&
rate swap math pricing
Understanding interest
January 2007
C D IA C # 06 -1 1&
rate swap math pricing
Understanding interest
January 2007
C D IA C # 06 -1 11 Intr
oduction
1 Basi
c Interest R
ate Swap M
echanics
3 Sw
ap Pricing
in Theory
8 Sw
ap Pricing i
n Practice
12 Fin
ding the Te
rmination V
alue of a Sw
ap
14 Sw
ap Pricing P
rocess
16 Co
nclusion
18 Refe
rences
p1
Introduction
As California local agencies are becoming involved in the interest rate swap market, knowledge of the basics of pric-ing swaps may assist issuers to better understand initial, mark-to-market, and termination costs associated with their swap programs.
This report is intended to provide treasury managers and staff with a basic overview of swap math and related pric-ing conventions. It provides information on the interest rate swap market, the swap dealer’s pricing and sales con-ventions, the relevant indices needed to determine pric-ing, formulas for and examples of pricpric-ing, and a review of variables that have an affect on market and termination pricing of an existing swap.1
Basic Interest Rate Swap Mechanics
An interest rate swap is a contractual arrangement be-tween two parties, often referred to as “counterparties”. As shown in Figure 1, the counterparties (in this example, a financial institution and an issuer) agree to exchange payments based on a defined principal amount, for a fixed period of time.
In an interest rate swap, the principal amount is not actu-ally exchanged between the counterparties, rather, inter-est payments are exchanged based on a “notional amount” or “notional principal.” Interest rate swaps do not generate
1 For those interested in a basic overview of interest rate swaps,
the California Debt and Investment Advisory Commission (CDIAC) also has published Fundamentals of Interest Rate
Swaps and 20 Questions for Municipal Interest Rate Swap Issu-ers. These publications are available on the CDIAC website at
www.treasurer.ca.gov/cdiac.
Figure 1
2
Municipal Swap Index.
far the most common type of interest rate swaps.
Index2
a spread over U.S. Treasury bonds of a similar maturity.
p2
Issuer Pays Fixed Rate to Financial Institution Financial Institution Pays Variable Rate to IssuerIssuer Pays Variable Rate to Bond Holders
Formerly known as the Bond Market Association (BMA)
new sources of funding themselves; rather, they convert one interest rate basis to a different rate basis (e.g., from a floating or variable interest rate basis to a fixed interest rate basis, or vice versa). These “plain vanilla” swaps are by Typically, payments made by one counterparty are based on a floating rate of interest, such as the London Inter Bank Offered Rate (LIBOR) or the Securities Industry and Financial Markets Association (SIFMA) Municipal Swap , while payments made by the other counterparty are based on a fixed rate of interest, normally expressed as The maturity, or “tenor,” of a fixed-to-floating interest rate swap is usually between one and fifteen years. By conven-tion, a fixed-rate payer is designated as the buyer of the swap, while the floating-rate payer is the seller of the swap. Swaps vary widely with respect to underlying asset, matu-rity, style, and contingency provisions. Negotiated terms
include starting and ending dates, settlement frequency, notional amount on which swap payments are based, and published reference rates on which swap payments are determined.
Swap Pricing in Theory
Interest rate swap terms typically are set so that the pres-ent value of the counterparty paympres-ents is at least equal to the present value of the payments to be received. Present value is a way of comparing the value of cash flows now with the value of cash flows in the future. A dollar today is worth more than a dollar in the future because cash flows available today can be invested and grown.
The basic premise to an interest rate swap is that the coun-terparty choosing to pay the fixed rate and the counterpar-ty choosing to pay the floating rate each assume they will gain some advantage in doing so, depending on the swap rate. Their assumptions will be based on their needs and their estimates of the level and changes in interest rates during the period of the swap contract.
Because an interest rate swap is just a series of cash flows occurring at known future dates, it can be valued by sim-ply summing the present value of each of these cash flows. In order to calculate the present value of each cash flow, it is necessary to first estimate the correct discount factor (df) for each period (t) on which a cash flow occurs. Dis-count factors are derived from investors’ perceptions of in-terest rates in the future and are calculated using forward rates such as LIBOR. The following formula calculates a theoretical rate (known as the “Swap Rate”) for the fixed component of the swap contract:
Theoretical Present value of the floating-rate payments Swap Rate =
Notional principal x (days
t/360) x df t
Consider the following example:
step example, follows:
Step 1 – Calculate Numerator
floating-rate payments.
on actual semi-annual payments.3
3
,
and the Financial Times of London.
