## 1 INTRODUCTION

My invitation to contribute to a conference on the topic of volatility included the suggestion that I should make my remarks ‘wide-ranging.’ However, on thinking about volatility I quickly realized that the topic was well worked over and I was unlikely to be able to make a worthwhile contribution. So instead, I decided to write about something quite different, that is, risk. On reading about risk it soon became clear to me that this was a very confused concept. There are, of course, many types of risk but there is still disagreement even when concentrating on just the risks faced by investors. The reason for this confusion may lie in the obvious fact that there are many types of people and researchers interested in the stock market. A brief list, roughly increasing in practicality, would include:

- (1)Continuous-time mathematicians (option pricing theory).
- (2)Uncertainty-theory economists (portfolio theory, diversification, CAPM).
- (3)Econometricians (ARCH).
- (4)Empirical statisticians (random walk, efficient market theory).
- (5)Journalists (rationalization of stock price movements)
- (6)Mutual funds/brokers' analysis—‘professionals’ (buy low, sell high)
- (7)Individual investors (include housing in portfolio)

(I have included some of the major discoveries associated with each group.)

Some of these groups could easily be further sub-divided and other groups added, such as financial engineers, financial physicists, financial biologists, and, who knows, financial theologians? Each group is inclined to have its own distinct attitude towards risk and to rather ignore the suggestions or requirements of other groups. It is well known that risk is a highly personal matter, the old dislike risk more than the young, women dislike it more than men, for example (see *Financial Times*, 7 April 2001, page xxiv). No simple definition or quantification will satisfy everyone.

It is convenient to start the discussion by considering a simple theory of risk proposed by R. Duncan Luce (1980, correction in 1981). If a return has density *f* Luce is concerned with finding an associated risk measure *R*(*f*). Let α > 0 represent a change of scale, so that *x*→*x*α with density

Two assumptions are made, the first is multiplicative:

**Assumption 1:**(2)where

*S*() is some increasing function with*S*(1) = 1; and**Assumption 2:**there is a non-negative function*T*, with*T*(0) = 0, such that for all density functions*f*,(3)It is shown in the correction that, using just these simple assumptions, it follows that

(4)with

*A*_{1},*A*_{2}both ≥0, and some θ > 0, where*r*is now a return or a mean adjusted return. It should be noted that the risk measure starts as a function of the whole distribution*f*, but ends with separate emphasis on each half-distribution.

Initially considering the symmetric case *A*_{1} = *A*_{2}, one has the class of volatility measures

which includes the variance, θ = 2, and the mean absolute deviation, θ = 1. In some previous studies (e.g. Ding, Granger and Engle, 1993; and Granger, Ding and Spear, 2000) the time series and distributional properties of these measures have been studied empirically and the absolute deviations found to have particular properties such as the longest memory.

It might be worth considering variance and mean absolute measures as alternatives given that returns from the stock market are known to come from a distribution with particularly long tails. Because the variance of a variance gets into the fourth moments of returns, which will be very unstable, and the variance of absolute returns is just the variance of a return, and thus be more stable, an argument can be made for absolute returns being preferable. A rather different argument comes from an old area of statistical literature. Suppose that one had the unlikely situation that a regression

is being considered and that the kurtosis of ε_{t} is, somehow, known. A statistician decided to use an *L*_{p} norm; that is, to minimize *E*[|ε_{t}|^{p}] for some *p* > 0 to perform the regression, what value of *p* is most appropriate? The is asymptotically normal with zero mean and a covariance matrix depending on *p*; so that *p* can be chosen to minimize this covariance matrix. A trio of papers (Nyquist, 1983; Money *et al*; 1982; Harter, 1977) find that the optimum *p*-values are *p* = 1 for the Laplace and Cauchy distribution for ε, *p* = 2 if ε is Gaussian, and *p* = ∞ (the min/max estimator) if ε has a rectangular distribution. Harter suggests the following approximate value based on the kurtosis *k* of ε:

It is rarely the case that the kurtosis is known when conducting an empirical exercise, but in finance the kurtosis of residuals can safely be thought of as being over 4, and so the *L*_{1} norm, corresponding to absolute returns, is preferred. This suggests that a regression for CAPM, for example, should use this *L*_{1} norm to get superior results.

The realization that the distribution of returns is very long tailed, and consequently not Gaussian, has been with us now for many years. It is worth being reminded that when the British and Dutch sent ships to the East Indies in the seventeenth century looking for spices such as pepper and nutmeg, roughly 50% of the ships did not return. Of those that did, half were unsuccessful in their search for spices. However, the successful trips were enormously profitable, particularly during periods when the Black Death was operative and nutmeg was thought to be the only successful cure available. This example of non-Gaussian returns is discussed in Milton (1999).

