Applied Financial - LECTURE 5

Applied FinancialInstruments &

Risk Management(FINM7041)

Lecture 5Swap

Overview of this lecture

I. Introduction

II. Main uses of Swaps

III. Valuation of Interest Rate Swap

IV. Currency Swap

V. Credit Risk

I. Introduction

• Swap is an agreement to exchange cash flows at

specified future times according to certain specified rules.

• A swap is equivalent to a coupon-bearing asset plus a coupon-bearing liability. Note that the coupons might be fixed or floating.

• Swaps can also be thought of as a package of forward contract.

• Two most common swaps: • Plain vanilla interest rate swaps • Fixed-for-fixed currency swaps

I. Introduction

• Mechanics of Interest Rate Swap (Plain Vanilla) Party A (buyer who longs the contract) agrees to pay fixed rate and receive floating rate, from counter-party B (seller who short the contract).

There is no initial exchange of principal

Define as the fixed rate. as the floating rate.

fixrfloatr

Receives variable cash flows

0 1 2 3 4 float float float float

fix fix fix fix fix

r r r r r

r r r r r

Pays fixed cash flows

I. Introduction

• Mechanics of Interest Rate Swap (Plain Vanilla) • One cash flows based on a fixed interest rate and another

referenced to an index that varies over time. • The principal (termed the notional principal) itself is not

exchanged. • At each settlement period, the two rates are compared

and the difference (times the notional principal) is paid by one counterparty to the other.

• The time t variable cash flow is based on the time t-1 floating interest rate, therefore • the first cash flow is known • all subsequent cash flows are unknown (but always

known one period in advance).

I. Introduction

• Mechanics of Interest Rate Swap (Plain Vanilla)• Example: A 3-year swap initiated on 5 March 2004 between Microsoft

and Intel. A notional principal of $100 million. Microsoft agrees to pay

to Intel an interest rate of 5% p.a. In return, Intel agrees to pay Microsoft

the six-month LIBOR rate. Payments are to be exchanged every six months. The 5% interest rate is quoted with semi-annual compounding

( )Analysisof cash Flows CFs to Microsoft

Date LIBOR

Floating

CF

Fixed

CF

Net

CF

05-Mar -04

05-Sep- 04

05-Mar- 05

05-Sep- 05

05 -Mar-06

05-Sep -06

05-Mar -07

0.042

0.048

0.053

0.055

0.056

0.059

-

-

$2.10 m

$2.40 m

$2.65 m

$2.75 m

$2.80 m

$2.95 m

-

-$2.5 m

-$2.5 m

-$2.5 m

-$2.5 m

-$2.5 m

-$2.5 m

-

-$0.40 m

-$0.10 m

$0.15 m

$0.25 m

$0.30 m

$0.45 m


• Converting a Liability

• Example: Suppose the following:

• Both companies want to borrow $10 million for 5 years

• Company A wants the liability to be in the floating rate

• Company B wants the liability to be in the fixed rate

A B Diff

(B-A)

Fixed Rate

Floating Rate

5.2%

LIBOR

+1.7%

2.0%

1.2%

7.2%

LIBOR+0.5%


• Converting a Liability • Do nothing: • Company A borrows at the floating rate of LIBOR + 0.5% • Company B borrows at the fixed rate of 7.2%

• Using swap: • In absolute term, • Company A has absolute advantage in both floating rate and

fixed rate markets. • But in comparative term, • Company B has a comparative advantage in the floating rate market

• Company A has a comparative advantage in the fixed rate market • Hence, an interest rate swap for 5 years between A and B but

how?

• Converting a Liability • Recall:

• Company B has a comparative advantage in the floating rate market but wants to borrow fixed rate.

• Company A has a comparative advantage in the fixed rate market but wants borrow floating

rate. • Suppose the term of the swap is that Company

A agreed to pay LIBOR + 1.7% to Company B

and Company B agreed to pay 6.8% to Company A.



• Converting a Liability That is, for A,

• Borrow from outside at a fixed rate of 5.2%• At the same time, swap with B

- pay LIBOR + 1.7% - receive 6.8%

→ Effectively, A borrows a floating rate contract

6.8%

LIBOR +1.7%

5.2% LIBOR + 1.7%

A B

II Main uses of Swaps• Converting a Liability

Analysisof cashFlows

Company A Company B

Pays outside

lenders

Pay under

the swap

Receives

under the

Swap

Net interest

If no swap

Net Effect

-(5.2%)

-(LIBOR + 1.7%)

6.8%

-(LIBOR + 0.1%)

-(LIBOR + 0.5%)

0.4% gain

-(LIBOR + 1.7%)

-(6.8%)

LIBOR + 1.7%

-(6.8%)

