31/1/2000 © K. Cuthbertson and D.Nitzsche Lecture Swaps (Interest and Currency) FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001)

31/1/2000

© K. Cuthbertson and D.Nitzsche

Lecture

Swaps (Interest and Currency)

FINANCIAL ENGINEERING:DERIVATIVES AND RISK MANAGEMENT(J. Wiley, 2001)

K. Cuthbertson and D. Nitzsche

Version 1/9/2001

31/1/2000


Topics Interest Rate Swaps

Introduction

Altering Cash Flows with a Swap

Cash Flows, Comparative Advantage and

Gains in the Swap

Valuation/Pricing a Swap (as bond portfolio)

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Introduction

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Introduction

• Swaps are privately arranged contracts in which parties agree to exchange cash flows.

• Swap contracts originated in about 1981.

• Largest markets is in interest rate swaps, but currency swaps are also actively traded.

• Most common type of interest rate swap is ‘Plain vanilla’ or fixed-for floating rate swap.

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Interest Rate Swaps

• Swaps can be used …

… to alter a series of floating rate payments (or

receipts). … to reduce interest rate risk of financial

institutions Swaps are used by some firms who can borrow

relatively cheaply in either the fixed or floating rate market.

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Interest Rate Swaps

A “plain vanilla” interest rate swap involves one party agreeing to pay fixed and another party agreeing to pay floating (interest rate), at specific time periods (eg. Every 6 months) over the life of the swap (eg 5 years).

Often a firm will borrow say“floating” from its bank and then go to a swap dealer who will agree to pay the firm “floating” , while the firm pays the swap dealer “fixed”

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Altering Cash Flows with a Swap

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Floating to Fixed: Liability

Fixed to Floating :Liability

Issue Floating Rate Bond or takes out bank loan at floating rate

Firm’s Swap LIBOR

LIBOR + 0.5

6% fixed

Net Payment for firm = 0.5 + 6.0 = 6.5% (= fixed)

Issue Fixed Rate Bondor take out bank loan at fixed rate

Firm’s Swap 6% fixed

6.2% fixed

LIBOR

Net Payment for firm = 0.2% + LIBOR (= floating)

Corporate Alters its (liability) cash flows with Swap

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Swap : Financial Intermediary

Financial Intermediary

FI’s Swap 11% fixed

12% fixed

LIBOR

After swapNet Receipts = (12 - 11) + LIBOR - (LIBOR-1) = 2% (fixed)

LIBOR-1%

Without swap if LIBOR>13% F.I. makes a loss

Mortgagees Depositor

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Reasons for Interest Rate Swaps

• 1) Hedge Risk• S&L (Building Soc) has fixed rate

mortgage receipts and pays out LIBOR on deposits

• (see above)

• 2) Lowers Overall Costs of bank loans • -for two (ie. Both corporate) borrowers

- (this is the “Principle of Comparative Advantage)

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Figure 1:Cash Flows in a Swap at t: Receive Fixed and Pay FloatingEquivalent to ‘long’ a fixed coupon bond and ‘short’ an FRN

ReceiveFixed

PayFloating

0 t 6m 12m n

...

0 t n

...t= 3-monthsA dashed line indicates an uncertain cash flowIn practice, the principal is not exchanged

18

6m 12m 18

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Figure 2: Ex-post Net Payments

Firm-B: Floating Rate Receiver (Fixed Rate Payer)

15th Sept(LIBOR = 10.0%)

15th March15th March(LIBOR=11%)

$ 100m(0.11-0.10)(1/2) = $ 5,000

$ 100m(0.10-0.10)(1/2) = $ 0

Fixed Rate = 10%

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Comparative Advantageand

Gains in the swap

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Comparative Advantage: Gains in the swap

A ultimately desires/wants to borrow floatingB ultimately desires/wants to borrow fixed

DIRECT BORROWING COSTS for A and BFixed Floating

• Firm-A 10.00 (Ax) LIBOR + 0.3% (AF)

• Firm-B 11.20 (Bx) LIBOR + 1.0% (BF)

Note that A can borrow at lower rate than B at both the fixed and floating rate (“A has absolute advantage”= higher credit rating). But the swap route will still be beneficial to BOTH A and B.

