© Paul Koch 1- 1 Chapter 7: Swaps I. Interest Rate Swaps. - Presentation

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© Paul Koch 1- 1 Chapter 7: Swaps I. Interest Rate Swaps.

© Paul Koch 1- 1 Chapter 7: Swaps I. Interest Rate Swaps. A. Mechanics of Interest Rate Swaps. 1. Example 1; Interest Rate SWAPs. a . Consider the following opportunities for companies A & B:

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© Paul Koch 1- 1 Chapter 7: Swaps I. Interest Rate Swaps.






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© Paul Koch 1-1 Chapter 7: Swaps I. Interest Rate Swaps. A. Mechanics of Interest Rate Swaps. 1. Example 1; Interest Rate SWAPs. a . Consider the following opportunities for companies A & B: Fixed Floating . Company A 10.0% LIBOR + 0.3 % (A wants floating.) Company B 11.2% LIBOR + 1.0 % (B wants fixed.) Difference: 1.2% 0.7% . b . Comparative Advantage : B has worse credit rating in both markets . However , B's risk premium over A is “less bad” in Floating market . Thus, B has comparative advantage in Floating market . i . Difference (1.2 - . 7) is margin to be captured in SWAP.

© Paul Koch 1-2 I.A. Mechanics of Interest Rate Swaps 1. Example 1 Fixed Floating . (continued) Company A 10.0% LIBOR + 0.3% Difference = 1.2 - 0.7 Company B 11.2% LIBOR + 1.0% = 0.5% Difference: 1.2% 0.7% . Margin to be captured. c. Mechanics: Company A Company B A borrows fixed: <10%> -- B borrows floating: -- <LIBOR+1.0%> A pays B: <LIBOR> LIBOR B pays A: 9.95% . <9.95%> . Net Borrowing Cost: <LIBOR+.05%> <10.95%> Margin Captured: 0.25% 0.25% . d. Or, through Bank: Company A Bank Company B A borrows fixed: <10.00%> -- -- B borrows floating: -- -- <LIBOR+1.0%> A pays Bank: <LIBOR> LIBOR -- Bank pays A: 9.90% < 9.90%> -- Bank pays B: -- <LIBOR> LIBOR B pays Bank: -- . 10.00% . <10.00%> . Net Borrowing Cost: <LIBOR+.10%> -- <11.00%> Margin Captured: 0.20% 0.10% 0.20% . 10% LIBOR LIBOR LIBOR+1% <──── ──────> ──────> ────> <────── <────── 9.9% 10% Company A FinancialInstitution Company B

© Paul Koch 1-3 I.A. Mechanics of Interest Rate Swaps 1. Example 1 Fixed Floating . (continued) Company A 10.0% LIBOR + 0.3% Difference = 1.2 - 0.7 (again) Company B 11.2% LIBOR + 1.0% = 0.5% Difference: 1.2% 0.7% . Margin to be captured. c. Mechanics: Company A Company B A borrows fixed: <10%> -- B borrows floating: -- <LIBOR+1.0%> A pays B: <LIBOR> LIBOR B pays A: 9.95% . <9.95%> . Net Borrowing Cost: <LIBOR+.05%> <10.95%> Margin Captured: 0.25% 0.25% .How do you determine the number, 9.95%? Each party gets 0.25% ( ½ of 0.5% ); Thus, A will pay (LIBOR + .30% - .25%) = (LIBOR + .05%); B will pay 11.2% - 0.25% = 10.95%. d. Or, through Bank: Company A Bank Company B A borrows fixed: <10.00%> -- -- B borrows floating: -- -- <LIBOR+1.0%> A pays Bank: <LIBOR> LIBOR -- Bank pays A: 9.90% < 9.90%> -- Bank pays B: -- <LIBOR> LIBOR B pays Bank: -- . 10.00% . <10.00%> . Net Borrowing Cost: <LIBOR+.10%> -- <11.00%> Margin Captured: 0.20% 0.10% 0.20% . How do you determine the numbers, 9.90% & 10.00%? Bank gets 0.10%; A & B get 0.20%. Thus, A will pay (LIBOR + .30 % - .20%) = ( LIBOR + .10% ); B will pay 11.2% - 0.20% = 11.00% .

