Divyesh Bakhda & Vlad Pavlov, Senior Financial Engineers, Numerix 

Credit valuation adjustment (CVA)

In the most general definition, counterparty risk arises due to losses precipitated by changes in the ability of the counterparty to meet future obligations – this could be in terms of either the net mark-to-market going more negative for the counterparty or its default probability going up. CVA is emerging as the standard CCE measure and is enshrined in the Basel III regulation. Intuitively CVA can be understood as the cost of insuring a portfolio of trades with a specific counterparty against losses due to its default. To understand the inputs into CVA calculations, consider an example where default is possible only at a single future point in time. While clearly a simplification, this assumption allows discussing the salient features and inputs into the calculation without overloading the article with technical details. 

Under the single default assumption, CVA can be estimated as the discounted value of the positive exposure weighted by the default probability. The exposure is the distribution of future values of the portfolio truncated at zero. While in general a portfolio (eg a portfolio of swaps) depending on future market conditions can represent either an asset or a liability for the firm, only the states where the value of the portfolio is positive contribute to credit risk. Only in these states the firm suffers a loss of value if the counterparty defaults. However, CVA does not account for the firm’s own default. Besides, the firm can take advantage of unfair valuations while calculating the charge. Hence Bilateral CVA (BCVA) was introduced to account for default of both parties and will thus emphasise on fair valuation from both parties as well.

CVA requires three intermediate inputs - discount factors, default probabilities and exposures. Discounting, while unquestionably important, relies on standard assumptions, methods and market data which will not be discussed here. We will concentrate on the calculation of exposures and default probabilities which are the more challenging aspects of CVA. 

Exposures

Exposure calculation involves taking a view on the future evolution of market risk factors. This view is usually expressed in the form of a quantitative statistical model. The selection of factors depends on the nature of the portfolio and the quality of data available to calibrate or estimate model parameters. For example, the value of a simple interest rate swap depends on future yields; the swap tends to be an asset to the party paying the fixed rate if rates go up or if the yield curve steepens and a liability otherwise. So to construct the distribution of future values for a swap requires a dynamic interest rates model. 

Once a model has been constructed, it can be used to produce future interest rate scenarios; revaluing the portfolio on each scenario results in a sample of an exposure distribution. This sample can be used to evaluate the expected exposure to default (expected positive exposure or EPE) and other relevant exposure metrics. 

Also, a typical portfolio of a firm is quite expansive and inclusive of different asset classes. The very nature of the banking business of lending across different markets exposes the firm to foreign interest rates and foreign exchange rate. This will require CVA to include exposures from transactions featuring different risk factors. This will result in the possibility of a negative mark-to-market for the client on a trade in one asset class being combined with a positive mark-to-market on another trade from a different sector, which would mitigate the aggregate counterparty exposure. Such an aggregate exposure will crucially depend on Cross asset correlation. This will necessitate a Hybrid modelling framework which would have the capability of jointly projecting various risk factors in a correlated fashion enabling calculation of future value of all the trades in a portfolio. This will also ascertain that a unified modelling framework will take care of scenario generation and pricing in a consistent manner.

Most trading banks have internal models for scenario generation as part of already installed risk systems. These systems are often re-used for exposure calculations. In the context of CVA estimation however this raises a number of issues which we revisit after discussing default probabilities. 

Default probabilities

Calculation of default probabilities is the most challenging and data demanding aspect of the CVA methodology. We briefly discuss some of the common methods for inferring default probabilities below.

1. Credit derivatives - Movements in the valuations of credit derivatives directly reveal changes in the market assessment of default probabilities. Whenever a liquid market exists, probabilities can be inferred from prices of credit derivatives – models used for such derivations are referred to as Reduced Form models. They do not explain the ‘why’ of default likelihood but only how market views individual credits. This is the most preferred method - unfortunately very little market information is available outside Europe and US.  

2. Credit spreads - Another financial contract directly sensitive to the creditworthiness of a firm is debt issued by the company. When negotiable debt instruments exist and market data are available, default probabilities can be inferred from the term structure of credit spreads. To infer default probabilities, changes in the sentiment on the counterparty creditworthiness have to be untangled from interest rate movements. This makes it necessary to introduce a benchmark curve and its choice and appropriateness. Also the credit spread may suffer from the same limitations as Yield to Maturity, as a measure, would.

3. Structural default models - The idea of all structural default models can be traced back to the original contribution by Merton. Equity is treated as a call option on the value of the firm. In this view, default is triggered if the value of firm’s assets falls below the value of liabilities. Limited liability then allows the shareholders to pass the firm to creditors without suffering any further economic loss. It can then be concluded that for public firms, part of the observed volatility of equity is due to the changes in the value of this option to walk away. If the value of the option can be separated from the value of default free equity, it can then be used to infer default probabilities. 

