| Математика
Key words: CVA, credit valuation adjustment, credit value adjustment, DVA, debt valuation adjustment, debt value adjustment

After the credit crunch of 2008 a significant amount of global attention was focused on the methodology of estimating counterparty credit risk (CCR) and utilizing it to adjust the P&L in order to reflect this risk.
Today CVA is a considerable topic described in the Basel III capital rules and it is an essential part of the IFRS 13 “Fair Value Management”.
We will analyze the purpose and the math behind CVA in this article.

The key acronyms we will be using are CVA and DVA. But what are CVA and DVA anyway?
After reading the first paragraph, it might not be hard to guess that CVA means Credit Valuation Adjustment, whereas DVA is Debit or Debt Valuation Adjustment, which depends on the entity’s own credit quality.
There is another type of adjustment that is also important in this topic – FVA (Funding Valuation Adjustment). Together all three compose the CCR.

CCR occurs primarily whenever an entity trades OTC derivatives (exchange-traded instruments have null CCR associated with them due to central counterparty regulations), or conducts securities financing transactions (for example, REPO).
In order to cope with the arising CCR the institutional participant is required to hold regulatory capital to manage the mark-to-market credit risk fluctuations.
Estimation of CCR has been a hot topic for debate because until recent years there was no widely accepted method of calculation.
One of the approaches to advanced CVA risk capital charge is the Internal Model Method (IMM).
It is focused on calculating the exposure at default (EAD) and relies on a scaling factor (α) which depends on the correlation between parties and external conditions (i.e. state of economy, model risk, etc.).
IMM requires a regulatory approval, a well-documented justification and certain conditions need to be met in order to be implemented.
A lot of banks pursue the IMM because of the capital savings and reputational benefit of the approval. We will look into the standardized approach to CVA.

The formulae for calculating the CVA and DVA described in Basel III are given below:

The above equations have several notations that are familiar to us from previous articles: LGD (Loss Given Default) is simply 1 – Recovery Rate; and PD (Probability of Default).
The subscript d is used for those values that are calculated for the bank’s own credit position.
And of course there are 2 more functions that remain unknown to us: EPE (Expected Positive Exposure) and ENE (Expected Negative Exposure):

where B(t) is money market account at time t. PE (positive exposure) and NE (negative exposure) are given by the maximum of either 0 or the differences between the risk free value of trades between counterparties V(t) and the value of posted collateral against the trades C(t):

As we can see the model depends on default probabilities.
Calibration of the standardized model requires the intensities to be bootstrapped from credit curves or other proxies on a regular basis.
Due to the fact that credit spreads can be volatile, the entity is required to hold regulatory capital against CVA VaR.
Management of these adjustments on a daily basis normally forces banks to have CVA trading desks.
These desks would normally operate by the following scheme (see chart below).

The CVA desk sells a CVA option to the swap trader, who has a credit risk exposure in an OTC derivative position. And on the other side of the deal, the CVA desk monitors the bank’s credit exposure and hedges the position by buying protection in the CDS market (can be single-name or index), going short in the bond market or delta-hedging using options. Certain side-effects may arise as a result of active CVA VaR management. This can be related to “wrong-way risk”, which occurs when exposure to counterparty is adversely correlated with its credit quality. It can lead to having a negative gamma or negative cross-gamma. And in case of hedging the CVA by purchasing protection in the CDS market, which is obviously the most popular, there can be a situation where all hedgers are positioned the same way. Actually the latter has occurred during the past couple of years: it can be observed in the values of credit spreads as they have been growing higher since 2008, whereas the swap rates have been indirectly lowered (negative credit gamma).

Nevertheless, should the CVA always be treated as a charge or can it be assessed as a gain? How are the main sensitivities that the CVA desks monitor are being hedged (exposure DV01, the credit SPV01 and gamma)? We will look deeper into these subjects in the next part on understanding CVA.