Quantifying Loss Cost Variability

Fill in the blanks:

The uncertainty inherent in the estimation of _________ implies that a range of ________ can be actuarially sound.

Those trained in actuarial science will likely answer “reserves.” That is, in effect, Principle 3 of the Casualty Actuarial Society’s Statement of Principles Regarding Property and Casualty Loss and Loss Adjustment Expense Reserves. However, reserves aren’t the only inherently uncertain actuarial estimates. Insurance rates or loss costs (the portion of a rate that covers expected losses and loss adjustment expenses) are also uncertain.  The actual loss cost we observe in historic data is only one possible realization of a random process. Just as we can quantify the uncertainty of a reserve estimate and create a range, we can also quantify the uncertainty of a loss cost estimate.

There are many methodologies for quantifying reserve uncertainty, but one common approach is “bootstrapping.” In essence, bootstrapping is a resampling technique that works as follows: We start with a data set that contains, for example, 100,000 records. We then create a new data set of 100,000 records by randomly selecting one record at a time from the original data set. Each time we sample a record, we “put it back,” so that some records from the original data set will get pulled multiple times, while others won’t get pulled at all. And while we often want our bootstrap sample to have the same number of records as the original data set, that doesn’t have to be the case. We can either down-sample (select fewer than 100,000 records for our bootstrap data set) or up-sample.

Here’s how we can use bootstrapping to quantify loss cost variability. Let’s say our expected average pure premium for a line of business is $1,000. In other words, we expect next year to see an average loss per policy of $1,000. We then take the data set we used to calculate the $1,000 average pure premium, create many bootstrap samples from it, and recalculate the average pure premium each time. Say we’ve created 1,001 bootstrap samples and thus have 1,001 estimates of average pure premium. We can then estimate the volatility of our original loss cost estimate by looking at the range and the percentiles of our 1,001 pure premium estimates. (The 11th smallest estimate is our 1st percentile, the 21st smallest is our 2nd percentile, and so on.) With enough bootstrap estimates, we can derive an entire distribution for pure premium.

There are many potential benefits to knowing the distribution of pure premium, such as:

  • Scenario analysis – An insurer looking to expand into a new set of territories can compare the expected variability of experience for its current book of business with that of its new book if it does expand. Perhaps the expansion will decrease pure premium variability because of the greater number of policies written, or perhaps it will increase variability because of the increased heterogeneity of the insurer’s book.
  • Reinsurance decisions – By assessing the variability of pure premium by limit retained, the insurer can decide which attachment points make the most sense.
  • Enterprise risk management – Underwriting risk is a principal source of uncertainty for insurers, and understanding the volatility of experience is vital in managing that risk.

The primary challenge in calculating pure premium percentiles is the data. Without a sufficient volume of data, you may not obtain a true distribution of possible outcomes. The bootstrap data sets will be, in effect, limited to experience that actually happened, without considering what could have happened. You reduce this problem if you have a large data set with which to work.