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Why not borrow directly in desired form ?

A ultimately desires to borrow floatingB ultimately desires to borrow fixed

Total Cost to A+B of DIRECT borrowing in desired form

BX + AF = 11.2 +(L+0.3) = L + 11.5Total Cost to A+B if initially borrow in “NON-DESIRED

form

AX + BF = 10.0 + (L + 1.0 ) = L + 11.0Hence TC is lower if initially borrow in “NON-DESIRED”

Net overall gain to A+B = (BX + AF) - (AX + BF) = 0.5Assume this is arbitrarily split 0.25 eachSwap provides mechanism to achieve this

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Table 1 : Borrowing Rates Facing A and B

Fixed Floating

Firm-A 10.00 (Ax) LIBOR + 0.3% (AF)

Firm-B 11.20 (Bx) LIBOR + 1.0% (BF)

Absolute difference (B-A) (Fixed) = 1.2 (Float) = 0.7Hence B has comparative advantage in borrowing at a

floating rate (“pays less more” )Hence Firm-B initially borrows at a floating rate

NCA/Quality Spread Differential NCA = (Fixed) - (Float) = 0.5

= (BX - AX ) - (BF - AF) - as on previous slide

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The Gain in the Swap

A ultimately desires to borrow floating

B ultimately desires to borrow fixed

1a)BUT B initially borrows “direct” at floating L+1.0

2) Assume B agrees in leg1 of swap to receive LIBOR

B’s Net payment so far is fixed 1.0

B’s (direct cost fixed - swap gain)

= 11.2-0.25=10.95

3) Hence in leg2 of swap B must pay 10.95-1.0 =9.95

( A will now also “fit” OK - see over )

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Figure 3 Interest Rate Swap (A and B)

3)B pays A fixed 9.95%

Firm BFirm A2)A pays B at LIBOR

1a)Issues(Borrows) Floating at LIBOR + 1%

1a)Issues(Borrows) Fixed at 10%

IN THE SWAP:

B is floating rate receiver and fixed rate payer

A is floating rate payer and fixed rate receiver

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Figure 4 Swap Dealer

Swap Dealer

Firm BFirm A

1a)Issues Floating at LIBOR + 1%

1b)Issues Fixed at 10%

2b)Floating LIBOR 2a)Floating LIBOR

3b)Fixed 10%3a)Fixed 9.9%

Note: Assume swap dealer makes 0.1 and A and B gain 0.2 each Note: Swap Dealer makes no profit on the floating rate leg

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Table 14.2 :Indicative Pricing Schedule for Swaps

Maturity Current T-bond rate

Bank pays fixed Bank receivesfixed

4 years 7.95 4 years TB + 40bp 4 year TB + 50 bp



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Valuation of Interest Rate Swaps

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• Pricing swaps using a synthetic bond portfolio

Valuing the floater (variable payments) at inception, all the receipts on a floating

rate bond have a value equal to the notional principal or par value, Q

immediately after a coupon payment on a floating rate bond, its value also equals the par value, Q.

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Fixed payments = fixed rate coupon bondFloating payments = floating rate bond

Fixed receipts-floating payerV(swap) = BX - BF

BX = price of coupon bond (using spot rates) - this is straightforward

BX = Ci e-ri.ti + Q e-r. n (tn)

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0 1 2 3

Q r1 Q f12

Q (1+f23)

f12 f12

r1

r2

r3

V( ALL future receipts at t=0 ) = Q (surprised?)

Value of Cash Flows on FRN at t = 0

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(Original time t = 0)

0 1 2

Q r1

Q (1+f12)

Note : We re-date end of year-1 as time t = 0.

V( ALL future receipts at t=1 ) = Q (more surprised?)

