# A new method for price and volume measurement of non-life insurance services: A statistical approach

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4 Figure 1. Chained value, price and volume indices for two types of nonlife insurances and for the number of claims incurred (1999 = 100). Motor vehicles Value Price Volume Claims 160 Property/fire Value Price Volume Claims measure quantities directly, while volume indices were previously obtained by deflating value indices with some price index. Direct volume measurement seems to be an appropriate starting point also for insurance services, since nominal values and prices are not directly observable. Direct volume measurement is one of the recommendations in the Eurostat handbook (2001) for price and volume measurement. Before setting up quantity measures, we first need a characterisation of insurance services in order to distinguish different product types. We suggest to distinguish non-life insurance services according to expected loss ( risk ) and the handling of risk by an insurer. So, we could imagine services to be characterised by pairs (r, μ), say, where r denotes expected loss, or risk, and μ is a parameter, or a set of parameters, which determine the handling cost of risk in order to cover the costs of administration, acquisition and claim handling, amongst others. We will work out and relate risk and handling cost in Section 4. Reasons for distinguishing services according to pairs (r, μ) could be that expected loss r is an indicator of the extent of the liability of insurers to policy holders, while μ represents the way in which risks are handled. Certain types of risks require greater care by insurers in establishing adequate coverage, for instance because of a higher probability of occurrence of large claims. This is reflected in our model in Section 4 by setting μ at smaller values. Smaller values of μ lead to larger handling costs and thus larger margins for an insurer. This has a positive effect on production volume ( heavier types of service have larger weights). In Section 4, we present a simple model expression, which combines expected loss, premiums and handling cost parameters μ, and a method for estimating expected loss and μ for five lines of insurance: 4

8 The handling cost parameters μ i P regarding premiums have the following values: for health and accident, for both motor vehicles and transport, and for both fire and other insurances. We thus have four free parameters in the expectations (3) and (4) (and two variances estimated from the data, one for investment income and one for loss). The maximum BIC-value obtained is The parameter values for direct investment income imply that the expected direct investment income to premiums ratio for year t is set at the value observed in year t 1, that is: ES t = s t 1. 7 This gives the model fit in Figure 2. The fit can be compared with the fit when using the expectation ES t 1 instead of s t 1. This unconditional expectation results in a lower BIC (267.1), even with two additional free parameters (i.e., α and β). Figure 2. Fits of expected direct investment income to premiums ratio (4) to data. Fits are shown for the expectation conditioned upon the value of the preceding year and for an unconditional expectation. (Data are from DNB) 0,15 0,12 0,09 0,06 0,03 0, Data Conditioned Unconditioned The handling cost parameters μ i S with regard to investment income are all equal to 1. This means that investment income is entirely used to cover risk, so that it does not enter nominal value. 8 The handling cost parameters μ i P regarding premium have different values for three lines of insurance. Insurers of healthcare and accident charge the lowest handling cost, namely 1 μ i P = per unit premium. The highest handling costs are found for fire and other insurances (0.504). This could be due to a higher probability of large claims for fire and liability, so that insurers charge relatively higher premiums in order to provide coverage against loss. The parameter values show that risks are dealt with in a different way across insurance types. The fits of expected loss to data are shown in Figure 3. The rapid decrease of expected and actual loss for health care and accident insurances in 2006 can be explained by changes in the Dutch healthcare insurance system, which was reformed in Nominal value and volume indices The models for expected investment income and expected loss, and the parameter estimates obtained for these models in the previous section, allow us to calculate the nominal values of the product groups 7 Application of the moving average model of Chen and Fixler (2003) instead of (4) gives the same result. 8 These handling cost parameters stay equal to 1 when leaving out years from the time series and refitting models (3) and (4). This strengthens the result that investment income is fully used to cover risk. 9 The distinction between base and supplementary insurances for health care, as referred to in footnote 3, is one of the main novelties of the reform. The distinction between public and private insurances, which existed before 2006, was different. Persons with an income exceeding a certain threshold could only take a private insurance. Insurance services for health care in the Dutch national accounts shifted from private to supplementary insurances in The boundary between government and insurance activity has shifted. 8

