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```									            Swaps

Finance 298
Analysis of Fixed Income Securities

Drake Fin 284
DRAKE UNIVERSITY
Drake
Introduction                         Drake University

Fin 284

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.
Drake
Mechanics of Swaps                     Drake University

Fin 284

The most common 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.
Drake
Notional Principal                  Drake University

Fin 284

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.
Drake
An Example                         Drake University

Fin 284

Company B agrees to pay A 5% per annum on a
notional principal of \$100 million
Company 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)
Drake
The Fixed Side                    Drake University

Fin 284

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                Drake
Drake University

for Firm B                       Fin 284

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
Drake
Swap Diagram               Drake University

Fin 284

LIBOR
Company A             Company B
5%
Drake
Offsetting Spot Position                      Drake University

Fin 284

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%
Company A                      Company B
Borrows (pays)     5.2%        Borrows (pays) LIBOR+.8%
Net           LIBOR+.2%        Net                5.8%
Drake
Swap Diagram                         Drake University

Fin 284

LIBOR
5.2%                                        LIBOR+.8%
Company A               Company B
5%
LIBOR +.2%                  5.8%

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.
Drake
Role of Intermediary                     Drake University

Fin 284

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.
Drake
Swap Diagram                      Drake University

Fin 284

LIBOR         LIBOR
5.2%                                        LIBOR+.8%
Co A            FI            Co B
4.985%        5.015%

A pays LIBOR+.215%
B pays 5.815%
The FI makes .03%
Drake
Day Count Conventions                   Drake University

Fin 284

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
Drake
Day Count Conventions II                  Drake University

Fin 284

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.
Drake
Role of the Intermediary                     Drake University

Fin 284

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.
Drake
Why enter into a swap?                 Drake University

Fin 284

Fixed     Floating
A          10%      6 mo LIBOR+.3
B          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.
Drake
Swap Diagram                      Drake University

Fin 284

LIBOR         LIBOR
10%                                        LIBOR+1%
Co A            FI            Co B
9.935%        9.965%

A pays LIBOR+.065% instead of LIBOR+.3%
B pays 10.965% instead of 11.2%
The FI makes .03%
Drake

Fin 284

Differences in business lines, credit history, asset
and liabilities, etc…
Drake
Valuation of Interest Rate Swaps                 Drake University

Fin 284

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.
Drake
Relationship of Swaps to Bonds           Drake University

Fin 284

In the examples above the same relationship
could have been written as
Company B lent company A \$100 million at the
six month LIBOR rate
Company A lent company B \$100 million at a
fixed 5% per annum
Drake
Bond Valuation                    Drake University

Fin 284

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.
Drake
Fixed portion                      Drake University

Fin 284

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:
n
B fix   keriti  Qe rntn
i 1

where Q is the notional principal and k is the
fixed interest payment
Drake
Floating rate valuation                    Drake University

Fin 284

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

 r1t1     *  r1t1
B fl  Qe            k e
where k* is the known next payment
Drake
Swap Value                         Drake University

Fin 284

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
Drake
Example                             Drake University

Fin 284

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)=98.24M

Bfloat = 100e -.1(.25) + 5.1e -.1(.25)          =-102.5M
-4.27 M
Drake
A better valuation                  Drake University

Fin 284

Relationship of Swap value to Forward Rte
Agreements
Since 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.
Drake
To do this you would need to:             Drake University

Fin 284

Calculate the forward rates for each of the LIBOR
rates that will determine swap cash flows
Calculate swap cash flows using the forward
rates for the floating portion on the assumption
that the LIBOR rates will equal the forward rates
Set the swap value equal to the present value of
these cash flows.
Drake
Swap Rate                        Drake University

Fin 284

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.
Drake
Example                         Drake University

Fin 284

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.
Drake
First floating payment                 Drake University

Fin 284

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

 93 
.038     50,000,000  490,833.3333
 360 
Drake
Second floating payment                  Drake University

Fin 284

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

 91 
.0395     50,000,000  499,236.1111
 360 
Drake
Example Floating side                  Drake University

