The objective is to improve your company’s current procedures for projecting frequency and severity trends. Your student project should include a proposed out-of-sample test to compare your company’s current procedures with the regression models or ARIMA models in your analysis. Readers of your report may not be persuaded unless you show that you can forecast more accurately.

Check how your company now projects claim frequency and severity trends. Many insurers use simple procedures that you can improve. Potential topics are:

Some insurers use internal claim severity or claim frequency numbers, which they fit to an exponential curve. Your student project may other ways to project claim costs.

Some insurers have extensively analyzed their claim severity and frequency trends. If your employer has analyzed its claim severity and claim frequency trends with regression and time series analyses, it probably has several data sets or time series that you can use for the student project, and you may get additional ideas for new analysis from the work already done.

Picking the inflation index the regression analysis is not always easy. The CPI comes in several flavors: medical, wage, urban, rural, seasonally adjusted or not seasonally adjusted, and dozens of sub-varieties. Rely on intuition and the fit of separate regressions.

Do not use two inflation indices as explanatory variables in a single regression. Some candidates think: "Let me use both wage inflation and medical inflation as explanatory variables. If the ordinary least squares estimators are 40% for wage inflation and 60% for medical inflation, I can use a 40% - 60% weighting of these two inflation indices to project claim severity trends.

The high multicollinearity distorts the ordinary least squares estimators. Random fluctuations in the inflation indices may cause large changes in the ordinary least squares estimators. The betas are not good indicators of the proper weight for each inflation index.

Industry claim frequency and severity trends are available from rating bureaus and other statistical agencies. Personal auto insurance has Fast Track industry data. With your own company’s claim severity time series, Fast Track industry severities, and CPI inflation indices, you can do several student projects.

Industry trends differ from company trends for three reasons, with different implications for forecasting.

All three of these reasons may contribute to the observed differences. Your student project determines the optimal forecasting method for future claim frequency and severity.

Many pricing actuaries give too much weight to internal (company) loss cost trends in comparison to external (industry) trends. These weights are often justified as "Our underwriting is stricter than that of other insurers, so we have lower loss cost trends." Strict underwriting does not necessarily cause lower loss cost trends, and most insurers say they have stricter underwriting.

If you say: "the industry loss cost trend is 8% per annum," your manager might say: "Yes, but what do our data show?" if the company’s own loss cost trend differs, the manager might give full weight to the company’s trend. The optimal weights of company vs industry trend depend on many items. Your student project uses the techniques in the on-line courses to improve the trend estimates.

The techniques depend on the course for which you do the student project. You can use ARIMA processes, regression analyses, and structural models.

Set up three models to forecast your insurer’s future claim severity or frequency.

(1) One model uses your own company’s claim severity data. Some insurers use a simple random walk. They fit their past claim severities to an exponential curve and apply the trend rate to the experience period severities. Some insurers uses four quarter moving averages; others use seasonally adjusted quarterly figures.

For the student project, you may form two other types of models:

Some actuaries use separate models from the average date of loss to the end of the inflation index and from the end of the inflation index to the average date of loss under the new rates.

For the first period (the historical period), use the change in the inflation index, adjusted by the parameters in the regression model. For example, if the regression model says that the claim severity trend is 20% greater than the inflation rate, and the inflation index increases 15% from the average date of loss in the experience period to the end of the inflation index, the average claim severity should increase 15% × 1.2 = 18%.

For the second period (the forecast period), you can use econometric forecasts of inflation or you can fit an ARIMA process to the inflation index. After forecasting inflation, you use your regression model to forecast average claim severity.

(2) A second model uses industry Fast Track data. Use the same type of models, though the parameters may differ. You may use an F test to see if the different parameters are just random fluctuation or a truly different model.

The important test is which model has the better out-of-sample goodness-of-fit for your own company’s average claim severities. It may be that the internal model has the better in-sample goodness-of-fit and the external data has the better out-of-sample goodness-of-fit.

(3) A third model uses the residuals of your company’s claim severity data regressed on industry Fast Track data. This model may be especially good at offsetting random loss fluctuations in your own data. Your hypothesis might be the following:

This is moving average MA(1) model, which you can easily test. Use both in-sample and out-of-sample goodness-of-fit tests.

In truth, we expect the moving average parameter to be no higher than 50%. The random loss fluctuations may be in 2007 or 2008 or a mix of both.

An autoregressive AR(1) parameter of 100% says: "If the company’s trend from 2007 to 2008 is two percentage points above the industry’s, we expect the same two percentage point difference next year as well."

The optimal ARIMA process depends on the insurer’s size and the stochasticity of the claim severities. A small insurer looking at uninsured motorist claims might have an MA(1) parameter of 50%. A large insurer looking at collision claims might have an MA(1) parameter of zero and a significant AR(1) parameter. You might compare two coverages or two lines of business.