© Paul Koch 1-4 I.A. Mechanics of Interest Rate Swaps 2. Exchange of Payments in an Interest Rate SWAP. a. Each payment date, one party owes the other - sends other party a check for difference between fixed & floating payments: i. e.g., If SWAP is based on 6-month LIBOR; every 6 months. ii. Payment is based on LIBOR 6 months earlier, paid in arrears. iii. First payment is made 6 months after start of contract. b. Example 2 : Face Value (Principal, L): $100 million; Maturity = 3 years; Company B receives floating (6-month LIBOR); pays fixed (5% p.a.); Payments every 6 months (Millions of $): point of view of Company B; Time LIBOR Floating Fixed Net (yr) rate Cash Flow Cash Flow Cash Flow . 0.0 4.20 (for B) 0.5 4.80 +2.10 -2.50 -0.40 1.0 5.30 +2.40 -2.50 -0.10 1.5 5.50 +2.65 -2.50 +0.15 2.0 5.60 +2.75 -2.50 +0.25 2.5 5.90 +2.80 -2.50 +0.30 3.0 6.40 +2.95 -2.50 +0.45 .** c. SWAP is a derivative! Value depends on interest rates; If rates increase, SWAP becomes more valuable to Company B.

© Paul Koch 1-5 I.A. Mechanics of Interest Rate Swaps d. If Notional Principal were exchanged, no change in net cash flow. Time LIBOR Floating Fixed Net (yr) rate Cash Flow Cash Flow Cash Flow . 0.0 4.20 (for B) 0.5 4.80 +2.10 -2.50 -0.40 1.0 5.30 +2.40 -2.50 -0.10 1.5 5.50 +2.65 -2.50 +0.15 2.0 5.60 +2.75 -2.50 +0.25 2.5 5.90 +2.80 -2.50 +0.30 3.0 6.40 +102.95 -102.50 +0.45 . This shows that SWAP is like an exchange of a fixed rate bond for a floating rate bond. For B: long floating rate bond; short fixed rate bond. For B: VSWAP = Bfloating - Bfixed; For A; VSWAP = Bfixed - Bfloating.

© Paul Koch 1-6 I.A. Mechanics of Interest Rate Swaps 3. Comparative Advantage argument. - Why is spread between fixed rates > spread between floating rates? - As SWAP markets develop, shouldn't this difference in spreads  0? a. Spread depends on nature of contracts available in fixed & floating mkts. i. Fixed rates typically quoted for 5-year T. notes; ii. Variable rates typically on 6-month ED deposits. b. During next 6 months, P(A will default) and P(B will default) are both small; But as we look further ahead, P(B will default) es faster than P(A will default). Thus, it should be true that the spread between 5-year rates > spread between 6-month rates. c. Furthermore, floating rate lender has option to increase spread, if creditworthiness of floating rate borrower  after 6 months. The fixed rate lender cannot do this until after 5-years; thus, fixed rate lender will charge larger premium on B for fixed.

© Paul Koch 1-7 I.A. Mechanics of Interest Rate Swaps 4. SWAP Quotes and Zero LIBOR Rates. a. Warehousing: Bank enters SWAP with one party without another party to offset trade. i. Then Bank hedges interest rate risk (with futures) for its overall exposure. b. SWAP Rates. Large banks are market-makers in SWAPs ; Quote bid & offer for fixed rate they will exchange for floating, for different maturities. i. Bid = fixed rate in SWAP where market maker will pay fixed & receive floating; Bid = 5-yr T.Note rate + xx bp, for SWAP that pays fixed & receives LIBOR, ii. Offer = fixed rate in SWAP where market maker will receive fixed & pay floating. Offer = Ask = 5-yr T.Note rate + (xx + 3 or 4) bp, for SWAP that receives fixed. iii. SWAP Rate = average of Bid and Offer fixed rates. c. SWAP Pricing Schedule gives Bank’s bid & offer fixed rates & SWAP rates. Example: Maturity Bid Offer SWAP Rate 2 6.03 6.06 6.045 3 6.21 6.24 6.225 4 6.35 6.39 6.370 5 6.47 6.51 6.490 7 6.65 6.68 6.665 10 6.83 6.87 6.850 . d. Implications: Bank will ‘buy’ SWAP @ (r + xx), or ‘sell’ SWAP @ (r + xx + 3 or 4). The xx bp vary with mkt conditions. The 3 or 4 bp are the bank's spread (profit).

