Chapter 9: XVAs

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What Are XVAs?

X-Value Adjustments (XVAs) are adjustments to the "risk-free" value of a derivatives portfolio to account for various costs and risks. The base mid-market value (no-default value) is adjusted by:

$value−CVA+DVA−FVA−MVA−KVA\text{Actual value} = \text{No-default value} - \text{CVA} + \text{DVA} - \text{FVA} - \text{MVA} - \text{KVA}$

CVA (Credit Valuation Adjustment)

CVA is the cost of counterparty credit risk — the risk that the counterparty will default while the trade has positive value:

$CVA=∑i=1n(1−R)⋅qi⋅vi\text{CVA} = \sum_{i=1}^{n} (1 - R) \cdot q_i \cdot v_i$

where:

$R$ = recovery rate (typically 40%)
$q_i$ = risk-neutral probability of counterparty default at time $t_i$
$v_i$ = expected positive exposure at time $t_i$

CVA reduces the value of the derivatives position. The higher the counterparty's credit risk and the greater the expected exposure, the larger the CVA.

Wrong-Way Risk

When exposure and default probability are positively correlated. Example: An oil producer hedging by buying puts from a bank whose creditworthiness is linked to oil prices. If oil prices drop, the puts are more valuable but the bank is more likely to default.

Right-Way Risk

When exposure and default probability are negatively correlated.

DVA (Debit Valuation Adjustment)

DVA is the counterparty's CVA — the benefit from your own default risk:

$DVA=∑i=1n(1−R)⋅q~i⋅v~i\text{DVA} = \sum_{i=1}^{n} (1 - R) \cdot \tilde{q}_i \cdot \tilde{v}_i$

where $q~i\tilde{q}_i$ is your own default probability and $v~i\tilde{v}_i$ is the expected negative exposure (counterparty's positive exposure).

DVA increases the value of your portfolio — if you might default, your liabilities are worth less. This is controversial since no company can "monetize" its own default.

Bilateral CVA

Accounts for both parties' default risk:

$BCVA=CVA−DVA\text{BCVA} = \text{CVA} - \text{DVA}$

FVA (Funding Valuation Adjustment)

FVA adjusts for the cost of funding uncollateralized derivatives. Two components:

FCA (Funding Cost Adjustment): Cost of funding the initial margin and variation margin for uncollateralized or partially collateralized trades
FBA (Funding Benefit Adjustment): Benefit when receiving collateral

FVA became important after the 2008 crisis when banks' funding costs increased significantly above risk-free rates.

MVA (Margin Valuation Adjustment)

The cost of posting initial margin in cleared and non-cleared derivatives. When initial margin must be posted, it ties up capital that has a funding cost:

$trade\text{MVA} = \text{PV of expected funding costs on initial margin over life of trade}$

KVA (Capital Valuation Adjustment)

The cost of holding regulatory capital against derivatives positions throughout their life. Banks must hold capital against:

Counterparty credit risk
Market risk
Operational risk

$trade\text{KVA} = \text{PV of expected costs of regulatory capital over life of trade}$

Practical Issues

XVAs are interconnected — CVA itself requires regulatory capital, creating interaction between KVA and CVA
XVAs are computed at portfolio level (netting sets reduce exposure)
Collateral agreements significantly reduce CVA and FVA
Computing XVAs requires modeling future exposures across the entire portfolio — computationally intensive