Long Tail Lines of Business

A professor examines the volatilities, correlations and risk capital allocation for long tail lines of businesses.

Ben Zehnwirth and David Odell | November 9, 2009

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Risk had become a buzz word in the last decade in our public and corporate discourse. People are required to make decisions on matters that increasingly depend on an understanding of this concept. In the course of this the concept itself undergoes change, becoming more precise in some respects and more elusive in others. Examples from the present time are almost too numerous to mention, but the most obvious being climate, chaotic financial markets, terrorism and the unpredictable effects of globalization. Our technology offers us unprecedented oversight and control over things but also adds rapid feedback so that a whole new level of unpredictability comes into play.

There’s unlikely to be a useful general theory of risk, but in some sectors the problem may be more tractable than others.

Reserve risk and underwriting risk for multiple LOBs of long-tail liability is one of these since the factors driving unpredictability are limited in scope and not self-referential as they are, for example, in tradable assets.

The concept of risk requires evaluation of future events and so mathematical probabilistic thinking, which originally evolved to understand gambling and insurance, is relevant. In simplest terms this means that the future (more generally, the unknown) is regarded as a set of outcomes and we can assign a numerical value to each one in such a way that these values reflect the best judgment of the likelihood of each individual outcome. If A is more likely to come about than B then the assigned probability should be higher, and to refine it further, if A is twice as likely to come about than B its probability should be represented by a number twice as large as that of B. Probability theory makes these concepts as precise as we are able.

A complete set of outcomes and probabilities is called a (probability) distribution, and so what is required to make this useful is a way of connecting a real situation with a distribution. The link is called a model. Sometimes the real situation and the mathematical model can be hard to tell apart. The model for a standard die is a thought-experiment die (or an abstract spinner with six outcomes). The model enables us to study a number of variants, such as (1) known fair die, (2) loaded die with known bias, (3) possibly loaded die, (4) loaded die with unknown bias. In (1) and (2) the passage to a distribution is immediate, (3) and (4) pose a very different problem where the distribution depends on past experience.

Most problems of practical relevance are of the general type of (3) and (4) above. Here the focus is on using the past data in the best possible way to derive a distribution. Modeling of data is an information problem.

From Insight to Foresight

We can think of a model as a mathematical structure which (1) describes the factors which determine the data and (2) quantifies the difference between the determining factors and what we actually see. Let’s call these pattern and noise. In other words a model is a way of reading the data as pattern plus noise. Distributional assumptions are made about the noise which is directly translatable into a distribution for the final outcomes. In the context of business modeling, such a distribution is the fundamental to allocation of risk capital, since Value at Risk, Expected Shortfall (T-VaR) or similar measures can be easily calculated. Accuracy of the distribution is paramount, however, especially if high percentiles are in view.

A model is thus a way of separating data into two parts, a (predictable) pattern and an (unpredictable) noise. A good model again is one that uses past data to do this in the best way. The split between the two parts is correctly placed when no more pattern remains in the noise and the distributional assumptions about it are well-supported, and when the elements comprising the pattern form a parsimonious intuitive picture of the situation being modeled. An over-parameterised model puts noise elements into the pattern with the result that forecasts drawn from it have no value. A good model provides insight into the forces driving the data, but an over parameterized model has little to tell us. An under- or badly- parameterised model leaves unmodeled structure in noise and so its forecasts tend to be systematically biased. Diagnostic testing is designed to reveal when any of these errors are present.

An example of this for long-tail liabilities in general insurance is that of models which treat the data as resulting from trends and levels (predictable and measurable pattern) and volatilities (about the trend structure) in distributional terms. The distributions on a long scale are normal with possible changing variances. In the Probabilistic Trend Family (PTF) modeling framework of Barnett and Zehnwirth, published in 2000, the three temporal axes of development period, accident (or underwriting) period, and calendar (or payment) period are allowed to contribute trends (development and calendar) and levels (accident), and the intensity of the volatility is allowed to vary along development or accident directions. Together these constitute a basic picture of the forces determining a loss development array. Loosely speaking, the development direction shows the intrinsic properties of the line of business (LOB), the accident direction, the company’s evolving market position and the calendar direction a combination of economic and social inflationary factors and less commonly changes in claim handling policies.

Figure: A model is understood by means of a model display chart which consists of four graphs, one for each of the trend directions and a fourth that shows the volatility in relation to development period. The grey bars represent the placement of model parameters.

