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Account for the pieces that make up the X-Value Adjustment (xVA) term.

FRM® Part II · Credit Risk Measurement and Management · Counterparty Risk and Beyond · FRM2.CR.15.h

The idea

X-value adjustment, xVA, is the umbrella term for the adjustments that move a derivative's price away from its risk-free value. CVA is the cost of the counterparty's possible default. DVA is the benefit from the dealer's own possible default, because it would not pay its liabilities in full. Bilateral CVA is CVA minus DVA. FVA is the cost or benefit of funding uncollateralised positions, split into a funding cost adjustment and a funding benefit adjustment, and the formula uses their net. ColVA arises from collateral terms such as eligible assets and the interest paid on collateral. MVA is the funding of initial margin and KVA the cost of capital. Seen from the dealer, the portfolio is worth the amount below. Value = risk-free value − CVA + DVA − FVA − ColVA − MVA − KVA. Exam point: watch the signs, since DVA adds value while the others reduce it.

Formulas

Value of a derivative after xVA

Value = RFV − CVA + DVA − FVA − ColVA − MVA − KVA

RFV
the risk-free value
CVA
the cost of the counterparty's possible default
DVA
the benefit of the dealer's own possible default
FVA
the net funding cost of uncollateralised positions
ColVA
the value from the collateral agreement's terms
MVA
the cost of funding initial margin
KVA
the cost of capital

Watch the signs: DVA adds value and the others reduce it. A funding benefit adjustment, where it exists, offsets part of FVA.

Bilateral credit value adjustment

BCVA = CVA − DVA

CVA
the cost of the counterparty's possible default
DVA
the benefit of the dealer's own possible default

A wider own spread raises DVA and so lowers BCVA, giving an accounting gain from the dealer's own weaker credit.

A practice question

From the question bank, with the reason each wrong option is wrong.

Bilateral credit value adjustment (BCVA) is correctly expressed as:

  1. CVA plus DVA.Why not: Adds the two terms; DVA is a benefit and reduces the adjustment.
  2. DVA minus CVA.Why not: Reverses the sign; bilateral CVA is a net cost in the usual convention.
  3. CVA minus DVA. - correct
  4. CVA multiplied by DVA.Why not: The adjustments are additive, not multiplicative.

BCVA nets the counterparty's default cost (CVA) against the benefit of own default (DVA), so BCVA = CVA - DVA.

A harder one

A dealer's own credit spread widens sharply and nothing else changes. What is the effect on the DVA component of its derivative valuations?

  1. DVA falls, because its own credit quality has worsened.Why not: Reverses the direction; a higher own default probability raises DVA.
  2. DVA rises, which increases the reported value of the dealer's derivative book and gives an accounting gain. - correct
  3. DVA is unchanged, because DVA depends only on the counterparty's credit spread.Why not: DVA depends on the dealer's own credit spread; the counterparty's spread drives CVA.
  4. DVA becomes a cash loss, because DVA is a cost the dealer must pay to the counterparty.Why not: DVA is a valuation benefit and not a cash payment.

DVA increases with the dealer's own default probability, as its liabilities are worth less. The accounting gain is counter-intuitive but is the mirror of CVA.

The hardest tier

A dealer's xVA desk reports the following for a portfolio (USD million): CVA 4.0, DVA 1.5, funding cost adjustment 2.0, funding benefit adjustment 0.5, KVA 3.0 and MVA 1.0. Using value reduction = CVA - DVA + FCA - FBA + KVA + MVA, the total reduction in the portfolio's value is:

  1. USD 12.0 millionWhy not: Adds every item as a cost, ignoring that DVA and FBA are benefits.
  2. USD 9.0 millionWhy not: Treats the funding benefit adjustment as a cost (adds 0.5) instead of subtracting it.
  3. USD 4.0 millionWhy not: Includes only CVA and drops the funding, KVA and MVA adjustments, which are part of xVA.
  4. USD 8.0 million - correct

4.0 - 1.5 + 2.0 - 0.5 + 3.0 + 1.0 = 8.0. DVA and the funding benefit are offsets; the other items increase the cost.

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