Swap Int Rate

8/8/2019 Swap Int Rate

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&rate swap math pricingUnderstanding interest

January 2007CDIAC

#06-11

California Debt and Investment Advisory Commission


2/24

&rate swap math pricingUnderstanding interest

January 2007CDIAC

#06-11

California Debt and Investment Advisory Commission


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1 Introduction 1 BasicInterestRat

eSwapMechanics 3 SwapPricinginTheory 8 SwapPricinginPractice12 Findin

gtheTerminationValueofaSwap

14 SwapPricingProcess 16 Conclusion 18

References


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p1

Introduction

As Caliornia local agencies are becoming involved in theinterest rate swap market, knowledge o the basics o pricing swaps may assist issuers to better understand initial,mark-to-market, and termination costs associated withtheir swap programs.

This report is intended to provide treasury managers and

sta with a basic overview o swap math and related pricing conventions. It provides inormation on the interestrate swap market, the swap dealers pricing and sales conventions, the relevant indices needed to determine pricing, ormulas or and examples o pricing, and a review ovariables that have an aect on market and termination

pricing o an existing swap.1

Basic Interest Rate Swap Mechanics

An interest rate swap is a contractual arrangement between two parties, oten reerred to as counterparties.As shown in Figure 1, the counterparties (in this example,

a nancial institution and an issuer) agree to exchangepayments based on a dened principal amount, or a xedperiod o time.

In an interest rate swap, the principal amount is not actually exchanged between the counterparties, rather, interest 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 o interest rate swaps,the Caliornia Debt and Investment Advisory Commission(CDIAC) also has published Fundamentals of Interest RateSwaps and20 Questions for Municipal Interest Rate Swap Issu-ers. These publications are available on the CDIAC website at

www.treasurer.ca.gov/cdiac.p1


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Figure 1

2

Municipal Swap Index.

ar the most common type o interest rate swaps.

Index2

a spread over U.S. Treasury bonds o a similar maturity.

p2

Issuer Pays

Fixed Rate

to

Financial

Institution

Financial

Institution

Pays

Variable Rate

to Issuer

Issuer PaysVariable Rate

to Bond Holders

Formerly known as the Bond Market Association (BMA)

new sources o unding themselves; rather, they convert

one interest rate basis to a dierent rate basis (e.g., roma foating or variable interest rate basis to a xed interestrate basis, or vice versa). These plain vanilla swaps are by

Typically, payments made by one counterparty are basedon a foating rate o interest, such as the London Inter

Bank Oered Rate (LIBOR) or the Securities Industry andFinancial Markets Association (SIFMA) Municipal Swap, while payments made by the other counterparty

are based on a xed rate o interest, normally expressed as

The maturity, or tenor, o a xed-to-foating interest rateswap is usually between one and teen years. By convention, a xed-rate payer is designated as the buyer o theswap, while the foating-rate payer is the seller o the swap.

Swaps vary widely with respect to underlying asset, maturity, style, and contingency provisions. Negotiated terms


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include starting and ending dates, settlement requency,notional amount on which swap payments are based, and

published reerence rates on which swap payments are

determined.

Swap Pricing in Theory

Interest rate swap terms typically are set so that the present value o the counterparty payments is at least equal tothe present value o the payments to be received. Present

value is a way o comparing the value o cash fows nowwith the value o cash fows in the uture. A dollar today isworth more than a dollar in the uture because cash fowsavailable today can be invested and grown.

The basic premise to an interest rate swap is that the counterparty choosing to pay the xed rate and the counterpar

ty choosing to pay the foating rate each assume they willgain some advantage in doing so, depending on the swaprate. Their assumptions will be based on their needs andtheir estimates o the level and changes in interest ratesduring the period o the swap contract.

