13.10.1 Present Value-at-Risk

The approach is as follows:

1. For each scenario, compute the accrued dynamic present value by dividing the (scenario-specific) present value by the Stochastic Discount factor; we do not take the new business into account.
2. For each scenario, compute VaR as market value minus accrued dynamic present value.
3. Sort VaR in ascending order and output it along with its normalized ranking (that is, the ranking divided by the total number of scenarios).

   The normalized ranking is an unbiased estimator of the probability that loss of value is less than VaR, that is, each couple of values (normalized ranking, VaR) is a point on the loss probability distribution curve.

Illustration of the Approach

We have only 10 scenarios. Today's market value is $80.

Table 13-9 Scenarios for Present Value-at Risk

Scenario Number	Stochastic Discount Factor	Present Value	Accrued Dynamic Present Value	VaR
1	0.99	81.6	82.4	-2.4
2	0.98	83.1	84.8	-4.8
3	0.97	81.5	84	-4
4	0.965	80.1	83	-3
5	0.95	79.9	84.1	-4.1
6	0.95	79	83.2	-3.2
7	0.949	79.2	83.5	-3.5
8	0.948	78.3	82.6	-2.6
9	0.947	75.1	79.3	0.7
10	0.946	70.1	74.1	5.9

After sorting we have:

Table 13-10 Scenarios after sorting

Scenario Number	Probability	VaR
2	0.1	-4.8
5	0.2	-4.1
3	0.3	-4
7	0.4	-3.5
6	0.5	-3.2
4	0.6	-3
8	0.7	-2.6
1	0.8	-2.4
9	0.9	0.7
10	1	5.9