A key step in the usual procedure for modeling a loss development pattern is to fit formulas to empirical age-to-age or age-to-ultimate factors. Having a fitted formula is useful because it provides an easy way to smooth the bumps found in most series of empirical factors. Also, if the fit is to age-to-ultimate factors, the formula usually provides a convenient way to interpolate the factors.
While the fitting is convenient and practical, it can hardly be said to have a substantive conceptual foundation. A formula is chosen because it is easy to compute and because it nicely fits the age-to-age factors. It is not derived from more basic assumptions in the sense that nothing is specifically built in to reflect that it is being fitted to data that represent ratios of loss for a particular exposure period as of given evaluation ages.
While a formula serves perfectly well for smoothing, it may not suffice, in and of itself, to handle other applications such as tail factor extrapolation, early age extrapolation or conversion of the factors from one exposure basis to another. Tail factor extrapolation is needed to get age-to-ultimate factors after a fit is obtained to age-to-age factors. Yet, an age-to-age factor formula may not immediately lead to the extrapolation. To obtain the desired age-to-ultimate factors the actuary may have to derive the product of an infinite series, make cut-off assumptions, or use a computerized numerical algorithm.
In early age extrapolation, the actuary is seeking factors at an evaluation age younger than the earliest evaluation age associated with the fitted factors. For example, the actuary may have accident year age-to-ultimate factors for evaluations at 12, 24, 36... months, yet may need to have factors at 6, 18, 30,... months. The problem is that the back extrapolation of a formula fit may or may not yield plausible results at earlier ages (i.e. the factor at 6 months). Some additional techniques may be needed to get reasonable factors at these ages.
Finally with regard to conversion, the actuary may have fitted accident year factors, but may want to have policy year factors. Yet a good fit to accident year factors may not directly lead to a good fit to the corresponding policy year factors. Actuaries have usually dealt with this conversion problem by using an average date of loss adjustment. Under this adjustment, the development factor for one type of exposure period at a given evaluation age is estimated by the development factor for the original type of exposure period at an adjusted evaluation age. The adjustment is equal to the difference in the 'average dates of loss for the different exposure periods. While this adjustment works well at mature ages after all exposures are earned, it goes awry at immature evaluation ages.
The conclusion i s that fitting with general formulas is a useful and flexible approach that must often be supplemented for extrapolation and conversion applications. The supplemental procedures may not be too difficult to implement. So, in the end, from a practical perspective, not too much should be made of the need to introduce them. However, it would be more convenient to have a model of loss development that would automatically handle extrapolation and conversion. Such a model would not start with a formula for age-to-age factors, but would instead be based on percent of ultimate or age-to-ultimate curves having an explicit dependence on the underlying exposure period.
Models such as this have been previously proposed. Yet they have not been widely adopted. Why? We speculate the reluctance stems from two essential areas of concern. First, there may be questions about the theoretical underpinnings of such models. Second, there may be doubts about whether the proposed models are practical.
In order to address these concerns, we will present a general, yet accessible, conceptual foundation for exposure dependent percent of ultimate models. We will start by relating an exposure period, such as an accident year or policy year, to an associated distribution of accident date lags. The accident date lag for a claim is defined as the length of time from the start of the exposure period to the accident date. We will show that the familiar parallelogram or rectangle diagram representation of an exposure period can be readily converted into a graph of the density of this accident date lag random variable. The cumulative distribution of the accident date lag may be identified with the percent of premium earned to date assuming the earning of premium corresponds exactly to the exposure to loss. We will argue that under certain conditions the percent of ultimate loss development curve may be expressed as the cumulative distribution of the sum of the accident date lag random variable plus another random variable that summarizes the claims process. The claims process in this context includes the delay between the accident date and report date, as well as the changes in the valuation of a claim and the time lags between these valuation changes. Perhaps the key insight underlying this construction is that exposure dependence can be isolated in the accident lag distribution.
We will then turn to applications. We will use the model to simulate patterns for different exposure periods, derive a convenient accident period development formula, fit and convert patterns, extend the average date of loss approximation, and approximate a converted pattern as the weighted sum of shifted versions of the original pattern. In the end we will hope to have shown that exposure dependent percent of ultimate models are not only pleasing to the theorist, but also useful to the practical actuary.
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