Folklore (if it is fair to attribute as folklore that which only dates back five years) tells us that VaR was developed to provide a single number which could encapsulate information about the risk in a portfolio, could be calculated rapidly (by 4:15), _and_ could communicate that information to nontechncial senior managers. Tall order, and not one that could be delivered upon without compromises.

Modern Portfolio Theory ("MPT"), as taught in business schools, tells us that the risk in a portfolio can be proxied by the portfolio standard deviation, a measure of spread in a distribution. That is, standard deviation is all you need to know in order to (1) encapsulate all the information about risk that is relevant, and (2) construct risk-based rules for optimal risk "management" decisions. [The more technically proficient will please forgive my playing somewhat fast and loose with the theory in the interests of clarity.] Strangely, when applied to the quest for the Holy Scale, standard deviation loses its appeal found in MPT. First, managers think of risk in terms of dollars of loss, whereas standard deviation defines risk in terms of deviations (!), either above or below, expected return and is therefore not intuitive. Second, in trading portfolios deviations of a given amount below expected return do not occur with the same likelihood as deviations above, as a result of positions in options and option-like instruments, whereas the use of standard deviation for risk management assumes symmetry.

An alternative measure of risk was therefore required. Why not measure the spread of returns, then, by estimating the loss associated with a given, small probability of occurrence. Higher spread, or risk, should mean a higher loss at the given probability. Then senior management can be told that there is 1 in 100, say, chance of losing X dollars over the holding period. Not only is this intuitively appealing, but it's easy to show that when returns are normally distributed (symmetric), the information conveyed is exactly the same as were standard deviation employed, it's just that the scale is different. This approach can be consistent with MPT. It seems, then that perhaps the Holy Scale has been found in VaR.

It's perhaps too easy to criticise efforts to implement the VaR concept. It takes some courage to venture into unfamilar terrain and missteps are inevitable. The VaR paradigm is still evolving (as is that of financial risk management in general) and experimentation should be encouraged. To speak of "best practices" is surely premature.

The general approaches to VaR computation have fallen into three classes called parametric, historical simulation, and Monte Carlo. Parametric VaR is most closely tied to MPT, as the VaR is expressed as a multiple of the standard deviation of the portfolio's return. Historical simulation expresses the distribution of portfolio returns as a bar chart or histogram of hypothetical returns. Each hypothetical return is calculated as that which would be earned on today's portfolio if a day in the history of market rates and prices were to repeat itself. The VaR then is read from this histogram. Monte Carlo also expresses returns as a histogram of hypothetical returns. In this case the hypothetical returns are obtained by choosing at random from a given distribution of price and rate changes estimated with historical data. Each of these approaches have strengths and weaknesses.

The parametric approach has as its principal virtue speed in computation. The quality of the VaR estimate degrades with portfolios of nonlinear instruments. Departures from normality in the portfolio return distribution also represent a problem for the parametric approach. Historical simulation (my personal favorite) is free from distributional assumptions, but requires the portfolio be revalued once for every day in the historical sample period. Because the histogram from which the VaR is estimated is calculated using actual historical market price changes, the range of portfolio value changes possible is limited. Monte Carlo VaR is not limited by price changes observed in the sample period, because revaluations are based on sampling from an estimated distribution of price changes. Monte Carlo usually involves many more repricings of the portfolio than historical simulation and is therefore the most expensive and time consuming approach.

It seems that VaR is being used for just about every need; risk reporting, risk limits, regulatory capital, internal capital allocation and performance measurement. Yet, VaR is not the answer for all risk management challenges.

No theory exists to show that VaR is the appropriate measure upon which to build optimal decision rules. VaR does not measure "event" (e.g., market crash) risk. That is why portfolio stress tests are recommended to supplement VaR. VaR does not readily capture liquidity differences among instruments. That is why limits on both tenors and option greeks are still useful. VaR doesn't readily capture model risks, which is why model reserves are also necessary.

Because VaR does not capture all relevant information about market risk, its best use is as a tool in the hands of a good risk manager. Nevertheless, VaR is a very promising tool; one that will continue to evolve rapidly because of the intense interest in it by practitioners, regulators and academics.

Page authored by: Barry Schachter.
(Courtesy: www.gloriamundi.org. Reproduced with permission. All trademarks belong to the author)

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