libor vs ois discounting

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LIBOR vs OIS Discounting

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1 LIBOR vs. OIS: The Derivatives Discounting Dilemma1 John Hull and Alan White March, 2012 This version July 2012 ABSTRACT Traditionally practitioners have used LIBOR and LIBOR swap rates as proxies for risk-free rates when valuing derivatives. This practice has been called into question by the credit crisis that started in 2007. Many banks now consider that overnight indexed swap (OIS) rates should be used for discounting when collateralized portfolios are valued and that LIBOR should be used for discounting when portfolios are not collateralized. This paper critically examines this practice. 1 We are grateful to Shalom Benaim, Christian Channel, Raphael Douady, Andrew Green, Jonathan Hall, Nicholas Jewitt, Paul Langill, Massimo Morini, Vladimir Piterbarg, Donald Smith, Elisha Wiesel, and Jun Yuan for comments on earlier versions of this paper. 2 LIBOR vs. OIS: The Derivatives Discounting Dilemma 1. Introduction The risk-free term structure of interest rates is a key input to the pricing of derivatives. Academic research usually assumes that the government borrowing rate is the risk-free rate. However, in the United States, Treasury rates tend to be low compared with the rates offered on other very-low-credit-risk instruments. One reason for this is the tax treatment of Treasury instruments. (Income from Treasury instruments is exempt from tax at the state level.) This has led researchers such as Elton et al (2001) to calculate credit spreads relative to the Treasury rate and attempt to identify a number of components, one of which is tax component. Derivatives dealers have traditionally used LIBOR and LIBOR swap rates as proxies for the risk-free rate. They calculate a risk-free zero curve using LIBOR rates, Eurodollar futures rates, and swap rates. LIBOR rates are the short-term borrowing rates of AA-rated financial institutions. Collin-Dufresne and Solnik (2001) show that swap rates are continually refreshed LIBOR rates and carry the same risk as a series of short-term loans to AA-rated financial institutions. When swap rates are used to bootstrap the LIBOR curve the resultant rates, in normal circumstances, are those that would apply to relatively low-risk (but not zero-risk) lending. For example, the credit risk inherent in a 5-year swap rate is the credit risk in 20 successive three-month loans to AA-rated financial institutions. The use of LIBOR as a proxy for the risk-free rate was called into question by the credit crisis that started in mid-2007. Banks became increasingly reluctant to lend to each other because of concerns about the possibility of default. As a result LIBOR quotes started to rise. The TED spread, which is the spread between three-month U.S. dollar LIBOR and the three-month U.S. Treasury rate, is less than 50 basis points in normal market conditions. Between October 2007 and May 2009, it was rarely lower than 100 basis points and peaked at over 450 basis points in October 2008. 3 Most derivatives dealers now use the OIS rate rather than LIBOR as the discount rate when valuing collateralized portfolios. LCH.Clearnet, a major central clearing party for OTC transactions, has also switched from discounting at LIBOR to discounting at the OIS rate for interest rate swaps. The reason often given for using the OIS rate as the discount rate is that it is derived from the fed funds rate and the fed funds rate is the interest rate usually paid on collateral. As such the fed funds rate and OIS rate and are the relevant funding rate for collateralized transactions. For non-collateralized transactions most dealers continue to use LIBOR as the discount rate. The argument here is that these transactions are not funded by collateral and LIBOR is a better estimate of the banks cost of funding than the OIS rate for these transactions. In this paper we argue that the OIS rate should be used as the discount rate for both collateralized and non-collateralized transactions. The reason has nothing to do with the dealers cost of funding. The OIS rate should be used as the discount rate because it is the best proxy for the risk-free rate and the risk-free rate should be used as the discount rate when risk-neutral valuation is used. Should an alternative superior proxy arise, it should be used for discounting. When valuing a derivative or a portfolio of derivatives with a particular counterparty, the first stage is to calculate a no-default value by assuming that there is no chance that either the dealer or the counterparty will default. As we will explain, this may have to be adjusted in situations where the interest on cash collateral is different from the risk-free rate. The second stage is to take the credit risk of the dealer and the counterparty into account by making credit value adjustment (CVA) and debit value adjustment (DVA) calculations. The CVA is an estimate of the