Electronic copy available at: http://ssrn.com/abstract=2036979

 

 

   

 

 

Boston  University  School  of  Management  Research  Paper  Series    

No.  2012-­‐11  

 

‘‘Valuing  Interest  Rate  Swaps  Using  OIS  Discounting’’    

 

Donald  J.  Smith  

 

 

 

Electronic copy available at: http://ssrn.com/abstract=2036979

Valuing Interest Rate Swaps Using OIS Discounting

March 2012

Donald J. Smith

Associate Professor of Finance

Boston University School of Management

595 Commonwealth Avenue

Boston, MA 02215

Phone: 617-353-2037

Email: donsmith@bu.edu

Electronic copy available at: http://ssrn.com/abstract=2036979

1

Valuing Interest Rate Swaps Using OIS Discounting

The financial crisis of 2007-09 precipitated a significant change in the practice of

interest rate swap valuation. Going from traditional LIBOR to OIS (overnight indexed

swap) discounting might not seem to be a profound event but it is more than just another

method to calculate fair values for over-the-counter derivative contracts. It embodies

newfound appreciation of counterparty credit risk and the role of collateral and central

clearing. Implementation of OIS discounting has created a cottage industry for risk

management consultants and trainers to deal with the technicalities of the new approach.

From my academic perspective, it is clear that many of our finance textbooks that cover

interest rate swaps need to be revised.

The first section of this note reviews interest rate swap valuation in principle, the

reasons for the move from LIBOR to OIS discounting, the implications for swap rates,

and the “winners and losers” that arise from the transition. The second section works

through a numerical example to illustrate the calculations. This entails bootstrapping a

sequence of discount factors that are consistent with interest rate swaps that have a

market value of zero. The implied LIBOR forward curve is derived (or, in general, the

forward curve for the money market reference rate). This curve becomes particularly

important under OIS discounting when valuing a swap as a combination of fixed-rate and

floating-rate bonds.

Fortunately for risk managers, OIS discounting uses the same types of analytic

techniques as the traditional approach. However, there are some differences that are

beyond the scope of this note, for instance, calculating the sensitivities of swap values to

changes in OIS rates and the LIBOR-OIS spread (i.e., working with dual curves rather

2

than a single curve for risk measurement) and dealing with cross-currency swaps. Also,

LIBOR is an interest rate that reasonably can be assumed to vary day by day in the

interbank market whereas OIS rates are more directly a tool of monetary policy,

suggesting that rate volatility depends on the pattern and timing of policy meetings and

actions.1

Interest Rate Swap Valuation in Principle

An interest rate swap can be interpreted as: (1) a series of forward contracts on the

reference rate, and (2) a long/short combination of a fixed-rate and a floating-rate bond.2

Consider a 2-year, USD 100 million notional principal, 5.26% fixed-rate, quarterly

settlement swap on 3-month LIBOR, as illustrated in Exhibit A. In the first interpretation,

the swap is a package of eight implicit forward rate agreements (FRAs), each exchanging

3-month LIBOR for a fixed rate of 5.26%. The counterparties to the series of FRAs are

the fixed-rate receiver on the swap (the “receiver”) and the fixed-rate payer (the “payer”).

In the second interpretation, the receiver has a long position in an implicit 2-year, 5.26%

quarterly payment bond and a short position in an implicit 2-year floating-rate note

(FRN) paying 3-month LIBOR with quarterly resets. The payer has the opposite bond

positions.

If this swap currently has a market value of zero, it is described as being “at-

market” or “at par”. Actual FRAs on 3-month LIBOR have a different fixed rate for each

future time period, reflecting the shape of the underlying forward curve. Unless that

curve is remarkably flat, the swap is a package of “off-market” FRAs because each has

the same fixed rate of 5.26%. However, the market values of the FRAs sum to zero. That

3

describes how the fixed rate on an at-market swap is determined at inception: It is set so

that the implicit FRAs, some of which have positive values and others negative values,

net to zero. In the bond interpretation, the implicit fixed-rate bond has a coupon rate set

so that its price matches that of the FRN paying 3-month LIBOR.

