Valuing derivatives used to be much simpler than it is today. For example, an interest rate swap could be valued by knowing nothing more than forward LIBOR rates. An interest rate cap could be valued by modeling the LIBOR short rate. Now, as explained in an earlier blog, it is necessary to worry about the behavior of OIS rates as well as LIBOR because OIS is generally accepted as being the correct discount rate for fully collateralized transactions.

Valuation complications are also created by what are known as valuation adjustments, the XVAs. These are adjustments to the valuation given by a basic model, for example the Black-Scholes-Merton model. In this blog we will discuss the credit valuation adjustment (CVA) and debit (or debt) valuation adjustment (DVA). Other valuation adjustments, specifically FVA, MVA, and KVA, will be covered in future blogs.

The credit valuation adjustment, CVA, has been recognized as an important element of pricing for a long time. It is the downward adjustment to the value of a derivative in a bilaterally cleared transaction because of the possibility that the counterparty will default. Because bilaterally cleared transactions are almost invariably governed by a master agreement that includes netting, the expected loss from a default by a particular counterparty depends on the whole portfolio of transactions a dealer has with a counterparty. It cannot be calculated on a transaction-by-transaction basis.

The CVA applied to a new transaction should in theory equal the incremental effect of the new transaction on the CVA for the portfolio of transactions with the counterparty. This can be positive or negative. If the value of the new transaction at future times is positively correlated with the total value of the other transactions in the portfolio, the incremental effect is likely to be positive. If it is negatively correlated with the total value of the other transactions, the incremental effect is likely to be negative.

To calculate CVA for a portfolio, analysts first determine $T$, the time until the end of the life of the longest derivative. They then divide the time between today and $T$ into a number of intervals and calculate

- The risk-neutral probability of a counterparty default in each time interval; and
- The present value of the expected loss if there is a counterparty default at the time represented by the midpoint of each interval

Suppose that the $i$^{th} interval runs from $t_{i-1}$ to $t_i$, the risk-neutral probability of default during the $i$^{th} interval is $q_i$, and the present value of the expected loss if there is a default in the $i$^{th} interval is $v_i$ If a default occurs in an interval, it is usually assumed to occur in the middle of the interval. The expected loss from default is

$$\sum_{i=1}^N (1-R)q_i v_i$$

where $R$ is the recovery rate realized by the dealer in the event of a default by the counterparty and $N$ is the number of intervals.

It is correct to use risk-neutral rather than real-world default probabilities because the calculation of CVA is a valuation exercise where expected future cash flows are estimated in a risk-neutral world and discounted at the risk-free rate. This means that the $q_i$ must be estimated from the counterparty’s credit spreads.

The procedure for determining the $q$’s is as follows. Suppose that $s_i$ is the credit spread for maturity $t_i$. The average hazard rate between time zero and time $t_i$ is, to a good approximation, $s_i/(1-R)$ so that the probability of no default between time zero and time $t_i$ is $\mbox{exp}[-s_i t_i/(1-R)]$. This means that

$$q_i=\mbox{exp}\left(-\frac{s_{i-1}t_{i-1}}{1-R}\right)-\mbox{exp}\left(-\frac{s_i t_i}{1-R}\right)$$

Typically credit spreads can be estimated directly for a few maturities and estimated for others using interpolation.

The $v_i$ are usually calculated using Monte Carlo simulation. The market variables determining the future value of the transactions that the dealer has with the counterparty are simulated in a risk-neutral world between time zero and time $T$. On each simulation trial the exposure of the dealer to the counterparty at the midpoint of each interval is calculated. These exposures are averaged and* *$v_i$ is set equal to the present value of the average exposure at the midpoint of the $i$^{th} interval discounted to today at the risk-free (OIS) rate.

Collateral agreements must be incorporated into the calculation of the $v_i$. Suppose that $C$ is defined as the collateral posted by the counterparty at the time of a default. (If $C$ is negative, $-C$ is the collateral posted by the dealer with the counterparty at the time of the default.) The exposure of the dealer to the counterparty is in all situations given by $\mbox{max}(V–C, 0)$ where $V$ is the market value of outstanding transactions to the dealer at the time of the default.

It is usually assumed that a period of time elapses between the time when a counterparty stops posting collateral and the close out of transactions. This period of time is known as the “cure period” or “margin period of risk”. It is typically 10 or 20 days. The effect of the cure period is that the collateral at the time of a default does not reflect the value of the portfolio at the time of the default. It reflects the value 10 or 20 days earlier.

