Value-at-Risk Based on Time-Varying Risk Tolerance Level *

The conventional judgement-based method for fixing the risk tolerance level in the Value-at-Risk (VaR) model might be a suboptimal method, because the procedure induces the possibility of bias in risk measurement. Conversely, a superior risk management practice might be one, where input parameters are determined by a quantitative process which is “non-subjective to the risk modeller’s preferences”. Based on this insight, we have improved on the VaR model. Our model allows time variation of the risk tolerance level and so is suitable for scenario-wise risk analysis.


Introduction
A class of risk measures, which are commonly referred to as "tail-related risk measures" in the economic literature, is based on basics of fixing ex-ante a risk tolerance level.Value-at-Risk is a common example of this class.Risk tolerance is the level of risk that an investor is willing to take.But, gauging risk appetite accurately can be a tricky task.In practice, the risk tolerance level is generally decided by judgement/or perception by a risk manager or a risk management committee or, in certain cases, an external regulatory body.For this purpose, it has been a common practice to follow recommendations by the BASEL committee of banking supervision.At present, BASEL guidelines are 99% and 99.9% confidence level for Value-at-Risk (VaR) and 97.5% confidence level for Expected Shortfall (ES) [1].Most probably, those recommendations are drawn on the basis of country-wise experiences of analysing large set of historical data.Al- ternatively, in certain cases, the risk modeller adopts commonly used percentages viz.99%, 95% and 90% for this purpose.Majumder [1], however, documented evidence from various developed and emerging equity markets of those incidents where a minor change in the risk tolerance level translated into a large difference in VaR.Nevertheless, such instances are not uncommon in financial markets.Similar observations were documented by Degennaro [2] who formed examples to establish that non-cooperative choices of the risk tolerance level by two investors were resulting in a substantial variation in their VaR estimates.Therefore, in many occasions, the risk modeller's preferences on the risk tolerance level could have large impacts on the tail measure.When those preferences are biased, being over concerned to the high volatile period/or stress or due to any other reason, the bias would be transfused into the tail measure.In this approach, the risk tolerance level, which was decided ex-ante during turbulence, maybe appropriate for the turbulent period.However, the same could be suboptimal for quiet periods.Logically, it is extremely difficult to get a risk tolerance level which is suited uniformly across scenarios and this is perhaps a reason for model risk in the conventional approach.
In an alternative approach, the present paper proposes that the risk tolerance level ought not to be pre-assigned, but may be determined by the model itself.In this framework, this parameter may vary with the shape of the loss distribution.
One way to determine the same might be using the Pickands-Balkema-de Haan theorem which essentially says that, for a wide class of distributions, losses which exceed the high enough threshold follow the generalized Pareto distribution (GPD) [3] [4].Using this theorem, it is easy to establish that the extreme right tail part of a distribution asymptotically converges to the tail of a generalised Pareto distribution (GPD).This hypothesis reveals that we can always find a region in the extreme right tail of the loss distribution, for which the equivalent region from a suitable GPD is available.Therefore, there exists a threshold, data above which shows generalized Pareto behavior.The threshold would essentially be reasonably large to cover all events which are "extreme" in nature.Naturally, events belonging to the rest of the distribution are "normal" or "non-extreme" in nature.The procedure gives us the opportunity to estimate simultaneously the tail size and the starting point of the tail.In other words, it allows simultaneous estimation of VaR and the risk tolerance level.The rest of the paper is organized as follows: Section 2 describes the model.Section 3 provides empirical findings and Section 4 concludes.

Behaviour of Losses Exceeding a High Threshold
Suppose 1 2 , , , n x x x are n independent realizations from a random variable (X) representing the loss with distribution function ) (truncated at the point u) can be defined as: Y F , we can define the distribution function of the excess over a high threshold u: Balkema and de Haan [3] and Pickands [4] showed that, for a large class of distributions, the generalised Pareto distribution (GPD) is the limiting distribution for the distribution of the excess, as the threshold (u) tends to the right endpoint.According to this theorem, we can find a positive measurable function where the distribution function of a two parameter generalised Pareto distribution with the shape parameter ( ξ ), and scale parameter ( ( ) ) has the following representation: and only if F belongs to the maximum domain of attraction of the generalised extreme value (GEV) distribution (H) [6].The equivalent representation of (2) could be in terms three parameter GPD: for 0 x u − ≥ , the distribution function of the three parameter GPD ( can be expressed as the limiting distribution function of the excess. ( ) with shape parameter ( ξ ), location parameter (u) and scale parameter ( σ ) has the following representation.
( ) This representation would provide us a theoretical ground to claim that there exists a threshold, the data above which would have generalized Pareto be haviour.

Identifying the Tail Region
Equations ( 1) and ( 2 level.Our empirical study based on S & P 500 composite index reveals that the tail risk of the loss distribution is well captured by the new risk measure in the normal as well as in the stress scenarios.The significance of the research is twofold: a) reduction of bias by minimising the scope of human intervention in risk measurement which is of practical as well as of social significance and b) gauging risk appetite methodically which is of academic significance.The approach may widen the applicability of tail-related risk models in institutional and regulatory policymaking.At this stage, however, it is not possible to provide the method for backtesting the new VaR model.This might be the topic for future research.

F
x with a finite or in- finite right endpoint (x 0 ).We are interested in investigating the behavior of this distribution exceeding a high threshold (u).In the line of Hogg and Klugman

Table 1 .
) suggest that for a sufficiently high threshold, it can be A comparison between VaR and VaR N-S based on S & P 500 Composite Index.VaR and VaR N-S are average based on 50 estimates.The standard error of the estimate is provided in the parenthesis.Data Source: Data Stream. Note: