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&rate swap math pricingUnderstanding interest
January 2007CDIAC
#06-11
California Debt and Investment Advisory Commission
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&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
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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
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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).
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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-
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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
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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.
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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.
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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-
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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
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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
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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,
8/8/2019 Swap Int Rate
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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
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California
Advisory
Debt and Inve
Commission
stment
915 CapitSacramen
ol Mall, Room 400to, CA 95814
phone| 916.653.3269
ww
fax | 916.6
54.7440
r.ca.govov/cdiac