An Option Pricing Framework for Valuation of Football Players
Order Statistics Applied to Risk Management
RADU TUNARU
Economics, Finance and International Business
London Metropolitan University
London EC3N 2EY
UK
Abstract: - Calculating VaR is in essence an estimation problem but confidence intervals of point estimates are needed for assessing their accuracy. Risk management for complex portfolios may consider simultaneously two or more VaR confidence levels. The order statistics representing empirically 1% and 5% VaR estimates are not independent and therefore they ought to be studied jointly. Moreover, quantile-based risk analytics using L-estimators require inference of a weighted average of order statistics. In this paper the joint probability distribution of order statistics is used for developing confidence regions for VaR estimates .
Keywords: order statistics, multivariate joint distributions; Value-at-Risk, expected tail loss, expected median given quantile
1 Introduction
VaR is widely used in the financial sector as a measure for market risk in normal conditions. This number has a strong influence on bank
capital, some of the major implications of this estimation process being described in [7]. Marshall and Siegel [10] found great errors in the estimation methods used in the industry. In the light of severe market disruptions the issue of how reliable is the model used for market risk is very important.
In this paper we consider mainly parametric models, but distribution-free results are also discussed. Focusing on the parametric models does not mean a narrow restriction, the probability distributions that may be used in the context investigated below varying from the classical normal Gaussian distribution to more suitable distributions for modelling the tails of the unknown distribution of financial returns like the extreme value distribution used in [12] .
The estimation of VaR is a statistical exercise and the risk manager or quant analyst has to consider the precision or reliability of the estimates proposed. Although there is a plethora of
models described in [5], the literature on the confidence associated to those estimators is
sparse.
In [8] the uncertainty associated with VaR models is considered but the main concern was model risk. While this is a valid concern that should not be overlooked by a practitioner, it is not the uncertainty that is dealt with in this paper.
In [9] it is suggested that it may be very hard to
determine statistically the accuracy of VaR estimates. This seminal paper was followed in [3] employing order statistics for assessing the VaR accuracy. Simulation methods can also be used as in [4] to build confidence intervals for VaR
estimates but only in some special cases linked to the normal distribution.
The rest of the paper is organized as follows. In the next section we outline how a player's performance can be measured with a performance index such as the Carling Opta Index. In sections 3 and 4 we develop the models and in section 5 we conclude.
2 Joint Probability Distributions for Order Statistics
From the probabilistic point of view the sample of our profit and loss (P/L) values is a random sample [pic] with cumulative distribution function [pic]
(1)
where the last equality follows from the i.i.d assumptions. For empirical calculations of VaR the sample re-ordered [pic] is of interest because the VaR at level ( equals the negative of the v-th lowest value, where v = 100(+1, which translates for empirical work based on the sample above into the negative of the v-th lowest value, where v = n(+1, or [pic].
If [pic] is the cumulative distribution function of the i-th order statistic then it is not difficult to see that [pic] and[pic]. As shown in [1] and used by Dowd ([3]), making use of the binomial distribution, it follows that
[pic]
(2)
Thus the cdf of any order statistic is
[pic](3)
In the following we shall denote [pic] by F(y), for simplicity. It can also be shown that
[pic] (4)
where [pic]
is the incomplete beta function and ((a,b) is the beta function. This helps to calculate the pdf function for those distributions that are absolute continuous with respect to a dominant probability
The pdf of the j-th order statistics is
[pic] (5)
where [pic].
The order statistics methodology can be easily implemented with the help of a good computational package such as Matlab. Table 1 contains the VaR and ETL as estimated via the
order statistics method for the normal distribution and the t-distribution for the series of P/L, at various confidence levels and sample sizes. In addition, the confidence intervals determined
as the 0.025% and 0.975% percentiles of the distribution of each risk measure are also included. For each sample size, the confidence intervals for both VaR and ETL are increasing with the increase in the level of confidence. Similar result are obtained for larger sample sizes and other distributions. For each level of confidence, the confidence intervals narrow with the increase in the sample size.
2.1 Distribution-free confidence intervals for VaR
From a practical point of view it seems safe to assume that the cdf F is strictly increasing. For any [pic] the equation
[pic] (6)
has a unique solution. This solution, the quantile of order (, refers to the entire population and it is
denoted by [pic]. The 95% VaR is[pic].
The order statistics provide a distribution-free confidence interval for the population quantiles. In [13] it is shown that
[pic](7)
This powerful result allows the construction of distribution-free confidence intervals for VaR. For given sample size n and VaR level (, there are many combinations of i and j that makes the quantity in (7) larger or equal to 1-(, the confidence level desired.
2.2 Bivariate order statistics
Our main interest lies on the bivariate joint
distribution of two order statistics that will provide the confidence regions (two-dimensional sets) for pairs of VaR estimates. For example, the confidence regions for 1%-VaR and 5%-VaR are recovered from the bivariate joint distribution of
[pic] where [pic] . This distribution is fully characterized by
[pic] (8)
with [pic]. The probability on the right side of equation(8) can be interpreted as the probability that at least i values from the entire sample [pic] are not greater than x and at least j values from the same sample are not greater than y. Thus
[pic]As in the univariate case developed in [1] it
follows that
[pic] (9)
for any x ................
................
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