9 Figure 3. Model fits of expected loss to data for the five insurance types separately and aggregated. Loss is in millions of euros. (Data are from DNB) Healthcare and accident Motor vehicles Data Model Data Model Transport Fire Data Model Data Model Other insurances All insurance types Data Model Data Model and index numbers for nominal value and production volume. Before presenting the expressions for value and volume indices and the results, we first summarise the data needed to calculate both. Data for production volume As was stated in Section 3, policy numbers are suggested as counting measures of production volume. These are grouped according to line of insurance. We have N = 5 of these product groups, with numbers of policies q i,k,t in product group i in quarter k of year t. These data are made available by the Verbond van Verzekeraars, the Dutch Association of Insurers. Policy numbers are available from year 2006 per line of insurance. 9

10 Data for nominal value, or turnover The five product groups are characterised by expected loss and the handling cost parameters μ i P and μ i S. These characteristics are estimated and then used to calculate nominal value. The following data are required for this purpose: Premiums earned; Loss incurred; Investment income; Possibly also one or more indicators of investment income, such as rent, but this depends on the type of model used for expected investment income. (For instance, no additional data are required when a moving average model is used.) Premium and loss data should be available for every product group and per year. If data are available on a finer time scale, for instance, per quarter, then the data at this level are preferred. Investment income data should be available per year and preferably also per line of insurance. In the next subsection, we use the notation P i,k,t for total premiums in product group i in quarter k of year t, and P i,t for total premiums in product group i summed over the quarters of year t. Value and volume indices The results of Section 4 imply that nominal value only depends on premium and its handling cost. Nominal value for product group i in year t is equal to (1 μ P i )P i,t. This leads to the following value index for non-life insurance services, over N = 5 product groups, in year t with respect to year t 1: N (1 μ ) i= 1 i Pi, t (6). N P (1 μ ) P P i= 1 i i, t 1 Laspeyres volume indices are used by convention at Statistics Netherlands. The volume index for year t with respect to year t 1 is calculated as follows: 4 N (1 μi ) Pi, k, t 1 qi, k, t (7). 4 N P k= 1 i= 1 (1 μ ) P qi, k, t 1 P m= 1 j= 1 j j, m, t 1 Year-on-year volume indices have been derived on a quarterly and yearly basis. Figure 4 shows the value and volume indices on a yearly basis. Nominal value increased with 1.21 percent per year on Figure 4. Volume and value index for non-life insurance services. Index numbers are multiplied by 100, by convention (2006 = 100) Volume index Value index 10

13 Figure 6. Value and volume indices for our (net premium) approach and the gross premium approach (2006 = 100) Value (net) Value (gross) Volume (net) Volume (gross) (3) Results for the Chen and Fixler (2003) approach We also studied the impact on the value and volume indices by replacing the models for expected loss and expected investment income by the models proposed in Chen and Fixler (2003). These models specify expected loss to premiums ratio and expected investment income to premiums ratio in year t as weighted sums of the actual and expected ratios in year t 1. Only the results are shown in this paper; the model expressions can be found in Chen and Fixler (2003). Chen and Fixler s models give a lower BIC than our models, so that the data are fitted less well with Chen and Fixler s models. The fits for investment income are the same, so the difference in fits is completely explained by the fits of expected loss model (3) and Chen and Fixler s model. A major difference between our expected loss model (3) and Chen and Fixler s model is that the latter model estimates expected loss from information of past years. Policies that are acquired in the present year may have different risk profiles, which would not be captured by moving averages. Our model considers the relation between expected loss, premiums and expected investment income for the same years, so it is updated. This could be a possible explanation for the better fits with our model (3). On the other hand, we should be cautious with such statements as we compared models for one data set. It would be interesting to consider other data sets in the future. Information criteria, such as the BIC, enable us to make a selection among competing models. The importance of having such selection criteria is illustrated by the differences in the results between our model and Chen and Fixler s model. The latter gives a larger variability in the yearly nominal value indices than our model, as can be noted in Figure 7. Nominal value increases in 2008 with 6.0 Figure 7. Volume and value index for the Chen-Fixler approach (CF) and the value index for our method (2006 = 100) Volume (CF) Value (CF) Value (our method) 13

