As discussed in Chapter 7, non-life insurance claims contain two aspects: inci- dence and intensity. This means that there is an additional complication compared with mortality risk since for most defined benefit pension schemes and insurance policies the intensity – more commonly known as the benefit – is known, either in absolute terms or is exactly defined by some other variables. The way in which the intensity of insurance claims is estimated is discussed below.
The incidence of non-life insurance claims is similar in nature to mortality and longevity risk. Volatility risk exists, because portfolios are of finite size. There is also level risk, which is dealt with by a combination of experience and risk rating.
Catastrophe risk is present as well – a concentration of policies by location or some other characteristic could leave an insurer open to an unexpectedly high incidence of claims. However, trend risk is more difficult to define and usually – but not always – less important. The change in the incidence of claims over time is more likely to follow the economic cycle than to trend in a particular direction. This means that changes should be modelled – probably through scenario analysis – consistently with other economically-sensitive risks. However, the short period of exposure for many insurance policies means that changes in the underlying risk are much less important than identifying the risk correctly in the first place.
There are two broad groups into which insurance classes can be placed. The first group includes those classes where there is a relatively high frequency of claims, such as motor or household contents insurance. Conversely, the second group in- cludes classes where the frequency of claims is very low and the size of claims
14.9 Non-life Insurance Risk 385 Table 14.14 Hypothetical Car Insurance Claim Probability for Two Risk Factors
Social, SDP and SDP and Domestic and Commuting Work Pleasure (SDP)
<1,000cc 0.10 0.15 0.22
Engine 1,000cc – 1,499cc 0.15 0.16 0.24
Size 1,500cc – 1,999cc 0.17 0.19 0.28
2,000cc – 2,999cc 0.23 0.24 0.33
>2,999cc 0.15 0.17 0.25
is greatly variable. Excess-of-loss reinsurance – where payments are made only if aggregate claims exceed a particular level – is a good example of this type of insurance.
14.9.1 Pricing High Claim Frequency Classes
Classes of insurance where the claim frequency is high tend to produce a significant volume of data which is relatively straightforward to analyse and model.
The most common way to model the incidence of claims in these classes of insurance for rating purposes is to construct multi-way tables covering all of the risk factors, so that the proportion of claims for any given combination of risk factors can be calculated. For example, the risk factors for motor insurance might include items such as the engine size and the use to which the vehicle would be put, as shown in Table 14.14.
A major drawback with this approach is as the number of rating factors increases, the number of observations in each ‘cell’ falls to levels that make statistical judge- ments difficult. For these reasons, generalised linear models (GLMs) are now often used to model the impact and interaction of the various risk factors.
When analysing the claim frequencies for a range of policies, there are a number of statistical issues that need to be addressed. The first relates to the fact that the dataset will often span a number of years. Given that different external factors such as the weather and the state of the economy will influence claims to different degrees in different years, the use of dummy variables will often be appropriate.
A subtler statistical issue arises from the fact that some policyholders will be included in the dataset for each year of the dataset, whilst others will not. In par- ticular, policyholders who move to a competitor in later years will be present only at the start of the period, whilst policyholders joining from a competitor will be present only at the end. This means that the data comprise what is known as an
‘unbalanced panel’. This is important, because if this factor is ignored, then the policyholder-specific nature of claim frequencies will also be ignored, meaning
Table 14.15 Claims per Year Year
2005 2006 2007 2008 2009
A 1 0 0 1 0
B n/a n/a 2 3 2
Policyholder C 1 0 3 n/a n/a
D n/a 0 0 0 n/a
E 3 0 1 0 n/a
that the standard errors for some explanatory variables might be too low. Consider, for example, a portfolio of five policyholders, with claim frequencies tracked over five years.
Whilst there is a total of 18 observations in Table 14.15, these observations are clearly not independent, and ignoring this fact in a regression will mean that policyholder-specific factors will artificially lower the variability in claim frequen- cies.
The way to counter this is to include an additional item of data in any regression, an indicator of the policyholder to whom the observations apply, which is used in the calculation of robust standard errors. Whilst the statistical methods behind this calculation are complex, the option to calculate robust standard errors exists in most statistical packages.
