Appendix 4: Valuation-date issues

We noted above that most VaR methodologies do not explicitly deal with the entire
evolution of the risk factors and the valuation function between the anchor date and
target date. Rather they just evaluate the valuation function as of the anchor date at
perturbed values of the risk factors for the anchor date. This simplification, while
done for strong practical reasons, can lead to surprising results.
Need for future valuation functions
Consider, for example, a cashflow in amount USD N in the base currency payable
one year after the anchor date. We would write the valuation function with respect
to the USD base currency in terms of our risk factors as
vt0óUSDND1(t0)
where D1(t0) denotes the 1-year discount factor at time t0. This equation was the
starting point for the analysis in the example on page 191, where we calculated VaR
for a 1-day time horizon. Scaling this example to a 1-year time horizon, as discussed
on page 201 would give a VaR approximately 16 times as large.
But this is crazy! The value of the bond in one year is N, so there is no risk at all
and VaR should be equal to zero. What went wrong? The problem is that we based our calculation on the valuation function as of the anchor date. But at the target
date, the valuation function is given simply by
vTóND0(T )
óN
Thus the formula that gives the value of the instrument in terms of risk factors
changes over time. This is due to the fact that our risk factors are fixed-maturity
assets. (Changing to risk factors with fixed-date assets is not a good solution to this
problem as these assets will not be statistically stationary, indeed they will drift to a
known value.)
The first idea that comes to mind to fix this problem is to use the spot valuation
function as of the target date instead of the anchor date. While this would fix the
problem for our 1-year cashflow, it causes other problems. For example, consider an
FRA that resets between the anchor and target dates and pays after the target date.
The value of this instrument as of the target date will depend on the risk-factor vector
between the anchor and target date and this dependence will not be captured by the
spot valuation function at the target date. For another example, consider a 6-month
cashflow. In most systems, advancing the valuation date by a year and computing
the value of this cashflow would simply return zero. It therefore appears that in order
to properly handle these cases, the entire trajectory of the risk factors and the
portfolio valuation function needs to be taken into account, i.e. we need to work with
future valuation functions are described in Appendix 1.