Capital Valuation Adjustment Research Articles

Structured XVAs: Diffusive Portfolios

In this paper, a continuous-time, structural model of a dealer-bank is presented to derive fair value equations for credit-risky financial products that are not perfectly hedged. The impact these contracts has on the dealer-bank's earnings volatility and, consequently, their solvency and financing costs, is taken into account. Explicit relationships between credit, debit, funding, and capital valuation adjustments (CVA, DVA, FVA, and KVA, respectively) are established, highlighting the interdependencies between unhedged credit risk and financing adjustments. To illustrate the practical application of the model, several straightforward numerical examples are provided.

The Two KVAs

The capital valuation adjustment (KVA) is one of the more recent additions to the XVA family. In this article, we examine two key questions surrounding KVA: What is the correct return on equity (RoE) that should be assigned to a derivatives portfolio and what impact should changes in a firm's equity capital levels or leverage have on valuation? Addressing these questions leads naturally to the definition of two distinct valuation adjustments we refer to as KVA1 and KVA2. KVA1 is a reflection of unpriced risk which can be present in incomplete markets. At the firm level it is proportional to the RoE but insensitive to changes in leverage. KVA2 on the other hand represents the shareholder cost of changes in leverage and is proportional to capital levels. However, the effective rate on the capital is the form's junior funding rate and not the RoE. This article is a high-level summary of two more detailed recent papers by the author.

Accounting mechanism for information signals about the imposition of sanctions in valuation of a company

Investors have to review regularly their forecasts and stock value estimates taking into consideration the imposition of sanctions against Russian companies. Due to the presence of behavioral effects, as well as to the incorrect accounting of incoming information, the risk of obtaining inaccurate stock value estimates increases. In this regard, a mechanism should be developed for adequate update of stock value estimates upon receipt of new information signals about the imposition of sanctions. The study is based on such methods as analysis, synthesis, the longitudinal method, as well as the method of discounted cash flows. The study is based on the dynamics of financial indicators of stock returns of the 40 most marketable Russian companies. A mechanism is proposed for accounting the informational signals about sanctions, which makes it possible to avoid behavioral effects and update adequately the intrinsic stock value estimates upon receipt of new information on the imposition of sanctions. This mechanism makes it possible to consider both changes in expectations regarding cash flows and changes in the cost of capital for companies. At the same time, the research in this field should be continued in order to develop a methodology for equity capital valuation adjustment that will be instrumental in consideration of variance in bias and emergence of the so-called “thick tails” in the distribution of stock returns under receipt of information signals on the imposition of sanctions.

KVA is Incomplete

In this article we derive a capital valuation adjustment for derivatives transactions due to market incompleteness. This is motivated by the fact that a return on equity (RoE) in excess of the riskless rate is the result of undiversifiable portfolio risk.The valuation adjustment represents the value of the excess return over the portfolio lifetime. More specifically, we show that the value of a derivative can be written as the sum of a risk-neutral base value and an adjustment which is the sum of a KVA and a FVA term.The KVA is proportional to the total shareholder equity as well as the RoE. We show that the RoE is function of the market price of the unhedgeable risk as well as the firms leverage. Our valuation adjustment is fundamentally risk based and has an alternative representation directly in terms of the residual diffusive and jump risk. This allows for a rigorous framework for allocating capital to sub-portfolios and individual trades.

KVA as a Transfer of Wealth

In this article we derive the shareholder loss due to a capital requirement associated to a derivatives transaction. This is a result of a transfer of wealth between shareholders and creditors of the firm. The charge required to negate this loss can be regarded as a capital valuation adjustment which we refer to as KVA2. Our approach does not assume a fixed hurdle rate on equity required by shareholders. Instead we derive the economic return on capital for a marginal derivatives trade. We provide two complementary derivations of the valuation adjustment. The first is based on a Merton single-period balance sheet model and the second on continuous time no-arbitrage arguments. Our resulting KVA expression is similar in structure to those proposed in the literature. We find however that the effective rate on capital that a shareholder should demand in a derivatives transaction is a junior funding rate as opposed to the return on equity. This is a consequence of the fact that the only risk a shareholder faces once a derivatives transaction is fully hedged is the default of the firm itself.