-(7.2%)

0.4% Gain


• Converting an asset

• Example: Suppose the following

• Company A wants the asset to be in the fixed rate

• Company B wants the asset to be in the floating rate

A B Diff

(A-B)

Fixed Rate

Floating Rate LIBOR

-0.25%

4.7% 0.5%

1.25%

5.2%

LIBOR -1.5%

II. Main uses of Swaps• Converting an asset

• Do nothing:

• Company A receives at the fixed rate of 5.2%

• Company B receives at the floating rate of LIBOR-1.5%

• Using Swaps:

• Company B has a comparative advantage in the fixed

rate market (but prefer floating rate)

• Company A has a comparative advantage in the floating

rate market (but prefer fixed rate)

• Enter to an interest rate swap with the following term of

the swap:

• Company A agreed to pay LIBOR-0.4% to Company

B and

• Company B agreed to pay 5.4% to Company A.



An interest benefit for both companies.

LIBOR-0.4%

LIBOR-0.25% 4.7%

A B

%5.4



Company A Company B

Receives from

outside borrowers

Pay under the swap

Receives under

The Swap

Net interest

If no swap

Net Effect

LIBOR-0.25%

-(LIBOR-0.4%)

5.4%

5.55%

5.2%

0.35% gain

4.7%

-(5.4%)

LIBOR – 0.4%

LIBOR-1.1%

LIBOR-1.5%

0.4% Gain

Analysisof cashFlows


• Role of financial intermediary

If the swap is intermediated by a swap dealer, the gains for the

counterparties will be a little bit lower.

In our case here,

• The intermediary enters into an offsetting contract with A and B

and nets 0.03% (spread)

• The spread depends on supply and demand

5.385%A B

Intermediary

(Gain=0.03%)

5.415%

%LIBOR- 0.4 %LIBOR- 0.4

%LIBOR- 0.25 4.7%


• Role of financial intermediary.

• Both parties to a swap do not contract the

intermediary at the same time.

• Intermediary can enter into the swap and hedge

its exposure until having an offsetting contract.

• This refers to as warehousing interest rate swap.


A. Valuation in terms of bond prices

• An interest rate swap can be valued as the difference

between value of fixed and floating rate bond.

• The position of a fixed rate payer/floating rate receiver

is equivalent to long a swap:

• Interest rate swap = Long floating rate note + short fixed

rate note

• if you long a swap (fixed rate payer), the value of swap is

swap float fixV B B



Long an interest rate swap:

= long floating rate note and short fixed rate note

0 1 2 3 4

float float float float

fix fix fix fix fix

r r r r r

r r r r r

Floating rate note:

Fixed rate note:

0 1 2 3 4

float float float float

fix fix fix fix fix

r r r r r

r r r r r

Combined

Principal

Principal

III. Valuation of Interest Rate SwapA. Valuation in terms of bond prices • Swaps is then the same as an agreement in which one party

lends principal amount at the variable rate (e.g. LIBOR) and

borrow principal amount at the fixed rate, and vice versa for

another party.

• Fixed rate debt is just ordinary coupon bond:

Where

is the time until ith payments are exchanged;

is the notional principal in swap agreement;

is the LIBOR zero rate corresponding to maturity ti;

is the fixed payment made on each payment date.

1

i i n n

nr t r t

fixi

B ke Le

it

L

ir

k



• Floating rate debt reprices to par immediately after each payment

• Par is the notional principal

• Next payment is known with certainty

• Hence, value of floating rate debt = PV of next cash flow + PV

of notional principal, or;

Where

is the floating rate payment that will be made on the next

the next payment date.

1 1( *) r tfloatB L k e

*K

III. Valuation of Interest RateSwap


• Example: Bank has agreed to pay 6-month LIBOR and

Receive 8% (semi-annual) on $100 million.

• Remaining life: 1.25 years

• LIBOR (continuous compounding)

• 10% for 3 months,

• 10.5% for 9 months

• 11% for 15 months

• The LIBOR 6-months rate at the last payment date (3 months

ago) was 10.2% (semi-annual)

• Calculate the value of the swap.



