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The theoretical foundations of XVAs

The theoretical foundations of XVAs

Bloomberg analyses the theoretical basis of valuation adjustments – known collectively as XVAs – focusing in particular on the works and findings of its head of quantitative XVA analytics, Mats Kjaer, who emphasises the role of the capital valuation adjustment as a major driver of derivatives trading profitability, and stresses the importance of dealer firms calculating and managing it

Mats Kjaer, Bloomberg
Mats Kjaer, Bloomberg

The past decade has been a time of fundamental change in the over-the-counter derivatives markets. Changes in market structure and regulatory requirements have significantly increased the funding, margin and capital requirements for these businesses, and there is now general acceptance of the importance of quantifying the associated costs into pricing and valuation. The use of an expanding series of XVAs has consequently become standard practice in derivatives trading.

As market use of these XVAs has become more extensive, research into their theoretical foundations by quantitative analysts has proceeded in parallel. Initial research focused primarily on the modelling of debt funding via funding valuation adjustments (FVA) and there were significant debates on that topic early in this decade. As modelling approaches to FVA have become broadly settled – and as the impact of the increased capital costs introduced in Basel III has become more pertinent – research focus has turned in recent years to similarly investigating the theory and costs of this regulatory capital through capital valuation adjustments (KVA).

In the balance redux, a paper by Mats Kjaer, head of quantitative XVA analytics at Bloomberg, aims to further general understanding of this area. The paper, which derives its title from the 2011 paper In the balance, co-authored with Christoph Burgard, addresses the nature of regulatory capital, how it can be viewed as a cost to dealers and how it interacts with the other XVA adjustments. It provides a consistent framework that addresses capital in conjunction with funding and counterparty credit from a balance sheet perspective.

As Kjaer discussed in a podcast hosted by Risk.net’s quant finance editor, Mauro Cesa, the origin of this paper is in his work to add a KVA to Bloomberg’s Multi-Asset Risk System – or Mars – XVA platform, the enterprise XVA product originally released by Bloomberg in 2017. A KVA metric based on the research in the paper has recently been added to the platform.

As outlined in the podcast and the paper, the approach Kjaer has adopted for KVA differs from the semi-replication method he developed in his seminal article on FVA, Funding strategies, funding costs, co-authored with Christoph Burgard.2 While other researchers had built on this method in initial papers addressing KVA, Kjaer instead built upon the corporate finance-inspired balance sheet approach used by Leif Andersen, Darrell Duffie and Yang Song in their 2016 paper, Funding value adjustments, published in The Journal of Finance.1 As Kjaer emphasises in the podcast, the appeal of their approach was that it was very transparent and allowed them to derive both firm and shareholder breakeven prices for a new derivatives contract. In his paper, he extends the approach to include the features of regulatory capital, equity financing and hedging that Andersen et. al’s did not address.

Starting with a single period model – by analysing the impacts on a dealer balance sheet and cashflow statement of entering into a new transaction financed by a combination of debt and equity regulatory capital – Kjaer derives model-independent expressions for the marginal breakeven prices from the perspective of a firm and its shareholders. The firm breakeven price represents what an external party should pay to acquire a derivative itself and its financing. The shareholder breakeven price represents the amount that needs to be charged to ensure the shareholders are not worse off from entering into the transaction when they account for all the costs of financing the derivative, including regulatory capital. Kjaer’s derivation highlights the importance of only considering scenarios where the dealer survives in the calculation of this shareholder breakeven price. 

Through considering a specific financing strategy that involves the marginal regulatory capital requirement, Kjaer further derives an intuitive expression for the marginal shareholder breakeven price that explicitly includes a KVA. This KVA accounts for the opportunity cost of assets liquidated at the time of trading to enter into the transaction and the reduction in debt financing that occurs from the use of equity financing. He demonstrates how the choice of management method for the KVA – namely, whether it is released immediately, held on the balance sheet instead of debt or reserved in retained earnings – impacts the value of the KVA itself.

Kjaer extends his single period model to a continuous time version that makes it applicable in practice. It is shown that, by choosing as the management method to reserve the KVA in retained earnings, it is possible to discount the underlying capital profile at the relatively higher return on equity. Reassuringly, when the XVAs are managed by dynamic hedging, the shareholder breakeven price equals the cost of setting up the hedge. This cost may then be calculated using the semi-replication method. 

Kjaer provides numerical examples of shareholder breakeven prices for sample swaps, with counterparties of different credit quality under both margined and unmargined scenarios. These examples demonstrate that the regulatory capital costs can be significant when trading either on an unmargined basis or with risky counterparties, but – by reserving the KVA in retained earnings – it is possible to reduce these costs. Regardless, these costs are a significant driver of derivatives trading profitability and it is important they are calculated and managed by dealer firms.

As Kjaer emphasises in the podcast: “To be able to discount your very big capital requirements with the return on equity will shrink your KVA quite significantly”. The caveat, however, remains: “You can only do that if you actually reserve it on the balance sheet. So, no pain, no gain.”



1. L Andersen, D Duffie and Y Song (2019), Funding value adjustments, The Journal of Finance 74 (1), pp. 145–192.

Learn more

Listen to the podcast, Mats Kjaer on how trades affect the balance sheet

Learn more about Bloomberg’s XVA solution


XVA – Special report 2019
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