14 percent and decreases with 5.3 percent in The respective value changes of +5.1 and 1.1 percent for our model are much more contained. The volume indices show small differences, as can be seen in Figure 4 and Figure 7. Annual volume growth is 0.06 percentage points larger on average for the Chen-Fixler approach during But the differences in the price indices are much larger. Results for refined product groups We also calculated value and volume indices by refining the product groups to 12 groups. We have policy numbers and information on premiums for subtypes of insurances within some of the five lines of insurance. Unfortunately, we do not have loss data at a more detailed level than the five lines of insurance, so we had to make assumptions for the subtypes of insurances in order to calculate expected loss and nominal values. The 12 subtypes of insurance are composed as follows: Health and accident insurances is not refined, as there are not enough data to do this. So this line of insurance remains one product group; The group of motor vehicles is subdivided into insurances for private cars, company cars and a remaining category, which includes motorbikes and caravans; The group of fire insurances is subdivided into insurances for private houses, companies and a smaller, residual group consisting of insurances for technical aspects, such as machine failure; The group of other insurances is subdivided into insurances for liability, legal aid, travelling and a remaining category, which includes financial loss; The group of transport insurances is not refined because of incomplete data, but this group has a small value share. In order to calculate expected loss for the subtypes of insurances within the groups of motor vehicles, fire and other insurances, we assumed that μ i P and μ i S have the same values for the subtypes within each group. The parameter values are equal to the values given in Section 4. Of course, the value indices are the same as in Figure 4 for the five main product groups. But the volume indices undergo slight changes: average annual volume growth increases with 0.97 percent under the refinement to 12 product groups, compared to the 0.86 percent under the coarser subdivision into 5 product groups. The additional volume growth of 0.11 percent is an overall effect, in which the larger volume growth occurs for subtypes of insurances with larger expected loss. In Figure 8, year-on-year volume changes per quarter are compared for three situations: for the cases with 5 and 12 product groups, and in the case where no subdivision into different product groups is made. The latter case implies that volume indices are calculated by simply dividing the total number Figure 8. Year-on-year volume indices per quarter, calculated for three situations: 12 product groups, 5 product groups, and without distinguishing different product types (year t 1 = 100) _1 07_2 07_3 07_4 08_1 08_2 08_3 08_4 09_1 09_2 09_3 09_4 12 product groups 5 product groups No product differentiation 14

