Economic Capital (EC hereafter), generally accepted as the amount of capital that is required to absorb unexpected losses(UL hereafter) during a time horizon at a portfolio manager’s level of confidence, is becoming a common currency for all such types of risks into a single enterprise-wide measure of aggregated risk. Practitioners commonly employ Value-at-Risk (VaR) as the standard risk measure for calculations in most aspects of EC. However, VaR does not satisfy the criteria of coherent risk measure (Atzner, 1997, 1999). Specifically, it may not have the sub-additive property which requires that the combined VaR of two sub-portfolios should not be greater than the sum of their respective VaR treated separately. Moreover, the amount of EC calculated by VaR merely tells us that a firm will exhaust the available capital when the upper bound of loss is reached rather than how much the additional capital that can cushion loses that exceeds it. In other words, the tail risk will disturb the EC impact on the unexpected losses and will further affect the rate of return that the shareholders demand on the EC.
Therefore, one such coherent risk measure called expected shortfall (ES hereafter) or conditional expectations shed lights on the alternative measure for calculating the economic capital. This measure, based on the extreme value theory (EVT hereafter), concentrates on the tail behavior of loss distribution. The main limitation of the method is in the selection process of the threshold. If a high value of threshold is selected, less observation is left for the estimation of the parameters of the tail distribution function. Thus, the simulation technique such as the Monte Carlo simulations can be used to rich the tail region. However, on the other hand, the tail region of the loss distribution represents the high severity low frequency events. Calculating EC on this area tends to overestimate EC and will further affect the efficiency of EC allocation.
One purpose of this paper is to define a risk measure for EC in a perspective of “full range of risk”. To achieve this goal, we utilize the concept of spectral risk measure and the original meaning of EC. Spectral risk measure is likewise a coherent risk measure with a highlight on its consideration of a user’s risk aversion. This measure employs a weighting function to estimate the weighted averages of the quantiles of a certain loss distribution. Therefore the more risk aversion of the user, the more EC will be allocated or the UL. On the other hand, the risk lover tends to allocate less EC for the same UL. However, according to the original meaning of EC, over-allocating EC will affect the return of the capital while under-estimating EC will suffer from insolvency. A proper EC should keep a balance for the risk and return. In this background, only a risk neutral user tends to allocate a suitable amount. Considering the fact that ES is a tail-risk neutral risk measure, we extend the “tail” threshold to the point of expected loss of the loss distribution, and ES for this loss region represents a risk neutral measure in a perspective of “full range of risk”. Therefore, we can obtain the risk-neutral EC, which is the embedded EC for the loss distribution.
The rest of the paper is organized as follows: Section 2 presents the VaR-based method to calculate EC and discuses its limitations via samples. Section 3 introduces the EVT and conducts an empirical analysis to calculate EC based on ES model. Section 4 discuses the concept of spectral risk measure, by which we define the risk-neutral EC and likewise employ EVT models to estimate the EC in the risk-neutral perspective. Finally, section 5 offers the main conclusions of the paper.
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