The upper panel of Exhibit B displays the balance sheets for the two

counterparties. The at-market swap having a value of zero is not shown—it is “off-

balance sheet” or, one can say, it is hidden on the line dividing assets from liabilities. The

only accounting item shown is OCI (other comprehensive income), a portion of

shareholders’ equity that is used to register changes in the value of derivative contracts.

Now suppose that time passes and market interest rates change. The middle panel shows

the balance sheets if swap market rates fall. The derivative appears as an asset to the

fixed-rate receiver—it has migrated off the line—and as a liability to the payer. The

offsetting accounting items are increases and decreases in OCI. The lower panel shows

results if instead swap rates rise. The accounting rules for derivatives (e.g, SFAS 133 and

IAS 39) determine whether the changes in swap values also need to flow through the

income statement before impacting OCI.

The methods to value an interest rate swap and to determine the size of boxes in

Exhibit B follow from the two interpretations. In the first, the exposure to the eight

implicit FRAs can be hedged by entering a “mirror” swap, thereby eliminating the risk of

further volatility in LIBOR. Suppose that the fixed rate on a 2-year, quarterly settlement,

at-market swap is 3.40% and that the 5.26% now off-market swap was entered several

years ago and currently has two years remaining. Rates have fallen (or, possibly, the

swap has just slid down a stable but steeply and upwardly sloped yield curve), as in the

4

middle panel. The fixed-rate receiver can enter, in principle, a pay-fixed mirror swap to

lock in the value of its asset; the payer can enter a receiver swap to stanch further losses.

This establishes a sequence of eight quarterly payments on the order of USD 465,000,

depending on the day-count convention and neglecting the bid-offer spread, flowing from

the payer to the receiver: (5.26% – 3.40%) * 100,000,000 * 0.25 = 465,000. The swap

valuation problem is to calculate the present value of this annuity.

In the combination-of-bonds interpretation, the value of the swap is driven by

changes in the implicit fixed-rate bond. As rates are assumed to have fallen, the 5.26%

fixed-rate bond is priced at a premium above par value. By design, an FRN has limited

price volatility. Said differently, its duration is low and often is figured to be the time

until the next rate reset date. Using this interpretation, the duration of the swap can

calculated as the difference in the durations of the two implicit bonds. A receiver swap

has a positive duration statistic. A payer swap in turn has negative duration, meaning its

market value is positively correlated to rates, as shown in the lower panel of Exhibit B.

The value of the swap is the premium price on the fixed-rate bond less the value of the

FRN, which typically is assumed to be par value on a reset date.

The key point is that swap valuation is all about discounting future cash flows.

The traditional approach has been to use discount factors that correspond to the reference

rate on the swap, e.g., 3-month LIBOR. That is a much stronger assumption that it might

seem. If the swap is not collateralized, it implies that the counterparty for which the swap

is a liability, here the fixed-rate payer, has credit quality consistent with the banks that

establish the LIBOR index. Presumably, this counterparty can borrow funds for two years

at LIBOR flat (meaning a margin of zero above or below the reference rate) on a

5

quarterly payment floating-rate basis or at 3.40% fixed. In sum, the LIBOR discount

factors are appropriate to get the present value of its unsecured future obligations.

Usually, this corresponds to an investment-grade borrower having a quality rating of A+

to AA– on its debt.

Suppose instead that the fixed-rate payer is a financially distressed company that

has had its credit rating lowered to non-investment grade. If the fixed-rate receiver

requested early termination of the swap, the payer would offer to settle the obligation for

an amount that reflects its cost of borrowed funds and not that of an investment-grade

issuer. While using default-risk-adjusted discount factors is appropriate in principle for an

early termination, it would be unwieldy for routine valuations carried out daily by swap

dealers having a multitude of open contracts.

The advantage to using the LIBOR swap curve is that there are good data

publically available for a full range of maturities. Importantly, the bootstrapped numbers

are “internal” to the valuation problem. In this traditional approach, the fixed rates on at-

market swaps (or the prices on 2-year fixed-rate bonds and FRNs having comparable

credit risk) can be used to bootstrap the discount factors needed to value the swap book.