Fin 284

Day    Futures   Fwd      Floating
Period
Count    Price    Rate    Cash 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
Drake
PV of Floating cash flows                 Drake University

Fin 284

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

490,833.3333
 486,061.8263
         93  
1  .038     
         360  
Drake
PV of Floating cash flows                      Drake University

Fin 284

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

499,236.1111
 489,495.4412
         93             91  
1  .038      1  .0395     
         360            360  
Drake
Example Floating side                      Drake University

Fin 284

Day    Fwd      Floating
Period                               PV of Floating CF
Count   Rate    Cash flow
91     3.80

1     93     3.95   490,833.3333    486,061.8263

2     91     4.45   499,236.1111    489,495.4412

3     90     4.72   556,250.0000    539,396.1423

4     91            596,555.5555    525,668.5915
Drake
PV of floating                     Drake University

Fin 284

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

\$2,040,622.0013
Drake
Swap rate                     Drake University

Fin 284

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
Drake
Example: Swap Cash Flows                Drake University

Fin 284

Day    Fwd      Floating
Period                                 Fixed CF
Count   Rate    Cash flow
91    3.80

1      93    3.95   490,833.3333   533,389.7003

2      91    4.45   499,236.1111   521,918.9541

3      90    4.72   556,250.0000   516,183.5810

4      91           596,555.5555   521,918.9541
Drake

Fin 284

The swap spread would then be the difference
between the swap rate and the on the run
treasury of the same maturity.
Drake
Swap valuation revisited                 Drake University

Fin 284

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.
Drake
Currency Swaps                     Drake University

Fin 284

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.
Drake
A simple example                                 Drake University

Fin 284

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 A
Dollar Cash              Sterling Cash
Flow (millions)          Flow (millions)
2/1/1999         -15.00                   +10.00
2/1/2000         +1.20                    -1.10
2/1/2001         +1.20                    -1.10
2/1/2002         +1.20                    -1.10
2/1/2003         +1.20                    -1.10
2/1/2004         +16.20                   -11.10
Drake
Intuition                       Drake University

Fin 284

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)
Drake

Fin 284

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

Dollars           AUD (Australian \$)
Company A 5%           12.6%
Company B 7%           13.0%
2% difference in \$US   .4% difference in AUD
Drake
The strategy                        Drake University

Fin 284

Company A borrows dollars at 5% per annum
Company B borrows AUD at 13% per annum
They enter into a swap
Result
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%
Drake
Swap Diagram                          Drake University

Fin 284

AUD 11.9%        AUD 13%
5%                                            AUD 13%
Co A               FI             Co B
5%              6.3%

A pays 11.9% AUD instead of 12.6% AUD
B pays 6.3% \$US instead of 7% \$US
The FI makes .2%
Drake
Valuation of Currency Swaps                Drake University

Fin 284

Using Bond Techniques
Assuming 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.
Drake
Swap valuation                    Drake University

Fin 284

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
Drake
Other swaps                             Drake University

Fin 284

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.
Drake
Beyond Plain Vanilla Swaps                Drake University

Fin 284

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.
Drake
Beyond Plain Vanilla                          Drake University

Fin 284

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.
Drake
Off Market Swap                    Drake University

Fin 284

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                     Drake
Drake University

Extension Swaps                    Fin 284

Forward swap – the payments are agreed to
begin at some point in time in the future
If 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 needed
Extension Swap – an agreement to extend the
current swap (a form of forward swap)
Drake
Basis Swaps                     Drake University

Fin 284

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
Drake
Yield Curve Swaps                    Drake University

Fin 284

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.
Drake
Rate differential (diff) swap            Drake University

Fin 284

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

Fin 284

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.
Drake
Flavored Currency Swaps                 Drake University

Fin 284

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
Drake
Circus Swap Diagram                   Drake University

Fin 284

LIBOR
Company A                     Company B
5% US\$
6% German Marks
Company A                     Company C
LIBOR
Drake
Circus Swap Diagram               Drake University

Fin 284

Company B
Company A    5% US\$
6% German Marks
Company C
Drake
Swapation                              Drake University