Because an interest rate swap is just a series o cash fows

occurring at known uture dates, it can be valued by simply summing the present value o each o these cash fows.In order to calculate the present value o each cash fow,it is necessary to rst estimate the correct discount actor(d) or each period (t) on which a cash fow occurs. Discount actors are derived rom investors perceptions o interest rates in the uture and are calculated using orward

rates such as LIBOR. The ollowing ormula calculates atheoretical rate (known as the Swap Rate) or the xedcomponent o the swap contract:

Theoretical Present value o the foating-rate paymentsSwap Rate =

Notional principal x (dayst/360) x d

t

p3


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Consider the following example:

step example, ollows:

Step 1 Calculate Numerator

foating-rate payments.

on actual semi-annual payments.3

3

,and theFinancial Times o London.

p4

A municipal issuer and counterparty agree to a $100 million plain vanilla swap starting in January 2006 that calls

or a 3-year maturity with the municipal issuer paying theSwap Rate (xed rate) to the counterparty and the counter-

party paying 6-month LIBOR (foating rate) to the issuer.Using the above ormula, the Swap Rate can be calculatedby using the 6-month LIBOR utures rate to estimate the

present value o the foating component payments. Payments are assumed to be made on a semi-annual basis (i.e.,180-day periods). The above ormula, shown as a step-by-

The rst step is to calculate the present value (PV) o the

This is done by orecasting each semi-annual paymentusing the LIBOR orward (utures) rates or the next threeyears. The ollowing table illustrates the calculations based

LIBOR orward rates are available through fnancial inorma-tion services including Bloomberg, the Wall Street Journal


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Annu

al

Semi-annualActualFloating

Floatin

gRate

PVofFloating

Time

Pe

riod

Daysin

Forw

ard

Forward

RatePayment

Forward

RatePayme

ntat

Period

Nu

mber

Period

Rate

PeriodRate

at

EndPeriod

Discou

ntFactor

EndofPeriod

(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

PVofFloatingRatePayments=

$12,816,6

63

ColumnDescrip

tion

A=Periodthe

interestrateisineect

B=Periodnumber(t)

C=Numberodaysintheperiod(sem

i-annual=180days)

D=Annualinterestrateortheuture

periodromnancialp

ublications

E=Semi-annu

alrateortheuturepe

riod(D/2)

F=Actualorecastedpayment(Ex$1

00,000,000)

G=Discountactor=1/[(orwardrateorperiod1)(orwardrateorperiod2)(orwardrateorperiodt)]

H=PVofoat

ingratepayments(FxG

)

p


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are used to di

year period. T

Step 2 Cal

As with the foating-rate pa

culate Deno

principal by the

minator

ments, LIBOR otional principal o

otional principal i

y

days in the

rward ratesr the three

s calculated

example:

by multiplyinperiod and the

The ollowing

g the notional

scount the no

he PV o the n

foating-rate

table illustra

o

tes the calculatio

rward discount actor.

ns or this

p6


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Annual

Semi-annual

Float

ingRate

Time

Period

Daysin

For

ward

Forward

Notional

Forw

ard

PVofNot

ional

Period

Number

Period

Rat

e

PeriodRate

Principal

Disco

untFactor

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

PVofNotional

Principal=

ColumnDescr

iption

A=Periodtheinterestrateisineect

B=Periodn

umber(t)

C=Number

odaysintheperiod(se

mi-annual=180days)

D=Annualinterestrateortheutu

reperiodromnancialpublications

E=Semi-an

nualrateortheutureperiod(D/2)

F=Notionalprincipalromswapco

ntract

G=Discountactor=1/[(orwardrat

eorperiod1)(orwardrateorperiod2)(orw

ardrateorperiodt)]

H=PVono

tionalprincipal[Fx(C/360)xG

]

p


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Step 3 Calculate Swap Rate

Using the results rom Steps 1 and 2 above, solve or thetheoretical Swap Rate:

Theoretical $12,816,663= = 4.61%Swap Rate $278,145,000

Based on the above example, the issuer (xed-rate payer)will be willing to pay a xed 4.61 percent rate or the lie o

the swap contact in return or receiving 6-month LIBOR.

Step 4 - Calculate Swap Spread

With a known Swap Rate, the counterparties can nowdetermine the swap spread.4 The market convention isto use a U.S. Treasury security o comparable maturity as a

benchmark. For example, i a three-year U.S. Treasury notehad a yield to maturity o 4.31 percent, the swap spread inthis case would be 30 basis points (4.61% - 4.31% = 0.30%).