Value of cash flows, FRN at t=1

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0 1 2 3

Q r1

t

Q f12 Q (1+f23)

r1 f12 f23

r1-t

r2-t

r3-t

Note : If t = 0.25 years into the swapthen 1-t = 0.75 years,, 2-t = 1.75 years, 3-t = 2.75 years

Value of cash flows FRN, between payment dates

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0 1 2 3

Q (1 + r1)

t

It can be shown that BF= V(FRN at t) = Q (1 + r1) / (1+r1-t)

Value of cash flows between payment dates :Equivalent Cash Flow

r1-t

r1

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End of

Interest Rate Swaps

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Currency Swaps

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Topics Currency Swaps

Reasons for Swap

Cash Flows, Comparative Advantage and

Gains in the Swap

Valuation of Currency Swap

as bond portfolio

as series of forward contracts

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Reason for undertaking a currency swap

• US firm (‘Uncle Sam’)with a subsidiary in France wishes to raise finance in French francs (FRF).

• The FRF receipts from the subsidiary in France will be used to pay off the debt.

• (This minimises foreign exchange risk)

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Reason for undertaking a swap

• French firm (‘Effel’) with a subsidiary in the US might wish to issue dollar denominated debt

• It will eventually pay off the interest and principle with dollar revenues from its subsidiary.

• This reduces foreign exchange exposure.

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• Assume Uncle Sam can raise finance (relatively) cheaply in dollars (say $100m) and

Assume Effel can raise funds cheaply in FRF (say FRF500)

• They might INITIALLY do so and then SWAP the payments of principal and interest.

• So the Effel ENDS UP paying dollars and the USam paying FRF

The Currency Swap

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Cash Flows in a FX Swap: Receive FRF and Pay USD

ReceiveFRF

PayUSD

0 t 6m 12m n

...

0 t n

...t= 3-monthsWe assume both USD and FRF are at fixed rates of interest

18

6m 12m 18

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Borrowing Costs

and

Comparative Advantage

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Dollar FRF

Uncle Sam 8% 11.5%

Effel 10% 12.0%

Absolute Difference 2% 0.5%

Effel:Comparative Advantage borrowing FRF

Net Comparative Advantage = 2 - 0.5 = 1.5%

T3: Borrowing Costs and Comparative Advantage

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Table 3 : Borrowing Rates (Contin)

• Effel has comparative advantage in borrowing in FRF.

• Hence Effel initially borrows in FRF

• Note ultimately Effel wants to borrow USD and Uncle Sam wants to borrow FRF’s. This is the motivation for the swap.

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Figure 5 Outset of a Currency Swap

French Bondholders

FRF500m

US Bondholders

$100

Effel Uncle SamSwap DealerFRF 500m

$ 100m

FRF 500m

$ 100m

$ 10

0m

8%

FR

F 5

00m

12%

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Effel initially borrows FRF at 12.0%

Uncle Sam initially borrows USD at 8%

However they then swap payments because:

Uncle Sam ultimately wants to borrow FRF

Effel ultimately wants to borrow dollars

Outset of a Currency Swap

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If USam and Effel were to (stupidly) initially borrow directly in their desired currency then

Total Cost (direct) = USam FRF + Effel $’s = 11.5 + 10 = 21.5

But by initially borrowing in their CA currenciesTotal Cost (CA) = USam $’s + Effel FRF

= 8 + 12 = 20

Hence Gain in the Swap = 21.5-20 = 1.5 (as before)

Source of gains in the Swap

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Assume (arbitrarily) the 1.5% gain is split)

Swap dealer gets 0.4% Uncle Sam gets 0.3%Effel gets 0.8%

Splitting the gains in the Swap

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USam gain of 0.3% impliesUSam pays 11.5 – 0.3 = 11.2% on the FRF leg (would have had to pay 11.5% directly)

Effel’s gain of 0.8% impliesits dollar payments in the swap are reduced from a direct cost of 10% (table 3) to 9.2%