amount by which the value of its derivatives portfolio with the counterparty should be reduced to reflect the possibility of the counterparty defaulting during the life of the derivatives portfolio. The DVA is an estimate of the amount by which the value of the derivatives portfolio should be increased to reflect the possibility of the dealer defaulting during the life of the portfolio. The value reported in the accounts should be the no-default value less the CVA plus the DVA. In Sections 2 and 3, we present some preliminary institutional material on overnight rates. Section 2 explains how overnight money markets work and why the fed funds rate is a good 4 proxy for the one-day risk-free rate. Section 3 then explains how the OIS rate is calculated and why a zero curve calculated from OIS rates provides the best proxy for the risk-free zero curve. In Section 4, we review the way CVA and DVA are calculated and used. In Section 5 we consider the relevance of funding costs to derivatives valuation. Section 6 discusses the valuation of collateralized portfolios and discusses the how valuations should take account of the interest paid on cash collateral. Section 7 covers the valuation of non-collateralized portfolios. It argues that the OIS (risk-free) rate is the correct discount rate to use when the no-default value of these portfolios are calculated. Using LIBOR as the discount rate gives correct answers, but only if certain adjustments are made to the calculation of CVA and DVA. Interest rates are used for a number of purposes when a portfolio of derivatives is valued. Section 8 argues that the risk-free (OIS) rate should be used in all situations. Using LIBOR in some situations and the OIS rate in others is liable to cause confusion. Section 9 illustrates how this confusion can lead to incorrect pricing. Conclusions appear in Section 10. 2. Overnight Rates Banks can borrow money in the overnight market on a secured or unsecured basis. Overnight U.S. dollar secured debt can be raised in the form of an overnight repurchase agreement (a repo) or at the Federal Reserves Discount Window.2 Unsecured U.S. dollar overnight financing comes in the form of federal funds and Eurodollars. A large fraction of the federal funds loans are brokered. Major brokers report the dollar amount loaned at each interest rate to the Federal Reserve Bank of New York (FRBNY) daily.3 The statistics reported by the brokers are used by FRBNY to calculate the average interest rate paid on federal funds loans each day. This average is called the effective federal funds rate. The FRBNY controls the level of the effective federal funds rate through open market transactions. 2 For a discussion of the repo market see Stigum (1990) or Demiralp et al (2004). For a discussion of the Discount Window see Furfine (2004). 3 See Demiralp et al (2004). Most federal funds transactions take place at interest rates that are an integral multiple of either one basis point or one-thirty-second of one per cent. 5 Money market brokers do not report the statistics for Eurodollar financing as they do for federal funds. As a result, the only easily available measure of the cost of overnight borrowing in the Eurodollar market is the level of overnight LIBOR reported by the British Bankers Association. On average, overnight LIBOR has been about 6 basis points higher than the effective federal funds rate except for the tumultuous period from August 2007 to December 2008. Given the substitutability of Eurodollar and federal funds, financing the apparent difference between the rates seems difficult to explain. This issue is addressed by Bartolini et al (2008). These authors attribute the observed differences to timing effects, the composition of the pool of borrowers in London as compared to New York, market microstructure differences between the dominant settlement mechanisms in London (CHIPS) and New York (Fedwire),4 and the difference between transaction prices (the brokered trades) and quotes which just provide the starting point for a negotiation. The overnight rate, whether federal funds or overnight LIBOR, is a rate on unsecured borrowing and as such is not a risk-free rate. Longstaff (2000) and other authors argue that the overnight repo rate is a better indicator of the risk-free rate since the borrowing is collateralized. Certainly a secured loan is subject to less credit risk than an unsecured loan. However there is substantial cross-sectional variation in repo rates related to the type of collateral posted. Up to mid-2007, rates for repos secured by U.S. federal government securities were 5 to 10 basis points below the federal funds rate while for repos secured by U.S. agency debt the rates were about one basis point below the federal funds rate. During the crisis, the rate for repos secured by federal government securities fell relative to the federal funds rate, but for other repos the rate rose relative to this rate. These cross-s

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