Swap Pricing in Practice

The interest rate swap market is large and ecient. Whileunderstanding the theoretical underpinnings rom whichswap rates are derived is important to the issuer, computer

programs designed by the major nancial institutions andmarket participants have eliminated the issuers need to

perorm complex calculations to determine pricing. Swappricing exercised in the municipal market is derived rom

three components: SIFMA percentage (ormerly known asthe BMA percentage).

4 The swap spread is the dierence between the Swap Rate andthe rate oered through other comparable investment instru-ments with comparable characteristics (e.g., similar maturity).

p8


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U.S. Trea

The choicecurve is bas

refect theirits own curr

sury Yield

ed on the arg

credit risk. Aency is assum

o the U.S. Treu

boed

ment that the yi

nd issued by a g

asury yield curve as the risk-reeelds on bonds

its yield shorates on U.S

participantes to supplto the econ

. Treasury sec

y

uld equal the rur

s views on a variety o actors incand demand or high quality credit relative

omic cycle, the eect o infation and investor

k-ree rate o interest. Interestities are infuenced by market

luding chang

isto have no credit risk so that

overnment in

expectations on interest rate levels, yield curve analysis,and changeity groups.

LIBOR Sp

s in credit spreads between xed-income qual-

read

LIBOR is t

London inteThe rate isLIBOR swathat the corisk inheren

rbank market

t in LIBOR, th

he interest ra

set or Eurodollar dp spread is a prunterparty must

b

e

te

orrow money rom each other.nominated

current supply/

charged when

eemium over thepay or the add

banks in the

deposits. Therisk ree rate

demand relaitional credit

tionship orvenience o

SIFM

The SIFMA

A P

xed versus fholding U.S. T

index is a t

ercentage

ore

ax

ating-rate swapsasury securities.

, and the con-

-exempt, weekly reset indexcomposedable-rate debenchmark

tax-exemptThe SIFMA

mand obligatior borrowers

obligations.

o 650 dierenoa

t high-grade, taxns (VRDOs). It isnd dealer rms o

percentage is set to approximate average mu-

exempt, varia widely usedvariable-rate

nicipal VRDVRDO rateBOR: [(1-M

O yields overs should equalarginal Tax R

tt

at

he long run. In theory, uturehe ater-tax eque) x LIBOR] plus a spread to

ivalent o LI-

p


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refect liquidity and other risks. Historically, municipalswaps have used 67 percentage o one-month LIBOR asa benchmark or foating payments in connection with

foating-rate transactions. The market uses this percentage based on the historic trading relationship between theLIBOR and the SIFMA index. There are a number o actorsthat aect the SIFMA percentage and they may maniestthemselves during dierent interest rate environments.The most signicant actors infuencing the SIFMA percentage are changes in marginal tax laws. Availability o

similar substitute investments and the volume o municipal bond issuance also play signicant roles in determining the SIFMA percentage during periods o stable rates.

The basic ormula or a SIFMA Swap Rate uses a comparablematurity U.S. Treasury yield, adds a LIBOR swap spread,then multiplies the result by the SIFMA percentage.

[Treasury yield o comparableSIFMA Swap Rate = maturity+ LIBOR Spread] x

SIFMA Percentage

Although pricing is generally uniorm, it is important toknow the components that comprise actual real-lie pric

ing and their eect on valuing the swap at any time duringthe contract period. Figure 2 below describes the SIFMASwap Rate calculation.

The Swap Yield Curve

As with most xed-income investments, there is a positivecorrelation between time and risk and thus required re

turn. This is also true or swap transactions.

Interest rates tend to vary as a unction o maturity. Therelationship o interest rates to maturities o specic security types is known as the yield curve.

p10


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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.STreasury 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


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tics, interest rFor municipal bonds and swaps o similar

or less pro

characterishigher or longer maturities

es. At dierent points in the

, this relationship may be morenounced, causing a more steeply sloped curve or a curve

ates tend to berelative to shorter maturiti

business cycle

est rates in the uture.

that is relativerefects investors expectatio

e Termina

nly fat. In general, the slope o th

s about the behae yield curvevior o inter-

Finding th tion Value of a Swap

component o

Once the swapket interest ra

the swap. As d

transaction ites will change

i

sthe payments ons

completed, chan

cussed in the Sthe foatingges in mar

wap Pricingin Theory section above, at the initiation o an interest rate

the xed-rate

swap the PV o

rate. I interesswap has beenare that the uswap will be h

cash fows wil

the foating-r

t rates increasinitiated, the

ture foatingigher than th

at

shortly ater an

e cash fows min

be zero at a specic int

current market expectationsrate payments due under theose originally expected when

le interest rate

us the PV o

erest

resent a cost t

the swap was priced. As shown

er.