Intensity for premium rating in these classes of insurance is generally mod- elled through multiple regression approaches. Clearly if the distribution of claim amounts is not normal (and for many classes it will not be), ordinary least squares regression is inappropriate and an alternative approach must be used.
As with claim incidence, there are statistical issues that should be faced when modelling claim intensity. As before, there are issues of seasonality and the fact that claim amounts will differ due to economic and environmental factors. Clus- tering is also an issue. However, there is another statistical issue faced here – the issue of censoring. It is tempting, and it seems logical, to calculate the influence of independent variables on claim amounts using the information only from policies where claims have occurred. However, there is also useful information in relation to policies where there have been no claims. This information can be used by carry- ing out what is known as a censored regression. As above, the statistics underlying this approach are complex, but the option to carry out censored regressions exists in most statistical packages.
Experience rating is also carried out in non-life insurance. This can be for an individual client (such as a car insurance policyholder), a corporate client (such as a firm with employer liability insurance) or for the entire portfolio of an insurance company. Particularly in this final case, it is important to consider the level of risk
14.9 Non-life Insurance Risk 387 within a number of homogeneous subgroups, so that any changes in the mix of risk types over time is properly reflected. As with life insurance, the results from experience- and risk-rating can be combined through the use of credibility.
14.9.2 Reserving for High Claim Frequency Classes
An important aspect of all insurance is calculating the amount of money needed to cover future outgoings. However, a particular issue for non-life insurance, espe- cially in relation to high claim frequency classes, is that there is a delay between claims being incurred and these claims being reported. Ignoring incurred-but-not- reported (IBNR) claims can significantly understate the reserves that must be held to cover future outgoings. There are a number of approaches that can be used to determine the level of outstanding claims. Three of the most common are:
• the total loss ratio method;
• the chain ladder method; and
• the Bornhuetter–Ferguson method.
All of these methods can be applied to aggregate claims or to loss ratios, and can be adjusted to allow for claim inflation.
The Total Loss Ratio Method
The total loss ratio method simply looks at the total premium that has been earned for a particular year and assumes that a particular proportion of those premiums will ultimately fund claims. The premium earned is the part of any premium cover- ing risk in a given time interval. So, for example, if a premium of£300 was received in respect of cover from 1 September 2008 to 31 August 2009, the premium earned in 2008 would be£100 and the premium earned in 2009 would be£200.
The loss ratio can be determined from historical data and adjusted as appropriate.
It can also be adjusted for changes in the underlying mix of business to give more accurate results.
Example 14.9 The table below gives the history of claims occurring in the last three years. All claims are notified no more than three years after happen- ing:
Earned Development Year (d)
Premium 1 2 3
Year 2007 250 150 160 200
of 2008 300 150 200 –
Claim (c) 2009 350 200 – –
What are the total estimated claims for 2007, 2008 and 2009 under total loss ratio approach, assuming a total loss ratio of 80%?
The loss ratio of 80% is consistent with that observed for claims occurring in 2007, being equal to 200/250. The total projected claims for 2008 and 2009 are therefore 80%×300=240 and 80%×350=280 respectively. The claim development can therefore be completed as follows:
Earned Development Year (d)
Premium 1 2 3
Year 2007 250 150 160 200
of 2008 300 150 200 240
Claim (c) 2009 350 200 – 280
The attraction of this approach is that it is simple, and can be applied with limited data. This also means that it can be applied to new classes of business. However, if historical loss ratios are used it is not clear how the method should be adjusted if it becomes clear that the rate of losses emerging is higher than has been experienced in the past.
The Chain Ladder Method
The chain ladder method is still the dominant approach for calculating the total projected number of claims. This considers the number of claims that have already been reported and uses the historical pattern of claim development to project the re- ported number of claims forward. This is done by calculating link ratios, the change in the total proportion of claims notified over subsequent periods historically, and applying these to years where the development of claims is incomplete. The ap- proach can also be applied to the cumulative value of claims or to the incremental value of claims in each year. The former approach places much greater weight on earlier claims, whilst the latter can become volatile when the period since claiming is great. The best way to explain the chain ladder method is by example.
