The Value-at-Risk (VaR) is probably the most known measure for quantifying and controlling the risk of a portfolio. The establishment of the VaR is of central importance for a credit institute, since it is the basis for a regulatory notification technique and for required equity investments. The description of risk is done with the help of an ``internal model'', whose job is to reflect the market risk of portfolios and similar risky investments over time. This often occurs though the choice of suitable portfolios of a specific risk factor, i.e., through principal components analysis (Chapter 19). With risks from option trading a linear transformation is often applied using the ``Greeks'' (Chapter 6).
The objective parameter in the model is the probability forecasts
of portfolio changes over a given time horizon. Whether the model
and its technical application correctly identify the essential
aspects of the risk, remains to be checked. The backtesting
procedure serves to evaluate the quality of the forecast of a risk
model in that it compares the actual results to those generated
with the VaR model. For this the daily VaR estimates are compared
to the results from hypothetical trading that are holden from the
end-of-day position to the end of the next day, the so called
``clean backtesting''. The concept of clean backtesting is
differentiated from that of ``mark-to-market'' profit and loss
(``dirty '') analyzes in which intra-day changes are also
observed. In judging the quality of the forecast of a risk model
it is advisable to concentrate on the clean backtesting.
The interest of an institute in a correct VaR calculation can be traced back to a rule of equity investing, which we will briefly describe here. Since 1997 (modification of the Basle market risk paper) institutes have been allowed to replicate specific risks using internal models. Included here under specific risks are those associated with daily price developments (``residual risks'') and others realized from rare occurrences (``event risks'' such as rating changes). Models that only consider residual risks are called ``surcharge model'', and those that consider event risks are called ``non-surcharge model''. For calculating capital investments the institutes have to use the following formula, Graumert and Stahl (2001):