Discounting Revisited: Valuation Under Funding, Counterparty Risk and Collateralization

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 make very few basic assumptions. We shed some light on the use of own credit risk in mark-to-market valuations, where the mark-to-market value of a portfolio increases as 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, excluding our own 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 present, i.e. if we use a replication strategy to value a portfolio, then counterparty risk and funding are tied together: 1) 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. 2) Buying protection against default has to be funded itself and we account for that. The valuation naturally accounts for wrong-way-risk (i.e., the correlation between counterparty default and counterparty exposure).