Counterparty Risk and CVA › The CVA Formula

Credit Valuation Adjustment represents the expected loss due to counterparty default, incorporated as an adjustment to the derivative's risk-free value. The intuition behind CVA is straightforward: if there is some probability that your counterparty will default and fail to make payments owed to you, then the derivative is worth less than its risk-free value. CVA quantifies exactly how much less the derivative is worth, and gives a fair price for counterparty credit risk.

To develop the CVA formula, we need to think about what happens at each possible default time. If the counterparty defaults at time $t$, and the derivative has positive value $V(t)$ at that moment, you will lose a fraction of that value determined by the Loss Given Default. We must consider all possible default times, weight each by its probability, and sum to find the total expected loss.

The fundamental CVA formula is:

$$\text{CVA} = \text{LGD} \cdot \int_0^T \text{EE}(t) \, d\text{PD}(t)$$

where:

• $\text{LGD}$: Loss Given Default, the fraction of exposure lost if default occurs
• $\text{EE}(t)$: Expected Exposure at time $t$
• $d\text{PD}(t)$: probability of default occurring in the infinitesimal interval around time $t$
• $T$: maturity of the contract

The formula calculates the total expected loss by integrating over the life of the contract. At any moment $t$, the expected loss conditional on default is $\text{LGD} \cdot \text{EE}(t)$. We weight this by the probability $d\text{PD}(t)$ that default happens exactly at that moment, and sum these potential losses across all times from 0 to $T$.

This continuous formula captures the essence of CVA, but continuous integration is not practical for implementation. In the real world, we observe exposure at discrete time points and must work with discrete default probabilities. The continuous formula is therefore typically discretized for practical computation:

$$\text{CVA} = \text{LGD} \cdot \sum_{i=1}^{n} \text{EE}(t_i) \cdot q(t_{i-1}, t_i)$$

where:

• $q(t_{i-1}, t_i)$: marginal probability of default between times $t_{i-1}$ and $t_i$
• $\text{EE}(t_i)$: expected exposure at time $t_i$
• $\text{LGD}$: Loss Given Default
• $t_i$: discrete time points where exposure is calculated
• $n$: number of time steps in the simulation

The discrete formula partitions the contract life into intervals and computes the contribution to CVA from each interval. For interval $i$, the contribution equals the expected loss if default occurs in that interval, multiplied by the probability of default in that interval. Summing across all intervals gives the total CVA.

As we discussed in Part II, Chapter 13, CDS spreads provide market-implied default probabilities. This connection allows us to use observable market prices to inform our CVA calculations, rather than relying solely on historical default rates or credit rating agency estimates. For a counterparty with CDS spread $s$ and recovery rate $R$, we can extract hazard rates and default probabilities that are consistent with market prices.

The hazard rate (also called the default intensity) represents the instantaneous probability of default conditional on survival to that point. Under a constant hazard rate assumption, the survival probability to time $t$ is:

$$S(t) = e^{-\lambda t}$$

where:

• $S(t)$: probability that the counterparty survives (does not default) until time $t$
• $\lambda$: constant hazard rate (default intensity)

This exponential form arises naturally from the definition of the hazard rate. If the probability of surviving an infinitesimal interval $dt$ is $1 - \lambda \, dt$, then the probability of surviving to time $t$ is the product of surviving all the small intervals from 0 to $t$. Taking the continuous limit of this product yields the exponential survival function.

The hazard rate $\lambda$ relates to the CDS spread approximately as:

$$\lambda \approx \frac{s}{1 - R}$$

where:

• $\lambda$: hazard rate (default intensity)
• $s$: CDS spread (premium paid for protection)
• $R$: recovery rate (fraction of face value recovered upon default)

This relationship emerges from the economics of a CDS contract, where the premium payments must compensate the protection seller for expected losses. We can derive this approximation by equating the expected cash flows of the two legs of a CDS contract over a short interval $\Delta t$:

1. Premium Leg: The protection buyer pays spread $s$ on the notional amount (set to 1). If survival probability is high ($e^{-\lambda \Delta t} \approx 1$), the payment is approximately $s \Delta t$.
2. Protection Leg: The protection seller pays the loss given default $(1-R)$ if a default occurs. The probability of default is approximately $\lambda \Delta t$, so the expected payment is $(1-R)\lambda \Delta t$.

Equating the two legs to find the fair spread:

$$\begin{aligned} \text{Premium Leg} &\approx \text{Protection Leg} && \text{(equate expected values)} \\ s \Delta t &\approx (1-R) \lambda \Delta t && \text{(substitute approximations)} \\ s &\approx \lambda(1 - R) && \text{(divide by } \Delta t \text{)} \\ \lambda &\approx \frac{s}{1 - R} && \text{(solve for } \lambda \text{)} \end{aligned}$$

This derivation shows that the CDS spread is approximately equal to the hazard rate multiplied by the loss given default. Intuitively, the protection buyer pays for the expected loss rate, which is the product of how often defaults occur (the hazard rate) and how much is lost when they do (the LGD).

The marginal default probability between $t_{i-1}$ and $t_i$ is:

$$q(t_{i-1}, t_i) = S(t_{i-1}) - S(t_i)$$

where:

• $q(t_{i-1}, t_i)$: probability of default occurring specifically in the interval $(t_{i-1}, t_i]$
• $S(t)$: cumulative survival probability to time $t$

This formula follows directly from the definition of survival probability. The probability of defaulting in a specific interval equals the probability of surviving to the start of the interval minus the probability of surviving to the end. This gives us the probability mass for default in that interval, which is exactly what we need for the discrete CVA formula.

A more precise formulation incorporates discounting, as the loss occurs at the time of default and must be expressed in present value terms:

$$\text{CVA} = \text{LGD} \cdot \sum_{i=1}^{n} D(t_i) \cdot \text{EE}(t_i) \cdot q(t_{i-1}, t_i)$$

where:

• $D(t_i)$: risk-free discount factor from time $0$ to $t_i$
• $\text{EE}(t_i)$: expected exposure at time $t_i$
• $q(t_{i-1}, t_i)$: marginal default probability for the interval
• $\text{LGD}$: Loss Given Default

This formulation aggregates the present value of expected losses across all time steps. The logic proceeds as follows: for each time period $i$, we compute the expected loss that would occur if default happens at that time, then discount that loss back to today and weight by the probability of default in that period. Specifically:

• $\text{LGD} \cdot \text{EE}(t_i)$ is the expected loss amount if default occurs at $t_i$
• $q(t_{i-1}, t_i)$ is the probability of default happening in this specific interval
• $D(t_i)$ discounts that future loss back to today's dollars

Summing these discounted components gives the total upfront price of counterparty risk. This is the amount that should be charged at trade inception to compensate for expected counterparty losses over the life of the contract.