KVA, Mind Your P's and Q's!

There is an increasing consensus within banks on the need to recognize the impact of rising capital requirements on their derivative business in the form of capital valuation adjustment (KVA). There are still concerns over how exactly KVA should be calculated, charged, and managed. We analyze the aspects of the costs arising due to holding of counterparty credit risk capital. This study demonstrates that banks which are careless with their KVA method and ignore actively managing it, can rake a significant loss and will eventually increase uncertainty in the shareholders' return on capital.

Stochastic approximation schemes for economic capital and risk margin computations

We consider the problem of the numerical computation of its economic capital by an insurance or a bank, in the form of a value-at-risk or expected shortfall of its loss over a given time horizon. This loss includes the appreciation of the mark-to-model of the liabilities of the firm, which we account for by nested Monte Carlo à la Gordy and Juneja [17] or by regression à la Broadie, Du, and Moallemi [10]. Using a stochastic approximation point of view on value-at-risk and expected shortfall, we establish the convergence of the resulting economic capital simulation schemes, under mild assumptions that only bear on the theoretical limiting problem at hand, as opposed to assumptions on the approximating problems in [17] and [10]. Our economic capital estimates can then be made conditional in a Markov framework and integrated in an outer Monte Carlo simulation to yield the risk margin of the firm, corresponding to a market value margin (MVM) in insurance or to a capital valuation adjustment (KVA) in banking parlance. This is illustrated numerically by a KVA case study implemented on GPUs.

CCR KVA Relief Through CVA: A Regression-Based Monte Carlo Approach

We present and examine, by example of a USD interest rate swap and a EUR/USD cross-currency basis swap, a regression-based Monte Carlo approach to counterparty credit default risk (CCR) capital and CCR capital valuation adjustment (KVA) calculations [assuming the standardized approach to counterparty credit risk for exposure-at-default (SA-CCR EAD) and the internal ratings-based (IRB) approach for CCR risk weights]. This approach allows to incorporate the capital lowering effect of credit valuation adjustment (CVA) in an efficient manner, without having to resort to lengthy nested Monte Carlo simulations. We find that the regression-based Monte Carlo approach works well in most situations. In other situations, the accuracy of the approach is sensitively controlled by the choice of explanatory variables. We discuss in detail the conditions and underlying dynamics under which this happens. In computing and presenting a selection of numerical examples, we also explore the impact of dynamic CCR risk weights on CCR KVA, and compare regression-based CCR KVA results with CCR KVA results from nested Monte Carlo, alternative frequently used CCR KVA simplifications, and standardized CVA KVA.

CCR KVA Relief Through CVA: A Regression-Based Monte Carlo Approach

We present and examine, by example of a USD interest rate swap and a EUR/USD cross-currency basis swap, a regression-based Monte Carlo approach to counterparty credit default risk (CCR) capital and CCR capital valuation adjustment (KVA) calculations [assuming the standardized approach to counterparty credit risk for exposure-at-default (SA-CCR EAD) and the internal ratings-based (IRB) approach for CCR risk weights]. This approach allows to incorporate the capital lowering effect of credit valuation adjustment (CVA) in an efficient manner, without having to resort to lengthy nested Monte Carlo simulations. We find that the regression-based Monte Carlo approach works well in most situations. In other situations, the accuracy of the approach is sensitively controlled by the choice of explanatory variables. We discuss in detail the conditions and underlying dynamics under which this happens. In computing and presenting a selection of numerical examples, we also explore the impact of dynamic CCR risk weights on CCR KVA, and compare regression-based CCR KVA results with CCR KVA results from nested Monte Carlo, alternative frequently used CCR KVA simplifications, and standardized CVA KVA.

KVA from the beginning

Understanding the interaction between a new derivative, its financing and the wider balance sheet during pricing is critical for dealer profitability. For this purpose we extend a single period structural balance sheet model developed in Andersen, Duffie and Song to include equity financing before deriving consistent firm and shareholder break even prices. The former is given by the risk neutral expectation of the derivative cash flow in all scenarios including dealer default, discounted at the risk-free rate. The latter on the other hand is given by a dealer survival measure expectation of the derivative cash flow, including at counterparty default, discounted at the dealer weighted average rate of capital. This discounting rate is partially determined by the minimum amount to equity funding required by regulators. We next show how the shareholder break even price can be split into funding and capital valuation adjustments. All results are valid whether the dealer chooses to hedge the derivative or not.

