  In this paper we discuss a general methodology to compute the market risk
measure over long time horizons and at extreme percentiles, which are the
typical conditions needed for estimating Economic Capital. The proposed
approach extends the usual market-risk measure, ie, Value-at-Risk (VaR) at a
short-term horizon and 99% confidence level, by properly applying a scaling on
the short-term Profit-and-Loss (P&L) distribution. Besides the standard
square-root-of-time scaling, based on normality assumptions, we consider two
leptokurtic probability density function classes for fitting empirical P&L
datasets and derive accurately their scaling behaviour in light of the Central
Limit Theorem, interpreting time scaling as a convolution problem. Our analyses
result in a range of possible VaR-scaling approaches depending on the
distribution providing the best fit to empirical data, the desired percentile
level and the time horizon of the Economic Capital calculation. After assessing
the different approaches on a test equity trading portfolio, it emerges that
the choice of the VaR-scaling approach can affect substantially the Economic
Capital calculation. In particular, the use of a convolution-based approach
could lead to significantly larger risk measures (by up to a factor of four)
than those calculated using Normal assumptions on the P&L distribution.
