drake drake university universite d’auvergne swaps

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Swaps


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Introduction

An agreement between two parties to exchange cash flows in the future.The agreement specifies the dates that the cash flows are to be paid and the way that they are to be calculated. A forward contract is an example of a simple swap. With a forward contract, the result is an exchange of cash flows at a single given date in the future. In the case of a swap the cash flows occur at several dates in the future. In other words, you can think of a swap as a portfolio of forward contracts.



Mechanics of Swaps

The most commonly used swap agreement is an exchange of cash flows based upon a fixed and floating rate. Often referred to a “plain vanilla” swap, the agreement consists of one party paying a fixed interest rate on a notional principal amount in exchange for the other party paying a floating rate on the same notional principal amount for a set period of time. In this case the currency of the agreement is the same for both parties.



Notional Principal

The term notional principal implies that the principal itself is not exchanged. If it was exchanged at the end of the swap, the exact same cash flows would result.



An Example

Company B agrees to pay A 5% per annum on a notional principal of $100 millionCompany A Agrees to pay B the 6 month LIBOR rate prevailing 6 months prior to each payment date, on $100 million. (generally the floating rate is set at the beginning of the period for which it is to be paid)



The Fixed Side

We assume that the exchange of cash flows should occur each six months (using a fixed rate of 5% compounded semi annually). Company B will pay:

($100M)(.025) = $2.5 Million to Firm A each 6 months.



Summary of Cash Flows for Firm B

Cash Flow Cash Flow Net

Date LIBOR Received Paid Cash Flow

3-1-98 4.2%

9-1-98 4.8% 2.10 2.5 -0.4

3-1-99 5.3% 2.40 2.5 -0.1

9-1-99 5.5% 2.65 2.5 0.15

3-1-00 5.6% 2.75 2.5 0.25

9-1-00 5.9% 2.80 2.5 0.30

3-1-01 6.4% 2.95 2.5 0.45



Swap Diagram

LIBORCompany A Company B

5%



Offsetting Spot Position

Company ABorrows (pays) 5.2%Pays LIBORReceives 5%Net LIBOR+.2%

Company BBorrows (pays) LIBOR+.8%Receives LIBORPays 5%Net 5.8%

Assume that A has a commitment to borrow at a fixed rate of 5.2% and that B has a commitment to borrow at a rate of LIBOR + .8%



Swap Diagram

Company A Company B

The swap in effect transforms a fixed rate liability or asset to a floating rate liability or asset (and vice versa) for the firms respectively.

5.2% LIBOR+.8%

LIBOR +.2%5%

LIBOR

5.8%



Role of Intermediary

Usually a financial intermediary works to establish the swap by bring the two parties together. The intermediary then earns .03 to .04% per annum in exchange for arranging the swap. The financial institution is actually entering into two offsetting swap transactions, one with each company.



Swap Diagram

Co A FI Co B

A pays LIBOR+.215%B pays 5.815%

The FI makes .03%

5.2% LIBOR+.8%

5.015%

LIBOR

4.985%

LIBOR



Day Count Conventions

The above example ignored the day count conventions on the short term rates.For example the first floating payment was listed as 2.10. However since it is a money market rate the six month LIBOR should be quoted on an actual /360 basis. Assuming 184 days between payments the actual payment should be

100(0.042)(184/360) = 2.1467



Day Count Conventions II

The fixed side must also be adjusted and as a result the payment may not actually be equal on each payment date. The fixed rate is often based off of a longer maturity instrument and may therefore uses a different day count convention than the LIBOR. If the fixed rate is based off of a treasury note for example, the note is based on a different day convention.



Role of the Intermediary

It is unlikely that a financial intermediary will be contacted by parties on both side of a swap at the same time.The intermediary must enter into the swap without the counter party. The intermediary then hedges the interest rate risk using interest rate instruments while waiting for a counter party to emerge. This practice is referred to as warehousing swaps.



Why enter into a swap?

The Comparative Advantage ArgumentFixed Floating

A 10% 6 mo LIBOR+.3B 11.2% 6 mo LIBOR +

1.0%

Difference between fixed rates = 1.2% Difference between floating rates = 0.7%

B Has an advantage in the floating rate.



Swap Diagram

Co A FI Co B

A pays LIBOR+.065% instead of LIBOR+.3%B pays 10.965% instead of 11.2%

The FI makes .03%

10% LIBOR+1%

9.965%

LIBOR

9.935%

LIBOR



Spread Differentials

Why do spread differentials exist?Differences in business lines, credit history, asset and liabilities, etc…



Valuation of Interest Rate Swaps

After the swap is entered into it can be valued as either:

A long position in one bond combined with a short position in another bond or A portfolio of forward rate agreements.



