  We compare two different bilateral counterparty valuation adjustment (BVA)
formulas. The first formula is an approximation and is based on subtracting the
two unilateral Credit Valuation Adjustment (CVA)'s formulas as seen from the
two different parties in the transaction. This formula is only a simplified
representation of bilateral risk and ignores that upon the first default
closeout proceedings are ignited. As such, it involves double counting. We
compare this formula with the fully specified bilateral risk formula, where the
first to default time is taken into account. The latter correct formula depends
on default dependence between the two parties, whereas the simplified one does
not. We also analyze a candidate simplified formula in case the replacement
closeout is used upon default, following ISDA's recommendations, and we find
the simplified formula to be the same as in the risk free closeout case. We
analyze the error that is encountered when using the simplified formula in a
couple of simple products: a zero coupon bond, where the exposure is
unidirectional, and an equity forward contract where exposure can go both ways.
For the latter case we adopt a bivariate exponential distribution due to Gumbel
to model the joint default risk of the two parties in the deal. We present a
number of realistic cases where the simplified formula differs considerably
from the correct one.
