On the White Board - April 2010

Apr 15, 2010

Credit Valuation Adjustment

Credit Valuation Adjustment (CVA) is by definition the difference between the risk-free portfolio value and the true portfolio value that takes into account the counterparty's default. In other words, CVA is the market value of the counterparty credit risk. Accurate pricing of the credit risk is an important determinant of mark to market earnings, and is the first line of defense in credit risk management.

For instance, the importance of accurately pricing credit risk may be briefly highlighted by the following points:

• ensure the firm's long-term soundness
• help reducing potential hidden future losses
• increase earnings and the shareholders' equity
• reduce the regulatory capital (for banks) and boost the return on the regulatory capital
• create more trading opportunities in the market (expanding derivative business)
• dynamically hedge the counterparty risk by buying credit protection on the counterparty

Some banks have set up internal credit risk trading desks charged with the responsibility for pricing, hedging, and bearing the credit risk components of all counterparty positions.

In practice, CVA can be measured in two ways: unilateral and bilateral. Under the unilateral approach, it is assumed that the valuation entity party is default-free. In this case CVA is the present value of future losses due to the counterparty's potential default. Unilateral CVA is also commonly called credit charge. Bilateral CVA takes into account that both parties can default. It is thus symmetric and results in an objective fair value calculation. Bilateral CVA is realized as the sum of credit charge and credit benefit, where the latter corresponds to the CVA seen from the counterparty point of view. In some relevant literature, the credit benefit is also referred as DVA, namely Debit Valuation Adjustment.

Let us assume that a bank has a portfolio of derivative contracts with a counterparty. We will denote the bank's credit exposure to the counterparty at any future time t by CE(t). This exposure takes into account the collateral arrangements, namely all netting and margin agreements, between the bank and the counterparty. If the counterparty defaults, the bank will be able to recover a constant fraction of exposure that we will denote by R. Denoting the time of counterparty default by τ, we can write the loss as

where T is the maturity. The (unilateral) CVA is the risk-neutral expectation of the discounted loss

where E is the risk neutral conditional expectation, D(t_a ,t) is discount factor at analysis date t_a , and PD(t_a ,t) is the risk-neutral probability of default between [t_a ,t].

The dependency between the counterparty default process and the exposure value is called wrong way/right way risk. In particular, the risk is wrong (right) way if the exposure tends to increase (decrease) when the counterparty credit quality worsens. Assuming independence between the exposure and the default process (counterparty credit quality), the equation above can be written as:

where EE is the risk-neutral expected exposure given by EE(t)= E[CE (t)].

Since the expected exposure is calculated at a fixed set of horizons h₁ , …, h_m , the above integral can be approximated by a finite sum. This approximation leads to the following crucial result:

The expected credit exposures EE(h_i) corresponds to the expected value on the distribution of credit exposures at horizon h_i and PD(t_a , h_i-1 ,h_i) is the time t_a (analysis date) probability of default between time h_i-1 and h_i.

The credit benefit can be expressed in an analogous way.

As mentioned before, the present value of the portfolio (from the bank point of view) is given by

In the bilateral approach CVA=credit charge + credit benefit. For the legal entities that are less risky than their counterparties, we will, broadly speaking, observe a credit charge (in absolute value) greater than the credit benefit.

A sometimes counter-intuitive implication with the bilateral CVA is that all things being equal, the present value from the bank’s point of view increases when its own spread widens. It is a point of debate to assess if this apparent profit can be monetized and who benefits from it.