• Example

Fixed rate bond

Semi-annual Coupon payment

$100 8% / 2 $4 *m m

0.1*3/12 0.105 9/12

0.11 15/12

4 4

($100 4)

$98.24

fix

fix

B m e m e

m e

B m

x x

x

x

x



• Example

Floating rate bond

The semi-annual LIBOR used = 10.2%/2 = 5.1%

Hence,

* $100 5.1% $5.1k *m m0.1 3($100 $5.1 ) $102.51floatB m m e m x / 12

swap fix floatV B B ( as the bank shorts the swap)

98.24 102.51 $4.27 m


B. Valuation in Terms of Forward Rate Agreements • A one-year swap with semi-annual payments is just a package

of two forward contracts • one with a six-month maturity • another with a 12-month maturity • An interest rate swap can also be viewed as a convenient package of forward rate agreements (FRA). • each exchange of payments (except the first payment) is

an FRA • The value of swap is the sum of the values of the forward rate

agreements underlying the swap • Note that the fixed rate in an interest rate swap is chosen so

that the swap is worth zero initially. • it does not mean that each forward contract underlying

a swap is zero initially


B. Valuation in Terms of Forward Rate Agreement

Recall: FRA

The value of FRA

where is the principal value; is the FRA rate; is the forward LIBOR rate for the period between and ; is the (continuously compounded) zero rate for a maturity .

Note that all the rates are measured with a compounding frequency

reflecting their maturity

2 22 1*( )*( ) R T

K FV L R R T T e

2R01T FR 2T

KR

L

RF

RK

2R 1T 2T2T


B. Valuation in Terms of Forward Rate Agreements

To value IRS in terms of FRA, we follow the following steps:

1. calculate each of the forward rates for each of the LIBOR rates that will determine swap cash flows.

2. calculate swap cash flows on the assumption that the LIBOR rates will equal to the forward rate (i.e.

expectations theory holds)

3. set swap rates equal to the PV of these cash flows


B. Valuation in Terms of Forward Rate Agreements • Example: Bank has agreed to pay 6-month LIBOR

and Receive 8% (semi-annual) on $100 million. • Remaining life: 1.25 years • LIBOR (continuous compounding)

• 10% for 3 months. • 10.5% for 9 months. • 11% for 15 months.• The LIBOR 6-month rate at the last payment date (3 months ago) was 10.2% (semi-annual)• Calculate the value of the swap.

III. Valuation of Interest Rate SwapB. Valuation in Terms of Forward Rate

Agreements

1. The first cash flow (in 3 months):

The cash flows for the payments 3 months have

already been set.

A rate of 8% will be the exchanged for a rate of 10.2%.

The value of the exchange is

0.1 3/120.5 $100 (0.08 0.102) $1.07NPV m e xx x

III. Valuation of Interest Rate Swap B. Valuation in Terms of Forward Rate Agreements

2. The second cash flow (in 9 months):

• Forward rate corresponding to 3 and 9 months is :

The rate of 10.75% obtained is a rate with continuous compounding

Need to convert it to a rate with semi-annual compounding.

• The value of the exchange is :

2 2 1 1

2 1

0.105 (9 /12) 0.10 3/120.1075

9 /12 3/12F

R T RTR

T T

x x

/ 0.1075/ 21 2 1 11.044%cR mmR m e e x

0.105 (9 /12)0.5 100 (0.08 0.11044) $1.41e m xx x


B. Valuation in Terms of Forward Rate Agreements

3. The third cash flows The value of the exchange is

4. Total value of the swap

0.11 15/120.5 100 (0.08 0.12102) $1.79e mxx x

($1.07 ) ( $1.41 ) ( $1.79 ) m m m

$4.27 m

IV. Currency Swap

A. Introduction • Currency Swap (CCS)

Exchanging principal and interest payments in one currency for principal and interest payments in another currency.

• In a currency swap, • two different currencies are periodically exchanged. • the principal is exchanged at the beginning and the

end of the swap • There are four types of basic currency swaps: • fixed for fixed • fixed for floating • floating for fixed • floating for floating

IV. Currency Swap

A. Introduction • Example of Currency Swap

• A five-year currency swap agreement between A and B. A pays a fixed rate of 11% in

sterling and receives a fixed rate of 8% in dollar from B (a fixed for fixed currency swap). Interest payments once a year. The principal amounts are $15m and £10m.

At origination:

At each annual settlement date:

At Maturity:

A B

10£ m

$15m

A B

*$15 0.08 $1.2m m

*10 11% 1.1£ £m m

A B

$15m

10£ m

IV. Currency Swap

A. Instruction

• Example: Suppose the following:

• GM has a comparative advantage in the USD market

but want to borrow in AUS.

• Qantas has a comparative advantage in the AUD market

but want to borrow in USD.

US AUD

GM 5.0% 12.6%

Qantas 7.0% 13.0%

Diff 2.0% 0.4%

IV. Currency Swap

A. Introduction

US6.3%

GM Bank QF

US 5%

AUD 11.9% AUD13%US 5% AUD 13%

IV. Currency Swap B. Valuation CCS in terms of Bond Prices

Long a currency swap: the domestic currency is received and a foreign currency

is paid

The value (in domestic currency) of a long position in swap is:

where is the value of the foreign-denominated bond underlying the swap (in the foreign currency);

is the value of home-denominated bond (in the home currency); is the spot exchange rate (number of units of domestic

currency per units of foreign currency).

