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In 1993 the Bank of International Settlement's (BIS) members met in Basle, Switzerland, and agreed that member banks and other financial institutions should hold in reserve enough capital to cover at least 10 days of potential losses. The measurement of these potential losses was to be based on the 95% 10-day VaR calculation.
The resulting Basle I accord came into force in 1998 and the subsequent Basle II, which was ratified in 2003, added the additional requirement to that operational risks also be quantified by 2008.
To ensure that this measure was sufficiently conservative, it was also multiplied by a scaling factor ranging from one to three times. Financial institutions, including SA's banks, are required to report their overall risk exposure on that basis.
VaR allows regulators and bank presidents to put a single number on their worst-case scenario and to plan for it accordingly. The appeal of such a single risk number is obvious but, as with all one-size-fits-all measures, potentially misleading for the uninformed.
VaR measures the expected loss of a portfolio over a specified holding period (HPR) for a set level of probability of confidence level (CL).
For example, a portfolio of R1m with an average expected return of 13%/year and a standard deviation of 20% would have a 10-day VaR of R60,370 at the 95th percentile. That means your losses over any 10-day period should only exceed R60,370 5% of the time or roughly one every year.
How big the losses actually are beyond that threshold, or during 5% of the time, is measured by a different risk measure known as the conditional VaR or expected shortfall.
Calculating Value at Risk
There are three main methods of calculating Value at Risk, they are:
1. The historical or empirical method 2. The parametric or analytic method 3. The simulation or Monte-Carlo method.
The historical method
The historical method involves simply taking the empirical P/L history and ordering it. Suppose we have 100 observations of the returns of our portfolio.
Using a spreadsheet we would simply order the returns from largest to smallest.
The Value at Risk for the 95th percentile would then be the 6th largest loss.
The advantage of the historical method is that it requires no assumption to be made about the nature or shape of the distribution of returns. The disadvantage is that we are thus implicitly assuming that the shape of future returns will be the same as those of the past.
For this to be statistically likely we need to ensure that we have a sufficient number of observations and that they're representative of all possible states of the portfolio (ie, incorporate data from both bull and bear markets).
Using monthly data, statistical certainty would typically require 34 years of data before we could be comfortable with that assumption. Since we seldom if ever have this much history, the empirical method isn't considered as accurate as either the parametric or simulation method.
The parametric or analytic method
The parametric or analytic method requires an assumption to be made regarding the statistical distribution (normal, log-normal, etc) from which the data is drawn. We can think of parametric approaches as fitting curves through the data and then reading off the VaR from the fitted curve.
The attraction of parametric VaR is that relatively little information is needed to compute it. The main weakness is that the distribution chosen may not accurately reflect all possible states of the market and may under- or overestimate the risk.
That problem's particularly acute when using VaR to assess the risk of asymmetric distributions, such as portfolios containing options and hedge funds. In such cases the higher statistical moments of skewness and kurtosis that contribute to more extreme losses ("fat tails") need to be taken into account. Fortunately, closed form formulas now exist for distributions, such as Student's t, the Extreme Value or Generalised Pareto distribution and for modifying the normal VaR to take account of excess skewness and kurtosis using the Cornish-Fisher expansion.
The Forsey-Sortino three-parameter log-normal distribution also gives a better fit to most financial time series.
So though some level of statistical sophistication is necessary, parametric methods exist for a wide variety of distributions.
The general form for calculating parametric VaR is:
Mean x HPR + (Z-Score x Std Dev x
SQRT(HPR))
Where:
Mean = Average expected return.
Std Dev = Standard deviation.
HPR = Holding period.
Z-Score = the number of standard deviations from the mean for a specified level of confidence, assuming a normal distribution.
The simulation or Monte Carlo method
The simulation or Monte Carlo method of calculating VaR has become increasingly popular in recent years due to the dramatic increase in the availability and power of desktop PCs. As the name implies, simulation VaR generates many thousand simulated returns drawn either from a parametric assumption regarding the shape of the distribution or, preferably, by re-sampling the empirical history and generating enough data to be statistically significant and then ordering them and reading off the desired percentile as in the historical calculation method.
All three methods have their place, with the Monte Carlo simulation, resampling or bootstrap methods probably being the best for the uninformed.
Particular care should be taken not to use the normal VaR calculation where the distribution is known or suspected to be asymmetric, as in the case of hedge funds and emerging markets.
Unfortunately, that's seldom done.
Another problem is the assumption by some calculations of a zero mean.
While that's generally appropriate for daily VaR calculations (the expected daily return of most markets being statistically insignificantly different from 0) it's certainly not the case for longer holding periods.
Perhaps the most serious problem with VaR is that it's not a mathematically fully coherent measure. Without getting too technical, that's because, excepting in the special case of the normal distribution, VaR doesn't satisfy the sub-additivity requirement for mathematical coherence. In other words, the sum of your component VaR's may be greater or less than that of the whole.
Apart from the obvious confusion that it may cause, it also results in the mean/VaR frontier of investments not always being a smooth convex shape.
That can cause problems when trying to find the optimal point, as the gradient point search methods used in most solver engines can get trapped in an SA minima or maxima and not always find the global maxima or minima you're looking for.
Used correctly, VaR can be an invaluable tool in putting a single number on the potential losses that might occur.
However, given the number of different calculation methods and widespread presence of "fat tails" in financial time series, great care should be taken to measure it correctly.
As with all quantitative measures it's also advisable to use experience and common sense. No amount of bootstrapping or simulating a small sample or data - such as the returns on the Nasdaq between 1998 and 2000 - will open your eyes to the true potential for loss.
Any sample period that's particularly short or which has an abnormally low standard deviation should be looked at twice.
It's also a good idea to stress test VaR calculations by looking at historical worst-case scenarios and using the correlation matrices from such periods to stress test your assumptions.
References: Value at Risk, Phillipe Jorion (1997); Coherent Measures of Risk, Artzner (1999); Conditional Value at Risk, Uryasev (2001).
Peter Urbani
PETER Urbani is a full-time risk consultant for KnowRisk Consulting. He was previously head of investment strategy at Fairheads Asset Managers and prior to that senior portfolio manager at Commercial Union Investment Management.