Relationship of Swaps to Bonds

In the examples above the same relationship could have been written asCompany B lent company A $100 million at the six month LIBOR rateCompany A lent company B $100 million at a fixed 5% per annum



Bond Valuation

Given the same floating rates as before the cash flow would be the same as in the swap example. The value of the swap would then be the difference between the value of the fixed rate bond and the floating rate bond.



Fixed portion

The value of either bond can be found by discounting the cash flows from the bond (as always). The fixed rate value is straight forward it is given as:

where Q is the notional principal and k is the fixed interest payment

nnii trn

i

trfix QekeB

1



Floating rate valuation

The floating rate is based on the fact that it is a series of short term six months loans.Immediately after a payment date Bfl is equal to the notional principal Q. Allowing the time until the next payment to equal t1

where k* is the known next payment

1111 * trtrfl ekQeB



Swap Value

If the financial institution is paying fixed and receiving floating the value of the swap is

Vswap = Bfl-Bfix

The other party will have a value of Vswap = Bfix-Bfl



Example

Pay 6 mo LIBOR & receive 8%3 mo 10%9 mo 10.5% 15 mo 11%Bfix = 4e .-1(.25)+4e -.105(.75)+104e -.11(1.25)=98.24MBfloat = 100e -.1(.25) + 5.1e -.1(.25) =-102.5M

-4.27 M



A better valuation

Relationship of Swap value to Forward Rte AgreementsSince the swap could be valued as a forward rate agreement (FRA) it is also possible to value the swap under the assumption that the forward rates are realized.



To do this you would need to:

Calculate the forward rates for each of the LIBOR rates that will determine swap cash flowsCalculate swap cash flows using the forward rates for the floating portion on the assumption that the LIBOR rates will equal the forward ratesSet the swap value equal to the present value of these cash flows.



Swap Rate

This works after you know the fixed rate.When entering into the swap the value of the swap should be 0.This implies that the PV of each of the two series of cash flows is equal. Each party is then willing to exchange the cash flows since they have the same value.The rate that makes the PV equal when used for the fixed payments is the swap rate.



Example

Assume that you are considering a swap where the party with the floating rate will pay the three month LIBOR on the $50 Million in principal.The parties will swap quarterly payments each quarter for the next year.Both the fixed and floating rates are to be paid on an actual/360 day basis.



First floating payment

Assume that the current 3 month LIBOR rate is 3.80% and that there are 93 days in the first period.The first floating payment would then be

3333.833,490000,000,50360

93038.



Second floating payment

Assume that the three month futures price on the Eurodollar futures is 96.05 implying a forward rate of 100-96.05 = 3.95Given that there are 91 days in the period.The second floating payment would then be

1111.236,499000,000,50360

910395.



Example Floating side

PeriodDay

CountFutures

PriceFwdRate

FloatingCash flow

91 3.80

1 93 96.05 3.95 490,833.3333

2 91 95.55 4.45 499,236.1111

3 90 95.28 4.72 556,250.0000

4 91 596,555.5555



PV of Floating cash flows

The PV of the floating cash flows is then calculated using the same forward rates.The first cash flow will have a PV of:

8263.061,486

36093

038.1

3333.833,490



PV of Floating cash flows

The PV of the floating cash flows is then calculated using the same forward rates.The second cash flow will have a PV of:

4412.495,489

36091

0395.136093

038.1

1111.236,499



Example Floating side

Period

DayCount

FwdRate

FloatingCash flow

PV of Floating CF

91 3.80

1 93 3.95490,833.333

3486,061.8263

2 91 4.45499,236.111

1489,495.4412

3 90 4.72556,250.000

0539,396.1423

4 91596,555.555

5525,668.5915



PV of floating

The total PV of the floating cash flows is then the sum of the four PV’s:

$2,040,622.0013



Swap rate

The fixed rate is then the rate that using the same procedure will cause the PV of the fixed cash flows to have a PV equal to the same amount.The fixed cash flows are discounted by the same rates as the floating rates.Note: the fixed cash flows are not the same each time due to the changes in the number of days in each period.The resulting rate is 4.1294686



Example: Swap Cash Flows

Period

DayCount

FwdRate

FloatingCash flow

Fixed CF

91 3.80

1 93 3.95490,833.333

3533,389.7003

2 91 4.45499,236.111

1521,918.9541

3 90 4.72556,250.000

0516,183.5810

4 91596,555.555

5521,918.9541



Swap Spread

The swap spread would then be the difference between the swap rate and the on the run treasury of the same maturity.



