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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.

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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.

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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 bank’s 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 dealer’s 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

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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 Reserve’s 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.

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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-sectional variations suggest that market microstructure issues

may play a larger role in explaining the difference between repo and federal funds rates than

credit risk does.5 As a result we believe that the repo rate is not a better measure of the risk-free

overnight rate than the effective federal funds rate.

4 CHIPS is the Clearing House Inter-Bank Payment System and Fedwire is the real time wire transfer system run by

the Federal Reserve 5 For example, it is possible that lenders in the repo market rather than making secured loans are using the market to

acquire ownership of securities that are otherwise difficult to acquire. See BIS Quarterly Review (December 2008).

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Ultimately, the magnitude of the credit spread in the effective federal funds rate is an empirical

question. A rough indication of the size of the credit spread can be inferred by looking at the

term structure of credit spreads based on the difference between USD LIBOR rates and the

Federal Reserve’s estimate of constant maturity Treasury bill rates for 1-, 3-, 6- and 12-month

maturities. In the period since January 2009 the shape of the term structure of credit spreads has

been consistent. The yearly average term structures of spreads are shown in Figure 1. Averaging

over all four years, the average difference between one-month USD LIBOR and one-month T-

bill rates is about 20 basis points while the average spread at one year is about 80 basis points.

Extrapolating this curve indicates a spread of about 11 basis points for the over-night LIBOR

rate. In the same period the average spread between over-night LIBOR and the effective federal

funds rate was about 6 basis points. This suggests that the credit spread for the effective federal

funds rate in this period is about 5 basis points, much of which can be attributed to non-credit

elements.

Our discussion of overnight rates has been couched in terms of U.S. dollar interest rates but it is

not unique to this currency. Similar overnight markets with similar characteristics exist in Euro,

Sterling and other major currencies.

3. Overnight Index Swaps

Overnight index swaps are interest rate swaps in which a fixed rate of interest is exchanged for a

floating rate that is the geometric mean of a daily overnight index rate. The calculation of the

payment on the floating side is designed to replicate the aggregate interest that would be earned

from rolling over a sequence daily loans at the overnight rate. In U.S. dollars, the index rate is

the effective federal funds rate. In Euros, it is the Euro Overnight Index Average (EONIA) and,

in sterling, it is the Sterling Overnight Index Average (SONIA).

OIS swaps tend to have relatively short lives (often three months or less). However, transactions

that last as long as five to ten years are becoming more common. For swaps of one-year or less

there is only a single payment at the maturity of the swap equal to the difference between the

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fixed swap rate and the compounded floating rate multiplied by the notional and the accrual

fraction. If the fixed rate is greater than the compounded floating rate, it is a payment from the

fixed rate payer to the floating rate payer; otherwise it is a payment from the floating rate payer

to the fixed rate payer. Similar to LIBOR swaps, longer term OIS swaps are divided into 3-

month sub-periods and a payment is made at the end of each sub-period.

In Section 1 we mentioned the continually-refreshed argument of Collin-Dufresne and Solnik

(2001). This shows that the 5-year swap rate for a transaction where payments are exchanged

quarterly is equivalent to the rate on 20 consecutive 3-month loans where the counterparty’s

credit rating is AA at the beginning of each loan. A similar argument applies to an OIS swap

rate. This rate is the interest that would be paid on continually refreshed overnight loans to

borrowers in the overnight market.

There are two sources of credit risk in an OIS. The first is the credit risk in fed funds borrowing

which we have argued is very small. The second is the credit risk arising from a possible default

by one of the swap counterparties. This possibility of counterparty default is liable to lead to an

adjustment to the fixed rate. The size of the adjustment depends on the slope of the term

structure, the probability of default by a counterparty, the volatility of interest rates, the life of

the swap, and whether the transaction is collateralized. The size of the adjustment is generally

very small for at-the-money transactions where the two sides are equally creditworthy and the

term structure is flat. It can also reasonably be assumed to be zero in collateralized transactions.6

Based on these arguments we conclude that the OIS swap rate is a good proxy for a longer term

risk-free rate.