For your ARIMA model, make sure you have a stationary process. For claim severity, take logarithms and first differences. For the student project, examine the graph of the data and the correlogram, and explain why logarithms and first differences are needed.

Your conclusions have strong implications for insurance pricing. Your company’s current procedure may indicate a trend factor of 10% and your model indicates a trend factor of 15% (or vice versa). A 5% change in rate levels affects market share, profitability, and corporate strategy.

If your student project suggests that the current trend procedure is not optimal and your proposed model does better, spend extra time on the text write-up. The ARIMA process fit to the residuals may show the value of a moving average or autoregressive model, but readers of your report want to understand why the model makes sense.

Some pricing actuaries point to differing past claim severity trends between their company and the industry to justify an internal model. A difference in observed trends proves that industry figures are not applicable to the company’s business. This argument sounds reasonable, but it is often misleading. The on-line courses give you the tools to test this justification. Test whether the industry trend vs your company’s trend is statistically significant. For many companies, the differences are not statistically significant, and your actuarial pricing is more accurate if you use a larger data base. Even if the difference is significant, you must test if the differences have positive or negative autocorrelations.

You can compare the models for different states, coverages, and time periods. Use a variety of states, coverages, and time periods to see if your results are consistent or are just random loss fluctuations.

An examination of state vs countrywide trends is similar to insurer vs industry trends. Use the same statistical procedures and hypotheses. Even states with different compensation systems, such as no-fault vs tort liability, may have the same claim severity trends. Test this empirically using the methods in the on-line courses.

Trends by class group are like trends by state. Examine each class group relative to the state (or countrywide) total.

Trends by coverage may differ, depending on the type of injury, such as bodily injury vs property damage.

Claim frequency has similar themes. Industry data may show a clearer relation to other macroeconomic variables, such as the cost of gas or weather conditions. You might use industry data to select autoregressive parameters and the residuals of your company’s data on the industry data to select a moving average parameter. It is harder to fit claim frequency to an ARIMA process, and improvements in trend procedures are valuable.

If you have quarterly data, you can use quarterly claim frequency and severity figures. Claim frequency and severity are often seasonal. You can de-seasonalize the data or you can use an AR(4) term. Examine the effects of the alternative methods and select the one that seems best.

Examine several models, such as ARIMA(1,1,0), ARIMA(2,1,0), ARIMA(0,1,1), and ARIMA(1,1,1). Use the in-sample goodness-of-fit tests to pick the optimal ARIMA process. If two models seem equally good, use out-of-sample tests to choose between them. Often a more complex model has a better in-sample fit, but a simpler model has better out-of-sample forecasts.

Set up an out-of-sample test between your ARIMA model and your company’s current ratemaking method. Make predictions for the next four quarters from each model, show them to your manager, and see which model forecasts better over the next year. It is likely – but not certain – that your model will do better.

In many cases (including claim frequency and severity projections), the ARIMA processes are better applied to the residuals. For a structural model, determine the optimal inflation index to use for claim severity and the optimal macroeconomic variables to use for claim frequency.

Develop a structural model, using inflation forecasts published by consulting firms. If you don’t have access to these forecasts, use your own forecasts of inflation by ARIMA modeling on the inflation indices. Regress your claim severity time series on the inflation index and apply an ARIMA process to the residuals.

Set up an out-of-sample test for the next four quarters. If your model does better than the company’s current procedures, you will get immediate credit (and perhaps a raise). If your model does not do better, you lose nothing. You will have received VEE credit and you learned much about claim frequency and severity.

Don’t wait four quarters to write up your student project. Send in your student project when you complete the modeling. A year later, you can write back to NEAS reporting the results of the comparison between your model and the company’s current procedures.

Readings

The following papers may give you more ideas for a student project on claim severities.

Masterson, Norton E., "Economic Factors in Liability and Property Insurance Claims Costs," ASTIN Bulletin International Actuarial Association - Brussels, Belgium, 1977: Volume 9, No 3, pages 278-280;

http://www.casact.org/library/astin/vol9no3/278.pdf

Masterson, Norton E., "Economic Factors in Liability and Property Insurance Claims Costs, 1935-1967," Proceedings of the Casualty Actuarial Society Casualty Actuarial Society - Arlington, Virginia, 1968: Volume LV, pages 61-89;

http://www.casact.org/pubs/proceed/proceed68/68061.pdf

Evans, Jonathan P., and Schmid, Frank, "Forecasting Workers Compensation Severities and Frequency Using the Kalman Filter," Casualty Actuarial Society Forum Casualty Actuarial Society - Arlington, Virginia, 2007: Winter, pages 43-66

http://www.casact.org/pubs/forum/07wforum/07w49.pdf