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The Actuary The magazine of the Institute & Faculty of Actuaries

Copulas for insomniacs?

It is difficult to model completely the financial market’s extremities, given its volatility and uncertainty. Even in companies with economic models in place, it would be surprising if actuaries were not having a few restless nights worrying about their company’s future.
The early stock growth models were developed using a simple thought process. The logic behind these were:
– stock growth (returns) has a memory-less property;
– future stock returns seem to follow a trending mean m; and
– the of the return has a defined variance s2.
This is the well-documented log-normal process for stocks. However, this model works well only in short-term analyses, when economic conditions are relatively stable.
For time series data spread over a longer term, it was assumed that the improper fit of the log-normal process was caused by the change of variance over time. The modelling thought process further evolved, using the idea of regimes. The regime-switching log-normal model developed and parameterised by an actuary, Mary Hardy, fits amazingly well with historical stock return data going back to (and including) the great depression. In her model, there are two possible regimes: one governed by a log-normal distribution with high volatility, and another governed by a log-normal distribution with much lower volatility; a probability process governs switches between the two.

Will the Hardy model work in future?
Although the statistical fit is very good for the two-regime log-normal, there are likely to be more than just two economic states or regimes. If we extend the idea to consider many regimes, the resulting model would have to consider the extremely good and extremely bad economic regimes. These types of situation are of particular importance to enterprise risk management.

So how do we extend the model?
We can assume that one economic indicator exists which completely defines an economic state or regime, including the correlation between processes. Such an indicator is unlikely to be either identifiable or measurable, however, the concept is useful in improving the model. If the parameter did exist and were known, the estimation of each economic process (eg stock, bond, or inflation) could be made accurately and independently of any other process. For a normal economic state, one process might be favourable and another neutral, whereas in a very poor economic state, all processes could take a turn for the worst the degree of correlation varies by economic state.
If this is a plausible model construct for understanding all economy-dependent variables, we have a copula a mathematical construct first introduced in 1959 in the study of probability. By postulating a latent variable that defines the economic state, a copula constructs the joint process (such as the financial markets in this example) from the univariate processes (in this example, one for stocks, one for bonds, one for inflation). The joint process implicitly attaches probabilities to the various possible values of the latent variable. In the general copula there may be several latent variables, though for simplicity, single-variable copulas are often used.

What’s the advantage of a copula?
In the standard model, correlation is estimated through a singular parameter (eg correlation = 0.3) which is only appropriate on average. For economic-dependent processes it is likely that correlation is closer to 1 (or 1) in the tails and closer to zero near the means. In copulas, the use of a probabilistic form for the correlation effect allows for the appropriate range of observed correlation effects.
The mathematical framework of the copula is also well suited for reducing computational time. Some forms of copulas can be constructed that produce well-known probability density functions, such as Gamma or Pareto, for the joint process. A known distribution is desirable when a closed-form solution is required, because Monte Carlo techniques can be avoided and replaced by mathematical formulae.

Why are copulas important?
As we saw, the two-regime model produced a very good fit to long-term series data and is a great improvement over the one-regime model. By natural extension, a model framed within a copula will improve the identification of the extreme events in the tail of each distribution, and provide a framework for understanding their correlation to one another.
So the copula is a useful mathematical construct that can be applied in financial models to help understand extreme outcomes in economy-dependent processes, eg stock growth, bond rates, and inflation. It is these extreme economic scenarios that are the main concern in determining the level of risk-based capital. With a better understanding of extreme scenarios, those responsible for corporate risk can rest a little easier.