The three-month LIBOR-OIS spread is the spread between three-month LIBOR and the three-

month OIS swap rate. This spread reflects the difference between the credit risk in a three-month

loan to a bank that is considered to be of acceptable credit quality and the credit risk in

continually refreshed one-day loans to banks that are considered to be of acceptable credit

quality. In normal market conditions it is about 10 basis points. However, it rose to a record 364

basis points in October 2008. By a year later, it had returned to more normal levels, but it rose to

6 Legislation requiring standard swaps to be cleared centrally means that swap quotes are likely to reflect

collateralized transactions in the future.

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about 30 basis points in June 2010 and to 50 basis points at the end of 2011 as a result of

European sovereign debt concerns.

The OIS zero curve can be bootstrapped similarly to the LIBOR/swap zero curve. If the zero

curve is required for maturities longer than the maturity of the longest OIS a natural approach is

to assume that the spread between the OIS zero curve and the LIBOR/swap zero curve is the

same at the long end as it is at the longest OIS maturity for which there is reliable data.

Subtracting this spread from the LIBOR zero curve allows it to be spliced seamlessly onto the

end of the OIS zero curve. In this fashion a risk-free term structure of interest rates can be

created.

4. CVA and DVA

Consider a portfolio of derivatives between a dealer and a counterparty. CVA is the expected

cost to the dealer of a default by the counterparty. DVA is the expected cost to the counterparty

of a default by the dealer. 7

In all situations the correct procedure for valuing the portfolio is to

first value it assuming neither side will default and then adjust for CVA and DVA. The value of

the portfolio to the dealer is

fnd − CVA+DVA (1)

where fnd is the no default value. This formula is proved rigorously by Burgard and Kjaer (2011)

who extend the hedging arguments of Black and Scholes (1973) and Merton (1973). As we

explain later some adjustments are necessary when the interest paid on collateral is different

from the risk-free rate.8

Some practitioners choose to subtract a funding value adjustment (FVA) from the value of the

portfolio. As we explain in the next section, this is not a practice we agree with. However, it does

not alter our main argument. This argument is that collateralized and non-collateralized

portfolios should be handled in the same way as far as equation (1) is concerned. (Of course,

CVA and DVA decrease as the level of collateralization increases.)

7 For a discussion of CVA and DVA, see Gregory (2009 ) or Hull and White (2012).

8 Adjustments for the interest paid on collateral has made it more difficult to transfer/novate derivatives

transactions. Work is underway to produce a standardized credit support annex for the industry.

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To calculate CVA and DVA, the life of the derivatives portfolio between the dealer and its

counterparty is divided into a number of time steps and the value of the derivatives portfolio is

simulated in a risk neutral world. On each simulation trial, the value of the portfolio to the dealer

and the collateral available to the dealer if there is a default by the counterparty are calculated at

the midpoint of each time step.9 This allows the net exposure conditional on default to be

calculated. The dealer’s credit cost is obtained by calculating the product of the probability of a

default and the present value of the net exposure for each time step, and then summing over all

time steps. The counterparty’s credit cost is calculated similarly by considering the dealer’s

probability of default and the counterparty’s exposure to the dealer. CVA is the average of the

dealer’s credit cost across all simulation trials; DVA is the average of the counterparty’s credit

cost across all simulation trials.

The exposure calculations take account of collateral posted by each side. Typically a “cure

period” is assumed. This is a period prior to default during which it is assumed that the

defaulting party posts no collateral and returns no collateral. For example, if the cure period is 20

days, and the collateral provided by defaulting party 20 days prior to a default is C, with a

negative C indicating that the non-defaulting party provided collateral to the defaulting party at

this time. The exposure of the non-defaulting party to the defaulting party is max(V − C, 0)

where V is the value of the portfolio to the non-defaulting party.

5. Funding Costs

In Hull and White (2012) we argued that funding costs should not be taken into account when

derivatives are valued. A dealer that funds existing transactions at the risk-free rate plus 100

basis points should, except for DVA adjustments, calculate similar values as a dealer that funds

at the risk-free rate plus 500 basis points.

