| Математика

Key words: Asset and Liability CVA, PD, LGD, EAD, DVA, EPE, ENE


CVA is the value of the expected losses from counterparty defaulting and can be formulated as well as:

СVA = Present Value of (PD * EAD * LGD)

CVA – adjustment to the price of derivative to account for counterparty credit risk. It is a price not a risk measure.
PD – default probability or how likely is counterparty to default
LGD – loss given default after recovery
EAD – exposure at default
Unilateral CVA – only the counterparty can default.

As banks themselves have become risky, counterparty risk must be analyzed from the bilateral perspective. Bilateral CVA is the adjustment which reflects the credit risks faced by both counterparties.
Bilateral CVA is the sum of the Asset and Liability CVA. Liability CVA is also known as debit valuation adjustment DVA. The DVA reduces the value of a liability derivative.

EAD can be predicted by EPE/ENE profiles.

EPE – Expected positive exposure
ENE – Expected negative exposure
The EPE and ENE profiles are central to the calculation of CVA and DVA. Simple CVA and DVA approximation formulas can be written as:

CVA = Present Value of (PD1 * EPE * LGD)
DVA = Present Value of (PD2 * ENE * LGD)

As can be seen, the mechanics of calculating CVA and DVA are almost identical but incorporating different PDs.
If there is a derivative deal between a Bank and a Corporate, the CVA of the Bank is the DVA of the Corporate and vice versa.

Working example. Suppose there is a 1-year plain vanilla interest rate swap with $1 billion nominal. Bank A pays a fixed rate of 0.6% and receive LIBOR3M from Counterparty_1. Day count convention is 30/360 basis.
Bank A valued the swap before CVA and DVA and it was -$875 262.

Bank A is expected to pay $874 453 in the first period, $249 532 in the second period and to receive $248 723 in the fourth period. Therefore, the Counterparty_1 is exposed to Bank’s A credit risk in the first two periods. The Bank A is exposed to default of Counterparty_1 in the fourth period.

CVA/DVA calculation is divided into the following steps:

Step №1. Calculate the present value (PV) of the EAD at each 3-months period.
Step №2. Calculate PD and LGD. For simplicity purposes, PD and LGD are taken as theoretical values.
More details about PD calculation can be found in the article Probabilities of Default.

In this example, it is assumed that forward rates will behave as expected as of 01.01.2016. Calculation of PV of the EAD for each period consists of defining Mark to market of the swap. It is a simple procedure. More details can be found in the CVADVA file

The end exposure as of 31.03.2016 is as follows:

The end exposure as of 30.06.2016 is as follows:

The end exposure as of 30.09.2016 is as follows:

Taking into account PD, LGD and EAD (to get calculation of EAD in the field «Bucket start» see Excel file) the final PV of expected loss (EL) can be calculated for each time period. The total sum representing CVA/DVA adjustment of $5 691.

The first Bucket_1 amount means that Counterparty_1 is exposed to the Bank’s A credit risk. Therefore, the PD and LGD of Bank A are taken into account. The amounts in the first and the second Buckets represent DVA amount.

The third Bucket_3 amount means that the Bank A is exposed to Counterparty_1 credit risk. That’s why the PD and LGD of Counterparty_1 are taken into account. The amounts in the third and the fourth Buckets represent CVA amount.