  We discuss the binary nature of funding impact in derivative valuation. Under
some conditions, funding is either a cost or a benefit, i.e., one of the
lending/borrowing rates does not play a role in pricing derivatives. When
derivatives are priced, considering different lending/borrowing rates leads to
semi-linear BSDEs and PDEs, and thus it is necessary to solve the equations
numerically. However, once it can be guaranteed that only one of the rates
affects pricing, linear equations can be recovered and analytical formulae can
be derived. Moreover, as a byproduct, our results explain how debt value
adjustment (DVA) and funding benefits are dissimilar. It is often believed that
considering both DVA and funding benefits results in a double-counting issue
but it will be shown that the two components are affected by different
mathematical structures of derivative transactions. We find that funding
benefit is related to the decreasing property of the payoff function, but this
relationship decreases as the funding choices of underlying assets are
transferred to repo markets.
