The year 2008 was not a good one for investors in corporate bonds, with record-breaking market value decreases as spreads widened. Insurance companies and regulators are keen to understand what 2008’s experience means for possible credit stresses in 2009 and beyond. This article is based on a paper and presentation that the Extreme Events Working Party discussed at this year’s Risk and Investment Conference.

The objective was to derive one-year stress tests for solvency capital calculations. We started our analysis on 10 years’ worth of daily data. There are practitioner rules of thumb for annualising daily stress tests, but we wanted something more sophisticated. We therefore tried generalised autoregressive conditionally heteroscedastic (GARCH) models, with results that were both alarming and informative.

We analysed daily IBOXX index data from December 1997 - April 2009 for portfolios of gilts and sterling corporate bonds. As the average duration of bonds in the indices was not always constant or the same, we converted the indices into daily yield changes using a simple approximation. Total return indices were used rather than published yields to include the effects of rating transitions, defaults, recoveries and income within the stress tests.

Figure 1 is a scatter plot of daily log returns recast as yield changes. We have also included the empirical percentiles at the 1-in-200 day level. Yield changes capture the pure asset risk. We also estimated the percentiles of the sum of yields, reflecting the risk for an investor with an initial 50% corporate/50% gilt portfolio. Finally, we examined the percentiles of the change in spreads, which might be relevant for an investor holding corporate bonds against annuity liabilities.

Having analysed daily data, the next challenge was to gross up the daily stress tests to derive annual stresses appropriate for statutory solvency work. Given an estimate of a 1-in-200 day move, what is the corresponding 1-in-200 year move? Practitioners often use the ‘square root of time’ rule to scale stress tests. For example, estimation of a 1-in-200 year spread widening could be as follows: Estimated 1-in-200 day spread widening: 0.08% Number of trading days per year: 250 Estimated 1-in-200 year spread widening: 0.08% x √250 = 1.26%

Unfortunately, this methodology fails on ‘back-testing’. We can ask whether this method, applied to data until March 2007, would have assigned a reasonable probability to what actually happened the following year. The answer is a clear ‘no’ — depending on exactly how you cut the data, the spread moves to March 2008 represent something between a 15 and 20 standard deviation event. Based on normal distributions, we should expect one of these at most once every 1050 years. To get one in a 10-year data period looks like very bad luck. Indeed, to get one of these in the lifetime of the universe — roughly 1010 years — looks like very bad luck. The need to bridge this gap makes the analysis of credit data both interesting and challenging.

What, then, is wrong with the square root of time calculation? It is based on a random walk model, in which returns are normal and independent from one day to the next. Visual inspection of daily returns shows a more complex pattern, with more extreme outliers than the normal predicts, as well as the phenomenon of volatility clustering, where periods of high volatility alternate with periods of low volatility.

There is an established tool to model volatility clustering, known as GARCH. In 1982, Robert Engle* proposed a class of models where volatility is itself a mean reverting process, depending on previous volatilities, a long-term average and a stochastic term. In 1986, Tim Bollerslev published the most commonly used generalisation of the model, adding a feedback mechanism so a spike in returns could feed back to the volatility process as well as the other way around. Engle’s work in this area earned him a share in the 2003 Nobel Prize for economics.

Avid readers of regulatory consultation papers may have noticed section 5.85 of the Committee of European Insurance and Occupational Pensions Supervisors’ recent CP 56. If adopted, this section will require undertakings to ‘keep track of the latest developments and trends in internal modelling, ...for instance by regular surveys of the scientific literature or communication with peers and the scientific community’. We think compliance would be difficult without consideration of models such as GARCH.

We found that GARCH holds some pitfalls for the unwary. For some parameter values, GARCH models behave well, with a stationary distribution that has finite variance. Other parameters may imply infinite variance, and for some choices a chance increase in volatility or a large observation can enter a cycle of positive feedback where volatility increases without limit. Compounding from daily to annual projections, the infinite variance models produce far more onerous stress tests.

For the data sets we examined, fitted parameters frequently lay close to the dividing line between finite and infinite variance models. Looking at different data series, or changing the start and end points, moves the fitted parameters to one or to the other side of the line. We considered rejecting the infinite variance models as economically implausible. Although this leads to more modest stress tests, this approach has three disadvantages.

First, although a particular data approach may produce a finite variance model this year, another year’s data may tip the parameters over the edge. Second, although our calculations are at an early stage, I guess that only the infinite variance models could pass a back-test on the critical 2008 data. And finally, we may be accused of rejecting infinite variance models simply because we dislike the answer. There are several possible ways forward. One approach is to blame the instabilities on lack of sophistication and seek to make the models even more sophisticated by, for example, allowing for fat-tailed residuals or higher orders of auto-regression. This feels a bit like throwing another ball into the tree to knock down the one that got stuck. So far our attempts in this direction have not provided the stability we had hoped for.

A more promising approach could be the use of structural models, relating the behaviour of equity and corporate bonds simultaneously to the value of a corporation’s underlying assets. Within these models, which make a number of controversial assumptions, bond holders have priority over equity holders. If, for example, a 1-in-200 year equity move is a 50% fall, then under most plausible calibrations the 1-in-200 year fall in corporate bonds would be less severe. If one accepts the underlying model, this creates an upper bound on spread-widening stresses which is still considerably stronger than used by most UK insurers, but excludes some of the infinite variance GARCH models.

A survey of the literature on volatility modelling reveals the wide application of GARCH models. When we embarked on fitting GARCH models, we expected to find a well-trodden path. In retrospect, our experience was more like walking a tightrope. Several more directions await exploration, so we encourage actuaries to look out for our session at November’s Life Convention for further updates.

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The views expressed are Mr Smith’s and not necessarily those of the working party or Deloitte. Mr Smith would like to express his thanks to Ralph Frankland, chair of the Extreme Events Working Party, and the other working party members both for the underlying calculations and for making amendments to earlier drafts of this article. R scripts for the GARCH model fitting are available on request from *elliot.varnell@kpmg.co.uk*

The Risk and Investment Conference presentations can be found at: *www.actuaries.org.uk/knowledge/publications_archive/conferences/*

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*Andrew Smith is a partner at Deloitte and a member of the Extreme Events Working Party*

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** Professor Robert Engle was interviewed by* The Actuary *in June 2009.*