Fin 284

An option on a swap that specifies the tenor,
notional principal fixed rate and floating rate
Price is usually set a a % of notional principal
The holder has the right to enter into a swap as the
Payer Swapation
The holder has the right to enter into a particular
swap as the fixed rate payer.
Swapation as                      Drake
Drake University

call (or put) Options                   Fin 284

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.
Drake
In-the-Money Swapations                    Drake University

Fin 284

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.
Drake
When to Exercise                      Drake University

Fin 284

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                       Drake
Drake University

swapation example                       Fin 284

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                     Drake
Drake University

value of the call                    Fin 284

The firm can sell a receiver swapation with terms
that match the call feature of the bond.
equal to the value of the call option.
Drake
Example                       Drake University

Fin 284

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.
Drake
Example continued                          Drake University

Fin 284

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.
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.
Drake
Example Continued                     Drake University

Fin 284

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.
Drake
Example Continued                     Drake University

Fin 284

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%
swapation
Extendible and Cancelable                Drake
Drake University

swaps                           Fin 284

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.
Drake
Extendible and Cancelable                        Drake University

Fin 284

Extendible pay fixed swap
= plain vanilla pay fixed plus payer swapation
Cancelable Pay Fixed Swap
= plain vanilla pay fixed swap + receiver swapation
=plain vanilla receive fixed swap + payable swapation
Creating synthetic                    Drake
Drake University

securities using swaps                   Fin 284

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.
Drake
The Parallel Loan Market                Drake University

Fin 284

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.
Drake
Parallel Loan Market                   Drake University

Fin 284

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

Fin 284

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).
Drake
Synthetic Fixed Rate Debt                   Drake University

Fin 284

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
Drake
Synthetic Floating Rate Debt                Drake University

Fin 284

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.
Drake
Synthetic Callable Debt                   Drake University

Fin 284

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.
Drake
Synthetic Callable Debt                  Drake University

Fin 284

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).
Drake
Synthetic non callable Debt                Drake University

Fin 284

Basically the earlier example swapations.
Drake
Synthetic Dual Currency Debt               Drake University

Fin 284

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
Drake
Synthetic Dual Currency Debt              Drake University

Fin 284

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.
Drake
All in Cost                     Drake University

Fin 284

The IRR for a given financing alternative, it
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                      Drake
Drake University

alternative proposals                 Fin 284

A 10 year semiannual 7% coupon bond with a
principal of \$40 million priced at par
A 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.
Drake
All in cost                    Drake University

Fin 284

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 Rabobank                    Drake
Drake University

An early swap example*                      Fin 284

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
Drake
Rabobank                        Drake University

Fin 284

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%
Drake
The Intermediary                      Drake University

Fin 284

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.
Drake
Problems                       Drake University

Fin 284

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.
Drake
The agreement                      Drake University

Fin 284

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
Drake
The swaps                     Drake University

Fin 284

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 discount
Rabobank received \$5,500,000 each year for 8
years and paid semi annul payments of LIBOR-x
Drake
The intermediary role                  Drake University

Fin 284

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.
Drake
BF Goodrich                     Drake University

Fin 284

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.
Drake
Rabobank’s Position                  Drake University

Fin 284

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.
Drake
Securing financing                   Drake University

Fin 284

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                                  Drake
Drake University

financing*                                            Fin 284

It is possible for swaps to increases accessibility
two the debt market
Mexcobre (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 risk
However 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
Drake
The original loan                  Drake University

Fin 284

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.
Drake
Drake University

Fin 284

Quarterly payments of 11.48%
Banks     interest plus principal  Escrow
Excess cash
\$210                  if it builds           Cash based
million               up                     on Spot Price
loan

4,000 tons of copper
Mexcobre                              SOGEM
per month
Drake
Swaps                        Drake University

Fin 284

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
Drake
Paribas
Drake University

Fin 284

Fixed       Floating                      \$2,000 Spot
per ton Price
Quarterly payments of 11.48%        per ton
Banks     interest plus principal    Escrow
Excess cash
\$210                     if it builds           Cash per
million                  up                    ton based
loan                                         on Spot Price

