XVA. Green Andrew

derivative valuation and the move away from the pure Black-Scholes-Merton framework. This is a controversial topic and will no doubt remain so for some time. The book can also be read as a practical guide to the calculation of valuation adjustments and it is therefore up to the reader what model elements are selected from those discussed.

      The book is organised into five main parts. Part I discusses models for counterparty credit risk and CVA, while Part II discusses FVA models as an extension to CVA model. The regulatory capital framework and KVA model are introduced in Part III. The implementation of XVA models is discussed in part IV and this section of the book is aimed to be a practical guide for those who are building bespoke internal models as well as those who may be buying third-party systems. Finally Part V discusses the management of XVA principally through active hedging programmes.

      PART One

      CVA and DVA: Counterparty Credit Risk and Credit Valuation Adjustment

      CHAPTER 2

      Introducing Counterparty Risk

      Take calculated risks. That is quite different from being rash.

– George S. PattonUS general (1885–1945)

      2.1 Defining Counterparty Risk

      Counterparty credit risk, sometimes known simply as credit risk or default risk, can be defined as the risk that a counterparty will fail to make payments that are due to another party. Consider a simple fixed rate loan in which the borrower makes annual fixed payments to the lender for five years before repaying the loan principal. The borrower defaults if they fail to pay any of the interest payments or the loan principal. The same is true for bonds and other securities, except the default will occur on payment to the securities holder. Derivatives also expose counterparties to credit risk. Some derivatives such as interest rate swaps involve bidirectional payments throughout their life which implies that both counterparties have default risk to each other, and that the direction of the risk changes during the life of the transaction.

      Counterparty credit risk is only present when one counterparty has an exposure to the other. The exposure at default (EAD)11 is the total amount owed by the defaulting party to the non-defaulting party,

(2.1)

where V is the total value. There is no exposure if the non-defaulting party owes the defaulting party money and this gives the max function in equation (2.1). If the exposure is positive and the defaulting party owes the non-defaulting party money, this value will form the basis of the claim the creditor will make against the defaulter through bankruptcy proceedings. The expected positive exposure12 (EPE) is the expected exposure of party A to their counterparty B at some future date with the expectation or average taken over all possible future outcomes on the date of interest as defined in equation (2.2).13

      The expected exposure profile is simply the expected exposure as a function of time. These exposure measures play a crucial role in the calculation of CVA. The expected negative exposure is the expected exposure that party B has to party A as seen from the perspective of A:

      (2.3)

The expected negative exposure enters the calculation of debit and funding valuation adjustments (DVA and FVA). The expected positive and negative exposure profiles for a five-year receivers interest rate swap are illustrated in Figure 2.1.

Figure 2.1 The expected positive and expected negative exposure profiles for a five-year interest rate swap with a notional of £100m at trade inception.

      Given the exposure at the time default occurs, the amount recovered is usually expressed as a percentage, R, the recovery rate.14 Hence the recovered amount is given by

      (2.4)

      The recovery rate is commonly used in the credit derivatives market. The loss-given-default or LGD is often used in traditional credit risk and is simply one minus the recovery rate,

      (2.5)

      2.1.1 Wrong-way and Right-way Risk

       Wrong-way risk is an important element of understanding credit exposure, although it is defined in a number of different ways. One traditional definition originating in credit risk modelling defines wrong-way risk as the risk that the exposure at default is larger than the exposure immediately prior to default. ISDA15 defines wrong-way risk in a much broader sense as occurring whenever the exposure to a counterparty is adversely correlated to the risk of the default of that counterparty (D’Hulster, 2001). This definition is the one most commonly applied in CVA modelling circles and is the one I use here. ISDA goes on to define specific wrong-way risk as associated with a particular transaction that has been poorly structured such as a repo agreement collateralised with the borrower's own bonds, and general or conjectural wrong-way risk as when credit quality and counterparty exposure are adversely linked through macro-economic factors. A simple example of this latter condition can be seen in the interest rate swaps market. A typical lending arrangement between a bank and a corporate counterparty involves a floating rate loan coupled with an interest rate swap in which the corporate pays fixed and receives a floating rate. The effective structure is then a fixed rate loan. In an economic recession interest rates typically fall at the same time that corporates experience a deterioration in their credit quality. The interest rate swap must be marked-to-market by the bank and as interest rates fall the value of the receiver swap increases, increasing the derivative exposure to the corporate. Hence this is an example of general wrong-way risk.

      Right-way risk is the inverse of wrong-way risk and occurs whenever exposure and credit quality are favourably correlated. A classic example of this is a trade with a gold producer in which the exposure is linked to the gold price. As the gold price increases the exposure increases; however, the value of gold reserves also increases and hence the credit worthiness of the gold miner also increases. A deeper investigation of the gold miner case study is revealing. If the gold miner has sold forward all the remaining reserves at a low price, the increasing gold price would have no impact on the credit worthiness of the gold miner. The transactions may be off balance sheet and not readily visible to external analysis. In practice assessing wrong-way and right-way risk is a complex process and the co-dependence between counterparty and macro-economic factors difficult to estimate.

      2.2 CVA and DVA: Credit Valuation Adjustment and Debit Valuation Adjustment Defined

      Credit Valuation Adjustment or CVA is the market price of credit risk on a financial instrument that is marked-to-market, typically an OTC derivative contract. Hence CVA is defined as the difference between the price of the instrument including credit risk and price where both counterparties to the transaction are considered free of credit risk,

(2.6)

Note that as defined in equation (2.6) CVA will always be positive if we consider only the credit risk of the

11

The regulatory EAD is a separate but related quantity that I will discuss later in Chapter 12.

13

is the filtration to time t. This is a technical element of stochastic calculus and the filtration simply represents all information such as asset prices up to time t. Where necessary I will include technical elements like this but they will be left out where appropriate and when it simplifies the presentation.

13

is the filtration to time t. This is a technical element of stochastic calculus and the filtration simply represents all information such as asset prices up to time t. Where necessary I will include technical elements like this but they will be left out where appropriate and when it simplifies the presentation.

15

International Swaps and Derivatives Association.