When using a model to forecast we are doing more than just relying on the assumption that the future will resemble the past. Statistical modeling has come under some criticism in recent years for a supposed blind assumption that the future will resemble the past. Some of this criticism is well deserved. The use of statistical modeling targeted here is like assuming that the pattern is fixed and the only thing that will vary is the noise. When results deviate more than expected from predictions the critic then says that the characterisation of the noise was too modest. But in reality the trend pattern may have changed. That is why it is important to treat the pattern, the predictable part of the model as representing an insight into the forces determining the data. Understood in this way we are in a position to project how the pattern might change in the future. We can draw on the model to some degree to help us make these projections by looking at the way the pattern may have changed in the past, but we also need to bring in everything we know about the factors affecting the business we are modeling.

Volatility (or residual) display for the model above. The mean of the residuals is represented by the dark shaded “river” at the centre of the three directional displays, the lighter shades are quantiles.

We are now in a position to consider the types of correlation that can occur between lines of business.

Multiple Lines of Business and Correlations

Figure: CVs of ten lines of long-tail business held by The Hartford, based on Schedule P 2006 data. The red line is the aggregate CV.

When a corporation is able to operate a shared risk capital pool for a number of lines of business there can be a significant reduction in the aggregate risk capital requirement.

A joint distributional model enables the estimation of the total risk capital needs for each individual line, in proportion to its contribution to aggregate volatility.  Within each line a trend structure model reveals the spread of risk capital need over future years, as in the example below.

Figure: From the illustrative analysis of The Hartford based on 2006 Schedule P, we see that the bulk of the risk capital required for WC will be needed 5-8 years into the future. CMP, however requires most of its risk capital in the next four years.

The presence of correlations between LOBs has an impact on the aggregate distribution of reserves across multiple LOBs. This naturally has ramifications for aggregate risk capital, and risk capital allocation by LOB and calendar year (cost of capital).

When multiple lines of business are involved we can say, as a first approximation, that the means are additive, but that the volatility components combine in more complex ways which depend on correlations. The basic principle of insurance (risk pooling) can be stated in these terms: the volatility of an aggregate is less than the sum of the individual volatilities, unless these volatilities are perfectly correlated.

The correlation confined to the volatility component is called Process Correlation.

The process correlation between LOB A and LOB B is 0.85

If we think of modeling, as suggested above, as a process of extracting pattern (signal) from noisy data, then the presence of a correlations means that there is a cross-over of information between two or more lines. This means that when the model parameters are determined by the data and a (process) correlation between lines is taken into account then the parameters in the individual models are also found to be correlated. This is called Parameter Correlation.

In addition, we might observe in the models for two more or less correlated lines that the (calendar) parameter changes tend to be in lock-step. This points to the case in which the lines are affected by common economic drivers.

All of the abovementioned correlations lead to reserve distribution correlation, which in turn has a calculable a reduction of diversification benefit in respect of Risk Capital Allocation.

To sum up there are four types of correlations:

1. Process Correlation (correlation between residuals, i.e. the volatility components)
2. Parameter Correlations
3. Same trend structure (especially along the calendar years)
4. Reserve distribution correlations

(1) induces (2). However,  (3) is the 'worst' kind of relationship you can have between two LOBs as it can result in very little, if any, risk diversification. It means that in terms of future calendar year trends the two LOBs move together, that is, a trend change in one LOB means a trend change in the other LOB, and is tantamount to the two LOBs having the same drivers.

Correlations of type (3) can be detected by inspection of the model display.  We have generally found this type to be quite rare unless the two segments are, for example net and gross data of the same LOB. However they might also need to be factored into scenario design, for example when a piece of pending legislation will adversely affect payment regimes in different lines, in this case the effects on aggregate reserve distribution can be estimated by the use of coordinated forecast scenarios.

By definition, correlation is a relationship between the unpredictable parts of a forecast, and its economic impact should be hedged by adequate risk provisions. This means that we must begin with a model which separates the trend structure from the volatility about the trend structure.  Two badly parameterized models which leave significant structure in their residuals may show high process correlation, when in fact there is none, the correlation being an artifact of similarly trended residuals. Hence it is important to recognise that you cannot measure the relationship between two LOBs unless you first identify the trend structure and process variability in each LOB.

Dr. Ben Zehnwirth, managing director, Insureware Pty Ltd., and professorial visiting fellow, School of Actuarial Studies. Dr. David Odell has a PhD in Pure Mathematics from Cornell University and a Graduate Diploma in statistics from Melbourne University, where he worked on mathematical finance and spatial statistics before becoming a senior statistician at Insureware.