The economic value is usually calculated using risk-neutral valuation where cash flows are

estimated in a risk-neutral world and discounted at the risk-free rate. It is important to note that

risk-neutral valuation is an artificial valuation procedure. It does not assume that the derivative

9 The collateral available typically depends on the value of the portfolio several days earlier when the counterparty is

assumed to stop paying collateral.

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can be funded at the risk-free rate. Like all valuation procedures, it implicitly assumes that the

incremental costs of funding the derivative correspond to its risks. When funding markets are

efficient this should always be the case.

Much to our surprise Hull and White (2012) proved to be quite controversial. Some practitioners

agree with us while others consider it correct to assume that an FVA cost reflecting the

difference between the dealers current funding costs and the risk-free rate should be included in

equation (1).

The arguments in this paper are related to the arguments in Hull and White (2012) but not

dependent on them. The key question considered in this paper is whether the discount rate for

collateralized transactions should be different from that for collateralized transactions. Whether

or not an FVA adjustment is made, risk-neutral valuation shows that the best proxy for the risk-

free rate should be used to determine no-default economic values for derivatives.

6. Collateralized Portfolios

We start by considering a portfolio that is subject to a CSA with

a) A two-way zero-threshold collateral agreement

b) No minimum transfer amounts

c) All collateral being required in the form of cash.

The value of the portfolio assuming no possibility of default by either side is calculated by

discounting risk-neutral cash flows using the risk-free (OIS) zero curve. If the interest paid on

cash collateral is the (risk-free) effective fed funds rate, this has no effect of the value of the

portfolio.

Suppose next that the rate paid on cash collateral is different from the effective fed funds rate.

Suppose the daily fed funds rate is fr and the daily rate paid on the collateral is .cr With the

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collateral arrangements we are considering, the situation is equivalent to one where a) the rate

paid on the collateral is fr and b) the derivatives portfolio provides a daily yield equal to fc rr .

This follows from the fact that from the point of view of each party, the value of the collateral

posted is always equal to the value of the derivative and in determining the economic value of

the derivative we act as if the derivative is expected to earn fr . The impact of the yield is to

reduce the return provided by the derivative by fc rr . This can be taken into account by

changing the discount curve from the OIS curve to one corresponding to .cr For example, if the

rate paid on collateral is x basis points below the fed funds rate, the discount curve should be x

basis points below the OIS curve. If a constant interest rate, y, is paid on collateral the discount

curve should be flat and equal to y.

Note that in the situation we are considering the impact of the interest rate on collateral being

different from fr can be considered on a deal-by-deal basis. When entering into a new transaction

a dealer does not need to take account of existing transactions with the counterparty. This is in

contrast to CVA and DVA which must be considered on a portfolio basis.

Our approach to calculating the no-default value is consistent with market practice. However, we

emphasize that our reason for using the approach is different from the reason usually given by

market participants. Many market participants would argue that the discount rate should

correspond to the funding rate. We argue that, when risk-neutral valuation is used, the discount

rate should always be the risk-free (OIS) rate. As it happens in the situation we are considering,

the discount rate can be used to allow for the interest rate on cash collateral being different from

the risk-free rate.

We now consider the effect of relaxing assumptions a), b), and c) given above. If there is a

minimum transfer amount that is not too large the analysis should still be approximately true. If

collateral can be provided in the form of marketable securities as well as cash and the interest on

cash collateral, rc, is different from the risk-free rate, rf, a reasonable assumption is that cash

collateral will be provided when rc < rf and marketable securities will be provided when rf > rc.10

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It can be argued that rf should here be the securities repo rate, not the fed funds rate.

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When securities are posted as collateral and the firm posting the collateral is assumed to bear all

the risks associated with the securities, the cost of posting the collateral is zero. In a competitive

market the economic value of the assets posted equals their purchase price. This is true no matter

how large a haircut is applied to the posted collateral.