Swap valuation revisited

The value of the swap will change over time.After the first payments are made, the futures prices and corresponding interest rates have likely changed. The actual second payment will be based upon the 3 month LIBOR at the end of the first period.Therefore the value of the swap is recalculated.



Currency Swaps

The primary purpose of a currency swap is to transform a loan denominated in one currency into a loan denominated in another currency. In a currency swap, a principal must be specified in each currency and the principal amounts are exchanged at the beginning and end of the life of the swap. The principal amounts are approximately equal given the exchange rate at the beginning of the swap.



A simple example

Assume that company A pays a fixed rate of 11% in sterling and receives a fixed interest rate of 8% in dollars.

Let interest payments be made once a year and the principal amounts be $15 million and L10 Million

Company ADollar Cash Sterling CashFlow (millions) Flow (millions)

2/1/1999 -15.00 +10.002/1/2000 +1.20 -1.102/1/2001 +1.20 -1.102/1/2002 +1.20 -1.102/1/2003 +1.20 -1.102/1/2004 +16.20 -11.10



Intuition

Suppose A could issue bonds in the US for 8% interest, the swap allows it to use the 15 million to actually borrow 10million sterling at 11% (A can invest L 10M @ 11% but is afraid that $ will strength it wants US denominated investment)



Comparative Advantage Again

The argument for this is very similar to the comparative advantage argument presented earlier for interest rate swaps.It is likely that the domestic firm has an advantage in borrowing in its home country.



Example using comparative advantage

Dollars AUD (Australian $)

Company A 5% 12.6%Company B 7% 13.0%2% difference in $US .4% difference in

AUD



The strategy

Company A borrows dollars at 5% per annumCompany B borrows AUD at 13% per annum

They enter into a swapResult Since the spread between the two companies is

different for each firm there is the ability of each firm to benefit from the swap. We would expect the gain to both parties to be 2 - 0.4 = 1.6% (the differences in the spreads).



Swap Diagram

Co A FI Co B

A pays 11.9% AUD instead of 12.6% AUDB pays 6.3% $US instead of 7% $US

The FI makes .2%

5% AUD 13%

6.3%

AUD 11.9%

5%

AUD 13%



Valuation of Currency Swaps

Using Bond TechniquesAssuming there is no default risk the currency swap can be decomposed into a position in two bonds, just like an interest rate swap.In the example above the company is long a sterling bond and short a dollar bond. The value of the swap would then be the value of the two bonds adjusted for the spot exchange rate.



Swap valuation

Let S = the spot exchange rate at the beginning of the swap, BF is the present value of the foreign denominated bond and BD is the present value of the domestic bond. Then the value is given as

Vswap = SBF – BD

The correct discount rate would then depend upon the term structure of interest rates in

each country



Other swaps

Swaps can be constructed from a large number of underlying assets. Instead of the above examples swaps for floating rates on both sides of the transaction. The principal can vary through out the life of the swap.They can also include options such as the ability to extend the swap or put (cancel the swap). The cash flows could even extend from another asset such as exchanging the dividends and capital gains realized on an equity index for a fixed or floating rate.



Beyond Plain Vanilla Swaps

Amortizing Swap -- The notional principal is reduced over time. This decreases the fixed payment. Useful for managing mortgage portfolios and mortgage backed securities.Accreting Swap – The notional principal increases over the life of the swap. Useful in construction finances. For example is the builder draws down an amount of financing each period for a number of periods.



Beyond Plain Vanilla

You can combine amortizing and accreting swaps to allow the notional principal to both increase and decrease.

Seasonal Swap -- Increase and decrease of notional principal based of f of designated plan

Roller Coaster Swap -- notional principal first increases the amortizes to zero.



Off Market Swap

The interest rate is set at a rate above market value.For example the fixed rate may pay 9% when the yield curve implies it should pay 8%.The PV of the extra payments is transferred as a one time fee at the beginning of the swap (thus keeping the initial value equal to zero)



Forward and Extension Swaps

Forward swap – the payments are agreed to begin at some point in time in the futureIf the rates are based on the current forward rate there should not be any exchange of principal when the payments begin. Other wise it is an off market swap and some form of compensation is neededExtension Swap – an agreement to extend the current swap (a form of forward swap)



Basis Swaps

Both parties pay floating rates based upon different indexes. For example one party may pay the three month LIBOR while the other pays the three month T- Bill.The impact is that while the rates generally move together the spread actually widens and narrows, Therefore the return on the swap is based upon the spread.



Yield Curve Swaps

Both parties pay floating but based off of different maturities. Is similar to a basis swap since the effective result is based on the spread between the two rates. A steepening curve thus benefits the payer of the shorter maturity rate. This is utilized by firms with a mismatch of maturities in assets and liabilities (banks for example). It can hedge against changes in the yield curve via the swap.