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A municipal issuer and counterparty agree to a $100 mil-lion “plain vanilla” swap starting in January 2006 that calls for a 3-year maturity with the municipal issuer paying the Swap Rate (fixed rate) to the counterparty and the counter-party paying 6-month LIBOR (floating rate) to the issuer. Using the above formula, the Swap Rate can be calculated by using the 6-month LIBOR “futures” rate to estimate the present value of the floating component payments. Pay-ments are assumed to be made on a semi-annual basis (i.e., 180-day periods). The above formula, shown as a
step-by-The first step is to calculate the present value (PV) of the This is done by forecasting each semi-annual payment using the LIBOR forward (futures) rates for the next three years. The following table illustrates the calculations based
LIBOR forward rates are available through financial informa-tion services including Bloomberg, the Wall Street Journal
A n n u al Se m i-an n u al Ac tu al F lo at in g F lo at in g R at e P V o f F lo at in g T im e P er io d D ay s i n Fo rw ar d Fo rw ar d R at e P ay m en t Fo rw ar d R at e P ay m en t a t P er io d Nu m b er P er io d R at e P er io d R at e at E n d P er io d D is co u n t F ac to r E n d o f P er io d (A ) (B ) (C ) (D ) (E ) (F ) (G ) (H ) 1/ 06 -6 /0 6 1 18 0 4. 00 % 2. 00 0% $2 ,0 00 ,0 00 0. 98 04 $1, 96 0, 80 0 7/ 06 -1 2/ 06 2 18 0 4. 25 % 2. 12 5% $2 ,1 25 ,0 00 0. 96 00 $2 ,0 40 ,0 00 1/ 07 -6 /0 7 3 18 0 4. 50 % 2. 25 0% $2 ,2 50 ,0 00 0. 93 89 $2 ,1 12 ,5 25 7/ 07 -1 2/ 07 4 18 0 4. 75 % 2. 37 5% $2 ,3 75 ,0 00 0. 91 71 $2 ,1 78 ,1 13 1/ 08 -6 /0 8 5 18 0 5. 00 % 2. 50 0% $2 ,5 00 ,0 00 0. 89 47 $2 ,2 36 ,7 50 7/ 08 -1 2/ 08 6 18 0 5. 25 % 2. 62 5% $2 ,6 25 ,0 00 0. 87 18 $2 ,2 88 ,4 75 P V o f F lo at in g R at e P ay m en ts = $1 2, 81 6, 66 3 C ol u m n D es cr ip ti on A = Pe ri od t he i nt er es t r at e i s i n e ff ec t B = Pe ri od n u m be r ( t) C = Nu m be r o f d ay s i n t he p er io d ( se m i-an nu al =1 80 d ay s) D = A n nu al i nt er es t r at e f or t he f ut u re p er io d f ro m fi n an ci al p ub lic at io n s E = Se m i-an nu al r at e f or t he f ut u re p er io d ( D /2 ) F = Ac tu al f or ec as te d p ay m en t ( E x $1 00 ,0 00 ,0 00 ) G = D is co u nt f ac to r=1 /[ (f or w ar d r at e f or p er io d 1 )( fo rw ar d r at e f or p er io d 2 )… (f or w ar d r at e f or p er io d t )] H = P V o f fl oa ti n g r at e p ay m en ts ( F x G )
p
are used to di year period. T
Step 2 – Cal
As with the floating-rate pa
culate Deno principal by the minator ments, LIBOR fo tional principal fo otional principal i y days in the rward rates r the three-s calculated example: by multiplyin period and the The following g the notional scount the no he PV of the n floating-rate table illustra fo
tes the calculatio rward discount factor.