Those calculations are demonstrated in the next section.

An important development in the interest rate swap market in recent years has

been widespread use of collateralization to mitigate counterparty credit risk. When the

market started in the 1980s, most swap contracts were unsecured and any imbalance in

the credit standings between the two counterparties was priced into the fixed rate or

managed by having the weaker party get some type of credit enhancement. In the 1990s,

after the introduction of the CSA (Credit Support Annex) to the standard ISDA

6

(International Swap and Derivatives Association) master agreement, posting collateral in

the form of cash or marketable securities became more common. Nowadays, bilateral

CSAs with a zero threshold, meaning only the counterparty for which the swap has

negative value posts collateral, is the industry norm. The specific terms of the CSA in the

ISDA document are complex and go beyond the scope of this note.

Johannes and Sundaresan (2009) contend that the fixed rate on a collateralized

swap should be higher than when it is uncollateralized and provide empirical evidence to

support that finding. The reasoning is similar to the convexity adjustment between

interest rates on exchange-traded futures and over-the-counter forwards. The idea is that

posting collateral is costly to the counterparty for which the swap is “underwater”,

meaning having a negative market value. Either the funds or the securities to satisfy the

collateral requirement need to be borrowed or are diverted from other uses.

The reason for the higher fixed rate on a collateralized swap is that the impact of

having to post costly collateral is not symmetric to the counterparties—the fixed-rate

receiver suffers from interest rate volatility while the payer benefits. Suppose the swap is

underwater to the receiver because swap rates have risen since inception. If rates rise

further, more costly collateral is needed; if rates fall, less is required. In contrast, suppose

the swap is underwater to the payer because rates have fallen. If rates then rise, less

collateral is needed; and if rates fall further, more is required. Systematically, the fixed-

rate receiver posts more costly collateral when interest rates go up; the payer posts more

when rates do down. This asymmetry, other things being equal, leads to a higher fixed

rate on the collateralized swap.

7

The main implication of collateralization is that the credit risk on the swap

becomes minimal, similar to exchange-traded futures contracts. To be sure, futures entail

initial margin accounts by both counterparties to provide an additional buffer, whereas

bilateral CSAs have a zero threshold so there still is some “tail” risk. In any case,

minimal credit risk means that the discount factors to get the present value of the annuity

for the difference between the contractual and at-market fixed rates (or to value the

implicit bonds) should be based on (near) risk-free interest rates.

Why then is it not market practice to use actively traded U.S. Treasury notes and

bonds, for which there are ample price data, to get the discount factors to value USD-

denominated derivatives that are nearly risk-free? The problem with Treasury yields is

that typically they are too low for this purpose. Treasuries are by far the most liquid debt

security and are in high demand as collateral in the repo market. Exemption from state

and local income taxes lowers their yields even more. Also, Treasury yields are more

volatile than swap rates because they are the first asset class to absorb fluctuations in

demand and supply arising from international capital flows, especially during flights to

quality.

The ideal discount factors to value collateralized contracts would come from

traded securities having the same liquidity, tax status, and volatility as the interest rate

swaps but credit risk approaching zero. Pre-2007, dealers as well as their regulators and

auditors viewed fixed rates on LIBOR swaps to be a reasonable and workable proxy for

the risk-free yield curve. However, in the post-2007 world the presence of a persistent

and sizable LIBOR-OIS spread exposes the “credit risk approaching zero” presumption.

8

An overnight indexed swap is a derivative contract on the total return of a

reference rate that is compounded daily over a set time period. In the U.S. dollar market,

the reference rate is the effective federal funds rate. It is calculated and released by the

Federal Reserve each day in its H.15 Report and is the weighted average of brokered

trades between banks for overnight ownership of deposits at the Fed (i.e., bank reserves).

The effective fed funds rate is not necessarily equal to the target rate set by the Federal

Open Market Committee (FOMC) and announced at regularly scheduled FOMC

meetings. The Fed merely aims to keep the effective rate close to its target via open

market operations of buying and selling securities. In the Euro-zone, the OIS reference

rate is EONIA (Euro Overnight Index Average, which essentially is the 1-day interbank

rate. In the U.K., the reference rate is SONIA (Sterling Overnight Index Average).

