Unilateral CVA as Expected Loss at the Stochastic Default Time
Summary
The document explains how unilateral credit value adjustment (CVA) can be written in continuous time as the expected discounted loss if a counterparty defaults before a deal matures. The loss depends on exposure at the moment of default and the loss given default, which is one minus the recovery rate. The default time is a random variable; it can be treated as infinite when no default occurs during the relevant horizon.
Under an independence assumption between default time and discounted exposure, the expectation can be expressed as an integral over default times, weighted by the default-time density. Dividing the horizon into time intervals turns this integral into a sum, with each interval weighted by its probability of containing the default. This recovers the familiar discrete-time calculation. The independence assumption excludes wrong-way risk, where exposure and counterparty credit quality move together; the document does not cover how to model that dependence.
Key ideas
- CVA is the expected present value of the loss incurred if default occurs before maturity.
- Exposure at default is evaluated at the random time the counterparty defaults.
- The default time can be represented as infinite when default does not occur within the horizon.
- With independence between default and discounted exposure, CVA integrates expected exposure against the default-time density.
- Discretizing the integral produces interval exposure terms weighted by interval default probabilities.
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# Credit Value Adjustment
# Credit Value Adjustment
i'm struggling with the idea of the time default random variable in the Unilateral CVA.
While the CVA in discrete model is only the sum of the discounted exposure of your financial position (the amount of what you must receive from the other part) multiplied by the probability of deafault. This is an easy expected value to calculate..
I'm not understanding why changes the approach in time continuos CVA calculation. In particular i'm not understanding what is a time of default.
Can you help whit a simple example? Thank You
## Answer by Antoine Conze (score 1, accepted)
https://quant.stackexchange.com/a/38076
The approach is the same in discrete and continuous time.
You have $$\text{CVA} = E\left[e^{-\int_0^{\tau} r_u du} \text{EAD}_{\tau} (1-R)\mathbf{1}_{\tau \leq T}\right]$$ where
- $\tau$ = stochastic time of default ($+\infty$ if the counterparty never defaults)
- $T$ = deal maturity
- $\text{EAD}_t$ = exposure at default = exposure (stochastic) to the counterparty if the counterparty defaults on time $t$
- $R$ = recovery rate
This formula simply states that CVA is the present value of a flow that represents the Loss Given Default upon default.
If you now assume that default time and discounted EAD are independent (no "wrong way risk") then $$\text{CVA} = \int_0^T E\left[e^{-\int_0^{t} r_u du} \text{EAD}_{t} (1-R)\right] \phi(t) dt$$ where $\phi()$ is the density of the distribution of $\tau$.
If you then discretize time with a discrete time line $t_i$, $t_0=0$, $t_N=T$, the integral is approximated as $$\text{CVA} = \sum_{i=1}^{N} E\left[e^{-\int_0^{t_i} r_u du} \text{EAD}_{t_i} (1-R)\right] P(t_{i-1} < \tau \leq t_i)$$ which is the discrete model CVA formula you are familiar with.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.