ns for this
A n n u al Se m i-an n u al F lo at in g R at e T im e P er io d D ay s i n Fo rw ar d Fo rw ar d No ti on al Fo rw ar d P V o f N ot io n al P er io d Nu m b er P er io d R at e P er io d R at e P ri n ci p al D is co u n t F ac to r P ri n ci p al (A ) (B ) (C ) (D ) (E ) (F ) (G ) (H ) 1/ 06 -6 /0 6 1 18 0 4. 00 % 2. 00 0% $1 00 ,0 00 ,0 00 0. 98 04 $4 9, 02 0, 00 0 7/ 06 -1 2/ 06 2 18 0 4. 25 % 2. 12 5% $1 00 ,0 00 ,0 00 0. 96 00 $4 8, 00 0, 00 0 1/ 07 -6 /0 7 3 18 0 4. 50 % 2. 25 0% $1 00 ,0 00 ,0 00 0. 93 89 $4 6, 94 5, 00 0 7/ 07 -1 2/ 07 4 18 0 4. 75 % 2. 37 5% $1 00 ,0 00 ,0 00 0. 91 71 $4 5, 85 5, 00 0 1/ 08 -6 /0 8 5 18 0 5. 00 % 2. 50 0% $1 00 ,0 00 ,0 00 0. 89 47 $4 4, 73 5, 00 0 7/ 08 -1 2/ 08 6 18 0 5. 25 % 2. 62 5% $1 00 ,0 00 ,0 00 0. 87 18 $4 3, 59 0, 00 0 $2 78 ,1 45 ,0 00 P V o f N ot io n al P ri n ci p al = C ol u m n D es cr ip ti on A = Pe ri od t he i nt er es t r at e i s i n e ff ec t B = Pe ri od n u m be r ( t) C = Nu m be r o f d ay s i n t he p er io d ( se m i-an nu al =1 80 d ay s) D = A n nu al i nt er es t r at e f or t he f ut u re p er io d f ro m fi n an ci al p ub lic at io n s E = Se m i-an nu al r at e f or t he f ut u re p er io d ( D /2 ) F = No ti on al p ri n ci pa l f ro m s w ap c on tr ac t G = D is co u nt f ac to r=1 /[ (f or w ar d r at e f or p er io d 1 )( fo rw ar d r at e f or p er io d 2 )… (f or w ar d r at e f or p er io d t )] H = P V o f n ot io n al p ri n ci pa l [ F x (C /3 60 ) x G ]
p
Step 3 – Calculate Swap Rate
Using the results from Steps 1 and 2 above, solve for the theoretical Swap Rate:
Theoretical $12,816,663
= =
4.61%
Swap Rate $278,145,000
Based on the above example, the issuer (fixed-rate payer) will be willing to pay a fixed 4.61 percent rate for the life of the swap contact in return for receiving 6-month LIBOR.
Step 4 - Calculate Swap Spread
With a known Swap Rate, the counterparties can now determine the “swap spread.”4 The market convention is to use a U.S. Treasury security of comparable maturity as a benchmark. For example, if a three-year U.S. Treasury note had a yield to maturity of 4.31 percent, the swap spread in this case would be 30 basis points (4.61% - 4.31% = 0.30%).
Swap Pricing in Practice
The interest rate swap market is large and efficient. While understanding the theoretical underpinnings from which swap rates are derived is important to the issuer, computer programs designed by the major financial institutions and market participants have eliminated the issuer’s need to perform complex calculations to determine pricing. Swap pricing exercised in the municipal market is derived from three components: SIFMA percentage (formerly known as the BMA percentage).
4 The swap spread is the difference between the Swap Rate and
the rate offered through other comparable investment
U.S. Trea
The choice curve is bas reflect their its own curr
sury Yield
ed on the arg credit risk. A ency is assum of the U.S. Tre
u bo
ed
ment that the yi nd issued by a g
asury yield curve as the risk-free elds on bonds
its yield sho rates on U.S participant es to suppl to the econ . Treasury sec y
uld equal the r ur
s’ views on a variety of factors inc
and demand for high quality credit relative omic cycle, the effect of inflation and investor k-free rate of interest. Interest ities are influenced by market luding chang-is
to have no credit risk so that overnment in
expectations on interest rate levels, yield curve analysis, and change
ity groups.