Until August 2007, the LIBOR-OIS spread was consistently narrow, typically just

5-10 basis points, thereby justifying the use of LIBOR discount factors to value

collateralized swaps. Some commentators date the onset of the financial crisis at August

9, 2007, which was the day when the LIBOR-OIS spread first spiked upward. It remained

high, oscillating between 50 and 100 basis points, and then jumped again in the fall of

2008, reaching its pinnacle at about 350 basis points after the announcement of the

Lehman bankruptcy on September 15, 2008. It then returned to more normal levels in

2009 only to go up again in 2011 reflecting concerns over the Euro-zone sovereign debt

crisis. A similar pattern appears in EURIBOR-EONIA spreads.

The OIS curve is now preferred by dealers to value collateralized interest rate

swaps because it removes the bank credit and liquidity risk that is being priced into

LIBOR. OIS rates, unlike LIBOR, now represent risk-free rates for banks and satisfy the

9

“credit risk approaching zero” criterion. Moreover, the Dodd-Frank Act of 2010

mandates that U.S. dealers use central clearing for standardized swaps and

collateralization for un-cleared transactions. In response, central clearers such the

CMEGroup and the London Clearinghouse, LCH.Clearnet, specifically use OIS

discounting to value interest rate swaps and to determine collateral requirements.

In general, fixed-rate receivers gain and payers lose following the switch from

LIBOR to OIS discounting when the swap yield curve is upward sloping and the LIBOR-

OIS spread is positive. Nashikkar (2011) points out that this could impact end-users who

have a large “directional book”, meaning they typically enter the same type of swap. For

example, life insurance companies and defined-benefit pension funds enter receive-fixed

swaps to reduce the duration mismatch between their assets, which are mostly equity and

corporate bonds, and their long-term liabilities. On the other hand, the GSEs

(Government-Sponsored Enterprises) like Fannie Mae and Freddie Mac tend to enter

pay-fixed swaps to hedge their positions in long-term fixed-rate mortgages. Other

impacts are on hedging strategies for swap dealers and end-users because the switch

introduces exposure to the LIBOR-OIS spread, on the pricing of other derivatives

because the implied LIBOR forward curve changes, and on possible implications for the

accounting for interest rate swaps (e.g., whether the swap qualifies for hedge accounting

treatment).

Interest Rate Swap Valuation in Practice

A numerical example is instructive to explore some of the nuances of swap

valuation using LIBOR and OIS discount factors.3 Consider again a 2-year, USD 100

10

million notional principal, 5.26% fixed versus 3-month LIBOR, quarterly settlement

swap at a time when the otherwise comparable at-market fixed rate is 3.40%. A cursory

value of USD 3,581,649 is obtained by discounting the 8-period annuity of USD 465,000

using the at-market fixed rate and neglecting the actual day-count convention:

465,000j(1+ 0.0340 /4)

= 3,581,649j=1

8∑

This calculation assumes a flat swap curve because each payment is discounted by the

same interest rate. In general, LIBOR discount factors for the full term structure are used

to integrate the shape of the swap curve, which typically is upward sloping.

The traditional valuation method uses cash market rates for LIBOR for the first 12

months and then at-market swap fixed rates beyond that. For this numerical exercise,

assume these observations for LIBOR deposits: 3-month, 0.50%; 6-month, 1.00%; 9-

month LIBOR, 1.60%; 12-month, 2.10%. In the USD market, the actual/360 day-count

convention and simple interest are used to determine cash flows. For 92 days in the first

3-month time period (n = 1), for instance, from March 15th to June 15th, the LIBOR

discount factor is:

1LIBORDF =

11+ 0.0050*92 /360( )

= 0.998724

For 92 days in the second quarter (n = 2) between June 15th and September 15th,

2LIBORDF is:

2LIBORDF =

11+ 0.0100*184 /360( )

= 0.994915

Calculated in the same manner,

3LIBORDF for 275 days is 0.987925, and

4LIBORDF for

365 days is 0.979152. To generalize, quarterly LIBOR discount factors based on money

11

market rates are determined with this formula in which Aj is the fraction of the year for

the jth period given the particular day-count convention (i.e., actual/360, actual/365,

30/360):

( )∑+=

=n

jn j

LIBORn

ALIBORDF

1 *11 , n = 1 to 4 (1)

For this exercise, quarterly discount factors suffice; in practice, daily discount factors are

needed to value the entire swap book.