0swap D FV B S B

FB

DB

0S

IV. Currency Swap B. Valuation CCS in terms of Bond Prices

The value of a swap where the foreign currency is received and

a domestic currency is paid is:

where

is the value of the foreign-denominated bond underlying

the swap (in the foreign currency);

is the value of home-denominated bond (in the home

currency);

is the spot exchange rate (number of units of domestic

currency per units of foreign currency).

0swap F DV S B B

FB

DB

0S

IV. Currency Swap

B. Valuation CCS in terms of Bond Prices

• Example: The Japanese interest rate is 4% p.a. and the US interest rate is 9% p.a. A US company enters a swap where it receives 5% p.a. in Yen with a principal of Yen 1200 million, and pays 8% p.a. with a principal of $10 million. The swap lasts for three years and the exchange rate is $0.00909/yen (or 110 yen=$1). Assume annual interest payments. What is the value of this swap?

IV. Currency SwapB. Valuation CCS in terms of Bond Prices

• Example:

• The value of foreign bond is:

• The value of domestic bond is:

• In our case, foreign currency is received and domestic

currency is paid short

• Hence,

0.04 1 0.04 2 0.04*360 60 (1200 60) 1230.55FB e e e Yen x x m

0.09 1 0.09 2 0.09*30.8 0.8 10 0.8 $9.644DB e e e x x m

0 1230.55*0.00909 9.644 $1.54swap F DV S B B m

IV. Currency SwapC. Valuation in terms of forward contracts

• The currency swap can also be viewed a series

of forward contracts.

• Hence, the value of the swap is the sum of the values of

the values of the forward contracts underlying the swap.

Example: The Japanese interest rate is 4% p.a. and the US interest rate is 9% p.a. A US company enters a swap where it

receives 5% p.a. in Yen with a principal of yen 1200 million, and pays 8% p.a. With a principal of $10 million. The swap lasts for three year and the exchange rate is $0.00909/yen. Assume annual interest payments.

What is the value of this swap?

IV. Currency Swap

C. Valuation in terms of forward contracts

• Recall :

where

is the value of the contract today;

is the forward price today;

is delivery price in the contract.

0( ) rTf F K e

f

0F

K

1

1 2 3

2

4

3

IV. Currency Swap Valuation in terms of forward contracts

To determine the we need forward exchange rate,

Recall:

The relationship between forward and spot exchange

rate is:

where

is the forward exchange rate;

is the spot exchange rate;

is the domestic risk-free interest rate;

is the foreign risk-free interest rate.

,tF

0FXF10S

r

fr

( )

0 0fr rFX FXF S e T

IV. Currency Swap

c. Valuation in terms of forward contracts

• One-year forward exchange rate:

• two-year forward exchange rate:

• three-year forward exchange rate:

• The exchange of interests involves: • Receive: Yen 1200 m * 0.05 = 60 million

Yen • Pay: USD 10 million * 0.08 = 0.8 million

USD

(0.09 0.04) 10.00909 0.009557e xx

(0.09 0.04) 20.00909 0.010047e xx

(0.09 0.04) 30.00909 0.010562e xx

IV. Currency Swap C. Valuation in terms of forward contracts

The values of forward contracts corresponding to

three exchanges of interests are:

The value of forward contract corresponding to the

exchange of principal at maturity is

Total = - 0.2071 – 0.1647 – 0.1269 + 2.0416

= 1.543 million USD

0.09 1

0.09 2

0.09 3

(60 0.009557 0.8 )

0.2071

(60 0.010047 0.8 )

0.1647

(60 0.010562 0.8 )

0.1269

e

m

e

m

e

m

m Yen m USD

USD

m Yen m USD

USD

m Yen m USD

USD

x

x

x

x

x

x

0.09 3(1200 0.010562 10 )

2.0416

e

m

xxm Yen m USD

USD

V. Credit RiskC. The value of swap is normally zero when it is first negotiated.

• this means that it costs nothing to enter into a swap

• it does not mean that the each forward contract underlying

a swap is worth zero initially.

• At a future time, its value is liable to be either positive or negative.

• Credit risk is risk resulting from uncertainty in a counterparty’s

ability or willingness to meet its contractual obligation.

• A financial intermediary has credit risk exposure from a

swap only when the value of the swap to the financial

intermediary is positive.

• Note that there is greater credit risk with a currency swap when

there will be a final exchange of principal, because there is a higher

probability of a large buildup in value, given one of the counterparties the incentive to default.

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Applied Financial - LECTURE 5