However, if one goes to some fairly old literature by the uncertainty economists, summarized in Machina and Rothschild (1987), and Levy (1992), they firmly reject use of all *V*_{θ} measures as useful quantifications of risk as they do not correspond to satisfactory utility functions. Levy (1992) states ‘while the variance provides some information it cannot serve as index for the risk involved for all utility functions’ (page 568). There are three specific cases where variance can serve as a risk index, if utility is quadratic or if the return distribution is normal or log-normal. It is curious why one group holds such firm views and another group, the econometricians, appear to ignore it. The uncertainty theorists would argue that one does not need a quantitative, that is, a cardinal, measure of risk as an ordinal measure will give you all that is needed. You do not need to say, risk of *A* is 10, risk of *B* is 8; you just want to know that *A* is riskier than *B* when making a decision. The obvious response is that you cannot perform Markowitz-style optimum portfolio selection techniques without a variance or similar measure, but many years ago Bawa (1975, 1978) showed how, at least in principle, stochastic dominance techniques can be used to rank portfolios. It is much less simple to use but is fairly straightforward if the estimation of Value-at-Risk is extended to the whole distribution. Conceptually, at least this is not difficult.

A major tool designed by the uncertainty theory economists is the mean preserving spread in which the original distribution of returns is manipulated to form a new distribution with the same mean but an increased risk. Two movements can take place below the mean, one ‘chunk’ can be moved from mid-lower distribution and placed nearer to the mean, and another chunk similarly moved from the mid-lower parts but towards the tail of the distribution. If balanced correctly, the mean will not change but the extra weight in the tail will increase risk. Any risk-adverse investor would prefer the original shape compared to the reshaped distribution after imposing the mean-preserving spread. However, it is interesting to note that the volatility measure (1.5), derived from the Luce results, has the property that it is unchanged by a one-sided mean preserving spread (MPS) if θ = 1 (corresponding to mean absolute difference), the measure is increased if θ > 1 (including variance) but is decreased if θ < 1. Thus these measures do not correspond to MPSs if θ ≤ 1. The same argument applies to the two-sided Luce result (4).

Statisticians and econometricians behave as though volatility, particularly variance, equates with risks yet uncertainty theorists reject this identity.

There is reason to believe that actual investors, both the professionals and others, do not equate risk with volatility. These investors have different attitudes towards the upper and lower parts of the distribution. Investors will agree that there is uncertainty in the upper part of the distribution but risk only occurs in the lower part. Portfolios are selected to reduce risk in the lower tail, but not uncertainty in the upper tail. The investor does not diversify to reduce the chance of an unexpected large positive return, only that of a large negative one. I have seen many studies of Value-at-Risk based on the lower tail but none so far on the upper tail. It is interesting that Luce, in his 1982 correction, produces a measure of risk with different weights for the two halves of the distribution. Markowitz (1959) suggested the use of a semi-variance at a very early stage. A generalization was proposed by Fishburn (1977) defined as

where *F*(*r*) is the distribution of returns and *t* is some target. It is related to a more recent risk measure called ‘expected shortfall,’ which appears to have superior properties to Value-at-Risk (VaR) according to Yamai and Yoshiba (2001). VaR_{α} at the 100 (1 − α)% confidence level is defined as the lower 100α percentile of the return distribution. The expected shortfall is defined as *R*_{1}(−VaR_{α})/(1 − α) + VaR_{α}. (Standard errors for a VaR can be estimated using standard binomial or Poisson distribution theory, or using a bootstrap method.) The early arguments that such techniques were too difficult to use for portfolio construct no longer hold with our modern computer techniques. It is not clear how the centre of the distribution should be defined, as zero, or the mean return, or the zero-risk interest return. The mean return is probably easiest to justify but the choice is not critical.

Decreasing the content of the upper part of the distributions reduces uncertainty rather than risk. This may be of importance if assets are being ‘sold short’ or with various types of derivatives.

Rather than just stay with mean and variance, one could move to inclusion and consideration of higher moments, such as measures of skewness and kurtosis. This has already been considered in the literature, e.g. El Bab Siri and Zakoian (2001).

The topics considered in this section have largely considered risk from the eyes of statisticians or theoretical economists rather than from the perspective of actual investors. The approaches taken are, possibly, too academic to be helpful, it is time to turn to possibly more realistic topics.