Beyond 12 months, discount factors are calculated by bootstrapping fixed rates on

at-market swaps. Typically, these quoted rates start at a tenor of two years. That creates a

problem for the risk manager and the need to interpolate for the span between year one

and year two. That often leads to a jump or a kink in the LIBOR forward curve and

discount factors. This example finesses that problem by assuming that at-market (or par)

swap fixed rates are: 15-month, 2.44%; 18-month, 2.76%; 21-month, 3.08%; 24-month,

3.40%. These swaps are for quarterly settlement versus 3-month LIBOR and use the

actual/360 day-count convention. The next discount factor,

5LIBORDF , comes from

solving this equation:

1 = (2.44% * 92 /360 * 0.998724) + (2.44%*92/360*0.994915) + (2.44%* 91/360 * 0.987925) + (2.44% *90/360*0.979152)

+ (2.44%* 90/360 +1) * 5LIBORDF , 5

LIBORDF = 0.969457

This treats the swap as a 15-month, 2.44% fixed-rate, non-amortizing (i.e., “bullet”) bond

priced at a par value of 1. [All of the reported results for this and the following equations

are calculated on a spreadsheet to preserve accuracy in the bootstrapping process, which

is sensitive to rounding. The rounded results from the calculations with full precision are

shown in the equations for consistent exposition.]

12

The discount factor for the sixth quarterly period,

6LIBORDF , uses

1LIBORDF

through

5LIBORDF along with the 18-month swap fixed rate of 2.76%.

1= (2.76%*92 /360 *0.998724) + (2.76%*92/360*0.994915) + (2.76%*91/360 *0.987925) + (2.76% *90/360*0.979152)

+ (2.76%*92/360 *0.969457) + (2.76% *92/360 +1)* 6LIBORDF ,

6LIBOR DF = 0.958690

Repeating the bootstrapping process for the seventh and eighth periods, which have 91

and 90 days in the quarter, obtains

7LIBORDF = 0.946531 and

8LIBORDF = 0.933045. The

general formula for bootstrapping LIBOR discount factors from at-market swap fixed

rates (SFRn) is:

nLIBORDF =

1− nSFR * j * jLIBORDFAj = 1

n − 1∑

1+ nSFR * nA( ), n > 4 (2)

The implied forward rate, sometimes called the projected rate, for 3-month

LIBOR between period n – 1 and period n based on the LIBOR discount factors is

designated

n − 1, nLIBORIFR . The sequence of implied forwards is bootstrapped with this

formula:

n − 1, nLIBORIFR = n − 1

LIBORDF

nLIBORDF

−1

*

1

nA (3)

For example, the “7 x 8” implied rate between months 21 and 24 is 5.7815%.

7, 8LIBORIFR =

0.9465310.933045

−1

*

190 /360

= 0.057815

The implied, or projected, LIBOR forward curve is particularly useful in pricing options

on swaps (i.e., “swaptions”) and non-standard interest rate swaps that have, for instance,

a deferred start date or a notional principal that varies over the lifetime of the contract.

13

The bootstrapped LIBOR discount factors and implied forward rates are summarized in

Exhibit C.

The 5.26%, 2-year, USD 100 million, off-market swap can be valued by

comparison to the at-market swap having a fixed rate of 3.40%. Its market value using

the LIBOR discount factors, denoted MVLIBOR, is USD 3,662,844.

LIBORMV = 0.0526 − 0.0340( )*100,000,000* j Aj = 18∑ * j

LIBORDJ = 3,662,844

The swap is an asset to the fixed-rate receiver, and an equivalent liability to the payer.