16 services from the handling of risks. The parameterisation therefore also gives insight into the allocation of financial sources to both types of activities. It turned out that expected investment income is fully assigned to risk coverage, so that nominal value is completely determined by premiums earned and their handling cost parameter. The results in Section 6 also showed that much of the heterogeneity in insurance services is already captured by distinguishing five product groups. A more refined product differentiation with 12 product groups resulted in an additional volume growth of about 0.1 percentage point per year on average. Despite the lack of loss data for the more detailed types of insurance, the results for the 12 product groups were found to be robust. The 12 product groups are characterised by different expected loss per policy and/or different handling cost parameters, so they should be distinguished as different product groups according to our volume framework of Section 3. The results for the most detailed level of product characterisation should therefore be used. Finally, it would be valuable to reproduce the analyses in this paper to other data and other countries as well and compare the findings with those reported here. Acknowledgements The author wants to thank Prof. Marcel Timmer (University of Groningen) for his useful remarks on a previous version of this paper, which was presented at the IARIW 2010 conference (Chessa, 2010), and various colleagues of Statistics Netherlands for their comments and for discussions on the subject. The author also wants to thank De Nederlandsche Bank and the Verbond van Verzekeraars for providing the data. References Aizcorbe, Ana M., Bonnie A. Retus, and Shelly Smith, BEA Briefing: Toward a health care satellite account, Survey of Current Business, 88, 24-30, May Berndt, Ernst R., David M. Cutler, Richard G. Frank, Zvi Griliches, Joseph P. Newhouse, and Jack E. Triplett, Price indexes for medical goods and services: An overview of measurement issues, in David M. Cutler and Ernst R. Berndt (eds), Medical Care Output and Productivity, Studies in Income and Wealth, Volume 62, University of Chicago Press, Chicago, , Chen, Baoline and Dennis J. Fixler, Measuring the services of property-casualty insurance in the NIPAs: Changes in concepts and methods, Survey of Current Business, 10-26, October Chessa, Antonio G., Measures of health care volume with applications to the Dutch health sector, Paper presented at the UKCeMGA and NIESR International Conference on Public Service Measurement, Cardiff, UK, November 2009 (see events/ukcemga-and-niesr-conference/index.html )., A new method for price and volume measurement of non-life insurance services for the Dutch national accounts, Paper presented at the 31 st General Conference of the International Association for Research in Income and Wealth, St. Gallen, Switzerland, August 2010 (see ). Claeskens, Gerda and Nils L. Hjort, Model Selection and Model Averaging, Cambridge University Press, UK, Denny, Michael, Measuring the real output of the life insurance industry: A comment, The Review of Economics and Statistics, 62, , Eurostat, Handbook on price and volume measures in national accounts, Fixler, Dennis J. Rethinking the NIPA treatment of insurance services for the comprehensive revision, Paper presented at BEA Advisory Committee Meeting, Bureau of Economic Analysis, 15 November

17 Hirshhorn, Ron and Randall Geehan, Measuring the real output of the life insurance industry, The Review of Economics and Statistics, 59, , 1977., Measuring the real output of the life insurance industry: A reply, The Review of Economics and Statistics, 62, , Hornstein, Andreas and Edward C. Prescott, Insurance contracts as commodities: A note, The Review of Economic Studies, 58, , Lal, Kishori, Measurement of output, value added, GDP in Canada and the United States: similarities and differences, Research paper, Statistics Canada, June OECD (Organization for Economic Cooperation and Development), Towards Measuring the Volume of Health and Education Services, Draft Handbook, October Sherwood, Mark, Output of the property and casualty insurance industry, The Canadian Journal of Economics, 32, , SNA (System of National Accounts), Annex IV: The treatment of insurance, social insurance and pensions, in System of National Accounts, 1993., Chapter 17: Cross-cutting and other special issues, Part 1: The treatment of insurance, in System of National Accounts,

18 Figure captions Figure 1. Chained value, price and volume indices for two types of non-life insurances and for the number of claims incurred (1999 = 100). Figure 2. Fits of expected direct investment income to premiums ratio (4) to data. Fits are shown for the expectation conditioned upon the value of the preceding year and for an unconditional expectation. (Data are from DNB) Figure 3. Model fits of expected loss to data for the five insurance types separately and aggregated. Loss is in millions of euros. (Data are from DNB) Figure 4. Volume and value index for non-life insurance services. Index numbers are multiplied by 100, by convention (2006 = 100). Figure 5. Nominal value of non-life insurance services according to the model and the data. (Data are from DNB) Figure 6. Value and volume indices for our (net premium) approach and the gross premium approach (2006 = 100). Figure 7. Volume and value index for the Chen-Fixler approach (CF) and the value index for our method (2006 = 100). Figure 8. Year-on-year volume indices per quarter, calculated for three situations: 12 product groups, 5 product groups, and without distinguishing different product types (year t 1 = 100). 18

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