# Correlation Between Years for Similar Population

A common problem in insurance is to predict the amount (number, severity, or aggregate total) of losses in the next year for a book of business (eg. all personal auto policies). To do this, one can look at the historical correlation between past years and use that to predict the next year. For example, a co-variance matrix between years can be used to perform simple OLS regression analysis.

Question: To determine the correlation between years, you can look at the sum total for each year. Let's say you have more detailed data (by claim or by policy) and each year, some policies will stay, some will leave, and some new ones will join. Can you use this to get a better estimate, or is the sum total the best you can do?

It sounds like what you want is something from the family of ARIMA models. Auto-regressive integrated moving averages. What those models do is take past means and the rate of change between means annually to predict a future value.

So you are asking if an additional predictor (e.g. attrition) will make your predicative model better? Yes it will. Will it make it much more predicative? probably not. Attrition is also auto correlated.

Unless you have an idea of what a "seasonal" influence would be, an ARIMA would probably be enough.

Prediction by looking at correlation is a bit vague I think, and correlation in time series is a tricky concept.

Let $N_i$ be the yearly number of losses each year. Then applying a linear regression is assuming the model $N_i = a N_{i - 1} + b + \text{error}_i$, where the errors are assumed to be normally distributed and independent. This is equivalent to a $AR(1)$ model, an autoregressive model with order $p$.

Now if $N_i$ is a sum of many different products, then it can be a good idea to model those separately. This is especially true if you have additional information on discontinuations and introductions of products, as you are now catching all these effects into a single error term.

As an example, say you are modelling the number of apples sold and the number of pears sold next year. If you know that pears will not be continued next year, you take that into account in your forecast.

This will get more tricky in particular if the different products affect each other (growth in one will imply growth or decline in another), in which case you cannot assume independent errors. In the case above, if apples have a bad year, it is likely that pears will also have a bad year and vice versa, so you get more extreme outcomes.

You are always making simplifications, and figuring out which are working well are mostly part of the job!