Discounting Revisited: Valuations under funding costs, counterparty risk and collateralization

by Christian P. Fries of DZ Bank AG

May 30, 2010

Abstract: Looking at the valuation of a swap when funding costs and counterparty risk are neglected (i.e., when there is a unique risk free discounting curve), it is natural to ask "What is the discounting curve of a swap in the presence of funding costs, counterparty risk and/or collateralization".

In this note we try to give an answer to this question. The answer depends on who you are and in general it is "There is no such thing as a unique discounting curve (for swaps)." Our approach is somewhat "axiomatic", i.e., we try to make only very few basic assumptions. We shed some light on use of own credit risk in mark-to-market valuations, giving that the mark-to-market value of a portfolio increases when the owner's credibility decreases.

We present two different valuations. The first is a mark-to-market valuation which determines the liquidation value of a product. It does, buy construction, exclude any funding cost. The second is a portfolio valuation which determines the replication value of a product including funding costs.

We will also consider counterparty risk. If funding costs are presents, i.e., if we value a portfolio by a replication strategy then counterparty risk and funding are tied together:

• In addition to the default risk with respect to our exposure we have to consider the loss of a potential funding benefit, i.e., the impact of default on funding.
• Buying protection against default has to be funded itself and we account for that.

JEL Classification: C15, G13.

AMS Classification: 68U20.

Keywords: Discounting, Valuation, Counterparty Risk, Collateral, Netting, CVA, DVA, Bond, Swap, Credit Risk, Own Credit Risk, Wrong Way Risk, Liquidity Risk.

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