Rate differential (diff) swap

Payments tied to rate indexes in different currencies, but payments are made in only one currency.



Corridor Swap

Payments obligation only occur in a given range of rates. For example if the LIBOR rate is between 5 and 7%.The swap is basically a tool based on the uncertainty of rates.



Flavored Currency Swaps

The basic currency swap can be modified similar to many of the modifications just discussed.Swaps may also be combined to produce desired outcomes. CIRCUS Swap (Combined interest rate and currency swap). Combines two basic swaps



Circus Swap Diagram

LIBORCompany A Company B

5% US$ 6% German Marks

Company A Company CLIBOR



Circus Swap Diagram

Company B

Company A 5% US$ 6% German Marks

Company C



Swapation

An option on a swap that specifies the tenor, notional principal fixed rate and floating ratePrice is usually set a a % of notional principal Receiver Swapation

The holder has the right to enter into a swap as the fixed – rate receiver

Payer SwapationThe holder has the right to enter into a particular swap as the fixed rate payer.



Swapation as call (or put) Options

Receiver swapation – similar to a call option on a bond. The owner receives a fixed payment (like a coupon payment) and pays a floating rate (the exercise price)Payer swapation – if exercised the owner is paying a stream similar to the issue of a bond.



In-the-Money Swapations

A receiver swapation is in the money if interest rates fall. The owner is paying a lower fixed rate in exchange for the fixed rate specified in the contract. Similarly a payer swapation is generally in the money if interest rates increase since the owner will receive a higher floating rate.



When to Exercise

The owner of the receiver swapation should exercise if the fixed rate on the swap underlying the swapation is greater than the market fixed rate on a similar swap. In this case the swap is paying a higher rate than that which is available in the market.



A fixed income swapation example

Consider a firm that has issued a corporate bond with a call option at a given date in the future. The firm has paid for the call option by being forced to pay a higher coupon on the bond than on a similar noncallable bond.Assume that the firm has determined that it does not want to call the bond at its first call date at some point in the future.The call option is worthless to the firm, but it should theoretically have value.



Capturing the value of the call

The firm can sell a receiver swapation with terms that match the call feature of the bond.The firm would receive for this a premium that is equal to the value of the call option.



Example

Assume the firm has previously issued a 9% coupon bond that makes semiannual payments and matures in 7 years with a face value of $150 Million.The bond has a call option for one year from today.



Example continued

The firm can sell a European Receiver Swapation with an expiration in one year. The Swapation terms are for semiannual payments at a fixed rate of 9% in exchange for floating payments at LIBOR.The firm receives a premium for the swapation equal to a fixed percentage of the $150 Million notional value (equal to the value of the call option). The firm can keep the premium but has a potential obligation in one year if the counter party exercises the swap.



Example Continued

In one year the fixed rate for this swap is 11% The option will expire worthless since the owner can earn a fixed 11% on a similar swap. The firm gets to keep the premium.



Example Continued

If in one year the fixed rate of interest on a similar swap is 7% the owner will exercise the swap since it calls for a 9% fixed rate. The firm can call the bond since rates have decreased. It can finance the call by issuing a floating rate note at LIBOR for the term of the swap. The floating rate side of the swap pays for the note and the firm is still paying the original 9% fixed, but it has also received the premium on the swapation



Extendible and Cancelable swaps

Similar to extension swaps except extension swaps represent a firm commitment to extend the swap. An extendible swap has the option to extend the agreement.Arranged via a plain vanilla swap an a swapation.



Extendible and Cancelable

Extendible pay fixed swap= plain vanilla pay fixed plus payer swapation

Extendible Receive-Fixed Swap= plain vanilla receive fixed swap + receiver

swapationCancelable Pay Fixed Swap= plain vanilla pay fixed swap + receiver

swapationCancelable Receive Fixed Swap=plain vanilla receive fixed swap + payable

swapation



Creating synthetic securities using swaps

The origins of the swap market are based in the debt market.Previously there had been restrictions on the flow of currency.A parallel loan market developed to get around restrictions on the flow of currency from one country to another, Especially restrictions imposed by the Bank of England.



The Parallel Loan Market

Consider two firms, one British and one American, each with subsidiaries in both countries.Assume that the free-market value of the pound is L1=$1.60 and the officially required exchange rate is L1=$1.44.Assume the British Firm wants to undertake a project in the US requiring an outlay of $100,000,000.