We now consider the situation where assumption a) is relaxed. Many different collateral

arrangements are observed in practice. For example, the threshold is sometimes non-zero;

collateral arrangements are sometimes one-way rather than two-way; sometimes an independent

amount is required. In all of these situations, the no-default value of a derivatives portfolio is

calculated using the risk-free (OIS) rate as the discount rate. The impact of rc being different

from rf can no longer be calculated on a deal-by-deal basis by adjusting the discount rate.

However, it can be conveniently calculated using Monte Carlo simulation at the same time as

CVA and DVA.

7. Non-Collateralized Portfolios

Non-collateralized portfolios, like collateralized portfolios, can be valued using equation (1). The

no-default value, fnd, should be calculated using the OIS rate as the discount rate. The role of

CVA and DVA is to adjust the no-default value of the derivative so that the impact of potential

defaults by the counterparty and the dealer is taken into account. (The discount rate used in the

calculation of CVA and DVA should also be the risk-free (OIS) rate of interest.)

Why then does the market continue to use the LIBOR rate for discounting when non-

collateralized portfolios are valued? Part of the reason may be a reluctance to move away from

pre-crisis practice for these portfolios. Another reason may be that non-collateralized trading is

often funded at LIBOR. (However, as explained in Section 5 funding issues should not affect the

discount rate used to calculate the no-default economic value.)

The use of LIBOR/swap rates for discounting has the effect of incorporating an adjustment for

credit risk into the discount rate. It cannot then be correct to calculate CVA and DVA using

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credit spreads for the counterparty and the dealer that reflect their total credit risk. If this were

done, there would be an element of double counting for credit risk.

The appendix shows that correct values are obtained if

a) The LIBOR/swap curve is used for discounting when the portfolio is valued;

b) CVA is calculated as the excess of the actual expected loss to the dealer from

counterparty defaults less the expected loss implicit in LIBOR/swap rates; and

c) DVA is calculated as the excess of the actual expected loss to the counterparty from a

default from a default by the dealer less the expected loss implicit in LIBOR/swap rates.

It is interesting to note that there is nothing special about the role of LIBOR/swap rates in this

result. Any yield curve can be used for discounting providing it is also used instead of

LIBOR/swap rates in b) and c) above. Note that the calculations in b) and c) are likely to give

approximately, but not exactly, the same result as using the LIBOR/swap curve as the risk-free

benchmark when credit spreads are calculated to the purposes of determining CVA and DVA.

A natural question is whether it is necessary to calculate CVA and DVA at all. Can we adjust for

credit risk by adjusting the discount rate? The appendix shows that this is possible in three

special cases. Specifically:

1. If portfolio will always have a positive value to the dealer, it can be correctly valued

using a discount curve determined from the counterparty’s borrowing rates.

2. If the portfolio will always have a negative value to the dealer, it can be correctly valued

using a discount curve determined from the dealer’s borrowing rates.

3. If the counterparty and dealer are equally creditworthy, any portfolio can be correctly

valued using a discount curve determined from their common borrowing rate.

The most widely traded derivative is a plain vanilla interest rate swap where LIBOR is

exchanged for a fixed rate. LIBOR/swap rates are used to determine expected payoffs on this

instrument. No asset prices are modeled. As a result, one of the attractions of using LIBOR

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discounting for non-collateralized trades may be that only one interest rate needs to be

considered when the trade is an interest rate swap. In reality however, this can only be an

attraction if a dealer’s portfolio with a counterparty consists only of interest rate swaps. Other

derivatives require interest rates to be used to define the risk-neutral growth rate of asset prices.

Also, risk-free (OIS) discount rates are necessary to calculate CVA and DVA correctly.

8. The Potential for Confusion

There are number of potential sources of confusion in the way discount rates are chosen. Interest

rates play two roles in derivatives valuation. They define asset returns in a risk-neutral world and

they are used for discounting expected payoffs. Traditional finance theory argues that the risk-

free rate should be used for both purposes. In the previous section we outlined an approach

where the LIBOR/swap curve is used for discounting and CVA and DVA are calculated in a

particular way. When this approach is adopted, it is tempting to also use the LIBOR/swap curve

to define the returns on assets in a risk-neutral world. This is not correct and is likely to lead to

lead to significant errors, particularly in stressed market conditions when the LIBOR-OIS spread

is large.