LIBOR Sp
s in credit spreads between fixed-income
qual-read LIBOR is t London inte The rate is LIBOR swa that the co risk inheren rbank market t in LIBOR, th he interest ra
set for Eurodollar d p spread is a pr unterparty must
b
e te
orrow money from each other. nominated
current supply/ charged when
e
emium over the pay for the add
banks in the deposits. The risk free rate demand rela-itional credit tionship for venience of SIFM The SIFMA A P fixed versus fl holding U.S. T index is a t ercentage o re ax ating-rate swaps asury securities.
, and the
con--exempt, weekly reset index composed able-rate de benchmark tax-exempt The SIFMA mand obligati for borrowers obligations. of 650 differen o a t high-grade, tax-ns (VRDOs). It is nd dealer firms of
percentage is set to approximate average mu-exempt, vari-a widely used variable-rate nicipal VRD VRDO rate BOR: [(1-M O yields over s should equal arginal Tax R t t at
he long run. In theory, future he after-tax equ
e) x LIBOR] plus a spread to
ivalent of
reflect liquidity and other risks. Historically, municipal swaps have used 67 percentage of one-month LIBOR as a benchmark for floating payments in connection with floating-rate transactions. The market uses this percent-age based on the historic trading relationship between the LIBOR and the SIFMA index. There are a number of factors that affect the SIFMA percentage and they may manifest themselves during different interest rate environments. The most significant factors influencing the SIFMA per-centage are changes in marginal tax laws. Availability of similar substitute investments and the volume of munici-pal bond issuance also play significant roles in determin-ing the SIFMA percentage durdetermin-ing periods of stable rates. The basic formula for a SIFMA Swap Rate uses a comparable maturity U.S. Treasury yield, adds a LIBOR “swap spread”, then multiplies the result by the SIFMA percentage.
[Treasury yield of comparable
SIFMA Swap Rate
=
maturity+ LIBOR Spread] xSIFMA Percentage
Although pricing is generally uniform, it is important to know the components that comprise actual real-life pric-ing and their effect on valupric-ing the swap at any time durpric-ing the contract period. Figure 2 below describes the SIFMA Swap Rate calculation.
The Swap Yield Curve
As with most fixed-income investments, there is a positive correlation between time and risk and thus required re-turn. This is also true for swap transactions.
Interest rates tend to vary as a function of maturity. The relationship of interest rates to maturities of specific secu-rity types is known as the “yield curve.”
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swap contract was initiated.
Figure 2
Example of 3 Year Generic SIFMA Swap
Treasury note
4.31%
+
Current 3 year LIBOR swap spread over 3 year U.S Treasury note
.30%
=
3 Year LIBOR Swap Rate
4.61%
Multiplied By
3 year SIFMA percentage
67%
=
3 Year SIFMA Swap Rate
3.09%
1 2 3
Figure 3
Swap Yield Curve
Using the example in Figure 2, Figure 3 graphically dis-plays a hypothetical “swap yield curve” at the time the Current Market Yield to Maturity on a 3 year U.S.
Time to Maturity ( Years) Treasury Yields
SIFMA Swap Rate LIBOR Swap Rate
5.00% 4.50% 4.00% 3.50% 3.00% 2.50% 2.00% R at e ( % ) 10 30
tics, interest r
For municipal bonds and swaps of similar
or less pro- characteris-higher for longer maturities es. At different points in the , this relationship may be more
nounced, causing a more steeply sloped curve or a curve ates tend to be
relative to shorter maturiti business cycle
est rates in the future. that is relative
reflects investors’ expectatio
e Termina
n
ly flat. In general, the slope of th s about the beha
e yield curve vior of
inter-Finding th
tion Value of a Swap
component of Once the swap ket interest ra
the swap. As d transaction i tes will change
i s the payments on s completed, chan cussed in the “S the floating ges in mar-wap Pricing in Theory” section above, at the initiation of an interest rate the fixed-rate
swap the PV of rate. If interes swap has been are that the fu swap will be h
cash flows wil the floating-r t rates increas initiated, the ture floating-igher than th at shortly after an e cash flows min be zero at a specific int current market expectations rate payments due under the ose originally expected when l
e interest rate us the PV of erest
resent a cost t
the swap was priced. As shown
er.
in Figure 4, this under the swap a o the floating-rate pay
If the new cash flows due under the swap are computed and if these are discounted at the appropriate new rate for each accrue to the fixed-rate payer nd will
rep-benefit will
flects how the future period and not the or increased from floating comp
value of the sw (i.e., reflecting iginal swap yi
the initial val onent has decl a t el u in p to the fixed-ra he current swap d curve), the pos e of zero and the ed from the init
te payer has yield curve itive PV re-value of the ial zero to a negative amount.