Parallel Loan Market

The cost of the project at the official exchange rate is 100,000,0000/1.44 = L69,444,000The cost of the project at the free market exchange rate is 100,000,0000/1.60 = L62,500,000The firm is paying an extra L7,000,000



Parallel Loan Market

The British firm lends L62,500,000 pounds to the US subsidiary operating in England at a floating rate based on LIBOR and The US firm lends $100,000,000 to the British firm at a fixed rate of 7% in the US the official exchange rate is avoided.The result is a basic fixed for floating currency swap. (In this case each loan is separate – default on one loan does not constitute default on the other).



Synthetic Fixed Rate Debt

A firm with an existing floating rate debt can easily transform it into a fixed rate debt via an interest rate swap.By receiving floating and paying fixed, the firm nets just a spread on the floating transaction creating a fixed rate debt (the rate paid on the swap plus the spread)



Synthetic Floating Rate Debt

Combining a fixed rate debt with a pay floating / receive fixed rate swap easily transforms the fixed rate. Again the fixed rates cancel out (or result in a spread) leaving just a floating rate.



Synthetic Callable Debt

Consider a firm with an outstanding fixed rate debt without any call option.It can create a call option. If it had a call option in place it would retire the debt if called. Look at this as creating a new financing need (you need to finance the retirement of the debt.)You want the ability to call the bond but not the obligation to do so.



Synthetic Callable Debt

Buying a receiver swapation allows the firm to receive a fixed rate, canceling out its current fixed rate obligation. It will pay a new floating rate as part of the swap (similar to financing the call with new floating rate debt).



Synthetic non callable Debt

Basically the earlier example swapations.



Synthetic Dual Currency Debt

Dual Currency bond – principal payments are denominated in one currency and coupon payments denominated in another currency.Assume you own a bond that makes its payments in US dollars, but you would prefer the coupon payments to be in another currency with the principal repayment in dollars. A fixed for fixed currency swap would allow this to happen



Synthetic Dual Currency Debt

Combine a receive fixed German marks and pay US dollars swap with the bond. The dollars received from the bond are used to pay the dollar commitment on the swap. You then just receive the German Marks.



All in Cost

The IRR for a given financing alternative, it includes all costs including administration, flotation , and actual cash flows.The cost is simply the rate that makes the PV of the cash flows equal to the current value of the borrowing.



Compare two alternative proposals

A 10 year semiannual 7% coupon bond with a principal of $40 million priced at parA loan of $40 million for 10 years at a floating rate of LIBOR + 30 Bps reset every six months with the current LIBOR rate of 6.5%. Plus a swap transforming the loan to a fixed rate commitment. The swap will require the firm to pay 6.5% fixed and receive floating.



All in cost

The bond has a all in cost equal to its yield to maturity, 7%Assuming the firm must pay $400,000 to enter into the swap so it only nest $39,600,000. Today. The net interest rate it pays is 6.8% implying semiannual payments of (.068/2)(40,000,000) = $1,360,000 plus a final payment of 40,000,000. This implies a rate of .034703 every six months or .069406 every year.



BF Goodrich and RabobankAn early swap example*

In the early 1980’s BF Goodrich needed to raise new funds, but its credit rating had been downgraded to BBB-. The firm needed $50,000,000 to fund continuing operations.They wanted long term debt in the range of 8 to 10 years and a fixed rate. Treasury rates were at 10.1 % and BF Goodrich anticipated paying approximately 12 to 12.5%

* taken from Kolb - Futures Options and Swaps



Rabobank

Rabobank was a large Dutch banking organization consisting of more than 1,000 small agricultural banks. The bank was interested in securing floating rate financing on approximately $50,000,000 in the Eurobond market. With a AAA rating Rabobank could issue fixed rate in the Eurobond market for approximately 11% and for a floating rate of LIBOR plus .25%



The Intermediary

Salomon Brothers suggested a swap agreement to each party.This would require BF Goodrich to issue the first public debt tied to LIBOR in the United States. Salomon Brothers felt that there would be a market for the debt because of the increase in deposits paying a floating rate due to deregulation.



Problems

Rabobank was interested in the deal,but fearful of credit risk. A direct swap would expose it to credit risk. Without an active swap market it was common for swaps to be arranged between the two counter parties.The two finally reached an agreement to use Morgan Guaranty as an intermediary.



