B. Hathi 29th Oct 2010.
Estimating tail risks (or worst case scenarios) is an important part of risk management for any company’s middle office or Treasury function. We can estimate the tail risks for any [randomly] varying parameter by using theoretical concepts of the well established continuous distributions. A starting point will be a raw data set, from which we construct a histogram and fit a distribution – which may be Normal (or Gaussian) if you want a quick, rough estimate or may be selected by looking at the results of goodness of fit tests such as ‘chi squared’ and ‘Kolmogorov-Smirnov’ tests, or simply by looking at the best fit by eye. In any case, once we are happy with our [chosen] fitted distribution, getting tail risks is easy: all we need to do is to infer quantiles from a cumulative distribution function CDF (or more precisely, an inverse CDF also known as ‘percent point function’ PPF). Typically, a lower quantile may be chosen to give a 5% probability (i.e. an integrated probability of finding the value between 0 to 5%) and an upper [5%] quantile of 95% (i.e. probability of finding the value between 95% to 100%). The Normal distribution is probably the most studied of the distributions and many of its properties are known, e.g. the 5% quantile is approximately equal to average +/- 1.65std (standard deviation); 1% quantile = average +/- 2std. This means a risk practitioner need not go through the hassle of evaluating CDF for a Normally distributed parameter – he/she can simply get the numbers by using such rules of thumb.
Let’s take Gbp/Usd data from the Fedral Reserve’s website as an example (please note, the last few days values are collected from other sources as the Fed only updates its database at weekly intervals.) Figure 1 (click on it to enlarge) shows the Gbp/Usd sample data set from 1971 to present (28th Oct 2010) in Excel, though I’ve imported the data in Python and done my calculations using Python 2.6.
Figure 1. GBP/Usd raw data set from 4th Jan 1971 to present, only the values highlighted in yellow are processed.
From the one-column of daily values of Gbp/Usd, we can calculate the average (or the mean) value to be 1.7824 (call it gbpAve) and its standard deviation to be 0.3192 (call it gbpStd) . Using these two numbers, gbpAve and gbpStd, and the above rule of thumb, we can estimate the lower 5% quantile for the Pound Sterling to be:
gbpAve - 1.65*gbpStd = 1.2556
and the upper 5% quantile to be:
gbpAve + 1.65*gbpStd = 2.3091
What we are saying is that the probability of finding the Gbp/Usd below 1.26 is 5% and similarly, 95% of the time the value of Gbp/Usd will be under 2.31 , assuming this FX pair (Gbp/Usd) to be Normally distributed. Sometimes these quantiles are referred to as ‘Value at Risk’ (VaR): in this case the VaR at 95% confidence level is ~ 1.26-2.31.
However, for a time-varying (or time series) quantity such as the Gbp/Usd, it is not sufficient to assign the upper and lower bounds as the value is not likely to jump from one end to the other around the expected (or mean) value. For example, the probability of finding the Pound 5% below its average value (i.e. below 1.74 according to the ‘PPF’, note this is not the same as gbpAve*0.95) is the same as the probability of finding the Pound 5% above (i.e. above 1.82), yet we don’t expect a jump from, say 1.74 to 1.82 in a day or two. Most people are interested in knowing how much could a current value (such as the Gbp/Usd) drift by, in say a year’s time? There is another rule of thumb for propagating a time series parameter over a given period (in number of days, in our case since the data set is sampled at 1 point per day) . According to the theory of ‘Random Walk’ (which is also based on Normal distribution), we can say that the variation in the value of the Pound will be:
Std * square root(days ahead).
Note, for this case we cannot use the above value of standard deviation, we have to first calculate the daily change and then its Std. If we look at the daily changes in Pound’s value, then its Std = 0.01037 (and its mean is close to zero, mean = - 8.1 E-5).
So in 21 days (which roughly equals a trading month), the value of Pound could be higher or lower by Std*sqrt(21) = 0.0475 . On 28th October we know that the Pound was worth 1.587 US dollars, so we can estimate that its value could drift by 1.587 +/- 0.0475 (i.e. between 1.635 and 1.540). For a full trading year (=252 days), we can say that the Pound is likely to change by +/- 0.165 (i.e. between 1.422 and 1.752). Another concept is to look at ‘Daily Earnings at Risk’ (DEaR), again this is simply a manifestation of the standard deviation. In our case the Pound’s DEaR, at 95% confidence level, is 0.0171, or in a more familiar [traders] language 171 pips.
It is possible to carry out more sophisticated distribution fits to our data.
Figure 2 shows probably a slightly tighter (or closer) fit in the form of a ‘Betaprime’ distribution, while Figure 3 shows a Cauchy distribution that gives slightly higher extreme tail risks. Gumbel Left (or Gumbel Min) is often useful to find the worst case lower quantile tail risk, while its counterpart, Gumbel Right (or Gumbel Max) is useful for worst case upper quantile risk.
Figure 2. Shows a 'Betaprime fit' (blue line) to the Gbp/Usd data and compares with the Normal distribution (red hashes).
Figure 3. Shows a 'Cauchy fit' (blue line) to the Gbp/Usd data, which has slightly broader (or fatter) tails compared to the Normal distribution (red hashes).
I’ve now developed software to fully reconstruct a histogram without reducing any intermediate outputs. For example, if we carry out 50 ‘guesses’ for daily value, then a 3-day forward prediction will consist of histogram of 50^3 = 125,000 guesses. Unfortunately due to the recursive nature of this program and the exponentially growing memory requirement, this method can only forecast a few days forward.
Source |
The Currency Forecasting Blog (CFB) | http://currency-forecasting.blogspot.com/2010/10/risky-business-calculating-fx-currency.html
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