Suppose that the midpoint of the $i$^{th} interval is $t_i^*(=(t_{i-1}+t_i)/2)$ and $c$ is the cure period. The Monte Carlo simulation to calculate the $v_i$ must be structured so that the value of the derivatives portfolio with the counterparty is calculated at times $t_i^*-c$ as well as at times $t_i^*-c$ as well as at times $t_i^*(i=1, 2..., N)$. On each simulation trial the value at time $t_i-c$ is used to calculate the collateral available at time* *$t_i^*$. The exposure at time $t_i^*$ is then calculated.[1]

The calculation of CVA by derivatives dealers is extremely computationally intensive. They typically have hundreds of transactions with each of thousands of different counterparties involving thousands of market variables. As transactions move to central counterparties (CCPs) or become collateralized in other ways so that expected default losses become negligible, this calculation burden will be eased. But there are always likely to be some transactions with end users where default losses have to be considered.

The incremental impact of a new transaction with a counterparty can be estimated by storing the results from the most recent CVA calculation. Specifically, the paths followed by the market variables and the values for the portfolio on each simulation trial are stored. When a new transaction is contemplated, it can be simulated on its own using the stored paths for the market variables and its incremental effect on the *v _{i}* calculated relatively quickly. It is not necessary to re-simulate all the other transactions with the counterparty.

Many dealers regard CVA itself as a complex derivative with payoffs dependent on a) whether counterparty defaults and b) the future behavior of the underlying market variables. They have desks whose job it is to hedge CVA exposure. Adjoint differentiation is a useful tool for calculating Greek letters for the CVA exposure.[2] Basel III requires market risk capital to be held for the CVA exposure related to credit spread movements.

A final point to note is that the analysis we have described assumes that the probability of default is independent of the exposure. Wrong-way risk occurs when the counterparty is more likely to default when the exposure to the counterparty is high than when it is low. Right-way risk occurs when the counterparty is more likely to default when the exposure to the counterparty is low than when it is high. Regulators allow for wrong-way risk by adding 40% to the results obtained when independence is assumed. In Hull and White (2012) we discuss a relatively simple way of estimating wrong-way risk.

**DVA**

A rather more controversial valuation adjustment than CVA is the debit (or debt) valuation adjustment (DVA). The DVA for a portfolio a dealer has with a counterparty is the incremental effect on the value of the portfolio as a result of the possibility that the dealer might itself default. CVA and DVA are opposite sides of the same coin. The dealer’s CVA is the counterparty’s DVA, and vice versa. Whereas CVA reduces the value of the portfolio, DVA increases it. The benefit to the dealer from DVA arises from the fact that when it defaults the dealer avoids payments that would otherwise be required on outstanding derivatives. Derivatives are zero-sum games. If the counterparty’s CVA (dealer’s DVA) is a cost to the counterparty, it must be a benefit to the dealer. The net value of transactions with a counterparty is

$$f_{nd}-\mbox{CVA}+\mbox{DVA}$$

where $f_{nd}$ is the no-default value, the value assuming that neither side will default.

DVA can be calculated at the same time as CVA. Equation (1) gives DVA rather than CVA if *R* is the recovery rate made by the counterparty in the event of a dealer default, $v_i$ is the value of a derivative that pays off the counterparty's exposure to the dealer at the midpoint of the $i$^{th} interval, and $q_i$ is the probability of a default by the dealer during the $i$^{th} interval. The counterparty's net exposure to the dealer, after taking the collateral posted by the two sides into account, can be calculated in an analogous way to the dealer's net exposure to the counterparty.

One surprising effect of DVA is that, when the credit spread of a derivatives dealer increases, DVA increases. This in turn leads to an increase in the reported value of the derivatives on the books of the dealer. Some banks reported several billion dollars of “profits” from this source in the third quarter of 2011. Accountants have recently decided that for some transactions (but not yet for derivatives) DVA should be an adjustment to equity rather than flow through the income statement. Regulators are generally uncomfortable with DVA and have excluded DVA gains and losses from the definition of common equity in the determination of regulatory capital.

In the next two blogs we shall discuss a number of newer valuation adjustments. These are the funding valuation adjustment (FVA), the margin valuation adjustment (MVA), and the capital valuation adjustment (KVA).

**References**

Gibbs, M. and R. Goyder, “Universal algorithmic differentiation,” FiNCAD, December 2014.

Giles, M. and P. Glasserman, “Smoking adjoints: fast Monte Carlo Greeks,” *Risk*, 19, No. 1 (2006): 88-92.

Henrard, M, “Adjoint algorithmic differentiation: calibration and the implicit function theorem,” *Journal of Computational Finance*, 17, No.4 (2014): 37-47.

Hull, J. and A. White, “CVA and wrong-way risk,” *Financial Analysts Journal*, Vol. 68, No. 5 (Sept/Oct 2012): 58-69.