Loss cost modeling has traditionally involved trend and loss development factors. Since claims can take a long time to settle, development allows the actuary to estimate their final settlement values as of the time of analysis. Trending allows the actuary to adjust for the changes in loss costs over time, particularly those due to changes in prices of insurance cost factors such as business mix, inflation, technology, and tort.
However, actuarial trend rates are not suitable for trending target loss variables of insurance predictive models. This is because they contemplate other dynamics—business mix—which are already contemplated in predictive models, and therefore introduce extraneous modeling complications and bias predictions. Given the thin margins of most insurance lines of business, such biases can have a leveraged effect on the profit and growth targets of the insurance carrier, and hence cannot be overlooked.
To understand how actuarial trends distort predictive model loss targets, the reader should note that the loss of insurance risks can be compartmentalized into two main factors: risk attributes and exogenous factors (such as technology, legal framework, prices of insurance commodities, just to mention a few). So, the expected pure premium of an insurance risk can be specified as:
Yit is ultimate pure premium for the ith risk in exposure period t;
β0 is the model parameter associated with the constant term;
βk is the model parameter associated with the kth model variable;
xikt is the kth model variable of the ith risk (such as insurance score) in exposure period t; and
α, the general inflation parameter, is the rate of change of loss costs over time due to changes in exogenous factors such as monetary inflation, technology, insurance laws, and other related factors; it does not, however, include changes in business mix or distribution. Those, rather, are accounted for by the risk attributes, xikt.
Therefore, it’s easy to see that the best way to estimate the expected pure premium of a given insurance risk is to run a GLM with non-trended ultimate pure premiums of the risk exposures on a trend term (policy year) and risk attributes for the respective exposure years. However, this is not what is commonly done in the industry. We instead model trended ultimate pure premiums on a trend term and risk attributes. Though popular, this practice is suboptimal. As previously mentioned, the actuarial trend factor, r, contemplates two trend effects: changes in business mix (denoted δ) and changes in exogenous factors (denoted α). Mathematically, r= α+ δ. While this makes it suitable to trend aggregate losses (as used in actuarial analysis), it renders the factor inappropriate for predictive modeling which occurs at the individual risk level.
To see this, notice that trended ultimate pure premium (trend-to year T) of an individual risk, denoted , has different expression from the untrended ultimate pure premiums shown in 1, which are the correct values we aim to predict:
The ratio of (2) to (1) is the bias in predictions which results when we model trended ultimate pure premiums on a trend term and risk attributes. The bias is calculated as:
A graph of the bias across exposure years is also shown below:
As the reader can easily observe, predictions under (2) are biased for all exposure years except for the trend-to year. The bias deteriorates with time, and even worse, will be elusive to the modeler who can easily remain unaware of the problem. The book of business will thus continue to bleed year after year until a cure is found, or at last, to death. We have called this problem the pricing hemorrhage.
In closing, the popular culture of modeling trended ultimate pure premiums seems intuitively appropriate but it however leads to a fatal bias in predictions. To avoid this pricing hemorrhage, we should instead model untrended ultimate pure premiums on a trend term and risk attributes.