The agreement

BF Goodrich issued a noncallable 8 year floating rate note with a principal value of $50,000,000 paying the 3 month LIBOR rate plus .5% semiannually. The bond was underwritten by Salomon.Rabobank issued a $50,000,000 non callable 8 year Eurobond with annual payments of 11%Both entered into a swap with Morgan Guaranty



The swaps

BF Goodrich promised to pay Morgan Guaranty 5,500,000 each year for eight years (matching the coupon on the Rabobanks debt). Morgan agreed to pay BF Goodrich a semi annual rate tied to the 3 month LIBOR equal to: .5(50,000,000)(3 mo LIBOR-x) x represents an undisclosed discountRabobank received $5,500,000 each year for 8 years and paid semi annul payments of LIBOR-x



The intermediary role

The two swap agreements were independent of each other eliminating the credit risk concerns of Rabobank.Morgan received a one time fee of $125,000 paid by BF Goodrich plus an annual fee of 8 to 37 Bp ($40,000 to $185,000) also paid by BF Goodrich.



BF Goodrich

Assuming that the discount from LIBOR was 50 Bp and that the service fee was 22.5 BP (the midpoint of the range). BF Goodrich paid an all in cost of 11.9488 % annually compared to 12 to 12.5% if they had issued the debt on their own.



Rabobank’s Position

At the time of financing it would have paid LIBOR plus 25 to 50 Bp. Given that it paid no fees and the fixed rate canceled out it ended up paying LIBOR - x.



Securing financing

BF Goodrich was able to secure financing via its use of the swaps market, this is a common use of the market.The example provides a good illustration of the idea of the comparative advantage arguments we discussed earlier.



A Second Example of securing financing*

It is possible for swaps to increases accessibility two the debt marketMexcobre (Mexicana de Corbre) is the copper exporting subsidiary of Grupo Mexico. In the late 1980’s it would have had a difficult time borrowing in international credit markets due to concerns or default riskHowever it was able to borrow $210 million for 38 months from a group of banks led by Paribas

* from Managing Financial Risk by Smithson, Smith and Wilford



The original loan

The banks lent the firm $210 Million at a fixed rate of 11.48%. The debt replaced borrowing from the Mexican government which had cost the firm 23%.A Belgian company Sogem agreed to buy 4,000 tons of copper per month at the prevailing spot rate from Mexcobre making payments into an escrow account in New York that was used to service the debt with any extra funds returned to Mexcobre.



Banks Escrow

Mexcobre SOGEM4,000 tons of copper per month

Cash based on Spot Price

Quarterly payments of 11.48% interest plus principal

$210millionloan

Excess cashif it buildsup



Swaps

Swaps were added between Paribas and the escrow account to hedge the price risk of copper and between Paribas and the banks to change the banks position to a floating rate

Banks Escrow

Mexcobre SOGEM4,000 tons of copper per month

Cash perton based

on Spot Price

Quarterly payments of 11.48% interest plus principal

$210millionloan

Excess cashif it buildsup

Paribas

SpotPrice per ton

FloatingFixed $2,000 per ton



Duration of Interest Rate Swaps*

A plain vanilla swap can be valued as a portfolio of two bonds, therefore the duration of the swap should equal the duration of the bond portfolio.The duration can be either positive or negative depending on the side of the swap

* Kolb, Futures Options and Swaps



Duration of Swaps

Duration of Receive Fixed Swap = Duration of Underlying coupon bond - Duration of underlying floating Rate Bond >0

Duration of Pay Fixed Swap = Duration of underlying floating Rate Bond- Duration of Underlying coupon bond <0



Example

Consider a swap with a semiannual fixed rate of 7% and a floating rate that resets each six months.The duration of the fixed rate side (assuming a 100 notional principal) is 5.65139 years

Duration of Receive Fixed Swap =5.65139-0.5=5.15369

Duration of Pay Fixed Swap =0.5-5.65139=-5.15369



Calculating Duration

Duration of floating rate security is equal to the time between resetting of the rate.Therefore the duration of the swap actually depends upon the duration of the fixed rate side.Receive Fixed rate swaps will then usually lengthen the duration of an existing position while pay fixed swaps will shorten the duration of an existing position.



Immunization with Swaps

Swaps can be used to hedge interest rate risk by impacting the duration of the assets and liabilities on the balance sheet.Going to look at a fictional financial services firm FSF



Balance Sheet for FSF

AssetsCash $7,000,000

Marketable Sec $18,000,000(6 mo mat Yield 7%)

Amortizing loans $130,467,133(10 yr avg matsemiannual8% avg yield)

Total Assets $1555,467,133

Liabilities6mo money mkt $75,000,000(avg yield 6%)

Floating Rate Notes $40,000,000(5 yr mat7.3% yld semi)