Setting this point aside, the use of the LIBOR/swap curve for discounting non-collateralized

portfolios can create a confusing situation for dealers when they develop systems for calculating

CVA and DVA. As explained in the previous section and the appendix, the correct calculation

methodology for CVA and DVA depends on whether OIS or LIBOR (or some other rate) has

been used as the discount rate. When LIBOR has been used as the discount rate, the calculation

of CVA and DVA should reflect the fact that some credit risk has already been taken into

account.

A further source of confusion lies in the discount rates used in CVA and DVA calculations.

These should be risk-free (OIS) rates even when the underlying portfolio is valued using the

LIBOR/swap rate as the discount rate. This means that the values that would be calculated by the

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CVA/DVA system may not correspond to the “no-default” values given by the company’s main

valuation system.

OIS rates must be used for discounting perfectly collateralized and partially collateralized

portfolios. It makes sense to use them for non-collateralized portfolios as well. The use of

LIBOR for discounting can in theory give the correct answer for non-collateralized transactions

if CVA and DVA are calculated in a particular way, but as has just been explained there are a

number of potential sources of confusion.

9. Numerical Results

To illustrate our results with a simple example, suppose that the dealer’s portfolio with the

counterparty consists of a non-collateralized forward contract to buy a non-dividend-paying

stock in one year. The stock price and delivery price are both $100, the volatility of the stock

price is 30% per annum, and the (OIS) risk-free rate is 3% for all maturities. We consider two

cases. Case 1 corresponds to what might be termed normal market conditions. LIBOR is 3.1%

for all maturities. The dealer’s and the counterparty’s adjusted credit spreads are 0.5% and 2.0%,

respectively, for all maturities. Case 2 corresponds to stressed market conditions. LIBOR is

4.5% for all maturities. The dealer’s and the counterparty’s adjusted credit are 2% and 3%,

respectively, for all maturities All rates are continuously compounded. As explained in the

appendix, the adjusted credit spread of a company is the credit spread it would have if its bonds

were treated in the same way as derivatives in the event of a default so that the claim equaled the

no-default value.

The integrals necessary to calculate CVA and DVA are evaluated by dividing the one year life of

the derivative into 200 equal time steps. Equations (A6), (A8) and (A10) in the appendix are

used to calculate expected losses between two times from credit spreads.11

Tables 1 and 2 show that OIS and LIBOR discounting give exactly the same results if CVA and

DVA are calculated appropriately. Table 3 shows the errors that are obtained when LIBOR

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When LIBOR loss rates are calculated it is the actual LIBOR-OIS credit spreads that should be used because it is

the actual LIBOR rates that are used for discounting. The counterparty and dealer credit spreads should be adjusted

credit spreads.

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discounting is mistakenly used to define the risk-neutral stock price return as well as the discount

rate. In Case 1, which corresponds to normal market conditions, the error is only 3.5%, but in

Case 2, which corresponds to stressed market conditions, it very high at 52.5%

10. Conclusions

There is no “perfect” risk-free rate. We have argued that the OIS rate is the best proxy currently

available.

We agree with the current practice of using OIS rates as discount rates when collateralized

portfolios are valued. However, our reasons for doing this are unrelated to funding costs and

therefore different from those of many market participants. We argue that the application of risk-

neutral valuation to calculate the no-default value of a derivative requires the discount rate used

to be the best proxy available for the risk-free rate. Contrary to current market practice, OIS

rates should be use to value non-collateralized portfolios as well as collateralized portfolios.

It is possible to use the LIBOR/swap zero curve (or any other zero curve) to define discount rates

when non-collateralized portfolios are valued providing CVA and DVA are defined

appropriately. However, this makes a bank’s systems unnecessarily complicated and is liable to

cause confusion.