4,000 tons of copper
Mexcobre                                 SOGEM
per month
Duration of Interest               Drake
Drake University

Rate Swaps*                      Fin 284

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
Drake
Duration of Swaps                        Drake University

Fin 284

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
Drake
Example                       Drake University

Fin 284

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
=5.65139-0.5=5.15369
Duration of Pay Fixed Swap
=0.5-5.65139=-5.15369
Drake
Calculating Duration                  Drake University

Fin 284

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.
Drake
Immunization with Swaps                      Drake University

Fin 284

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
Drake
Balance Sheet for FSF                                Drake University

Fin 284

Assets                           Liabilities
Cash                    \$7,000,000   6mo money mkt       \$75,000,000
(avg yield 6%)
Marketable Sec    \$18,000,000
(6 mo mat Yield 7%)                  Floating Rate Notes \$40,000,000
(5 yr mat7.3% yld semi)
Amortizing loans    \$130,467,133
(10 yr avg mat
Coupon Bond         \$24,111,725
semiannual
(10 yr semi 6.5% coup
8% avg yield)
\$25,000,000 par, 7% YTM

Total Assets       \$1555,467,133     Net worth           \$16,355,408
Total Liab & NW    \$155,467,133
Drake
Basic Duration                           Drake University

Fin 284

\$ Weighted Duration              N
 DA   w i Da i
of Asset Portfolio             i 1

Asset i
where w i 
Market Value of All Assets
Da i  Macaulay Duration of asset i
\$ Weighted Duration            N
 DL   w jDl j
of Liability Portfolio        j1

Asset j
where w j 
Market Value of All Liabilitie s
Dl j  Macaulay Duration of Liability j
Drake
Duration                                 Drake University

Fin 284

Assets                            Liabilities
Duration                              Duration
Cash                       0.00    6mo money mkt               0.5000
Marketable Sec            0.500    Floating Rate Notes         0.5000
Amortizing loans       4.604562    Coupon Bond               7.453369

Total Duration                      Total Duration
(7,000,000/155,467,133)0.000      (75,000,000/155,467,133)0.500
+(18,000,000/155,467,133)0.500     +(40,000,000/155,467,133)0.500
+(130,467,133/155,467,133)4.605    +(24,111,725/155,467,133)7.45337
3.922013                           1.705202
Hedging the                         Drake
Drake University

portfolios separately                    Fin 284

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
Drake
Swap notional value                  Drake University

Fin 284

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:
=5.65139-0.5=5.15369
Duration of Pay Fixed Swap
=0.5-5.65139=-5.15369
Drake
Asset Hedge                     Drake University

Fin 284

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
Drake
Liability Hedge                     Drake University

Fin 284

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                  Drake
Drake University

Liabilities together                  Fin 284

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

(The repricing model)                            Fin 284

Repricing GAP
The 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.
Drake
GAP Analysis                      Drake University

Fin 284

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.
Drake
Interest Sensitive GAP                      Drake University

Fin 284

GAP  Rate Sensistive Assets - Rate Sensistive Liabilitie s
Given the Gap it is easy to investigate the
change in the net interest income (NII) of the
financial institution.

Change in NII  (GAP)(Chan ge in Rates)
NII  (GAP)( R)
Drake
Example                             Drake University

Fin 284

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
Drake
GAP Analysis                      Drake University

Fin 284

Asset sensitive GAP (Positive GAP)
RSA – RSL > 0
If interest rates h NII will h
If interest rates i NII will i
Liability sensitive GAP (Negative GAP)
RSA – RSL < 0
If interest rates h NII will i
If interest rates i NII will h
Would you expect a commercial bank to be
asset or liability sensitive for 6 mos? 5 years?
Drake
Important things to note:                  Drake University

Fin 284

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.
Drake
Steps in Calculating GAP                  Drake University

Fin 284

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.
Drake
Alternative measures of GAP                       Drake University

Fin 284

Cumulative GAP
Totals 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                       Drake
Drake University

