Some further musings about loss reserving.
1. Why do we develop reported losses? By definition, they’re correlated to paid (Reported = Paid + Case). Does a projection of reported losses convey anything new and meaningful to us? Here’s a simple experiment: project reported and paid using whatever means you think are appropriate. Take the projected difference between the two and tell me how often you get case reserves that are either negative or make little sense whatsoever. There are models (Munich Chain Ladder, Halliwell’s seemingly unrelated regression equations) which attempt to resolve this, but they’re not often used in practice. Moreover, even when they may be used, they’re merely one of a set of estimates which, individually, probably have conflicting assumptions.
I suppose what I’m really trying to say is this: we should only model atomic variables. It’s appealing to model reported losses, but we should ensure that the individual components have been handled properly. If they’re not, we should question what additional value- if any- is gained when modeling a composite variable like reported losses.
2. Following on from that thought and as part of my sinking further into MRMR, I’m prepared to divide the world of reserving models into three categories:
1. Models which use static predictors. This will be the additive method. Here the predictor is typically something like on-level earned premium or exposure. The predictors are (within the context of a loss reserve model) non-stochastic. They’re BLUE and have a number of other convenient properties.
2. Models with autoregressive stochastic predictor. This is the multiplicative chain ladder. Here, the response is used to generate the next predictor variable. Because the predictor variables are themselves stochastic, we have to treat the variability with a bit more caution.
3. Models with dependent stochastic predictors. This is analogous to frequency/severity methods. Here, the response from one variable is used as the predictor for another. There is an order of operations which enables the fit and projection, but it’s one which has some appealing intuition. So, earlier when I pose the question of why we model reported losses at all, what I would propose is the following: model the case reserves separately and add them to the modeled paid losses. To incorporate a relationship between the two, regress incremental paid against prior outstanding case. that ought to make a fair bit of sense. Depending on how mature the losses are, paid losses ought to bear a strong relationship to the outstanding reserves. In turn, the case reserves may be modeled using a static predictor (case 1) such as earned premium. Or, they may be modeled dependent on another stochastic predictor such as open claims. Open claims may either use static predictors or in turn depend on a function of reported and closed claims.
I’ll admit that this all starts to get a bit crazy, but I also feel that it gets a bit closer to reality. Decomposition of aggregate losses into more manageable components allows us to focus on elements that are easier to think about an explain. I love the additive model, but recognize that there are inherent limitations. It doesn’t speak to the issue of rising frequency or changes in severity of claims. In that way, it’s about as dumb as multiplicative chain ladder, with its slow, inexorable march toward ultimate.
That third class of model is something which will soon find its way into MRMR. (And by “soon”, I mean in about a year or so. Finding time to work on this stuff is a real struggle.)
As always, thoughts are welcome.