3.5 Mark-to-Model Backtesting

A backtesting procedure compares the VaR prediction with the observed loss. In a mark-to-model backtesting the observed loss is determined by calculation of the present value before and after consideration of the actually observed risk factor changes. For $ t_0\le t\le t_1$ the present value at time $ t+h$ is calculated with the yield $ R(t+h,T-t)$, which is obtained from observed data for $ R_i(t+h)$ by linear interpolation, according to

$\displaystyle PV(t)={1\over\big\{{1+R(t+h,T-t)}\big\}^{T-t}}.$ (3.23)

This corresponds to a loss $ L(t)=PV(t)-PV(t+h)$, where, again, the shortening of the time to maturity is not taken into account.

The different frameworks for the VaR estimation can easily be integrated into the backtesting procedure. When we, e.g., only consider changes of the benchmark curve, $ R(t+h,T-t)$ in (3.23) is replaced with $ B(t+h,T-t)+S(t,T-t)$. On an average $ (1-\alpha)\cdot 100$ per cent of the observed losses in a given time interval should exceed the corresponding VaR (outliers). Thus, the percentage of observed losses is a measure for the predictive power of historical simulation.