This amount is higher than the cursory value calculated at the beginning of the section

because the actual/360 day-count is used, and the discount factors reflect lower rates due

to the upward slope to the swap curve.

The combination-of-bonds approach and LIBOR discounting produce the same

value for the swap. The implicit FRN has a market value of USD 100 million. Often par

value on a rate reset date is simply assumed for a floating-rate note, but it is useful in

understanding the implications of OIS discounting to calculate its price by applying the

LIBOR discount factors to the sequence of implied, or projected, LIBOR forward rates.

FRNMV =100,000,000*j

j − 1, jLIBORIFR * Aj = 1

8∑ * jLIBORDF

+100,000,000* 8LIBORDF =100,000,000

The 5.26% fixed-rate note is priced at a premium above par value, USD 103,662,844,

because current swap market rates are lower than the contractual rate—in terms of bond

valuation, its coupon rate exceeds the yield to maturity.

FIXEDMV =100,000,000*0.0526* j Aj = 18∑ * j

LIBORDF

+100,000,000* 8LIBORDF =103,662,844

14

When the implicit bonds are priced using LIBOR discount factors, the difference in the

bond prices determines the market value of the swap.

844,662,3000,000,100844,662,103 =−=−= MVMVMV FRNFIXEDLIBOR

The key assumption behind this valuation using LIBOR discounting is either: (1)

the swap is uncollateralized and the fixed-rate payer, for which the swap is a liability, is a

LIBOR flat credit risk, or (2) the swap is collateralized (or centrally cleared) and LIBOR

discount rates are a reasonable proxy for the risk-free yield curve. Nowadays,

collateralization is the norm and the LIBOR-OIS spread is not insignificant. Therefore,

LIBOR discounting is no longer appropriate for collateralized or centrally cleared

transactions. That explains why OIS discounting is becoming the new standard for

interest rate swap valuation.

Suppose the 3-month fixed rate is 0.10% on an overnight indexed swap for a

notional principal of USD 100 million. At settlement, the settlement payoff will be based

on the difference between the fixed and floating legs on the swap. Assuming 92 days for

the quarter and an actual/360 day-count, the fixed leg is USD 25,556.

100,000,000* 92360

*0.0010 = 25,556

The floating leg depends on the sequence of realized daily reference rates.

100,000,000* 1+ 1EFF360

* 1+ 2EFF

360

* ...* 1+ 92EFF

360

−1

EFF1, EFF2,…, EFF92 are the reported daily observations for the effective fed funds

rate.4 Net settlement on the OIS is the difference between the two legs.

15

OIS fixed rates out to 12 months use simple interest, following market practice for

LIBOR in the money market. In the same manner as equation (1), OIS discount factors

for the first four quarters are calculated as:

( )∑+=

=n

jn j

OISn

AOISDF

1*11 , n = 1 to 4 (4)

Suppose that the OIS rates are: 3-month, 0.10%; 6-month, 0.60%; 9-month, 1.20%; 12-

month, 1.70%. These time periods translate to 92, 184, 275, and 365 days. Using

equation (4), the first two OIS discount factors are:

1OISDF =

11+ 0.0010*92 /360( )

= 0.999745,

2OISDF =

11+ 0.0060 *184 /360( )

= 0.996943

In the same manner,

3OISDF = 0.990917 and

4OISDF = 0.983056.

Consistent with LIBOR swaps, OIS contracts for tenors longer than 12 months

entail periodic settlement payments. Assume that these are quarterly settlements for an

actual/360 day-count. The OIS fixed rates are: 15-month, 2.00%; 18-month, 2.30%, 21-

month, 2.60%; 24-month, 2.90%. These assumed swap rates track a LIBOR-OIS spread

in the 40-50 basis point range. The general formula for bootstrapping the OIS discount

factors beyond 12 months has the same structure as equation (2):

nOISDF =

1− nOIS * j * jOISDFAj = 1

n − 1∑

1+ nOIS * nA( ), n > 4 (5)

The OIS discount factor for the fifth quarter (n = 5) is 0.974724.

