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Enhanced CVA in the Cube
The upshot of our post on Wrong-Way Risk, is that it is entirely possible to need multiple views of CVA. The challenge is now that CVA, an already onerous calculation, may need to be calculated twice. Practically, we strive to reuse as much calculation between the two methods as possible.
First, we deal with the regulatory method of CVA calculation. Recall the usual formula :
Under this method, a full set of simulations is run for the future time steps. Expected exposure is determined through aggregation, per counterparty, for each time step. The CVA is then, simply, this fixed expected exposure value multiplied by the relevant probability of default, and loss given default, for the counterparty (or counterparty category), with the proviso that a seperate expected exposure be calculated with stressed parameters. CVA VaR is calculated across the relevant scenarios for PD and LGD.
Next, we consider simulating the credit process alongside the underlying valuation variables. This corresponds to a monte carlo calculation of :
Barring an analytic solution to this problem, a monte carlo simulation is now required. We re-use the value simulations that were used to determine the expected exposure and, for each counterparty, simulate a credit loss variable , indicating whether counterparty has defaulted, under scenario , at time . () is if default has occurred, and 0 if it has not. One then simulates counterparty contract default losses as
The CVA contributing scenarios are then those scenarios under which default occurs. If one has an aggregation function, , which calculates expected exposure, for a given counterparty, given the scenario values across all that counterparties positions, under portfolio hierarchy and netting hierarchy , this function can be reused to calculate a CVA charge based on simulation of the counterparties credit process, that is :
As the cube is already populated with , it is a simple matter to multiply each relevant contract value by the relevant credit variable to obtain .
The aggregation, per counterparty will still need to be performed twice, but the same aggregation function can be used in both cases to obtain the CVA under each method.
A CVA solution too focused on meeting regulatory requirements, and not considerate enough of the potential trading challenges of CVA may not have the ability to simulate the credit process of the relevant counterparty, along with the underlying contract values, under a suitable distribution, and to calculate a CVA charge off the back of this simulation. With a suitable, open mark-to-future cube structure it is easy to remedy this shortcoming. One simply simulates the counterparties credit loss process, multiplies the contract values by this process to obtain the relevant CVA contributing loss values, and finally, applies the standard CVA aggregation to obtain a CVA charge for the relevant counterparty. The important consideration here is that either both CVA methods be native functionality, or that the CVA calculation take place in a relatively open, powerful analytic engine (ActivePivot is a good example) so that calculations are easy to enhance or extend.