The approach we propose is consistent with the hedging arguments of Burgard and Kjaer (2011)

and others. Its advantage is that it clearly separates three aspects of the valuation of a derivatives

book with a counterparty:

1. The calculation of the risk-free value of the book assuming no collateral;

2. The impact of the credit risk of the dealer and the counterparty; and

3. The impact of the interest paid on collateral being different from the risk-free rate.

17

In the particular case where there is a two-way, zero-threshold, low-minimum-transfer-amount

CSA, the impact of the interest rate paid can be taken into account by adjusting the discount rate.

18

References

Bank for International Settlements “International Banking and Financial Market Developments,”

Quarterly Review, December 2008.

Bartolini, Leonardo, Spence Hilton, and Alessandro Prati, “Money Market Integration,” Journal

of Money, Credit and Banking, Vol. 40, No. 1 (February 2008), pp. 193-213.

Brigo, Damiano and Massimo Morini (2011) “Close Out Convention Tensions,” Risk, 24, 12

(December 2011), 74-78.

Burgard, Christoph and Mats Kjaer, “Partial Differential Equation Representations of

Derivatives with Bilateral Counterparty Risk and Funding Costs,” The Journal of Credit Risk, 7,

3, Fall 2011.

Collin-Dufresne and Bruno Solnik, "On the Term Structure of Default Premia in the Swap and

Libor Market," The Journal of Finance, 56, 3, June 2001.

Demiralp, Selva, Brian Preslopsky, and William Whitesell (2004) “Overnight Interbank Loan

Markets,” Manuscript, Board of Governors of the Federal Reserve.

Elton, Edwin J., Martin J. Gruber, Deepak Agrawal, and Christopher Mann, “Explaining the Rate

Spread on Corporate Bonds,” Journal of Finance, 56, 1 (February 2001), 247-77.

Furfine, Craig H., “Standing Facilities an Interbank Borrowing: Evidence from the Federal

Reserve’s New Discount Window,” Federal Reserve Bank of Chicago working paper 2004-1.

Gregory, Jon, Counterparty Credit Risk: the New Challenge for Financial Markets, John Wiley

and Sons, 2009.

Hull, John and Alan White, “Valuing Credit Default Swaps I: No Counterparty Default Risk,”

Journal of Derivatives, Vol. 8, No. 1, (Fall 2000), pp. 29-40.

Hull, John and Alan White, “CVA and Wrong Way Risk,” forthcoming, Financial Analysts

Journal, 2012.

19

Hull, John and Alan White, “Is FVA a Cost for Derivatives Desks,” forthcoming, Risk

Longstaff, Francis A., “The Term Structure of very Short-term Rates: New Evidence for the

Expectations Hypothesis,” Journal of Financial Economics, Vol. 58 (2000), pp. 397-415.

Stigum, Marcia, The Money Market, 3rd ed., Homewood, Illinois, Dow Jones-Irwin, 1990.

20

Appendix

Consider a non-collateralized portfolio of derivatives between a dealer and a counterparty. The

value today of the derivatives position to the dealer is

nd DVA CVAf f (A1)

where ndf is the no-default value of the position and CVA and DVA are defined as follows:12

0CVA (1 ( )) ( ) ( )

T

c cR t f t q t dt (A2)

0DVA (1 ( )) ( ) ( )

T

d dR t f t q t dt (A3)

In these equations, T is the longest maturity of the derivatives in the portfolio, )(tf is the value

today of a derivative that pays off the dealer’s exposure to the counterparty at time t, )(tf is the

value today of a derivative that pays off the counterparty’s exposure to the dealer at time t ,

( )dq t t is the probability of the dealer defaulting between times t and t+t, and ( )cq t t is the

probability of the counterparty defaulting between times t and t+t.13

The variables ( )dR t and

( )cR t are the dealer’s and the counterparty’s expected recovery rate from a default at time t. The

amount claimed on an uncollateralized derivatives exposure in the event of a default is the no-

default value. The recovery rates, ( )dR t and ( )cR t , can therefore be more precisely defined as

the percentage of no-default value of the derivatives portfolio that is recovered in the event of a

default.