( ) 974724.0360/92*0200.01

*0200.01 4*1

5 =+

∑−=

=j OISOIS

A DFDF

jj

16

The remaining OIS discount factors are:

6OISDF = 0.965259 ,

7OISDF = 0.954878, and

8OISDF = 0.9433. The assumed OIS fixed rates and the corresponding discount factors are

tabulated in Exhibit 4.

Now suppose that both the 5.26% off-market and the 3.40% at-market interest

rate swaps are collateralized (or centrally cleared) so that OIS discounting is appropriate.

The market value is USD 3,681,873.

OISMV = 0.0525 − 0.0340( )*100,000,000* j Aj = 18∑ * j

OISDJ = 3,681,873

This is higher than the market value using LIBOR discount factors, USD 3,662,844. The

size of the difference is a function of the gap between the contractual and mark-to-market

fixed rates, the tenor, the LIBOR-OIS spread, and the shape of the underlying yield curve.

What matters is that this market value better captures the minimal credit risk on a

collateralized (or centrally cleared) interest rate swap.

An important point is that care needs to be taken in valuing the swap using the

combination-of-bonds approach, which is common in academic textbooks, because both

implicit bonds need to be priced as risk-free securities. The market value for the implicit

2-year, 5.26% fixed-rate bond using the OIS discount factors is USD 104,750,723.

FIXEDMV =100,000,000*0.0526* j Aj = 18∑ * j

OISDF

+100,000,000* 8OISDF =104,750,723

It is tempting to get the value of the implicit FRN by using the OIS discount factors on

the implied LIBOR forward curve, as reported in Exhibit C. The market value for the

FRN would be USD 101,078,899, as expected above par value because of its risk-free

status.

17

FRNMV =100,000,000 * j

j − 1, jLIBORIFR * Aj = 1

8∑ * jOISDF

+100,000,000 * 8OISDF =101,078,899

The value for the swap using OIS discounting and the combination-of-bonds approach

then is calculated to be USD 3,671,824.

OISMV =104,750,723 −101,078,899 = 3,671,824

The problem is that this does not match the value using OIS discounting and the market-

to-market approach, for which the result is USD 3,681,873.

The resolution of the discrepancy is that a new implied, or projected, LIBOR

forward curve is needed—one that is consistent with pricing LIBOR deposits and at-

market LIBOR swaps with OIS discount factors. For the money market segment of the

swap curve, for which observations on LIBOR deposits are used to get the discount

factors, this is the formula for n = 1 to 4:

DFA

DFAIFRA DFLIBORIFR

nOISn

jjjjnnj

OISOISjj

nj OIS

OISnn

*

* * 11 ,11

,1

** ∑−∑=

−= −=

− (6)

For n > 4, the at-market LIBOR swap fixed rates are used:

n − 1, nOISIFR =

nSFR * j * jOISDFAj = 1

n∑ − jj − 1, j

OISIFR * A * jOISDFj = 1

n − 1∑

nOIS

nA * DF (7)

The new sequence of implied LIBOR forwards is included in Exhibit D. The

important result is that the market value of the implicit FRN for OIS discounting is USD

101,068,849.

FRNMV =100,000,000 * j

j − 1, jOISIFR * Aj = 1

8∑ * jOISDF

+100,000,000 * 8OISDF =101,068,849

18

The value of the 5.26%, 2-year collateralized swap using the combination-of-bonds

approach is USD 3,681,873, now matching the result for the mark-to-market method.

OISMV =104,750,723 −101,068,849 = 3,681,873

Pricing and valuing LIBOR swaptions and non-standard swaps are other applications for

equations (6) and (7). These procedures require a LIBOR forward curve—and it is the

implied LIBOR forward curve consistent with OIS discounting that is relevant.

Conclusion

The switch from LIBOR to OIS discounting in the valuation of collateralized (or

centrally cleared) interest rate swaps is not a technical advance coming out of financial

engineering or math finance research projects. In fact, the same bootstrapping procedures

are used, albeit with some adjustments to address the differences in the factors driving the

volatility in LIBOR and OIS rates and data availability. The switch is more conceptual in

that it establishes that counterparty credit risk and collateralization are significant

elements in valuation and that LIBOR swap discount factors no longer are a reasonable

proxy for the risk-free yield curve. Risk managers need to be aware of this switch even if

their swaps are not collateralized or centrally cleared; financial educators need to

introduce OIS discounting into their swap training materials and textbooks.

