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Transcript of 31/1/2000 © K. Cuthbertson and D.Nitzsche Lecture Swaps (Interest and Currency) FINANCIAL...
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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
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© K. Cuthbertson and D.Nitzsche
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
5 years 8.00 5 years TB + 46bp 5 year TB + 56 bp
6 years 8.05 6 years TB + 58bp 6 year TB + 68 bp
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Valuation of Interest Rate Swaps
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Valuation of Interest Rate Swaps
• 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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Valuation of Interest Rate Swaps
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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