conditions (comorbidities). In the plots below, the development of the share in the (hospital) patient population and the status of the observed mortality frequency are represented for a number of important age groups and gender.

Figure 5

Figures 5 and 6: development in the share of hospital admissions per group (bars) and development in total mortality risk (labels and line).

These plots provide insight (into fluctuations) in the mortality probability within a risk group. To predict future mortality, these data must be viewed on an incremental basis: can the daily mortality be explained by the number of hospital admissions from X number of days earlier?

In order to be able to make an optimal prediction model for future mortality, we would like to combine information on important risk factors from the admitted patients: for example, does it make a difference whether someone has underlying suffering at a younger age and do older men have a lower chance of survival than older women? To provide insight, these risk categories should be available per patient. Unfortunately, RIVM only publishes tables without relationship between gender, age and comorbidities: we can only make models on the three separate series.

Since we can only predict mortality with these three risk factors as separate explanatory variables, the question is what the impact of a split might be. Statistics Netherlands’ Statline provides information about the Dutch population as a whole. Here you will find data on age, gender and syndromes. On the one hand, there are more women in the older age groups (risk factor), but on the other hand it is known that cardiovascular diseases, hypertension, diabetes1 and lung diseases are more common for men than for women. Since this selection of comorbidities is highly relevant in terms of mortality, we see that the gender ratio strongly correlates with comorbidities. We also note that the incidence of comorbidities is strongly correlated with age. If we combined data for the three risk factors, there was a high probability of autocorrelation. Fitting three separate models prevents autocorrelation and estimation differences may indicate that certain trends may or may not be better captured by one or more models.

To arrive at a mortality forecast for the two most recent days and one day ahead, we optimize GLM models from the quasi Poisson family. Generalized Linear Models (GLM) are frequently used in the actuarial profession to quantify risk factors. The explanatory variables in the model capture the effect of the mean hospital stay of a patient in a certain risk class on the probability of dying X days later. With these models we can make a fairly accurate prediction of mortality a few days ahead. Because we also have to make a forecast of the two most recent days that cannot be projected with Chain Ladder, we initially project the mortality for three days in advance. By adding the extra day (the third day ahead) an entire diagonal can be built up in the Chain Ladder model with the next day’s position. The estimates for mortality as of 13-4-2020 are shown in the table and graph below.

https://www.volksgezondheidenzorg.info/onderwerp/diabetes-mellitus/cijfers-context/huidige-situatie#node-aandeel-diabetes-type-1-totaal-naar-leeftijd-en-geslacht

Figures 7 and 8: Mortality prediction by model, including estimated bandwidth.

By averaging the model results, a weighed picture of the total number of deaths in the past days is obtained. With this full estimate for the two most recent days based on the average model results, the pattern of the Chain Ladder model for the two most recent days can be scaled-up. Subsequently, the development in daily mortality (ultimates) can also be displayed. An attempt was made to capture the trend in the data using various “best fitting” functions.

Figure 9: Development of mortality by event day. Red line: exponential relationship, blue line: sigmoid function, green line: Gaussian curve (normal distribution).

The above figure shows a clear downward trend for the number of deaths in the recent week. This is a slightly delayed image that is also visible in Figures 3 and 4 for the number of hospital admissions. The green line (the Gaussian curve) seems to best capture the trend in mortality. We cannot build entirely on an extrapolation of this curve for a mortality forecast. After all, the models recognize that it is very relevant whether there is an influx of new hospital patients and in which risk groups they fall (on average).

Figure 10: Triangle number of deaths by event and development date.

Figure 11: Cumulative mortality and mortality forecast, 75 days ahead. Model based on expected inflow of new patients in age cohorts. Blue: Historical Cumulative Development. Red: predicted future development.

A limitation in these predictions is that they assume unchanged circumstances. These results will have to be adjusted in the event of policy changes, such as more testing or removal of restrictions in society. 9

At the same time, observations by Statistics Netherlands show that the RIVM data probably underestimated the national excess mortality as a result of Covid-19. The projection model described in this article does not capture this missing mortality either. The deaths missing in the RIVM data mainly concern deaths at home or in care institutions where people could not be tested for Covid-19. Therefore, the model results should be seen as only indicative: the number of hospital admissions is likely to be (strongly) correlated with the number of unregistered deaths. This is true in the case that within an age and / or comorbidity group, people are just as likely to recover or die quickly or be ill long enough to be hospitalized. At the moment, at least there is no scientific literature that rejects this assumption.

Actuarial models

All the models mentioned in this article are prediction models widely used in the actuarial profession. The RIVM figures on Covid-19 touch on actuarial topics such as IBNR, GLM pricing models, seasonal patterns (time series) and forecasting. The predictive power of the (combined) models can be tested daily. Back testing and sensitivity analysis are important to provide a robust estimate for the future impact of Covid-19 on Dutch society. As still many developments are expected (for example the reversal of a number of quarantine measures) which may affect the spread of Covid-19, it will be necessary to continuously monitor the development of the hospital admissions and mortality. We are happy to accept the challenge to further develop the actuarial models and to update you on the insights these models provide.

Melchior Mattens MSc AAG  
melchior.mattens@arcturus.nl 

Melchior Mattens MSc AAG
melchior.mattens@arcturus.nl 

Places in the Netherlands where hail with a diameter of at least 2 cm has fallen since 2006. Source: European Severe Weather Database

Minimum- en maximumtemperatuur voor poolgebied, Noord-Amerika, Europa en Rusland – 10-daagse projectie vanaf 24 juli 2018. Record hitte in California (>50oC), Noordelijk Afrika (Algerije >50oC), Japan en Korea (>40oC) en noordelijk Scandinavië (>30oC). Bron: University of Maine (ClimateReanalyzer.org)

 Average nationwide precipitation deficit as measured in 2018. Source: KNMI

Angela van Heerwaarden 
angela.vanheerwaarden@arcturus.nl

Angela van Heerwaarden
angela.vanheerwaarden@arcturus.nl

Michiel van Dellen  
michiel.vandellen@arcturus.nl 

Michiel van Dellen
michiel.vandellen@arcturus.nl 

Example of how different reinsurance contracts can stabilize an insurer’s claims ratio.

More profit and volatility, or less risk? Which scenario is most suitable for you?