Portfolio KVA: I Theory

Abstract

XVA models for the calculation of CVA, FVA (see for example (Burgard and Kjaer 2013)), KVA(Green, Kenyon, and Dennis 2014), MVA (Green and Kenyon 2014) and TVA (Kenyon and Green 2014a) have frequently been formulated at the counterparty level. However, it is clear that some elements of the Regulatory Capital Framework are calculated at entity level rather than per counterparty (BCBS-128 2006; BCBS-189 2011) and hence the corresponding KVA term includes elements that are calculated across all counterparties. The Leverage Ratio for example, is clearly a function of all bank positions. Furthermore, in the KVA model of (Green, Kenyon, and Dennis 2014) capital was assumed to be readily available in any quantity at treasury level to be allocated down to individual businesses in return for the payment of a return\gamma_K. In practice, however, the central capital pool cannot be increased or decreased on demand. Additional equity capital is normally obtained through a rights issue, a relatively rare an expensive operation for a company, while equity capital can be returned to shareholders through stock buy backs, although these operations are highly regulated. The cost of capital at portfolio level should depend on the overall capital requirement in some way. Furthermore the act of trading and hedging can no longer be done at single counterparty level. A trade and hedge as a pair will imply two counterparties and capital changes due to both transactions. The hedge attracts capital and hence to hedge to overall portfolio will require an iterative process to neutralise the risk of the whole portfolio including KVA across all counterparty positions. Hence there are entity level aspects of XVA that have hitherto been ignored. This paper rectifies this problem by extending the Burgard-Kjaer semi-replication framework to multiple counterparties and multiple underlying assets, and allowing for entity-level capital requirements and effects. The work is presented in two parts, this paper covering theoretical and algorithmic developments and paper II covering practical examples and impact analysis.

Keywords: KVA, CVA, FVA, TVA, XVA, Leverage Ratio, Regulatory Capital, Return on Capital, Basel II, Basel III, CRD IV, Regulation, Pricing, Derivative Valuation

JEL Classification: E22, G11, G12, G13, G18, G21, G28, G31, G38, H77, K23, L51, M20, M41