# The major XVA components

We wish to go beyond the regulatory calculation formulas to better understand the impact of accurately modelling and efficient computation for huge banking portfolios and different stress periods. In a variety of earlier research projects, the academic beneficiaries have developed prototype answers for related research questions appearing in risk management. Together with the industry we aim to generalize these research findings to the real world practice, with data and relevant portfolios from the industrial partners. An innovation of this EID is that industry and academia will work closely together on state-of-the-art methodologies with mathematics and data sciences for real-life XVA problems. The **major components of XVA **and their mathematical aspects are briefly reviewed below with the specific innovative research tasks being indicated. Starting some years ago with Credit Valuation Adjustment (CVA), a number of new regulatory XVA components have risen to prominence.

** Credit Valuation Adjustment (CVA): **aims to capture the impact of counterparty default in an uncollateralised OTC trade. CVA can be viewed as the market value of the counterparty credit risk. It is computed under the risk- neutral measure as an integral over time of the product of the three key elements, the discounted expected exposure, counterparty default probability and loss-given-default. In real-life situations, however, these three elements are typically not independent and

*wrong-way risk (WWR)*occurs when the expected exposure is adversely correlated with the credit quality of the counterparty. We will develop accurate models for WWR. Another important aspect in CVA is the detailed understanding of the impact of specific market features in stressful periods, like the occurrence of so-called steep smile and skew implied volatility market patterns. Potential Future Exposure (FPE) is a related quantity to CVA, which requires real world measure economic scenarios for the future, augmented with risk neutral valuation at certain future time points (Q in P simulations). This is also a highly nontrivial mathematical modelling issue we will address. The numerical issues will also be tackled.

** Debt Valuation Adjustment (DVA): **While CVA accounts for the default probability of the counterparty, DVA covers the default of the party making the calculation. As for CVA, the impact of smile/skew dynamics is important. To reduce the credit risk in a derivatives contract, the parties can add a

*credit support annex*(CSA), requiring one or both parties to post collateral. Valuation under CSA with DVA is a nontrivial research topic.

** Collateral Valuation Adjustment (CollVA): **Collateral is an asset that is passed (without ownership changing) from one party to another, as a means to reduce counterparty default risk. For a collaterized trade, in the event of default, the non-defaulting party may seize the collateral and use it to offset any losses. Posting collateral against a derivatives position significantly alters the credit risk and funding profile of that position. A perfectly collateralised derivative has no credit risk, although in practice this is rare due to the particulars of collateral posting. CollVA captures this cost or benefit, and is closely related to funding. Detailed insight in hedging under Counterparty Credit Risk (CCR) and collateral posting is important in this EID, as well as optimizing the posting of collateral.

** Funding Valuation Adjustment (FVA): **FVA represents the cost of funding uncollateralised OTC derivatives. It can be interpreted as a cost associated to the hedge of market risk of an uncollateralized transaction. FVA is divided into Funding Benefit Adjustment (FBA) and Funding Cost Adjustment (FCA). A funding benefit arises for a party when it acquires a derivative with a negative market value (a liability). In this case, the party will accept this liability in exchange for cash. This amount of cash can be used to fund other transactions, in lieu of raising external funding. A funding cost arises in the opposite scenario, i.e. when the derivative has a positive market value. In the representative formula, the funding spread, i.e. the net costs for lending or borrowing the collateral, and the joint survival probability require accurate modelling. FVA may appear in the hedge portfolio that is set up within this EID.

** Capital Valuation Adjustment (KVA): **Basel III requires banks to hold counterparty risk-related regulatory capital against OTC derivative transactions, in order to be able to absorb unexpected losses when markets are in stress. KVA represents the expected cost of holding capital which will be consumed by a trade over its life time. There are still concerns on how KVA should be computed and managed. Under the IMM approach, the CCR capital for a portfolio depends on the future exposure profile.

*Expected exposure*is the time conditional expected value, under the real world measure (P), of the positive exposure. It would involve simulation of market risk factors under P and computing the risk- neutral portfolio value for each of these market states, under Q measure. This is a highly nontrivial modelling and computational task.

** Margin Valuation Adjustment (MVA): **Since September 2016, banks are required to post initial margin for bilateral trades. The cost of posting the initial margin over the length of the contract is known as MVA. The aggregate value of the initial margin worldwide is estimated to exceed €250 Billion. It is crucial to accurately capture a portfolio’s lifetime funding cost of the initial margin. A single calculation for this involves thousands of portfolio

*sensitivities*, so that a direct simulation is not feasible. However, a novel application of adjoint algorithmic differentiation (AAD) to calculate forward portfolio sensitivities makes a direct forward simulation attainable. This technique will be developed, as it is transparent and maintains consistency with other XVA’s.

We will develop a **hedge framework **in which a credit default event should be hedged, with collateralized and uncollateralized OTC transactions. The difference between the value of a (default risky) portfolio with (XVA) valuation adjustments and a risk-free portfolio is ** Total Valuation Adjustment (TVA)**, TVA = CVA + DVA + CollVA + KVA + MVA + FVA. Accurate modelling under WWR and skew/smile volatility dynamics and efficient computation make this a highly relevant project for the industry, as well as for the academic research.