Coupon Bond $24,111,725(10 yr semi 6.5% coup$25,000,000 par, 7% YTM

Net worth $16,355,408Total Liab & NW $155,467,133



Basic Duration

iasset ofDuration Macaulay DaAssets All of ValueMarket

Asset wwhere

DawDAPortfolioAsset of

Duration Weighted$

i

ii

i

N

1ii

jLiability ofDuration Macaulay DlsLiabilitie All of ValueMarket

Asset wwhere

DlwDLPortfolioLiability of

Duration Weighted$

j

jj

j

N

1jj



Duration

AssetsDuration

Cash 0.00Marketable Sec 0.500Amortizing loans 4.604562

Total Duration(7,000,000/155,467,133)0.000

+(18,000,000/155,467,133)0.500

+(130,467,133/155,467,133)4.605

3.922013

LiabilitiesDuration

6mo money mkt 0.5000Floating Rate Notes 0.5000Coupon Bond 7.453369

Total Duration(75,000,000/155,467,133)0.500

+(40,000,000/155,467,133)0.500+(24,111,725/155,467,133)7.4533

71.705202



Hedging the portfolios separately

It is easy to use duration to hedge the interest rate risk of the portfolio.The idea is to construct a portfolio with a duration of zero.Let MVi be the market value and Di be the Duration of the assets (A), liabilities (L) or hedge vehicle (H) then

MVA(DA)+MVH(DH) = 0 and

MVL(DL)+MVH(DH) = 0



Swap notional value

Given the duration of the hedge (a swap) it is then possible to solve for a notional value (or market value) of the swap that would make the portfolio duration zero. Previously we found the duration of a swap:

Duration of Receive Fixed Swap =5.65139-0.5=5.15369

Duration of Pay Fixed Swap =0.5-5.65139=-5.15369



Asset Hedge

The asset can then be hedged by solving for the notional value (MVH) of the pay fixed swap

MVA(DA)+MVH(DH) = 0

155,467,133(3.922)+(-5.15369)(MVH) =0

MVH=$118,365,451



Liability Hedge

The liabilities can then be hedged by solving for the notional value (MVH) of the receive fixed swap

MVL(DL)+MVH(DH) = 0

(-139,111,725)(1.705202)+(5.15369)(MVH) =0

MVH=$46,048,651



Hedging Assets and Liabilities together

The entire balance sheet can be hedged with one interest rate swap by using GAP analysis.



Static GAP Analysis(The repricing model)

Repricing GAPThe difference between the value of interest sensitive assets and interest sensitive liabilities of a given maturity. Measures the amount of rate sensitive (asset or liability will be repriced to reflect changes in interest rates) assets and liabilities for a given time frame.



GAP Analysis

Static GAP-- Goal is to manage interest rate income in the short run (over a given period of time)

Measuring Interest rate risk – calculating GAP over a broad range of time intervals provides a better measure of long term interest rate risk.



Interest Sensitive GAP

Given the Gap it is easy to investigate the change in the net interest income (NII) of the financial institution.

sLiabilitie Sensistive Rate - Assets Sensistive Rate GAP

R)(GAP)( NII

Rates)in ge(GAP)(Chan NIIin Change



Example

Over next 6 Months:Rate Sensitive Liabilities = $120 million

Rate Sensitive Assets = $100 Million

GAP = 100M – 120M = - 20 Million

If rate are expected to decline by 1%

Change in net interest income = (-20M)(-.01)= $200,000



GAP Analysis

Asset sensitive GAP (Positive GAP)RSA – RSL > 0If interest rates NII will If interest rates NII will

Liability sensitive GAP (Negative GAP)RSA – RSL < 0If interest rates NII will If interest rates NII will

Would you expect a commercial bank to be asset or liability sensitive for 6 mos? 5 years?



Important things to note:

Assuming book value accounting is used -- only the income statement is impacted, the book value on the balance sheet remains the same.

The GAP varies based on the bucket or time frame calculated.

It assumes that all rates move together.



Steps in Calculating GAP

Select time Interval

Develop Interest Rate Forecast

Group Assets and Liabilities by the time interval (according to first repricing)

Forecast the change in net interest income.



Alternative measures of GAP

Cumulative GAPTotals the GAP over a range of of possible maturities (all maturities less than one year for example).Total GAP including all maturities



Other useful measures using GAP

Relative Interest sensitivity GAP (GAP ratio)GAP / Bank SizeThe higher the number the higher the risk that is present

Interest Sensitivity Ratio

SensitiveAsset 1

SensitiveLiability 1

sLiabilitieSensitive Rate

Assets Sensitive Rate



What is “Rate Sensitive”

Any Asset or Liability that matures during the time frameAny principal payment on a loan is rate sensitive if it is to be recorded during the time periodAssets or liabilities linked to an indexInterest rates applied to outstanding principal changes during the interval



Unequal changes in interest rates

So far we have assumed that the change the level of interest rates will be the same for both assets and liabilities.If it isn’t you need to calculate GAP using the respective change.Spread effect – The spread between assets and liabilities may change as rates rise or decrease

)R(RSL)(-)R(RSA)( NII liabiltiesassets



Strengths of GAP

Easy to understand and calculate

Allows you to identify specific balance sheet items that are responsible for risk

Provides analysis based on different time frames.