The probabilities cq and dq can be estimated from the credit spreads on bonds issued by the

counterparty and the dealer. A complication is that the claim in the event of a default for a

derivatives portfolio is in many jurisdictions different from that for a bond. The credit spreads on

bonds issued must be used in conjunction with claim procedures applicable to the bonds and

12

We assume that the recovery rate, default rate, and value of the derivative are mutually independent. 13

Some adjustment for the possibility that both parties will default during the life of the derivatives portfolio may be

necessary. See, for example, Brigo and Morini (2011).

21

their expected recovery rates to estimate default probabilities. Hull and White (2000) indicate

how these calculations can be carried out.

We define the “adjusted borrowing rate” of a company as the borrowing rate it would have if a

zero-coupon bond issued by the company were treated in the same way as derivatives in the

event of a default. We similarly define the “adjusted credit spread” of a zero-coupon bond issued

by a company as the credit spread it would have if it were treated in the same way as a derivative

in the event of a default. Once default probabilities have been estimated as just described, a term

structure of adjusted credit spreads for a company can be calculated.

For convenience, we define the loss rate for the dealer as ( ) ( )(1 ( ))d d dL t q t R t and the loss rate

for the counterparty as ( ) ( )(1 ( ))c c cL t q t R t so that equations (A1), (A2), and (A3) become

nd0 0

( ) ( ) ( ) ( )T T

d cf f f t L t dt f t L t dt (A4)

As a first application of equation (A4) suppose that the dealer has a portfolio consisting only of a

zero-coupon bond issued by the counterparty and promising a payoff of $1 at time T. The zero-

coupon is treated like a derivative in the event of a default. In this case, 0)( tf and

nd)( ftf . Furthermore

TTsff c )(expnd

where )(tsc is the adjusted credit spread for a zero coupon bond with maturity t issued by the

counterparty. It follows that

TTsdttL c

T

c )(exp)(10

(A5)

and

))(exp())(exp()( 2211

2

1

ttsttsdttL c

t

tcc (A6)

22

Similarly when the portfolio consists of a zero-coupon bond issued by the dealer, nd)( ftf ,

0)( tf and

TTsff d )(expnd

where )(tsd is the adjusted credit spread for a zero coupon bond with maturity t issued by the

dealer. It follows that

TTsdttL d

T

d )(exp)(10

(A7)

and

))(exp())(exp()( 2211

2

1

ttsttsdttL c

t

tcc (A8)

There are three special cases where it is possible to use the discount rate to adjust for default risk

1. The portfolio promises a single positive payoff to the dealer (and negative payoff to the

counterparty) at time T. In this case 0)( tf and nd)( ftf so that from equation

(A4)

T

c dttLff0

nd )(1

Using equation (A5) we obtain

TTsff c )(expnd

This result shows that the derivative can be valued by using a discount rate equal to the

risk-free rate for maturity T plus sc(T).

2. The portfolio promises a single negative value to the dealer (and positive value to the

counterparty) at time T. In this case nd)( ftf and 0)( tf so that

23

T

d dttLff0

nd )(1

Using equation (A7) we obtain

TTsff d )(expnd

This result shows that the derivative can be valued by using a discount rate equal to the

risk-free rate for maturity T plus sd(T).

3. The derivatives portfolio promises a single payoff, which can be positive or negative at

time T, and the two sides have identical loss rates: Lc (t) = Ld (t) = L(t) for all t.

Because nd)( ftff

equation (4) becomes

T

dttLff0

nd )(1

From equation (A5) or (A7) we obtain

TTsff )(expnd

where s(T) is the common adjusted credit spread for the dealer and the counterparty. This

result shows that the derivative can be valued by using a discount rate equal to the risk-

free rate for maturity T plus s(T).