19

Notes

1. For an illustration of these technicalities, see the articles by Justin Clarke of Edu-Risk

International, which are available at www.edurisk.ie.

2. A third interpretation, which is not typically used in practice, is that an interest rate

swap is a long/short combination of an interest rate cap and floor that have strike rates

equal to the fixed rate on the swap; see Brown and Smith (1995).

3. This example of swap valuation using LIBOR discounting is based on Smith (2011).

4. This description of the calculation of the floating leg on an OIS contract is an

abstraction. In practice, weekends and holidays are handled with an odd mix of simple

and compound interest. Suppose that for a 5-day OIS, the effective fed funds rate is

0.09% on Thursday, 0.10% on Friday, and 0.11% on Monday. The Friday rate is used for

Saturday and Sunday, however, on a simple interest basis. The floating leg would be

calculated as:

100,000,000 * 1+0.0009

360

* 1+

3*0.0010360

* 1+

0.0011360

−1

As formulated in the text, the Friday rate would be compounded for the three days:

100,000,000* 1+0.0009

360

*

3

1+0.0010

360

* 1+

0.0011360

−1

20

Exhibit A: 2-year, 5.26% Interest Rate Swap

Fixed-Rate Payer

“The Payer”

Fixed-Rate Receiver

“The Receiver”

5.26% Fixed Rate

3-Month LIBOR

21

Exhibit B: Counterparty Balance Sheets

Upper Panel: At-market Interest Rate Swap

Middle Panel: Off-market Interest Rate Swap, Swap Rates Fall

Lower Panel: Off-market Interest Rate Swap, Swap Rates Rise

Fixed-Rate Receiver Fixed-Rate Payer

OCI

OCI

OCI OCI

Fixed-Rate Receiver Fixed-Rate Payer

Swap Swap

OCI

OCI

Fixed-Rate Receiver Fixed-Rate Payer

Swap Swap

22

Exhibit C: LIBOR Discounting

Period Number of

Days

LIBOR Deposits and Swap Fixed

Rates

LIBOR Discount Factors

Implied LIBOR Forward Rates

1 92 0.50% 0.998724 0.5000% 2 92 1.00% 0.994915 1.4981% 3 91 1.60% 0.987925 2.7989% 4 90 2.10% 0.979152 3.5840% 5 92 2.44% 0.969457 3.9132% 6 92 2.76% 0.958690 4.3949% 7 91 3.08% 0.946531 5.0818% 8 90 3.40% 0.933045 5.7815%

23

Exhibit D: OIS Discounting

Period Number of

Days OIS

Fixed Rates OIS

Discount Factors Implied LIBOR

Forward 1 92 0.10% 0.999745 0.5000% 2 92 0.60% 0.996943 1.5014% 3 91 1.20% 0.990917 2.8223% 4 90 1.70% 0.983056 3.6477% 5 92 2.00% 0.974724 3.8138% 6 92 2.30% 0.965259 4.3888% 7 91 2.60% 0.954878 5.0717% 8 90 2.90% 0.943385 5.7658%

24

References

Brown, Keith C. and Donald J. Smith, Interest Rate and Currency Swaps: A Tutorial, Research Foundation of the Institute for Chartered Financial Analysts, 1995, available at the CFA Institute website. Clarke, Justin, “Swap Discounting & Pricing Using the OIS Curve”, Edu-Risk International, available at www.edurisk.ie. Clarke, Justin, “Constructing the OIS Curve”, Edu-Risk International, available at www.edurisk.ie. Johannes, Michael and Suresh Sundaresan, “The Impact of Collateralization on Swap Rates”, Journal of Finance, Vol. LXII, No. 1, February 2007. Nashikkar, Amrut, “Understanding OIS Discounting”, Barclays Capital Interest Rate Strategy, February 24, 2011. Smith, Donald J., Bond Math: The Theory Behind the Formulas, Wiley Finance, 2011.