Swap dealer: assume (for simplicity)Pays Uncle Sam 8% in dollarsPays Effel 12% in FRF

- so that the two firms payments and receipts are matched (ie. no FX risk for them)

Gains in the Swap

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Figure 6: Interest Flows on Currency Swap

French BondholdersFRF 500m

US Bondholders

$ 100 m

Effel Uncle SamSwap Dealer

($ 9.2m)9.2%

(FF 60m)12%

$ 8m 8%

FR

F 6

0m

12%

($ 8m)8%

(FF 56m)11.2%

Swap Dealer: $Gain = 9.2 - 8 = 1.2%

FRF loss = 12 - 11.2 = 0.8%.

Net position = 1.2 - 0.8 = 0.4%

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Valuation of Currency Swaps

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Valuation of Currency Swaps

• Holding (long) a dollar denominated bond and issuing a FRF denominated bond. Receives USD and pays out FRF

• Payments/liability in French francs for ‘Uncle Sam’. Hence, appreciation of FRF (depreciation of USD) implies loss on swap.

• Two methods :

– Currency swap as a bond portfolio– Currency swap as a set of forward contracts

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Figure A14.5 : Currency Swap

Timet 1 2 3 n

F1

Cd Cd Cd Cd Cd

Cf1 Cf2 Cf3 Cfn

F2

F3

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Value of swap in USD at time t :

$V = BD - (S)BF

BF is the FRF value of French (foreign) bond underlying the swap,

BD is the $ value of US bond underlying the swap, S is the exchange rate ($/FRF)

Suppose the swap deal of FRF 500m for $100m has been in existence for 1 year with another 3 years to run

Valuing Currency Swaps as a Bond Portfolio

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Valuing Currency Swaps as a Bond Portfolio

Exchange rates moved from S = 0.2($/FRF) to

S = 0.22($/FRF), r($) = 9%, r(F) = 8%

‘Uncle Sam’ $ coupon receipts in the swap = 0.08 ($ 100m) = $8m

‘Uncle Sam’ FRF coupon payments in the swap = 0.112 (FRF 500m) = FRF 56m.

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Valuing Currency Swap as Set of Forward Contracts

‘Uncle Sam’ receives

annual USD C$ = $8m and principal M$ = 100m

pays out CF = FRF 56m and principal MF = FRF 500m.

This is a series of forward contracts Value of forward cash flows : $(C$ - FiCF)

Forward rate today is : Fi = Ste(r($)-r(F))t

Each net cash flow : $(C$ - FiCF)e-r($)t

Example : Let S = 0.22($/FRF), r($) = 9%, r(F) = 8% V = -$21.66m (see textbook p. 376)

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Other Types of Swap

• Basis swap floating-floating swap yield curve swap

• Amortising swap• Accreting swap• Rollercoaster swap• Diff swaps or quanto swaps• Forward swap• Swap option or swaption

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Swap are over-the-counter (OTC) instruments.

Interest rate swap in practice involves the exchange only of the interest payments

Currency swap involves the exchange of principal (at t=0 and t=T) and interest payments.

Swap dealers (usually banks) take on one side of a swap contract

Summary Swaps

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If Swap dealer cannot immediately find a matching counterparty ,may hedge the risk in the swap using futures or options

Swap dealers earn profits on the bid-ask spread of the swap deal

The cash flows on one side of a swap contract are equivalent to that party taking a long and short position in two bonds. This synthetic swap enables one to value a swap contract.

All swaps have a zero value at inception (this is how the fixed rate in the swap is determined).

Summary Swaps

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Subsequently changes in the fixed interest rate on an interest rate swap lead to an increase or decrease in the value of the swap to a particular party.

(The value of the floating leg remains (largely) unchanged at par, Q).

A currency swap changes value due to changes in the fixed interest rate and in the exchange rate.

Summary Swaps

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END OF SLIDES

Documents

31/1/2000 © K. Cuthbertson and D.Nitzsche Lecture Swaps (Interest and Currency) FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001)