Using the tabl value of the sw
e below, the fol ap based on a
lo 5
wing example ca
0 basis point increase in the lculates the
current SIFMA swap rate. The contract was written for a 3-year, $100,000,000 SIFMA swap that was initiated one year ago. The contract has 2 additional years to run before maturity.
This calculation shows a PV for the swap of $948,617, which reflects the future cash flows discounted at the cur-rent market 2-year SIFMA swap rate of 3.59 percent. If the floating-rate payer were to terminate the contract at this point in time, they would be liable to the fixed-rate payer for this amount. Issuers typically construct a “termination matrix” to monitor the exposure they may have based on different interest rate scenarios.
Change in Swap Value to Issuer
as Rates Change
Figure 4Rates Rise Rates Fall
Issuer Pays Fixed
+
–
Issuer Receives Fixed
–
+
The counterparties will continuously monitor the market value of their swaps, and if they determine the swap to be a financial burden, they may request to terminate the con-tract. Significant changes in any of the components (e.g., interest rates, swap spreads, or SIFMA percentage) may cause financial concern for the issuer. It is also important to note that there are other administration fees and/or contractual fees associated with a termination that may influence the decision whether to end the swap.
Notional Amount: $100,000,000
Existing Fixed Rate Paid by Issuer: 3.09%
Current Market Fixed Rate for 2-year SIFMA swap: 3.59% Annual Fixed Annual Fixed
Payments @ Payments @ Present
year 3.09% 3.59% Difference Value
2 $3,090,000 $3,590,000 $ 500,000 $ 482,672
3 $3,090,000 $3,590,000 $ 500,000 $ 465,945
Swap value= $ 948,617
Swap Pricing Process
The interest rate swap market has evolved from one in which swap brokers acted as intermediaries facilitating the needs of those wanting to enter into interest rate swaps. The broker charged a commission for the trans-action but did not participate in the ongoing risks or ad-ministration of the swap transaction. The swap parties were responsible for assuring that the transaction was successful.
In the current swap market, the role of the broker has been replaced by a dealer-based market comprised of large commercial and international financial institu-tions. Unlike brokers, dealers in the over-the-counter market do not charge a commission. Instead, they quote “bid” and “ask” prices at which they stand ready to act as counterparties to their customers in the swap. Because dealers act as middlemen, counterparties need only be concerned with the financial condition of the dealer, and not with the creditworthiness of the other ultimate end user of the swap.
Administ The price of a fixed inter terest rate i rative Conventions a fixed-to-floa est rate and an s based. The flo
i
ating rate can b ting swap is quote ndex on which t d in two parts: he floating in-e basin-ed on an rity of LIBO set “flat;” th no margin a to quote th which mean R) plus or mi dded. The co e fixed interest
s that the fixe at is, the floati
d u n nv
rate as an “all-i
interest rate is quoted relative index of short-term market rates (such as a given
matu-n s a givematu-n margimatu-n, or it camatu-n be g interest rate index itself with ention in the swap market is n-cost” (AIC), to a flat floating-rate index.
The AIC typ securities w swap. For e ically is quote ith a maturity xample, a swa d co p as a spread over rresponding to t dealer might qu U.S. Treasury he term of the ote a price on a three-year plain vanilla swap at an AIC of “72-76 flat,” which mean
(that is, ent basis points U.S. Treasur dexed to a and “sell” (r s the dealer s er into the sw over the preva ies while rece specified matu eceive a fixed t a il iv r ra
ands ready to “buy” the swap p as a fixed-rate payer) at 72 ing three-year in
ing floating-rate ity of LIBOR wi te and pay the fl
terest rate on payments in-th no margin, oating rate) if the other pa over U.S. Tr market vary The spread three-year p
rty to the swa greatly depen lain vanilla sw easury securiti may be less th p d a e an agrees to pay 7 ing on the type p, while spread s. Bid-ask spread
five basis point
6 basis points of agreement. s for nonstan-s in the nonstan-swap s for a two- or m-tailored swa dard, custo Timing of A swap is two days lat
Payments negotiated on er on its initia a p l “ “trade date” an s tend to be high settlement date. d takes effect er. ” Interest be-gins accruin usually coin ing-rate pay based on th g on the “effe cides with th ments are adj e prevailing m
c u a e
tive date” of the swap, which sted on periodic “reset dates”
rket-determine initial settlemen
d value of the t date.