GAP                                       Fin 284

Relative Interest sensitivity GAP (GAP ratio)
GAP / Bank Size
The higher the number the higher the risk that is
present
Interest Sensitivity Ratio
Rate Sensitive Assets
Rate Sensitive Liabilitie s
 1  Liability Sensitive
 1  Asset Sensitive
Drake
What is “Rate Sensitive”                 Drake University

Fin 284

Any Asset or Liability that matures during the
time frame
Any principal payment on a loan is rate sensitive
if it is to be recorded during the time period
Assets or liabilities linked to an index
Interest rates applied to outstanding principal
changes during the interval
Unequal changes in interest                 Drake
Drake University

rates                              Fin 284

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.
liabilities may change as rates rise or decrease

NII  (RSA)( R assets ) - (RSL)( R liabilties)
Drake
Strengths of GAP                        Drake University

Fin 284

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.
Drake
Weaknesses of Static GAP                          Drake University

Fin 284

Market Value Effects
Basic 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 aggregation
Repricing 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.
Drake
Weaknesses of Static GAP                         Drake University

Fin 284

Runoffs
Periodic 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)
Drake
Weaknesses of GAP                             Drake University

Fin 284

Off Balance Sheet Activities
Basic 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
Drake
Basic Duration Gap                       Drake University

Fin 284

Duration Gap

\$ Weighted Duration \$ Weighted Duration
Basic DGAP                       
of Asset Portfolio   of Libaility Portfolio
Basic DGAP  DA  DL
Drake
Basic DGAP                             Drake University

Fin 284

If the Basic DGAP is +
If Rates h
i in the value of assets > i in value of liab
Owners equity will decrease
If Rate i
h in the value of assets > h in value of liab
Owners equity will increase
Drake
Basic DGAP                             Drake University

Fin 284

If the Basic DGAP is (-)
If Rates h
i in the value of assets < i in value of liab
Owners equity will increase
If Rate i
h in the value of assets < h in value of liab
Owners equity will decrease
Drake
Basic DGAP                       Drake University

Fin 284

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.
Drake
DGAP                         Drake University

Fin 284

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

MVL
DA  DL
MVA
MVL
DGAP  DA  DL
MVA
Drake
DGAP calculation           Drake University

Fin 284

MVL
DGAP  DA  DL
MVA
139,111,725
DGAP  3.922013  1.705202
155,467,133
 2.396201
Drake
Hedging with DGAP                     Drake University

Fin 284

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.
Drake
Hedging with DGAP                      Drake University

Fin 284

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
Drake
DGAP and owners equity                  Drake University

Fin 284

Let MVE = MVA – MVL
We can find MVA & MVL using duration
From our definition of duration:
Δi
ΔP  D          P Applying the formula
(1  i)
Δy
MVA  -DA       MVA
1 y
Δy
MVL  -DL      MVL
1 y
Drake
Drake University

Fin 284
ΔMVE  ΔMVA - ΔMVL
Δy              Δy      
 -DA      MVA - - DL      MVL 
1 y            1 y     
Δy
 -(DA)MVA - (DL)MVL 
1 y
            MVL  Δy
 -(DA) - (DL)      1  y MVA
            MVA 
Δy
ΔMVE  -DGAP       MVA
1 y
Drake
DGAP Analysis                     Drake University

Fin 284

If DGAP is (+)
An h in rates will cause MVE to i
An i in rates will cause MVE to h
If DGAP is (-)
An h in rates will cause MVE to h
An i in rates will cause MVE to i
The closer DGAP is to zero the smaller the
potential change in the market value of equity.
Drake
Weaknesses of DGAP                        Drake University

Fin 284

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
curve
Must continually monitor and adjust duration
It is difficult to measure duration for non interest
earning assets.
Drake
More General Problems                             Drake University

Fin 284

Interest rate forecasts are often wrong
To 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.
Drake
Changing Duration                        Drake University

Fin 284

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.

    MVH *

 Total Assets 
DG  DG  DS 
*

              
Decreasing Duration GAP                  Drake
Drake University

to One year                         Fin 284

    MVH *

 Total Assets 
D G  D G  DS 
*

              
    MVH *

 \$155,467,133 
1.0  2.396201  5.15369              
              
MVH  42,137,025
*

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

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