For the ease of exposition, we assume that all the trades with the counterparty are nettable and that collateral if there is any can be described by the instantaneous model. Consider, for example, the EE contributions given by Equation The following calculations are added to Steps The algorithm above assumes independence between the exposure and the counterparty's credit quality. More generally, there may be dependence between them which can come from two sources:. Let us introduce a stochastic default intensity process without specifying its underlying dynamics.
The counterparty-level exposure may depend either on the intensity value at time , or on the entire path of the intensity process from zero to. We can use the intensity process to convert the expectation conditional on default at time in Equation 21 to an unconditional expectation so that the conditional EE contribution becomes. As described for unconditional EE contributions, the calculations for conditional EE contributions can be performed during a Monte Carlo simulation of exposures.
In this case, given the dependence of exposures on the counterparty credit quality, the intensity process needs to be simulated jointly with the market risk factors that determine trade values. This joint simulation is done path-by-path: simulated values of the intensity and of the market factors at time are obtained from the corresponding simulated values at the earlier time points. It is also useful in practice to estimate EE and CVA contributions quickly outside of the simulation system.
To facilitate such calculations, we derive analytical EE contributions, for the case when trade values are normally distributed. For simplicity, and to avoid dealing with stochastic discounting factors, we assume that, at time the distribution of trade values is given under the forward to time probability measure.
Under this measure, the discounted conditional EE in Equation 9 can be written as. Assume that the value of trade at each future time is normally distributed with expectation and standard deviation under the forward to probability measure:.
Since the sum of normal variables is also normal, the discounted portfolio value is normally distributed:. We first calculate counterparty-level EE and EE contributions assuming independence between exposures and counterparty credit quality.
We obtain the results for the general case of a netting agreement with a margin agreement. The simpler case, with no margin agreement, is obtained as the limiting case when the threshold goes to infinity.
For the clarity of exposition, we assume the instantaneous collateral model. In the presence of a margin agreement, the counterparty-level stochastic exposure is given by Equation Substituting Equation 46 into Equation 22 and taking the expectation, we obtain.
Let us consider now the EE contributions given by Equation 32 type B allocations. Removing the discounting and substituting Equations 45 and 46 into Equation 32 , we obtain. For the case without a margin agreement, we take the limit of Equations 50 - This leads to the counterparty-level EE. Note that the EE contributions obtained in the previous section are contributions of the trades in the portfolio to the counterparty-level unconditional discounted EE.
We need to modify the approach to obtain the contributions to the counterparty-level EE conditional on the counterparty defaulting at the time when the exposure is measured.
An obvious approach is to define an intensity process and compute the conditional EE contributions as the expectation over all possible paths of the intensity process See Appendix 1 , but this requires a Monte Carlo simulation. In this section, we develop an alternative, simpler approach that results in closed form expressions for the conditional EE contributions. For this purpose, we define a Normal copula 13 to model the codependence between the counterparty's credit quality and the exposures.
If the portfolio contains trades with non-zero , the standard normal risk factor , which drives the portfolio value, also depends on :. In this section we briefly comment on the properties and interpretation of the analytical contributions derived in this section. Equation 55 can be understood from the incremental viewpoint of the CMC method. According to Equation 17 , the EE contribution of trade is determined by the infinitesimal change of the counterparty-level EE resulting from an infinitesimal increase of the weight of trade in the portfolio.
The effect of an increase of the weight of a trade on the portfolio value distribution can be viewed as the sum of two effects:. If the weight of trade is increased by , the expectation of portfolio value changes by. Let us first ignore the change of the standard deviation and consider how a uniform shift of the entire distribution by affects the counterparty-level EE.
Scenarios with positive portfolio value contribute the same amount to the exposure change, while scenarios with negative portfolio value contribute nothing. Therefore, the increment of the EE will be given by the product of the magnitude of the shift and the probability of the portfolio value being positive. It is straightforward to verify that.
Thus, the first term in the right-hand side of Equation 55 describes the increment of the counterparty-level EE resulting from the infinitesimal uniform shift of the portfolio value distribution associated with an increase of the weight of trade. The second term of Equation 55 describes the change of the width of the portfolio value distribution. The change of the standard deviation of the portfolio value resulting from increasing the weight of trade by can be calculated as.
It appears that Equation 55 has simple linear dependence on and the product. However, this is only part of the true dependence. Since trade is part of the portfolio, depends on and depends on and the correlation of trade with the rest of the portfolio. Moreover, correlation is the correlation between the values of trade and the portfolio that includes trade itself. Because of this, depends on the ratio see Equation Thus, unless trade represents a negligible fraction of the portfolio, the true dependence of EE contribution on trade parameters is non-linear.
In this case, only the first two terms of Equation 52 allow interpretation from the incremental viewpoint of the CMC method: the first term can be explained as the effect of the uniform shift and the second term as the effect of the widening or narrowing of the portfolio value distribution. The third term results from the allocation of exposure when the portfolio value is above the threshold. If the value of trade is correlated with the counterparty's credit quality, its value distribution at time conditional on the counterparty's default at time differs from its unconditional value distribution.
If the correlation is positive right-way risk , the distribution shifts down; if the correlation is negative wrong-way risk , the distribution shifts up. In both cases, the distribution becomes narrower. Under the normal approximation, the shift of the distribution is described by Equation 64 , and the narrowing is described by Equation An interesting property of Equation 64 is its dependence on the counterparty's PD.
To understand this, let us consider the bank entering into the same trade with an investment-grade counterparty A and with a speculative-grade counterparty B. We are interested in the trade value distribution conditional on the counterparty's default at the time of observation. For the case of wrong right way risk, the deterioration of the counterparty's credit quality to the point of default pushes trade values higher lower.