in Figure 4, thisunder the swap a

o the foating-rate pay

I the new cash fows due under the swap are computed andi these are discounted at the appropriate new rate or each

accrue to the xed-rate payer nd will rep-benet will

fects how the

uture periodand not the or

increased romfoating comp

value o the sw

(i.e., refectingiginal swap yi

the initial valonent has decl

a

tel

uin

p to the xed-ra

he current swapd curve), the pos

e o zero and theed rom the init

te payer has

yield curveitive PV re

value o theial zero to a

negative amount.

Using the tablvalue o the sw

e below, the olap based on a

lo5wing example ca0 basis point increase in the

lculates the

p12


17/24

current SIFMA swap rate. The contract was written or a3-year, $100,000,000 SIFMA swap that was initiated one

year ago. The contract has 2 additional years to run beore

maturity.

This calculation shows a PV or the swap o $948,617,which refects the uture cash fows discounted at the current market 2-year SIFMA swap rate o 3.59 percent. I thefoating-rate payer were to terminate the contract at this

point in time, they would be liable to the xed-rate payer

or this amount. Issuers typically construct a terminationmatrix to monitor the exposure they may have based ondierent 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 marketvalue o their swaps, and i they determine the swap to bea nancial burden, they may request to terminate the contract. Signicant changes in any o the components (e.g.,interest rates, swap spreads, or SIFMA percentage) maycause nancial concern or the issuer. It is also importantto note that there are other administration ees and/or

contractual ees associated with a termination that mayinfuence the decision whether to end the swap.

p13


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Notional Amount: $100,000,000

ExistingFixedRatePaidbyIssuer:3.09%

CurrentMarketFixedRatefor2-yearSIFMAswap:3.59%AnnualFixed AnnualFixed

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 rom one inwhich swap brokers acted as intermediaries acilitatingthe needs o those wanting to enter into interest rateswaps. The broker charged a commission or the transaction but did not participate in the ongoing risks or administration o the swap transaction. The swap parties

were responsible or assuring that the transaction wassuccessul.

In the current swap market, the role o the broker hasbeen replaced by a dealer-based market comprised olarge commercial and international nancial institutions. Unlike brokers, dealers in the over-the-countermarket do not charge a commission. Instead, they quotebid and ask prices at which they stand ready to act ascounterparties to their customers in the swap. Becausedealers act as middlemen, counterparties need only beconcerned with the nancial condition o the dealer,and not with the creditworthiness o the other ultimateend user o the swap.

p14


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Administ

The price oa xed interterest rate i

rative Conventions

a xed-to-foaest rate and ans based. The fo

iating rate can b

ting swap is quotendex on which t

d in two parts:he foating in-e based on an

rity o LIBOset fat; thno margin ato quote th

which mean

R) plus or mi

dded. The coe xed interests that the xe

at is, the foati

d

unnv

rate as an all-iinterest rate is quoted relative

index o short-term market rates (such as a given matun s a given margin, or it can be

g interest rate index itsel withention in the swap market is

n-cost (AIC),

to a fat foating-rate index.

The AIC typsecurities wswap. For e

ically is quoteith a maturityxample, a swa

dcop

as a spread overrresponding to tdealer might qu

U.S. Treasuryhe term o theote a price on

a three-year plain vanilla swap at an AIC o 72-76 fat, which mean

(that is, entbasis pointsU.S. Treasurdexed to aand sell (r

s the dealer s

er into the swover the prevaies while rece

specied matueceive a xed

t

ail

ivr

ra

ands ready to buy the swap

p as a xed-rate payer) at 72ing three-year ining foating-rateity o LIBOR wite and pay the f

terest rate onpayments inth no margin,oating rate) i

the other paover U.S. Trmarket varyThe spreadthree-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

ve basis point

6 basis points

o agreement.

s or nonstan

s in the swap

s or a two- or

m-tailored swadard, custo

Timing of

A swap istwo days lat

Payments

negotiated oner on its initia

a

p

l trade date an

s tend to be high

settlement date.d takes eect

er.