These three cases can be generalized. A derivative promising positive payoffs to the dealer at

several different times is the sum of derivatives similar to those in 1 and can be valued using

discount rates that reflect the counterparty’s adjusted borrowing rates. Similarly, a derivative

promising payoffs negative payoffs to the dealer at several different times is the sum of

derivatives similar to those in 2 and can be valued using discount rates that reflect the dealer’s

adjusted borrowing rates. Finally, Case 3 shows that when the two sides have identical credit

risks, any derivative can be valued by using discount rates that reflect their common adjusted

borrowing rates.

24

We now return to considering the general situation. We define ( )L t as the loss rate at time t for

a company whose adjusted credit spread is the LIBOR-OIS credit spread. This means that,

similarly to equations (A5) to (A8)

TTsdttLT

)(exp)(10

(A9)

and

))(exp())(exp()( 2211

2

1

ttsttsdttLt

t (A10)

where )(ts is the zero-coupon LIBOR-OIS spread for maturity t.

If the LIBOR/swap curve defined the loss rate for both the dealer and the counterparty, equation

(A4) would show that the value of the portfolio to the dealer is given by

nd0 0

( ) ( ) ( ) ( )T T

f f f t L t dt f t L t dt (A11)

From equations (A4) and (A11)

T

c

T

d dttLtLtfdttLtLtfff00

)()()()()()( (A12)

From the extension of the third result given above we know that f is the value of the derivatives

portfolio when LIBOR/swap curve is used for discounting. It follows that we obtain correct

valuations of any derivatives portfolio if the LIBOR/swap curve is used for discounting and

CVA and DVA are calculated using loss rates for the dealer and counterparty that are the excess

of the actual loss rates over the loss rates that would apply for an entity able to borrow at

LIBOR/swap rates.

This result can be generalized. There is nothing special about the LIBOR/swap zero curve in our

analysis. Any yield curve can be used as the “risk-free” zero curve when a derivative is valued

providing CVA and DVA are calculated using loss rates that are the excess of the actual loss rate

over the loss rate implied by the yield curve.

25

Figure 1: Relationship Between Credit Spreads and Maturity

0

20

40

60

80

100

120

0.00 0.25 0.50 0.75 1.00

Cre

dit

Sp

read

(B

asis

Po

ints

)

Maturity (Years)

Credit Spread Term Structure: LIBOR - T-Bill

2009 2010 2011 2012

26

Table 1

Value to dealer of a non-collateralized long position in a one-year forward contract to buy a non-

dividend paying stock.

Stock price = 100, delivery price =100, volatility = 30%, OIS rate = 3%, LIBOR = 3.1%, dealer

adjusted credit spread = 0.5%, counterparty adjusted credit spread = 2%

OIS LIBOR

Discounting Discounting

No-Default Value 2.955 2.952

CVA 0.187 0.178

DVA 0.032 0.026

Total 2.801 2.801

Table 2

Value to dealer of a non-collateralized long position in a one-year forward contract to buy a non-

dividend paying stock.

Stock price = 100, delivery price =100, volatility = 30%, OIS rate = 3%, LIBOR = 4.5%, dealer

adjusted credit spread = 2%, counterparty adjusted credit spread = 3%

OIS LIBOR

Discounting Discounting

No-Default Value 2.955 2.911

CVA 0.279 0.138

DVA 0.128 0.032

Total 2.805 2.805

27

Table 3

Percentage price errors in the calculated value to dealer of a non-collateralized long position in a

one-year forward contract to buy a non-dividend paying stock when LIBOR discounting is used

and LIBOR is also incorrectly used to define the growth rate in the stock price

Case 1: Stock price = 100, delivery price =100, volatility = 30%, OIS rate = 3%, LIBOR = 3.1%,

dealer adjusted credit spread = 0.5%, counterparty adjusted credit spread = 2%

Case 2: Stock price = 100, delivery price =100, volatility = 30%, OIS rate = 3%, LIBOR = 4.5%,

dealer adjusted credit spread = 2%, counterparty adjusted credit spread = 3%

Case 1 Case 2

No-Default Value 3.052 4.400

CVA 0.179 0.152

DVA 0.026 0.029

Total 2.899 4.277

Error 3.5% 52.5%