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 © 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)

31/1/2000 © K. Cuthbertson and D.Nitzsche Introduction

31/1/2000 © K. Cuthbertson and D.Nitzsche Introduction Swaps are privately arranged contracts in which parties agree to exchange cash flows. Swap contracts originated in about 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.

31/1/2000 © K. Cuthbertson and D.Nitzsche Interest Rate Swaps Swaps can be used … y… to alter a series of floating rate payments (or receipts). y… to reduce interest rate risk of financial institutions ySwaps are used by some firms who can borrow relatively cheaply in either the fixed or floating rate market.

31/1/2000 © K. Cuthbertson and D.Nitzsche 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”

31/1/2000 © K. Cuthbertson and D.Nitzsche Altering Cash Flows with a Swap

31/1/2000 © K. Cuthbertson and D.Nitzsche 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 % fixed Net Payment for firm = = 6.5% (= fixed) Issue Fixed Rate Bond or 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

31/1/2000 © K. Cuthbertson and D.Nitzsche Swap : Financial Intermediary Financial Intermediary FI’s Swap 11% fixed 12% fixed LIBOR After swap Net Receipts = ( ) + LIBOR - (LIBOR-1) = 2% (fixed) LIBOR-1% Without swap if LIBOR>13% F.I. makes a loss MortgageesDepositor

31/1/2000 © K. Cuthbertson and D.Nitzsche 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)

31/1/2000 © K. Cuthbertson and D.Nitzsche Figure 1:Cash Flows in a Swap at t: Receive Fixed and Pay Floating Equivalent to ‘long’ a fixed coupon bond and ‘short’ an FRN Receive Fixed Pay Floating 0 t6m12mn... 0 tn t= 3-months A dashed line indicates an uncertain cash flow In practice, the principal is not exchanged 18 6m12m18

31/1/2000 © K. Cuthbertson and D.Nitzsche Figure 2: Ex-post Net Payments Firm-B: Floating Rate Receiver (Fixed Rate Payer) 15th Sept (LIBOR = 10.0%) 15th March (LIBOR=11%) $ 100m( )(1/2) = $ 5,000 $ 100m( )(1/2) = $ 0 Fixed Rate = 10%

31/1/2000 © K. Cuthbertson and D.Nitzsche Comparative Advantage and Gains in the swap

31/1/2000 © K. Cuthbertson and D.Nitzsche Comparative Advantage: Gains in the swap A ultimately desires/wants to borrow floating B ultimately desires/wants to borrow fixed DIRECT BORROWING COSTS for A and B Fixed Floating Firm-A10.00 (A x ) LIBOR + 0.3% (A F ) Firm-B11.20 (B x ) LIBOR + 1.0% (B F ) 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.

31/1/2000 © K. Cuthbertson and D.Nitzsche Why not borrow directly in desired form ? A ultimately desires to borrow floating B ultimately desires to borrow fixed Total Cost to A+B of DIRECT borrowing in desired form B X + A F = (L+0.3) = L Total Cost to A+B if initially borrow in “NON-DESIRED form A X + B F = (L ) = L Hence TC is lower if initially borrow in “NON- DESIRED” Net overall gain to A+B = (B X + A F ) - (A X + B F ) = 0.5 Assume this is arbitrarily split 0.25 each Swap provides mechanism to achieve this

31/1/2000 © K. Cuthbertson and D.Nitzsche Table 1 : Borrowing Rates Facing A and B Fixed Floating Firm-A10.00 (A x ) LIBOR + 0.3% (A F ) Firm-B11.20 (B x ) LIBOR + 1.0% (B F ) Absolute difference (B-A)  (Fixed) = 1.2  (Float) = 0.7 Hence 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 = (B X - A X ) - (B F - A F ) - as on previous slide

31/1/2000 © K. Cuthbertson and D.Nitzsche 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) = = ) Hence in leg2 of swap B must pay =9.95 ( A will now also “fit” OK - see over )

