What is the impact or what is the potential difference between the regulatory method and whatever model an institution may use to calculate their CVA VaR (as well as their CVA)? For the purposes of a demonstration and investigation, we consider Merton’s Model, where the equity value of the firm is modelled as the assets less the liabilities.

If the asset value is less than the firms liabilities , then the equity is worthless and there is a default event. We introduce a contrivance, that their is some residual value to the firm in event of default, allowing there to be call options on the companies equity which may still be in the money at the time of default.

The probability of default, assuming is log-normally distributed then :

If we then assume that there is some residual value in the event of default , and that value is eroded for any further decline in the asset value  below the critical level. The expected exposure of a call warrant, struck at is :

And the payoff of the warrant at maturity will simply be :

being the residual equity value of the firm.

For our example, we consider an option that is maturing in 1 day, and it’s 1 day probability of default.

• – Asset value is 10.3
• – Liability value is 10
• – Residual value is 0.4
• – Strike value is 0.2
•  – Asset volatility is 0.4

So, for each unit exceeds -0.4, the firm will retain a  unit of equity value, and the warrant on this equity is struck at 0.2. The option can then be modelled as an option on , struck at 9.8.

To maintain consistency with a practical implementation of the CVA calculation we proceed as follows :

• Simulate 1000 potential terminal values for the option tomorrow.
• Calculate the expected exposure as the average of positive payoffs across these thousand paths.
• CVA is the PD (calculated per the provided formula) across 250 market data scenarios.
• Simulate CVA across the 250 market data VaR scenarios, and calculate a CVA VaR charge.

We apply 3 different methodologies :

• Regulatory methodology – calculated a base expected exposure based on current market values. Simulate PD across a number of market scenarios, and so, a market scenario for CVA.
• 50% Model – As above, but expected exposure is recalculated across each scenario, maintaining the co-dependence between probability of default and expected exposure (though not modelling the full degree of co-dependence).
• Full Model – The credit process is simulated, along with contract value, and default only occurs if the asset value falls below 10 (per our model). This models full co-dependence between probability of default, and expected exposure.

The resultant market simulations of CVA resulted in the following CVA scenarios across the 3 different methodologies :

The credit process and the process driving the value of the option are essentially the same. All warrants will have some sort of similar codependent relationship. Whenever the options exposure is high, the probability of default will be low, and vice versa. The model has a buffer effect, which reflects that holders of the options have low exposure to default, and, as a result the CVA charge is less volatile across market scenarios, under the model.

Clearly the regulatory method results in the most volatile CVA. The partial model gives the next most volatile result, and finally, the full model gives the most stable result.

The resulting CVA VaR results are as follows :

Clearly the risk charge would be lower were CVA to be claculated using a full simulation and modelling of the credit process and the underlying asset. However, the higher charge mandated under Basel III is inevitable, and can be thought of as a conservative “model-risk” charge that avoids any optimistic modelling which relies on assumptions about co-dependent structures.

However, the demonstration. although rather contrived, shows the benefits, from a risk management perspective, of having a co-dependent model for exposure and probability of default, where there is compelling evidence for a relationship between the two.

As an aside, it is no coincidence that we chose a short dated trade for the purposes of demonstration. For longer dated trades, realised exposure will be more uncertain, proxying the exposure to the expected exposure calculated under current (or stressed) market conditions is appropriate to avoid calculating 250,000 scenarios (1000 per each of the 250 VaR scenarios), per instrument per time step.  For shorter dated trades, expected exposure converges to the current exposure value, and this exposure is driven more by current market rates. It may even be appropriate to proxy expected exposure to current exposure, and avoid scenarios altogether for those very close maturity dates.

The excel file containing relevant calculations is accessible here : WrongWayRiskDemo. These calculations can easily be manipulated to show that the model would introduce more volatile CVA scenarios in the presence of Wrong-Way Risk (try replacing the call option with a put).