# The Impact of Discounting on Asset Swap Levels

In Chapter 2 we discussed at great length the role of discounting. In Section 2.4.3 in particular we tried to gauge the impact of discounting on the value of a swap. We have concluded that a par swap (which we have also called at the money) is affected less than an out-of-the-money swap by a change in discount curve. We could ask the same question in relation to the asset swap spreads.

If an entity is issuing a bond and the bond price will result in a corresponding funding level/asset swap spread, wouldn't the choice of discounting for the swap part of the structure (i.e., the left-hand side of Equation 6.1) have an impact on the asset swap spread itself?

Let us imagine that, following the lead of the major financial institutions and clearinghouses in which all swaps are discounted with overnight index swaps, Equation 6.1 would be rewritten as

(6.4)

where indicates a discount factor driven by the overnight index swap and indicates a discount factor driven by the LIBOR (we have left the correctionuntouched since it is a credit-driven correction). The spread is the new asset swap spread, which could be different from *s*A. From Equation 6.4 we notice something that is important, perhaps obvious, but often forgotten when the reasons behind a choice of discounting is forgotten: the move toward OIS discounting only affects swaps and leaves bonds unchanged. OIS discounting is a discounting that takes into account the posting of collateral, in the case of bonds there is of course no such thing. To be precise we should not be writingeither because, as we discussed at great length when dealing with bond cash flow discounting, it does not make much sense to try to put too precise a label on the interest rate component of a bond discount factor. If we need to break the unity of such a discount factor it is better to be vague and say that there is some interest rate element and some credit element contributing to it. In the above we wrote simply to stress that it is definitively not an OIS-driven discount.

In Equation 6.4, is identical to *D*i in Equation 6.1; we have only written it differently to stress its origin. This means that the right-hand side of the two equations is the same, which is comforting since it corresponds to a traded price, that is, the bond's price. However, the left-hand sides of the two equations, although overall they need to equal each other (by being equal to the right-hand sides), they are not the same in essence because of the different choice of discounting. In each we have only one free parameter, the asset swap spread, and the structural difference between the two might mean that the two asset swap spreads, sA and , might be different.

We have already mentioned the interesting discussion on asset swap sensitivities found in O'Kane [66]. Here it is slightly different because we are not wondering what the effect of interest moves is on the swap level, rather we are decoupling the index and discount curve and, holding the former constant, we ask ourselves what the impact is on the asset swap spread of a change in the latter. It turns out the impact of discounting on the asset swap spread is not very significant. If we assume that the asset swap we enter into is at the money (which is not always the case in a normal asset swap but it is often the case in the asset swap made against a bond in the type of transaction carried out by a treasury desk) then the impact of discounting is limited as it is for all at-the-money swaps.

By estimating the sensitivity to a change in discount factor in the left- hand side of 6.4, we show this in detail in Appendix E.