31/1/2000 © K. Cuthbertson and D.Nitzsche Figure 3 Interest Rate Swap (A and B) 3)B pays A fixed 9.95% Firm BFirm A 2)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

31/1/2000 © K. Cuthbertson and D.Nitzsche 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

31/1/2000 © K. Cuthbertson and D.Nitzsche Table 14.2 : Indicative Pricing Schedule for Swaps

31/1/2000 © K. Cuthbertson and D.Nitzsche Valuation of Interest Rate Swaps

31/1/2000 © K. Cuthbertson and D.Nitzsche Valuation of Interest Rate Swaps Pricing swaps using a synthetic bond portfolio Valuing the floater (variable payments) yat inception, all the receipts on a floating rate bond have a value equal to the notional principal or par value, Q yimmediately after a coupon payment on a floating rate bond, its value also equals the par value, Q.

31/1/2000 © K. Cuthbertson and D.Nitzsche Valuation of Interest Rate Swaps Fixed payments = fixed rate coupon bond Floating payments = floating rate bond Fixed receipts-floating payer V(swap) = BX - BF BX = price of coupon bond (using spot rates) - this is straightforward BX =  C i e -ri.ti + Q e -r. n (tn)

31/1/2000 © K. Cuthbertson and D.Nitzsche 0123 Q r 1 Q f 12 Q (1+f 23 ) f 12 r1r1 r2r2 r3r3 V( ALL future receipts at t=0 ) = Q (surprised?) Value of Cash Flows on FRN at t = 0

31/1/2000 © K. Cuthbertson and D.Nitzsche (Original time t = 0) 012 Q r 1 Q (1+f 12 ) 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

31/1/2000 © K. Cuthbertson and D.Nitzsche 0123 Q r 1 t Q f 12 Q (1+f 23 ) r1r1 f 12 f 23 r 1-t r 2-t r 3-t Note : If t = 0.25 years into the swap then 1-t = 0.75 years,, 2-t = 1.75 years, 3-t = 2.75 years Value of cash flows FRN, between payment dates

31/1/2000 © K. Cuthbertson and D.Nitzsche 0123 Q (1 + r 1 ) t It can be shown that BF= V(FRN at t) = Q (1 + r 1 ) / (1+r 1-t ) Value of cash flows between payment dates : Equivalent Cash Flow r 1-t r1r1

31/1/2000 © K. Cuthbertson and D.Nitzsche End of Interest Rate Swaps

31/1/2000 © K. Cuthbertson and D.Nitzsche Currency Swaps

31/1/2000 © K. Cuthbertson and D.Nitzsche 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

31/1/2000 © K. Cuthbertson and D.Nitzsche 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)

31/1/2000 © K. Cuthbertson and D.Nitzsche 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.

31/1/2000 © K. Cuthbertson and D.Nitzsche 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

31/1/2000 © K. Cuthbertson and D.Nitzsche Cash Flows in a FX Swap: Receive FRF and Pay USD Receive FRF Pay USD 0t6m12mn... 0tn t= 3-months We assume both USD and FRF are at fixed rates of interest 18 6m12m18

31/1/2000 © K. Cuthbertson and D.Nitzsche Borrowing Costs and Comparative Advantage

31/1/2000 © K. Cuthbertson and D.Nitzsche Dollar FRF Uncle Sam 8%11.5% Effel10%12.0% Absolute Difference2%0.5% Effel:Comparative Advantage borrowing FRF Net Comparative Advantage = = 1.5% T3: Borrowing Costs and Comparative Advantage

31/1/2000 © K. Cuthbertson and D.Nitzsche 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.