Since counterparty A is "further away" from default than counterparty B, the deterioration of credit quality to the point of default is larger for counterparty A. Therefore, trade values conditional on default of A are shifted more than trade values conditional on default of B. Note that this is not specific to the normal approximation, but is a general property not related to any model.
In this section, we present some simple examples that illustrate the behavior of exposure and hence CVA contributions. For ease of exposition, we assume that trade values are Normal, as well as market and credit independence.
However, as discussed earlier in Section 7. We first present an example when there is no collateral agreement in place, and then show the impact of adding a collateral agreement to the portfolio. As a first step to understand this behavior, consider Equations 54 and 55 , which give the counterparty-level EE and the EE trade contributions, in the case when there is no margin agreement in place:. The EE contribution of instrument is a function of:.
Both the counterparty-level EE and the trade contributions can be seen as the sum of two components: a mean value component first term in the equations , and a volatility component second term.
These components weigh the mean value or mean value contributions and the volatility volatility contribution , respectively, by the Normal distributions and density evaluated at the ratio for the entire counterparty portfolio. Thus, the overall level of the counterparty portfolio's mean value and volatility determine how the individual instrument's mean and volatility contribution are weighted to yield the EE contributions.
Figure 1 plots these weights as a function of. A low ratio weighs the volatility contribution much higher; while a high ratio weighs mean values much more. To illustrate the impact of various parameters on EE contributions, consider now the simple counterparty portfolio, which comprises of 5 transactions over a single step. The portfolio has a mean value and variance of We assume that trade values are independent. The portfolio is constructed so that for each trade, its mean value and volatility are inversely related; thus the first instrument, P1, has the lowest mean 0 and largest volatility 2 , while position 5 has the highest mean value 4 , and lowest volatility 0.
This may not only be reasonably realistic, but it will also help highlight some of the points below. Using Equations 54 and 55 , we compute the EE and contributions for the portfolio. The EE for the portfolio is The trade contributions to EE are fairly close to the contributions to the mean values in Table 1 0.
Now, we vary the overall mean value of the portfolio, , while leaving intact the volatility, , as well as the percent contributions of each instrument to the mean exposure and volatility in table 1. This allows us to express the trade contributions in terms of how deep in- or out-of-the-money the counterparty portfolio is relative to its volatility.
Figure 2 plots the EE, as well as its mean and volatility components, as functions of the portfolio's. For large negative portfolio mean values, the EE red line is zero. In this case, the mean value component of EE is actually negative, and the volatility component compensates for this to generate positive EEs.
As increases beyond zero, the volatility component decreases and, once 2 , the EE is completely dominated by the mean value. Figure 3 shows the EE contributions for each of the 5 trades as a function of.
There is a clear shift in dominance between the mean and volatility components as the portfolio's mean value increases. At one side of the spectrum, when the mean portfolio values are negative, trades 4 and 5, which have the largest negative mean values and lowest volatilities, produce very large negative EE contributions. The opposite occurs for trades 1 and 2 with low negative means and large volatilities. As the portfolio's increases, trades 4 and 5 end up dominating the contributions, with the EE contribution converging to the mean value contributions themselves.
We consider now the case when there is a margin agreement, and demonstrate the impact of the collateral on the trade contributions. In very general terms:. As the threshold is increased, the EE reductions decrease, as expected. Also, the collateral thresholds become more effective at reducing EE as the portfolio is deeper in-the-money i.
For example, a normalized threshold of 2 does not reduce EE until the portfolio's mean value is positive. The presence of the collateral affects each instrument's contributions differently. We believe it should be. Then what does this mean for the exposure value that feeds the CVA per Article One interpretation would suggest the CVA would be applied to column on template C So, this would include the incurred CVA referred to in Article 6.
Likewise for the c. The consultation runs until 25 September The proposed draft RTS do not deal directly with the Value at Risk VaR spread methodology but specify the criteria this methodology has to satisfy to allow for a proxy spread to be used in the calculation of the advanced CVA adjustment.
In particular, they specify how the rating, industry and region criteria should be incorporated in a proxy spread.
The requirements contained in these draft RTS are mainly addressed directly to institutions and only in some cases to competent authorities.
Comments can be sent to the EBA by clicking on the "send your comments" button on the consultation page. Please note that the deadline for the submission of comments is 25 September All contributions received will be published following the close of the consultation, unless requested otherwise.
The European Banking Authority EBA launches today a consultation on Draft Regulatory Technical Standards RTS for credit valuation adjustment risk on the determination of a proxy spread and the specification of a limited number of smaller portfolios.
The consultation runs until 15 September. Main features of the RTS. In particular they specify:. Skip to main content. Follow us on:. Regulatory Technical Standards in relation to credit valuation adjustment risk Adopted and published on the Official Journal These Regulatory Technical Standards RTS specify certain elements of the calculation of own funds requirements for credit valuation adjustment CVA risk. News Press Release Consultation Papers.
Through the proposed amendments the EBA expects to ensure a more adequate calculation of own funds requirements for CVA risk. The consultation runs until 06 July Against this backdrop, policy recommendations 7 and 8 of the CVA report concluded that the RTS should be amended to address the difficulties associated with the determination of proxy spreads for large numbers of counterparties, for which spreads may never be observed on markets, as well as issues linked with LGDMKT.
The proposed amendments are expected to lead to a more adequate calculation of own funds requirements for CVA risk, including in some cases a reduction of own funds requirements for CVA risk, thus partially remedying the over-estimation of current own funds requirements for counterparties in the scope of the CVA risk charge in the EU.
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