Weaknesses of Static GAP

Market Value EffectsBasic repricing model the changes in market value. The PV of the future cash flows should change as the level of interest rates change. (ignores TVM)

Over aggregationRepricing may occur at different times within the bucket (assets may be early and liabilities late within the time frame)Many large banks look at daily buckets.



Weaknesses of Static GAP

RunoffsPeriodic payment of principal and interest that can be reinvested and is itself rate sensitive.You can include runoff in your measure of rate sensitive assets and rate sensitive liabilities.Note: the amount of runoffs may be sensitive to rate changes also (prepayments on mortgages for example)



Weaknesses of GAP

Off Balance Sheet ActivitiesBasic GAP ignores changes in off balance sheet activities that may also be sensitive to changes in the level of interest rates.

Ignores changes in the level of demand deposits



Basic Duration Gap

Duration Gap

DLDADGAP Basic

PortfolioLibaility of

Duration Weighted$

PortfolioAsset of

Duration Weighted$ DGAP Basic



Basic DGAP

If the Basic DGAP is +If Rates in the value of assets > in value of liabOwners equity will decreaseIf Rate

in the value of assets > in value of liabOwners equity will increase



Basic DGAP

If the Basic DGAP is (-)If Rates in the value of assets < in value of liabOwners equity will increaseIf Rate

in the value of assets < in value of liabOwners equity will decrease



Basic DGAP

Does that imply that if DA = DL the financial institution has hedged its interest rate risk?

No, because the $ amount of assets > $ amount of liabilities otherwise the institution would be insolvent.



DGAP

Let MVL = market value of liabilities and MVA = market value of assetsThen to immunize the balance sheet we can use the following identity:

MVA

MVLDLDADGAP

MVA

MVLDLDA



DGAP calculation

396201.2

133,467,155

725,111,139705202.1922013.3DGAP

MVA

MVLDLDADGAP



Hedging with DGAP

The net cash flows represented on the balance sheet have the same properties as a long position in a bond with a duration of 2.396201.We can hedge using our equation from before and the duration of the interest rate swap.



Hedging with DGAP

Since the duration of our position is positive we want the duration of the hedge to be negative. This requires the pay fixed swap from before with a notional value equal to MVH below.

MVi(Di)+MVH(DH) = 0

$155,467,725(2.396201)+(-5.151369)MVH=0

MVH=$72,316,800



DGAP and owners equity

Let MVE = MVA – MVLWe can find MVA & MVL using duration

From our definition of duration:

MVLy1

Δy-DL MVL

MVA y1

Δy-DAMVA

formula theApplying Pi)(1

ΔiDΔP



MVAy1

Δy-DGAPΔMVE

MVAy1

Δy

MVA

MVL(DL)- (DA)-

y1

Δy(DL)MVL-(DA)MVA -

MVLy1

ΔyDL--MVA

y1

Δy-DA

ΔMVL-ΔMVA ΔMVE



DGAP Analysis

If DGAP is (+)An in rates will cause MVE to An in rates will cause MVE to

If DGAP is (-)An in rates will cause MVE to An in rates will cause MVE to

The closer DGAP is to zero the smaller the potential change in the market value of equity.



Weaknesses of DGAP

It is difficult to calculate duration accurately (especially accounting for options)Each CF needs to be discounted at a distinct rate can use the forward rates from treasury spot curveMust continually monitor and adjust durationIt is difficult to measure duration for non interest earning assets.



More General Problems

Interest rate forecasts are often wrongTo be effective management must beat the ability of the market to forecast rates

Varying GAP and DGAP can come at the expense of yield

Offer a range of products, customers may not prefer the ones that help GAP or DGAP – Need to offer more attractive yields to entice this – decreases profitability.



Changing Duration

You can also manipulate the duration of your cash flows. This allows you to lower your interest rate sensitivity instead of eliminating it.Let DG

* be the desired duration gap, DG be the current duration gap, DS be the duration of the Swap, and MVH* be the notional value of required for the swap.

Assets Total

MVDDD

*H

SG*G



Decreasing Duration GAP to One year

The negative sign just indicate that we need a pay fixed swap (the duration would then be negative making the MV positive)

025,137,42MV

33$155,467,1

MV15369.5396201.20.1

Assets Total

MVDDD

*H

*H

*H

SG*G

drake drake university universite d’auvergne swaps

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