31/1/2000 © K. Cuthbertson and D.Nitzsche Figure 5 Outset of a Currency Swap French Bondholders FRF500m US Bondholders $100 EffelUncle SamSwap Dealer FRF 500m $ 100m FRF 500m $ 100m 8% FRF 500m 12%

31/1/2000 © K. Cuthbertson and D.Nitzsche 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

31/1/2000 © K. Cuthbertson and D.Nitzsche If USam and Effel were to (stupidly) initially borrow directly in their desired currency then Total Cost (direct) = USam FRF + Effel $’s = = 21.5 But by initially borrowing in their CA currencies Total Cost (CA) = USam $’s + Effel FRF = = 20 Hence Gain in the Swap = = 1.5 (as before) Source of gains in the Swap

31/1/2000 © K. Cuthbertson and D.Nitzsche Assume (arbitrarily) the 1.5% gain is split) Swap dealer gets0.4% Uncle Sam gets 0.3% Effel gets 0.8% Splitting the gains in the Swap

31/1/2000 © K. Cuthbertson and D.Nitzsche USam gain of 0.3% implies USam 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% implies its 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 dollars Pays Effel 12% in FRF - so that the two firms payments and receipts are matched (ie. no FX risk for them) Gains in the Swap

31/1/2000 © K. Cuthbertson and D.Nitzsche Figure 6: Interest Flows on Currency Swap French Bondholders FRF 500m US Bondholders $ 100 m EffelUncle SamSwap Dealer ($ 9.2m) 9.2% (FF 60m) 12% $ 8m 8% FRF 60m 12% ($ 8m) 8% (FF 56m) 11.2% Swap Dealer: $Gain = = 1.2% FRF loss = = 0.8%. Net position = = 0.4%

31/1/2000 © K. Cuthbertson and D.Nitzsche Valuation of Currency Swaps

31/1/2000 © K. Cuthbertson and D.Nitzsche 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

31/1/2000 © K. Cuthbertson and D.Nitzsche Figure A14.5 : Currency Swap Time t 123 n F1F1 CdCd CdCd CdCd CdCd CdCd Cf1Cf1 Cf2Cf2 Cf3Cf3 CfnCfn F2F2 F3F3

31/1/2000 © K. Cuthbertson and D.Nitzsche Value of swap in USD at time t : $V = B D - (S)B F B F is the FRF value of French (foreign) bond underlying the swap, B D is the $ value of US bond underlying the swap, S is the exchange rate ($/FRF) ySuppose 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

31/1/2000 © K. Cuthbertson and D.Nitzsche Valuing Currency Swaps as a Bond Portfolio yExchange rates moved from S = 0.2($/FRF) to S = 0.22($/FRF), r($) = 9%, r(F) = 8% y‘Uncle Sam’ $ coupon receipts in the swap = 0.08 ($ 100m) = $8m y‘Uncle Sam’ FRF coupon payments in the swap = (FRF 500m) = FRF 56m.

31/1/2000 © K. Cuthbertson and D.Nitzsche Valuing Currency Swap as Set of Forward Contracts ‘ Uncle Sam’ receives annual USD C $ = $8m and principal M $ = 100m pays out C F = FRF 56m and principal M F = FRF 500m. This is a series of forward contracts yValue of forward cash flows : $(C $ - F i C F ) yForward rate today is : F i = S t e (r($)-r(F))t yEach net cash flow : $(C $ - F i C F )e -r($)t Example :  Let S = 0.22($/FRF), r($) = 9%, r(F) = 8%  V = -$21.66m (see textbook p. 376)

31/1/2000 © K. Cuthbertson and D.Nitzsche Other Types of Swap Basis swap yfloating-floating swap yyield curve swap Amortising swap Accreting swap Rollercoaster swap Diff swaps or quanto swaps Forward swap Swap option or swaption

31/1/2000 © K. Cuthbertson and D.Nitzsche 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

31/1/2000 © K. Cuthbertson and D.Nitzsche 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

31/1/2000 © K. Cuthbertson and D.Nitzsche 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

31/1/2000 © K. Cuthbertson and D.Nitzsche END OF SLIDES