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Wrong Way Risk in CVA Calculations
By John Hull PhD | August 16, 2016 The credit value adjustment (CVA) is an adjustment to the way a dealer values a portfolio of derivatives with a counterparty to allow for the possibility that the counterparty might default. The CVA reduces the value of the portfolio by the amount that is expected to be lost if the counterparty defaults. We described how it is calculated in an earlier blog: “Valuation Adjustments 1.”  The life of the outstanding derivatives is divided into a number of intervals and CVA is set equal to

\begin{equation}
\sum_{i=1}^{N} \left( 1- R \right) q_i \nu_i
\end{equation}

where $q_i$ is the (risk-neutral) probability of a counterparty default in the ith interval, $\nu_i$ is the present value of the dealer’s expected exposure to the counterparty at the midpoint of the ith interval, R is the assumed recovery rate, and N is the number of intervals.

When calculations are carried out in the usual way the probability of default, $q_i$ is assumed to be independent of the expected exposure, $\nu_i$.  This means that the probability of a counterparty default in the ith interval when the dealer’s potential loss is high is the same as that when it is low. The situation where $q_i$ is positively dependent on $\nu_i$ is referred to as “wrong-way” risk. In this case, when the dealer’s exposure to the counterparty is relatively high the probability of a default by the counterparty is higher than normal. The situation where $q_i$ is negatively dependent on $\nu_i$ is referred to as “right-way” risk. In this case, the counterparty is more likely to default when the dealer’s exposure is relatively low. Wrong-way risk increases CVA while right-way risk reduces CVA.

Both wrong-way and right-way risk are observed in practice. Consider the case of a hedge fund that has a large unhedged portfolio of derivatives with a dealer. If the market variables that drive the value of the derivatives move in such a way that the value of the portfolio to the hedge fund is negative, the hedge fund is losing money. If the losses are sufficiently large the hedge fund will default. At the same time the value of the portfolio to the dealer is positive so the dealer’s exposure is large. This results in wrong-way risk for the dealer. The probability of default is large when the dealer’s exposure is large.

Alternatively, consider the case of a gold producer that has used derivatives to hedge 50% of its production by entering into forward contracts in which the dealer agrees to buy gold in the future at a specified price. In this case there is likely to be right-way risk. The gold producer is likely to default when the price of gold is low and it is losing money on the unhedged part of its production. But when the price of gold is low the value of the forward contracts to the dealer is also low. In this case, the derivatives have a positive value to the gold producer and a negative value to the derivatives dealer.  The dealer’s exposure is therefore likely to be low when the gold producer defaults.

A simple way of dealing with wrong-way risk is to multiply the expected exposure $\nu_i$ by a factor “alpha” in the version of the model in which $\nu_i$ and $q_i$ are assumed to be independent. The effect of this is to increase CVA by the alpha multiplier. The Basel II rules set alpha equal to 1.4 or allow banks to use their own models, with a floor for alpha of 1.2. This means that when wrong-way risk exists, at minimum, the CVA has to be 20% higher than that given by the model where $q_i$ and $\nu_i$ are independent. If a bank does not have its own model for wrong way risk, the CVA has to be 40% higher. Estimates of the increase in CVA due to wrong-way risk calculated by the models used by banks range from 1.07 to 1.10. Another similar approach to calculating wrong-way risk is to set $\nu_i$, equal to the present value of the exposure that is k standard deviations above the expected exposure for some k.

We propose a different approach, which we first suggested in Hull and White (2012). Instead of changing the calculation of the expected exposures, the v’s, we change the calculation of the probability of default, the q’s. The probability of default between time t and time $t + \delta t$, the hazard rate $h(t)$, depends on the evolution of the market variables in the Monte Carlo simulation used to calculate CVA up until time t. We model a relationship between the hazard rate of a counterparty and a variable (or variables) that can be calculated in the Monte Carlo simulation and may affect the dealer’s exposure to the counterparty. This relationship can be either deterministic or stochastic. There are three ways of proceeding:

1. Assume a relationship between the counterparty’s hazard rate and a variable, x, that is closely related to the dealer’s exposure to the counterparty. A reasonable approach here is to set x equal to the value of the dealer’s portfolio with the counterparty. If there is no relationship between the hazard rate and the portfolio value then there is no wrong way or right way risk. A positive relationship is indicative of wrong way risk and a negative relationship is indicative of right-way risk.

2. Assume a relationship between the counterparty’s hazard rate and a variable, x, that a) affects the value of the counterparty’s portfolio and b) has a big effect on the counterparty’s health. This variable, because it affects the value of the counterparty’s portfolio, is already part of the Monte Carlo simulation. The variable chosen for a gold producer might be the price of gold. For another company, it could be an exchange rate or interest rate.

3. Assume a relationship between the counterparty’s hazard rate and a variable x that does not affect the value of the counterparty’s portfolio, but potentially has a big effect on its health. Possible choices for x are the counterparty’s five-year credit spread, its stock price, or the Moody’s KMV distance to default for the counterparty. Daily historical data must be used to estimate correlations between the variable chosen and the other variables in the Monte Carlo simulation so that the process assumed for the variable is appropriately modeled.

The relationship between the hazard rate h and the variable x has the form

\begin{equation}
h(t) = f \left( x \left( t \right) \right)
\end{equation}

for some function f. The function can involve a noise term (although in practice we find this makes very little difference).  The function chosen may depend on the nature of the variable x and the nature of the company. The function f must have the property that $h(t) \ge 0$ for all possible values of x(t) and the noise term, if any. It must also have the property that for all times t the expected probability that there is no default before time t based on the random hazard rates equals the probability of no default before time t that is inferred from credit spreads at time zero. This means that the expected value of  $\exp \left( - \bar h t \right)$ must equal $\exp \left( -st / \left( 1-R \right) \right)$ where $\bar h$ is the average hazard rate between time zero and time t, s is the counterparty’s credit spread for maturity t,  and expectations are taken over all possible paths that x may follow. To allow the condition to be satisfied, the function f must involve a parameter which is a function of time. A simple model which we have used is

\begin{equation}
h(t) = \exp \left[ a(t) + b x(t) \right]
\end{equation}

Here b is a parameter determining the impact the changes in x on proportional changes in the hazard rate and a(t) is a function of time chosen to match credit spreads.