This idea was further enhanced by Black and Cox and practically implemented by CreditGrades to put to commercial use. In public domain it is called the Finkelstein-Lardy model. It estimates an occurrence of a default when the asset value of the firm crosses a default barrier – hence equity is treated as a barrier style option on the firm value. The following diagram summarises the model.

Model description and its salient features

Based on the capital structure of a firm, its asset value can be broken down into its equity and debt components. Vo = E + L*D. Vo is the initial asset value. The asset value V is believed to follow a stochastic process with some asset volatility (this is generally proxied by Equity volatility – either implied or realised). The process is shown as above in the diagram.

By definition, when a firm approaches a default, its equity value E approaches zero. Thus the value of the firm approaches the value of outstanding debt of the firm. Mathematically V = L*D – which could be intuitively thought of as an average Recovery Rate of the firm’s debt. L*D can be treated as a barrier and the firm value process becomes that of a Barrier option. When the barrier is breached it is considered as a default.

Also because of the nature of Recovery rate, the barrier L*D is not treated constant. A default during a bearish economy can result in a very low recovery. Hence it is derived out of some distribution which is characterised by some mean value L and standard deviation (variability) Lambda.

Thus the most important things that determine the survival of a firm are:

a) Initial distance to default – This is the distance between Vo and L*D – the distance between them, acting as a cushion, would be the value of Equity. If it is too large then regular diffusion process of asset value will not be able to breach it in a short period of time. This will overestimate the short term survival probabilities. Thus something like a Jump-diffusion process will have to be considered. However this is non-trivial so the model depends on the variability of the barrier.

b) Asset volatility – This is the volatility of the asset value process and very crucial to determine the future path which the asset value will assume.

c) Variance of the recovery rate barrier – As mentioned before, this will control the short term survival probability and also model the recovery rate in a more realistic fashion treating it as variable whose value would be derived based on the market condition.

While the original Merton formulation relies on the estimates of asset volatilities and ultimately proved unsuitable for practical use, a number of successful proprietary methodologies (notably Moody’s KMV model and Finkelstein-Lardy (CreditGrades) model based on the same premise are available. The key to these methodologies is in estimating the ‘default point’ or the value of assets at which the shareholders no longer have an incentive to keep the equity. This estimation requires data on fundamental variables usually obtained from accounting statements.

4. Statistical models - Statistical approaches infer default probabilities based on factor analysis and historical incidence of defaults. To construct a statistical model, company data such as accounting ratios, industry and market information, etc are combined with a database of defaults. For example, a default probability can be inferred from the rating based on the historical default rates of companies with the same rating. This can be further stratified by industry, state of the business cycle, etc. 

The limitation of these models is that they require considerable amount of inputs to generate probabilities. Another limitation is that they produce ‘real-world’ default probabilities which are not directly suitable for pricing calculations.

Right way/wrong way risk

Right/wrong way risk refers to the situation where the default is not independent of the exposure. If the value of the portfolio correlates positively with the counterparty’s credit standing, there exists a wrong way risk; the counterparty is most likely to default exactly when the value of the contracts to the firms is high thus triggering larger losses. Right/wrong way correlation can have a material impact on the CVA estimate and the current consensus is that CVA modelling is not complete if correlations between exposure and the credit process are not accounted for. Basel III in particular, while allowing the standard CVA calculation under the assumption of independence, assigns a safety multiplier that scales up the resulting capital requirement to account for the potential effect of the correlation. This multiplier can be reduced if the bank can demonstrate accurate internal modelling of the risk providing an additional incentive for development of CVA systems. 

Problems with CVA calculations

Technical aspects aside, both the probability and exposures for CVA calculations have to be generated under so called ‘risk-neutral’ measures. The difference between ‘risk-neutral’ and ‘real-world’ measures is that the former takes into account the market price of risk while the latter only reflects the observed or ‘real’ probabilities. Typically risk-neutral probabilities are recovered directly from market prices and are used for derivatives valuations. In contrast, statistical models produce real-world probabilities which are used for market risk (VAR) analysis. 

CVA challenges

Market data is hard to come by in Asia.

Market risk systems are generating exposures under real-world measures. These exposures are not appropriate for CVA calculations which require simulations under risk-neutral measures but the distinctions are usually ignored in practice. This means that a full CVA system would typically require a technology upgrade. 

Conclusions

Firms participating in OTC markets are exposed to counterparty risk. They manage their risk by setting credit limits at counterparty level and by calculating and allocating economic capital. Modelling counterparty risk is more difficult than modelling lending risk, because of the uncertainty of future credit exposure. In this article we have discussed two main modelling issues: modelling credit exposure and calculating CVA. Modelling exposure is crucial for an improved risk management application, while modelling CVA is a necessary step for pricing and hedging counterparty credit risk.