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rateswapmath pricing

Understanding interest

January 2007

CDIAC #06-11

California Debt and Investment Advisory Commission rateswapmath 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 nancial institution and an issuer) agree to exchange payments based on a dened principal amount, for a xed 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

MunicipalSwapIndex.

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

FixedRate

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 oating or variable interest rate basis to a xed interest rate basis, or vice versa). These “plain vanilla" swaps are by Typically, payments made by one counterparty are based on a oating 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 xed rate of interest, normally expressed as The maturity, or “tenor," of a xed-to-oating interest rate swap is usually between one and fteen years. By conven tion, a xed-rate payer is designated as the buyer of the swap, while the oating-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.

SwapPricinginTheory

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 ows now with the value of cash ows in the future. A dollar today is worth more than a dollar in the future because cash ows available today can be invested and grown. The basic premise to an interest rate swap is that the coun terparty choosing to pay the xed rate and the counterpar ty choosing to pay the oating 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 ows occurring at known future dates, it can be valued by sim ply summing the present value of each of these cash ows. In order to calculate the present value of each cash ow, it is necessary to rst estimate the correct discount factor (df) for each period (t) on which a cash ow 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 xed component of the swap contract: Theoretical Present value of the oating-rate payments

Swap Rate = Notional principal

x (days t /360) x df t p3

Consider the following example:

stepexample,follows:

Step 1 - Calculate Numerator

oating-ratepayments. onactualsemi-annualpayments. 3 3 and the

Financial Times

of London. p4

SwapRate( xedrate)tothecounterpartyandthecounter-

The rststepistocalculatethepresentvalue(PV)ofthe

This is done by forecasting each semi-annual payment LIBOR forward rates are available through nancial informa tion services including Bloomberg, the Wall Street Journal

Annual

Semi-annual Actual Floating Floating Rate PV of Floating Ti me Period Days in Forward Forward Rate Payment Forward Rate Payment at Pe riod Number Period Rate Period Rate at End Period Discount Factor End of Period (A)(B)(C)(D)(E)(F)(G)(H) 7/ 1/ 7/ 1/ 7/

PV of Floating Rate Payments= $12,816,663

Column Description

A=Periodtheinterestrateisineffect

B=

Periodnumber(t)

C= D=Annualinterestrateforthefutureperiodfrom nancialpublications

E=Semi-annualrateforthefutureperiod(D/2)

F=Actualforecastedpayment(E

x $100,000,000)

H=PVofoatingratepayments(Fx

G) p5 are used to di year period. T

Step2-Cal

As with the oating-rate pa

culateDeno principal by the minator ments, LIBOR fo tional principal fo otional principal iy 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 oating-rate table illustrafo tes the calculatiorward discount factor. ns for this p6

Annual

Semi-annual Floating Rate

Ti me Period Days in Forward Forward Notional Forward PV of Notional Pe riod Number Period Rate Period Rate Principal Discount Factor Principal (A)(B)(C)(D)(E)(F)(G)(H)

1/06-6/0611804.00%2.000%$100,000,000 0.9804$49,020,000

7/

06-12/0621804.25%2.125%$100,000,000 0.9600$48,000,000

1/

07-6/0731804.50%2.250%$100,000,000 0.9389$46,945,000

7/

07-12/0741804.75%2.375%$100,000,000 0.9171$45,855,000

1/

08-6/0851805.00%2.500%$100,000,000 0.8947$44,735,000

7/

08-12/0861805.25%2.625%$100,000,000 0.8718$43,590,000

$278,145,000 PV of Notional Principal=

Column Description

A=Periodtheinterestrateisineffect

B=

Periodnumber(t)

C= D=Annualinterestrateforthefutureperiodfrom nancialpublications

E=Semi-annualrateforthefutureperiod(D/2)

F=

Notionalprincipalfromswapcontract

G=

H=PVofnotionalprincipal[F

x (C/360) x G] p7

Step 3 - Calculate Swap Rate

theoreticalSwapRate:

Theoretical $12,816,663 = =

4.61%

Swap Rate $278,145,000

Basedontheaboveexample,theissuer( xed-ratepayer)

wi llbewillingtopaya xed4.61percentrateforthelifeof th

Step 4 - Calculate Swap Spread

With a known Swap Rate,the counterparties can now

determine the

“swapspread."

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

Theinterestrateswapmarketislargeandef cient.While

swaprates programsdesigned bythemajor nancialinstitutionsand theBMApercentage). 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

Thechoice

curveisbas reecttheir itsowncurr sury Yield edonthearg creditrisk.A encyisassumoftheU.S.Tre u bo edmentthattheyi ndissuedbyag as ury yield curve a s 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 ination and investor k-free rate of interest. Interest ities are inuenced by market luding changis to have no cred it risk so that overnment in expectation s on interest rate levels, yield curve analysis, and change ity groups.

LIBOR Spsincreditspreads between xed-income qual-

read

LIBOR is t

London

inte The rateis

LIBORswa

thattheco riskinherenrbank market t in

LIBOR, thhe 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 addbanks in the deposits. The risk free rate demand relaquotesdbs_dbs28.pdfusesText_34

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