An Indifference Approach to the Cost of Capital Constraints: KVA and Beyond

The strengthening of capital requirements has induced banks and traders to consider charging a so called capital valuation adjustment (KVA) to the clients in OTC transactions. This roughly corresponds to charge the clients ex-ante the profit requirement that is asked to the trading desk. In the following we try to delineate a possible way to assess the impact of capital constraints in the valuation of a deal. We resort to an optimisation stemming from an indifference pricing approach, and we study both the linear problem from the point of view of the whole bank and the non-linear problem given by the viewpoint of shareholders. We also consider the case where one optimises the median rather than the mean statistics of the profit and loss distribution.

Credit, funding, margin, and capital valuation adjustments for bilateral portfolios

We apply to the concrete setup of a bank engaged into bilateral trade portfolios the XVA theoretical framework of (Albanese and Crépey2017), whereby so-called contra-liabilities and cost of capital are charged by the bank to its clients, on top of the fair valuation of counterparty risk, in order to account for the incompleteness of this risk. The transfer of the residual reserve credit capital from shareholders to creditors at bank default results in a unilateral CVA, consistent with the regulatory requirement that capital should not diminish as an effect of the sole deterioration of the bank credit spread. Our funding cost for variation margin (FVA) is defined asymmetrically since there is no benefit in holding excess capital in the future. Capital is fungible as a source of funding for variation margin, causing a material FVA reduction. We introduce a specialist initial margin lending scheme that drastically reduces the funding cost for initial margin (MVA). Our capital valuation adjustment (KVA) is defined as a risk premium, i.e. the cost of remunerating shareholder capital at risk at some hurdle rate.

KVA, Mind Your P's and Q's!

There is an increasing consensus within banks, on the need to recognize the impact of rising capital requirements on their derivative business in the form of capital valuation adjustment (KVA). However, because of varied reasons, there are still concerns over how exactly KVA should be computed, charged, and managed. The focus of our analysis here is on the numerical aspects of costs arising due to holding of counterparty credit risk (CCR) capital. CCR capital for a portfolio today under the internal model method (IMM) approach is based on its future exposure profile, usually computed by Monte Carlo methods. For the corresponding KVA, we need (outer) Monte Carlo simulation of the future capital, which ideally would then involve (inner) Monte Carlo simulation. Additionally, the measures under which the outer and inner scenarios should be simulated can be different, ideally real-world (P) in risk-neutral (Q). In this work we, first, propose the use of stochastic grid bundling method (SGBM), an American Monte Carlo based computational technique to circumvent the problem of nested simulation working under hybrid measures to efficiently solve the P-in-Q problem. Second, we demonstrate that hybrid measures can substantially affect the KVA values. Third, through a simple hedging example we demonstrate the implication of hedging KVA under various approximations and hybrid measures.

Practical Applications of A Conversation with Danske Bank’s Jesper Andreasen

The world of quantitative finance is facing new demands on its tools, technologies and talents. Driven by regulatory requirements, banks must take capital requirements and valuation adjustments seriously, but the solutions to many of the new mandates are time consuming and labor intensive. Quantitative talent and technology can help, argues Jesper Andreasen of Danske Bank . In a conversation with Institutional Investor Journals , Andreasen offers his views on the challenges and opportunities created by an increasingly compliance-driven environment. He is speaking on the subject of capital valuation adjustments (KVA) and time series estimation of stochastic models at Global Derivatives .

Uncollateralised Interest Rate Swap xVA

Derivative valuation adjustments (xVA) have been an evolving topic over the past decade, now giving us a rich theory of what was traditionally referred to as the relatively opaque "credit spread" imposed by banks. As independent risk advisors, we are regularly positioned between clients and their banks, considering the impacts of both parties' xVA when entering into and terminating positions. This paper documents our implementation of xVA modelling at the time of writing, specifically for the derivatives which our clients trade the most: uncollateralised interest rate swaps. It is also one of the first open demonstrations of an independent party modelling banks' xVA at levels actually observed by them. We focus on the most significant credit (CVA), funding (FVA) and capital (KVA) valuation adjustments. Our full modelling framework and its calibration are described and examples are given for swaps under the Standardised Approach (SA-CCR) capital framework, effective 1 January 2017.

