Counterparty Risk and CVA › Other Valuation Adjustments (XVA)

CVA was the first of many valuation adjustments that have transformed derivative pricing. The full family of "XVAs" includes several additional adjustments that capture costs and benefits beyond counterparty credit risk.

DVA is the mirror image of CVA; it represents the benefit to you if you default on your obligations. When your own credit spreads widen, your liabilities are worth less in present value terms. While controversial (you benefit from your own deteriorating credit), DVA is required under accounting standards and creates symmetric treatment of bilateral credit risk.

$$\text{DVA} = \text{LGD}_{\text{own}} \cdot \sum_{i=1}^{n} D(t_i) \cdot \text{ENE}(t_i) \cdot q_{\text{own}}(t_{i-1}, t_i)$$

where:

• $\text{LGD}_{\text{own}}$: your own Loss Given Default
• $\text{ENE}(t_i)$: Expected Negative Exposure at time $t_i$ (value to counterparty)
• $q_{\text{own}}(t_{i-1}, t_i)$: your own marginal default probability
• $D(t_i)$: discount factor

The formula mirrors CVA: it aggregates the present value of the debt relief ($\text{LGD}_{\text{own}} \cdot \text{ENE}$) you would receive if you defaulted, weighted by the probability of your own default in each period. This creates the paradox that a deterioration in your own credit quality increases DVA, improving your P&L, a feature that many find counterintuitive and has sparked ongoing debate in accounting and risk management circles.

FVA reflects the cost of funding uncollateralized derivative positions. If you have a positive exposure to a counterparty without collateral, you must fund that asset at your borrowing rate, which exceeds the risk-free rate. FVA captures this funding cost:

$$\text{FVA} = \sum_{i=1}^{n} D(t_i) \cdot \text{EE}(t_i) \cdot s_{\text{funding}} \cdot \Delta t_i$$

where:

• $s_{\text{funding}}$: funding spread above risk-free rates
• $\Delta t_i$: length of time interval $i$
• $\text{EE}(t_i)$: expected exposure requiring funding
• $D(t_i)$: discount factor

FVA recognizes that uncollateralized derivatives tie up capital that could otherwise be deployed elsewhere. The funding spread represents the premium you pay to borrow unsecured funds, and FVA accumulates this cost over the expected exposure profile of the trade.

KVA accounts for the cost of regulatory capital required to support derivative positions. Regulators require banks to hold capital against potential losses, and this capital has an opportunity cost:

$$\text{KVA} = \sum_{i=1}^{n} D(t_i) \cdot K(t_i) \cdot r_{\text{capital}} \cdot \Delta t_i$$

where:

• $K(t_i)$: required regulatory capital at time $t_i$
• $r_{\text{capital}}$: cost of capital (hurdle rate)
• $D(t_i)$: discount factor
• $\Delta t_i$: length of time interval $i$

For each period, the term $K(t_i) \cdot r_{\text{capital}} \cdot \Delta t_i$ represents the cost of holding the required regulatory capital. Summing the discounted values of these periodic costs yields the total KVA. This adjustment ensures that trades are priced to deliver adequate returns on the capital they consume.

For cleared trades and bilateral trades under new margin rules, initial margin must be posted. This margin has a funding cost captured by MVA, similar in structure to FVA but applied to margin requirements rather than exposure. Initial margin, unlike variation margin, does not offset exposure dollar for dollar; instead, it provides an additional buffer against potential losses during the closeout period. The funding cost of posting this margin must be incorporated into trade pricing.