Interest begins accruinusually coining-rate paybased on th

g on the eecides with thments are adje prevailing m

c

ua

etive date o the swap, which

sted on periodic reset datesrket-determine

initial settlemen

d value o the

t date. Float-

p1


20/24

a sequence o

requency or

foating-rate i

dates) specie

interest-rate i

payment date

the foating-r

ndex, with subs (

d by the agree

atndex itsel. For

sequent payment

the reset

s made onalso known as settlement

ment. Typically,

e index is the term o theexample, the foating rate

Fixed interest

on a plain vanwould, in mosment dates ol

six months, or

payment inte

illa swap indext cases, be resetlowing six mo

one year. Semi

r

eevery six months

nths later.

vals can be three

annual payment

with pay-

months,

d to the six-month LIBOR

intervals

vals between iFloating-rate

are most common because they coincide with the inter-

oten do.

ts on U.S. Treasury bonds.vals need not coincide with

ment intervals, although theyt intervals coincide, it is common practice

nterest paymenpayment inter

xed-rate payWhen paymen

Conclusio

to exchange o

rate and foati

The goal o th

n

nly the net di

ng-rate payme

is report has be

nts.

erence between the xed-

basic un

to oer the requestions to

prior to enteri

derstanding o municipal inte

dvisor or un

en to provide arest rate swap p

on to ask relevant phis/her nancial ang into an interest rate swap.

ader a oundatiderwriter

ricing and

ricing

Pricing municipal interest rate swaps is a multi-aceted

variables to demarket has evolved, pricing

exercise incor

which allows

porating econotermine a air a

m

transparency has

ic, market, tax,nd appropriate r

and creditate. As theincreased,

interest rate swap(s).determine a

As shown abodetermine inte

ve, small chanrest rate swap

air initial andthe issuer to us

t

ges in the components thatpricing can have a nancial

ermination pricee many analytical tools to

or their

p16


21/24

eect on thcan be timeresources to

I an issuer i

consuming anthe analysis a

s contemplati

e issuer. Also,dn

ad

ng

requires the issud monitoring o the contract.

ministering a s

entering into a

er to dedicatewaps program

swap transaction, thesecontext o tbe able to id

speculative

heir overall entiy risks in

purposes.

issues and oth

alternative nancing methh

ena

erent in swaps, reods, and avoid us

ncial plan. Thers should be eva

issuer shouldluated in the

cognize othering swaps or

p1


22/24

p18

References

F. Fabozzi. Th

(Seventh Edition), The McGr

e Handbook of F

aw-Hill Companies, 2

ixed Income Securities

005.

A. Kuprianov,Derivatives, FeEconomic Qu

D. Rubin, D. Gthe Historical

Over-the-Counderal Reserve

arterly Volume

oldberg, and I.Relationship B

tB7

Get

er Interest Rateank o Richmon9, No. 3, Summer 1

reenbaum.Report onween SIFMA and LIBOR,

993.d

CDR Financial Products, August 2003.

February 6, 2002.

Credit ImpactMunicipal Fin

s of Variable Rance, Standard

ate Debt and Swaps iand Poors Ratings D

n

over the Past QFinance Special Issue 2005, 125-153.

irect,

W. Bartley Hil

the State and Local Governm

dreth and C. K

uarter Century,

,Interest Rate Swaps, Caliornia

u

ent Municipal DebtPublic Budgeting &

rt Zorn, The Evolution of

Market

Bond Logistix

C. UnderwoodMunicipal Tre

LLC, Januaryasurers Assoc

25,iation Advanced

2006.Workshop,


23/24

AcknowledgementsThis docum

Doug Sk

byKrist

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

t

o t

on (CDIAC).

his report may be rep

he Caliornia Debt an

roduced

d

OSP 07 101009


24/24

California

Advisory

Debt and Inve

Commission

stment

915 CapitSacramen

ol Mall, Room 400to, CA 95814

phone| 916.653.3269

ww

fax | 916.6

[email protected]

54.7440

r.ca.govov/cdiac

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Swap Int Rate