Example 14.10 Considering the case in Example 14.9, what are the total estimated claims for 2008 and 2009 under the chain ladder approach?
Earned Development Year (d)
Premium 1 2 3
Year 2007 250 150 160 200
of 2008 300 150 200 –
Claim (c) 2009 350 200 – –
14.9 Non-life Insurance Risk 389 Each cell,Xc,d, contains the claims that occurred in yearcand were reported by the end of thedth year. To calculate an estimate for X2008,3, the link ratio must be calculated using data from the claims that occurred in 2007. In partic- ular, this link ratio,l2,3is calculated as 200/160=1.25. The estimated value ofX2008,3is therefore 1.25×200=250.
To calculate X2008,3, two years’ worth of data are available, so the link ra- tio,l1,2is calculated as(160+200)/(150+150) =1.20. This means that the estimated value ofX2009,2 is 1.20×200=240. The link ratiol2,3can then be applied to this value to estimateX2009,3as 1.25×240=300. The claim devel- opment can therefore be completed as follows:
Earned Development Year (d)
Premium 1 2 3
Year 2007 250 150 160 200
of 2008 300 150 200 250
Claim (c) 2009 350 200 240 300
One recent augmentation to the chain ladder approach is the treatment of the development as stochastic. Such an approach could be carried out by considering the statistical distribution of the link ratios in each year, obtaining not just an av- erage ratio but also the extent to which they vary from year to year. In order to do this, a number of years of claim development history would be needed, and as the length of the history rises its relevance falls. However, a potential advantage of this approach is that it allows the linkage of claim levels to economic, market and other variables.
The Bornhuetter–Ferguson Method
Another way to assess the ultimate claim level is to use the Bornhuetter–Ferguson method. This is essentially the total loss ratio approach adjusted for claims reported to date. The chain ladder approach is used to derive link ratios, and these are used to calculate the proportion of the expected loss ratio that should develop in each period. Then, for any period that the loss is actually known, the prediction from the combined loss ratio and chain ladder approach is replaced with the known figure.
Again, this is best demonstrated by example.
Example 14.11 Considering the case in Example 14.9 and assuming a loss ratio of 80%, what are the total estimated claims for 2008 and 2009 under the Bornhuetter–Ferguson approach?
Earned Development Year (d)
Premium 1 2 3
Year 2007 250 150 160 200
of 2008 300 150 200 –
Claim (c) 2009 350 200 – –
The link ratios l1,2 and l2,3 have already been calculated as 1.20 and 1.25 respectively. This means that the claims reported by the end of the third year – which are also the total claims – are 1.25 times the claims reported by the second year, and they are 1.25×1.20=1.50 the claims reported by the end of the first year. This means that 1/1.50=0.67, or 67% of claims are paid by the end of the first year, whilst 1/1.25=0.80 or 80% are paid by the end of the second year.
The first part of the Bornhuetter–Ferguson estimate for the claims arising in 2008 is therefore the value of claims already reported, 200. The second part is the product of the premiums earned, the loss ratio and the chain ladder propor- tion of claims outstanding, i.e., 300×0.80×(1−0.80) =48. This means that the 2008 claims estimate is 200+48=248.
Using the same approach, the first part of the Bornhuetter–Ferguson esti- mate for the claims arising in 2009 is the value of claims already reported, again 200. The second part is the product of the premiums earned, the loss ratio and the chain ladder proportion of claims outstanding, 350×0.80× (1−0.67) =93. This means that the 2009 claims estimate is 200+93=293.
The table can therefore be completed as follows:
Earned Development Year (d)
Premium 1 2 3
Year 2007 250 150 160 200
of 2008 300 150 200 248
Claim (c) 2009 350 200 – 293
14.9.3 Low Claim Frequency Classes
The second group of classes is that where the frequency of claims is very low and the size greatly variable. This means that these classes of insurance are particularly amenable to modelling using extreme value theory.
Copulas can also play an important part in modelling of these classes, since many large aggregate claims will arise as a result of some sort of concentration of risk. For example, hurricane damage will affect a large number of properties in a particular area. This means that if an insurer receives a large claim in respect