&
rate swap math pricing
Understanding interest
January 2007
CDIAC #06-11
California Debt and Investment Advisory Commission
&
rate swap math pricing
Understanding interest
January 2007
CDIAC #06-11
California Debt and Investment Advisory Commission
1 Introduction
1 Basic Interest Rate Swap Mechanics
3 Swap Pricing in Theory
8 Swap Pricing in Practice
12 Finding the Termination Value of a Swap
14 Swap Pricing Process
16 Conclusion
18 References
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 pricing, 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.
p1
Figure 1
2
Municipal Swap Index.
far the most common type of interest rate swaps.
Index
2
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 Issuer
Issuer 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 payments 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
p3
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.
p4
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
Annual Semi-annual Actual Floating Floating Rate PV of Floating
Time Period Days in Forward Forward Rate Payment Forward Rate Payment at
Period Number Period Rate Period Rate at End Period Discount Factor End of Period
(A) (B) (C) (D) (E) (F) (G) (H)
1/06-6/06 1 180 4.00% 2.000% $2,000,000 0.9804 $1,960,800
7/
06-12/06 2 180 4.25% 2.125% $2,125,000 0.9600 $2,040,000
1/
07-6/07 3 180 4.50% 2.250% $2,250,000 0.9389 $2,112,525
7/
07-12/07 4 180 4.75% 2.375% $2,375,000 0.9171 $2,178,113
1/
08-6/08 5 180 5.00% 2.500% $2,500,000 0.8947 $2,236,750
7/
08-12/08 6 180 5.25% 2.625% $2,625,000 0.8718 $2,288,475
PV of Floating Rate Payments= $12,816,663
Column Description
A= Period the interest rate is in effect
B=
Period number (t)
C= Number of days in the period (semi-annual=180 days)
D= Annual interest rate for the future period from fi nancial publications
E= Semi-annual rate for the future period (D/2)
F= Actual forecasted payment (E
x
$100,000,000)
G= Discount factor=1/[(forward rate for period 1)(forward rate for period 2)…(forward rate for period t)]
H= PV of fl oating rate payments (F
x
G)
p5
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
p6
Annual Semi-annual Floating Rate
Time Period Days in Forward Forward Notional Forward PV of Notional
Period Number Period Rate Period Rate Principal Discount Factor Principal
(A) (B) (C) (D) (E) (F) (G) (H)
1/06-6/06 1 180 4.00% 2.000% $100,000,000 0.9804 $49,020,000
7/
06-12/06 2 180 4.25% 2.125% $100,000,000 0.9600 $48,000,000
1/
07-6/07 3 180 4.50% 2.250% $100,000,000 0.9389 $46,945,000
7/
07-12/07 4 180 4.75% 2.375% $100,000,000 0.9171 $45,855,000
1/
08-6/08 5 180 5.00% 2.500% $100,000,000 0.8947 $44,735,000
7/
08-12/08 6 180 5.25% 2.625% $100,000,000 0.8718 $43,590,000
$278,145,000 PV of Notional Principal=
Column Description
A= Period the interest rate is in effect
B=
Period number (t)
C=
Number of days in the period (semi-annual=180 days)
D= Annual interest rate for the future period from fi nancial publications
E= Semi-annual rate for the future period (D/2)
F= Notional principal from swap contract
G=
Discount factor=1/[(forward rate for period 1)(forward rate for period 2)…(forward rate for period t)]
H= PV of notional principal [F
x
(C/360)
x
G]
p7
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 instru-
ments with comparable characteristics (e.g., similar maturity).
p8
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 LI-
p9
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 during periods of stable rates.
The basic formula for a SIFMA Swap Rate uses a comparable
ma
turity 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]
x
SIFMA Percentage
Although pricing is generally uniform, it is important to
know the components that comprise actual real-life pric-
ing and their effect on valuing the swap at any time during
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.”
p10
p11
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%
Rate ( %)
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
p12
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 4
Rates 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.
p13
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 @ Pr
esent
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.
p14
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 based 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 given margin, or it can 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 swap
s for a two- or
m-tailored swadard, 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. Float-
p15
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 floati
The 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
p16
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
p17
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 Financia
l 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-Mo
t
Pr
or
en 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
OSP 07 101009
California
Advisory
Debt and Inve
Commission
stment
915 Capit
Sacramen
ol Mall, Room 400
to, CA 95814
phone | 916.653.3269
ww
fax | 916.6
cdiac@treasure
w.treasurer.ca.g
54.7440
r.ca.gov
ov/cdiac