Capturing Initial Margin in Counterparty Risk Calculations

This paper compares a range of alternative approaches to incorporate Initial Margins (IMs) in the modelling of counterparty credit risk exposures. With the rise of Central Counterparties to clear OTC derivatives and the incoming legislation requiring bilateral margining for uncleared derivatives between financial counterparties the reflection of IMs becomes an essential model component that drives exposures, associated regulatory capital requirements and valuation adjustments such as CVA and MVA. The influence of the modelling choices is explored by means of typical derivatives portfolios. For the actual estimation of a path-dependent (“stochastic”) IM through time the use of quantile regression is suggested as an econometrically reliable approximation. Banks’ internal counterparty risk models will likely exhibit a basis vis-a-vis the actual IM mechanisms in practice (for example, owing to different risk factor representations and/or calibrations). In this context, the paper suggests that a simplified representation in the form of a “dynamic IM” can approximate most of the quantities of interest to a reasonable degree.

Gap Risk KVA and Repo Pricing: An Economic Capital Approach in the Black-Scholes-Merton Framework

Although not a formal pricing consideration, gap risk or hedging errors are the norm of derivatives businesses. Starting with the gap risk during a margin period of risk of a repurchase agreement (repo), this article extends the Black-Scholes-Merton option pricing framework by introducing a reserve capital approach to the hedging error's irreducible variability. An extended partial differential equation is derived with two new terms for expected gap loss and economic capital charge, leading to the gap risk economic value adjustment and capital valuation adjustment (KVA) respectively. Practical repo pricing formulae is obtained showing that the break-even repo rate decomposes into cost of fund and economic capital charge in KVA. At zero haircut, a one-year term repo on main equities could command a capital charge as large as 50 basis points for a 'BBB' rated borrower.

Capital Valuation Adjustment and Funding Valuation Adjustment

Capital Valuation Adjustment and Funding Valuation Adjustment

Accounting for KVA Under IFRS 13

We explore the accounting view on KVA for derivatives, where KVA is defined as the lifetime cost of capital, i.e. the Capital Valuation Adjustment. Accounting uses ‘fair value’ in most circumstances. IFRS 13 (Fair Value Measurement) defines fair value and it explicitly replaces previous requirements in individual standards. This fair value is based on financial management and management accounting models and concepts. These concepts are embodied in economic decision making models that use discounting, relevant cash flows, and any incremental changes consequent upon the project or investment. These economic decision making models naturally include the cost of allocating scarce capital that cannot then be used elsewhere. In short, IFRS 13 specifically requires preparers of accounts to use economic models. In this case the project or investment is the derivative, and regulations (e.g. Basel III) set minimum capital requirements. Thus by IFRS 13 accounts should reflect the costs of allocated capital, i.e. KVA.

KVA: Capital Valuation Adjustment

Credit (CVA), Debit (DVA) and Funding Valuation Adjustments (FVA) are now familiar valuation adjustments made to the value of a portfolio of derivatives to account for credit risks and funding costs. However, recent changes in the regulatory regime and the increases in regulatory capital requirements has led many banks to include the cost of capital in derivative pricing. This paper formalises the addition of cost of capital by extending the Burgard-Kjaer (2013) semi-replication approach to CVA and FVA to include an addition capital term, Capital Valuation Adjustment (KVA). The utilization of the capital for funding purposes is also considered. The use of the semi-replication approach means that the exibility around the treatment of self-default is carried over into this analysis. The paper further considers the practical calculation of KVA with reference to the Basel II (BCBS-128 2006) and Basel III (BCBS-189 2011) capital regimes and their implementation via CRD IV (EU 2013b; EU 2013a). The paper also assesses how KVA may be hedged, given that any hedging transactions themselves lead to regulatory capital requirements and hence capital costs. Finally a number of numerical examples are presented to gauge the cost impact of KVA on vanilla derivative products.