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The Actuary The magazine of the Institute & Faculty of Actuaries
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A switch in time can help align

The sharp stockmarket falls in the last year have highlighted the fact that equity investment can be very risky. During the first nine months of 2002 the FTSE-100 index fell more than a quarter, on top of falls in the previous two years. This is leading to questions about the solvency of insurance companies.

Measuring equity risk
Many life insurers are now beginning to look at stochastic modelling in order to measure solvency risk more seriously, and are wanting their models to reflect the possibility of events we have seen. Traditionally, a log-normal model has been used to model equity returns, but this may not be appropriate. During October 1987 the FTSE-All Share index fell by 30%. Using a log-normal model, there is an extremely small probability of this occurring unless the parameters are set so that they are unrealistic under normal conditions. Many criticisms have been levelled at the log-normal distribution for modelling equities; it has been suggested that it fails to capture extreme price movements and does not reflect changes in volatility over time. Maybe we can better model UK equity markets by selecting a different model.
We have investigated two other possible distributions. The first, a ‘GARCH’ model, is used by many financial modellers. The second, the ‘regime switching log-normal’ model is used by many life insurance companies in Canada (which has a stochastic modelling element for setting some reserves).

A GARCH model
First, what is a GARCH model? GARCH stands for ‘generalised autoregressive conditionally heteroskedastic’. Put more simply, it means that past volatility and returns are used to calculate tomorrow’s volatility (there are many different forms of GARCH model we have selected a model where the volatility is dependent on just the last period’s volatility and return). It is a bit more complex than a log-normal model, but it does have the advantages of ‘fatter tails’ and allowing the volatility to vary over time.

Regime-switching log-normal model
This model is widely used for modelling equity-linked guarantees overseas as it provides a very good fit to North American equity markets (this is discussed further in ‘A Regime Switching Model of Long-Term Stock Returns’ by Hardy, North American Actuarial Journal, April 2001).
The model assumes that while equity returns are log-normally distributed, the actual log-normal distribution used can vary from one time period to the next. For instance, for a two-regime model, returns will use one log-normal distribution together with a probability of switching to a different distribution for future time periods (and a different probability of switching back again to the current distribution in periods after that). Using a two-regime model for the FTSE-All Share, we found one regime had high returns and fairly low volatility (and returns stayed in this regime about 90% of the time). The second regime had negative returns and much higher volatility. If this is a good model for the UK markets, it is possible that we are in this second regime at the moment.
One further advantage of using models where the volatility changes over time is that it allows correlation between markets to rise when returns are very poor.

Which model is better for UK markets?
We have investigated whether these models are a good fit for the UK FTSE-All Share Index. There are several problems that can be encountered along the way in fitting distributions. Good quality data are hard to obtain, and even when they have been obtained, a decision has to be made whether it is useful for what we are trying to do.
Figures 1 and 2 show the different distributions using data from the last 30 years. Even over this period, there are changes occurring in the data monthly volatility was lower in the later years than the earlier years. Correlation with overseas markets has increased. To do any form of modelling we need to strike a balance between using lots of data and data that is more recent and relevant.
Looking at figure 1 (previous page), we can see that the regime-switching log-normal model provides a good fit around the median point of the distribution (taking the squared differences showed this was the best fit overall). The GARCH model is not quite so good (but better than a log-normal model). This is important as well as getting stochastic modelling right at the extremely poor returns where there are lots of costs, we want to get it right for ‘average’ scenarios because there are likely to be a lot of them.
However, our real test for a stochastic model is whether it can reflect markets when returns are extremely poor. This is difficult with limited data and there are very few months when the market has fallen by more than 10%. Judgement will normally have to be used as to whether the model shows movements that are extreme without showing completely implausible results. Figure 2 (previous page) below shows that the regime- switching log-normal model provides a better fit than the log-normal distribution and the GARCH at the extreme end (where monthly return falls by more than 15%).
So which model should be used? There is no definitive answer to this. The log-normal model, while not providing a perfect fit to the real world, is still very attractive for several reasons. First, it is very well known and understood by many people, so results can be more easily explained. Second, it is considerably easier to manipulate mathematically; for example calculating derivative prices is much easier when returns are assumed to follow a standard log-normal model. Last, the log-normal model is a lot easier to implement, so saving time and money.
However, for more sophisticated applications, the extra effort in using more complex return distributions will lead to more confidence in the model’s results and fewer nasty surprises when market conditions are exceptionally poor. For the example shown here, the regime-switching model shows a better fit than the GARCH model, but there will be cases where a GARCH (or, indeed, other models) will be appropriate.

How does this help us manage equity risks?
Using more sophisticated modelling for equity risks leads to more accurate modelling and more confidence in any conclusions. This will mean better pricing of insurance products and more effective management of the risk that equity exposures might lead to life insurers becoming insolvent.

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