a sequence of frequency for floating-rate i dates) specifie interest-rate i payment date the floating-r ndex, with sub
s ( d by the agree
at ndex itself. For
sequent payment
the reset s made on also known as settlement ment. Typically,
e index is the term of the example, the floating rate
Fixed interest on a plain van would, in mos ment dates fol six months, or
payment inte illa swap index t cases, be reset
lowing six mo one year. Semi
r e
every six months nths later.
vals can be three annual payment
with pay-months, d to the six-month LIBOR
intervals vals between i
Floating-rate
are most common because they coincide with the
inter-often do. ts on U.S. Treasury bonds. vals need not coincide with ment intervals, although they
t intervals coincide, it is common practice nterest paymen payment inter fixed-rate pay When paymen
Conclusio
to exchange o rate and floatiThe goal of th
n
nly the net di ng-rate payme
is report has be nts.
fference between the
fixed-basic un-to offer the re
questions to prior to enteri
derstanding of municipal inte
dvisor or un en to provide a rest rate swap p on to ask relevant p his/her financial a
ng into an interest rate swap. ader a foundati
derwriter ricing and ricing
Pricing municipal interest rate swaps is a multi-faceted variables to de
market has evolved, pricing exercise incor which allows porating econo termine a fair a m transparency has ic, market, tax, nd appropriate r
and credit ate. As the increased,
interest rate swap(s). determine a f
As shown abo determine inte
ve, small chan rest rate swap air initial and the issuer to us
t
ges in the components that pricing can have a financial
ermination price
e many analytical tools to for their
effect on th can be time resources to If an issuer i consuming an the analysis a s contemplati e issuer. Also, d n ad ng
requires the issu
d monitoring of the contract. ministering a s entering into a er to dedicate waps program swap transac-tion, these context of t be able to id speculative heir overall fi entify risks in purposes. issues and oth
alternative financing meth h e na
erent in swaps, re ods, and avoid us ncial plan. The rs should be eva
issuer should luated in the cognize other ing swaps for
p18
References
F. Fabozzi. Th
(Seventh Edition), The McGr e Handbook of F
aw-Hill Companies, 2
ixed Income Securities
005. A. Kuprianov, Derivatives, Fe Economic Qu D. Rubin, D. G the Historical Over-the-Coun deral Reserve arterly Volume oldberg, and I. Relationship B t B 7 G et er Interest Rate ank of Richmon 9, No. 3, Summer 1 reenbaum. Report on
ween SIFMA and LIBOR,
993. d
CDR Financial Products, August 2003.
February 6, 2002.
Credit Impact Municipal Fin
s of Variable R ance, Standard
ate Debt and Swaps i
and Poor’s Ratings D
n over the Past Q
Finance Special Issue 2005, 125-153.
irect, W. Bartley Hil
the State and Local Governm
dreth and C. K
uarter Century,
, Interest Rate Swaps, California u
ent Municipal Debt
Public Budgeting & rt Zorn, The Evolution of
Market Bond Logistix C. Underwood Municipal Tre LLC, January asurer’s Assoc 25, iation Advanced 2006. Workshop,
Acknowledgements
This docum
Doug Sk by Krist
Special t
ent was writ
arr, Research
hanks to
in Szakaly-Mot
Pr oren by
ogram Specialist, e, Director of Pol and reviewed icy Research. Kay Cha Ken Ful Debora Tom W ndler, Chandl lerton and Rob h Higgins, Higg alsh, Franklin T er e in Asset Manageme empleton; and rt Friar, Fullerto s Capital Manag nt;n & Friar, Inc.; ement;
Chris W
for thei
inters, Winter
r review and comments.
s and Co., LLC
© All Rights without w
Investment Advisory Commis Reserved. No part o ritten credit given t si f t o t
on (CDIAC). his report may be rep he California Debt an
roduced d
