Advantages of the heavy tail method
Portfolio analysis and value-at-risk (VAR) assessment usually proceed upon the assumptions that returns on financial assets are normally and independently distributed over time.
These assumptions come in handy since the implied additivity property allows one easily to aggregate weighted returns on particular assets into a portfolio. Moreover, in this case only three parameters are needed to calculate the VAR of a particular investment strategy: the mean, variance and the covariance of the returns on the assets. Alas, time and again, experience and scientific research have shown us that returns are not normally distributed and that the time independence assumption is overly strong.
It is now widely recognised that, while returns are uncorrected over time, so that one cannot forecast directions of market movements, returns are not independent due to the fact that there are clusters of volatility. In other words, if the market is turbulent it is likely to remain turbulent for a while, and vice versa. The popular RiskMetrics product implements this market feature by using an exponential weighted average of past volatilities to predict tomorrow's volatility. Option traders frequently update their measure of volatility for the same reason.
The other deviation from the standard assumptions is the non-normality of returns. The empirical return density is more peaked in the middle and has fatter tails than the standard bell-shaped curve. This means that on quiet days the market is less volatile than would be expected if returns were normally distributed, while in times of turbulence the movements are more dramatic. While the market is aware of this heavy tail feature, witness the smile effect, the market has not responded to this in the way it has dealt with volatility clusters. The primary reason is probably the difficulty in handling non-normal heavy tail distributions due to the lack of the additivity property. Nevertheless, pension funds with concern for underfunding, and risk managers and oversight agencies who care about insolvency, should be aware that traditional VAR products give a distorted picture of the risk. The reason is that pension funds and risk managers are not concerned with day-to-day risk of realising a negative return. Their concern is more for the dramatic losses that occur as infrequently as once every 10 years. At this frequency, volatility clusters measured with RiskMetrics or GARCH-type methods have no predictive content. However, a large number of realisations and appropriate statistical methods can yield a lot of information.
As it turns out, all heavy tail distributions display the same tail behaviour which is captured by a single number, the so-called tail index. It is quite straightforward to calculate the tail index, and it permits smoothing and extension of the tail outside the sample . In the figure we show losses from the FT All Share index using daily data from 1990 to 1997. The step function gives the empirically realised losses on the x-axis with the frequency of the losses on the y-axis. The small insert gives the entire empirical distribution function. The dotted line gives the estimated tail probabilities by using the tail index. Note that the curve extends beyond the sample, and therefore we can get estimates of the frequency and magnitudes of losses for a much larger time interval than we have data for. For reference we also plot with the dashed line the tail probabilities if it is assumed that the data are normal distributed. Note that the normal seriously underestimates the VAR. For accurate estimation of uncommon events the length of the sample should be as large as possible. By using daily data from 1976 to 1997 we can forecast the maximum expected daily loss in the FT index in 10 years as 6.3%, and predict that a single-day loss of the same magnitude as the 12% loss on October 19, 1987 would only occur once every 76 years. For example, on an investment of 100 million in the index, once every 10 years there is a day where one loses 6.3 million, while if one assumes normality the predicted loss is only 2.9 million.
There are other advantages of using the heavy tail model for calculating the VAR. For example, RiskMetrics calculates the VaR over multiple periods by using the square-root-of-time rule. This is the appropriate procedure if returns are normal distributed. For fat tailed data the square-root-of-time rule is too high. Therefore, normal extrapolation produces too conservative VAR measures, while the heavy tail method will provide more accurate answers.
Jon Danielsson is Professor of Economics at the University of Iceland and a visiting lecturer in finance at the London School of Economics, and Casper de Vries is a Professor of Economics at the Tinbergen Institute and Erasmus University Rotterdam. Their e-mail is firstname.lastname@example.org and email@example.com respectively. Some of their research papers on financial extremes and VaR can be downloaded from http://www.hag.hi.is/~jond/research.