© Paul Koch 1-8 I.A. Mechanics of Interest Rate Swaps 5. Complication : a. 6-month LIBOR quoted with semiannual compounding for 360-day yr; T. Note Rate quoted with semiannual compounding for 365-day yr. b. To make 6-mo. LIBOR rate compatible with 5-yr T. Note Rate, multiply 6-mo. LIBOR rate by (365 / 360). (Annualize LIBOR Rate)6. CBOT trades futures on the 3-year and 5-year SWAP rates. a. Contracts settled in cash to a SWAP rate that is the median of the average of the bid & offer quotes of seven dealers randomly selected from a list. b. CBOT also trades options on SWAP futures ( SWAPtions ). [option to buy or sell futures (promise) to buy or sell SWAP.]

© Paul Koch 1-9 I.A. Mechanics of Interest Rate Swaps 7. Determining the LIBOR Zero Curve. a. Consider SWAP where fixed rate = SWAP rate. Can assume value of this SWAP = 0.*** (Why else would mkt-maker choose bid & ask centered around SWAP rate?) b. Since VSWAP = Bfixed - Bfloating = 0 in this case, Bfixed = Bfloating . Banks usually discount OTC cash flows at LIBOR. The floating rate bond underlying SWAP pays LIBOR. As a result, the value of this bond, Bfloating = SWAP principal (coupon rate = mkt rate) A SWAP rate is therefore a LIBOR par yield; (bond worth face value) the coupon rate on the LIBOR bond that makes it worth par. c. ED futures can be used to determine LIBOR zero rates. SWAP rates can also help determine LIBOR zero rates. As in b., SWAP rates define a series of LIBOR par yield bonds. These can be used to bootstrap a LIBOR Zero Curve in the same way that T. bonds are used to bootstrap the Treasury Zero Curve.

© Paul Koch 1-10 I.A. Mechanics of Interest Rate Swaps d. In the U.S ., Spot LIBOR rates are typically used to define the LIBOR Zero Curve for maturities up to 1 year. ED futures rates are used for maturities between 1 & 2 years, and sometimes up to maturities of 5 years. SWAP rates used to calculate Zero Curve for longer maturities. Similar procedure used to get LIBOR zero rates in other countries. e.g., EURO LIBOR zero rates are determined from Spot EURO LIBOR rates, 3-mo Euro futures, & Euro SWAP rates. e. May use LIBOR & Swap rates to bootstrap LIBOR Zero Curve. OR, may use Overnight Indexed Swap (OIS) rates instead. See Hull for details.Note: LIBOR used to price SWAPs; OIS Rates used to discount cash flows.

© Paul Koch 1-11 I.A. Mechanics of Interest Rate Swaps 8. Following credit crisis of 2007, there is international agreement that standard SWAPs be traded on electronic platforms & cleared thru CCPs. The SWAPs are then treated like futures contracts: Initial and variation margin are posted by both sides. In U.S., this rule does not apply when one of the parties in a SWAP is an end user whose main activity is not financial (i.e., not a bank), and who is using SWAPs to hedge. Occasionally, banks may be lucky to enter 2 offsetting SWAPs with non-bank counterparties -- natural hedge + rule does not apply. But normally a bank enters a SWAP with no offsetting trade, so the BANK must manage risk by entering offsetting trade with another BANK. Trade with other BANK must be executed on electronic platform, and cleared through a CCP. So initial trade with non-financial has no collateral or margin required, but offsetting trade with another BANK must have collateral & margin.

© Paul Koch 1-12 I.B. Valuation of Interest Rate Swaps 1. Valuing Interest Rate SWAP in terms of Bond Prices. a. Suppose you receive fixed and pay floating in a SWAP. Same effect as if you: - lend @ fixed rate; borrow @ floating rate; - buy fixed rate bond (Bfix); sell floating rate bond (Bfl). - Value of SWAP (VSWAP) is difference in values of 2 bonds. b. Let k = fixed payments received at times ti (1  i  n); L = notional principal in SWAP agreement Bfix = value of fixed rate bond underlying SWAP; Bfl = value of floating rate bond underlying SWAP; Then: VSWAP = Bfix - Bfl . c. Discount rates should reflect riskiness of underlying cash flows. i. Discount cash flows using LIBOR Zero Rates (or OIS Zero Rates). ii. Given this assumption, Bfl = L when SWAP is entered. (value of floating rate bond = notional principal). If coupon rate = discount rate, then Bfl = L = par value.

© Paul Koch 1-13 I.B. Valuation of Interest Rate Swaps d. Consider SWAP where fixed rate = SWAP rate ( avg of bid & ask). i. Can assume VSWAP = 0 when SWAP is entered.Again -- (Why else would mkt-maker choose bid & ask ctred around SWAP rate?). ii. Since VSWAP = Bfix - Bfl = 0, it also follows that Bfix = Bfl = L, when SWAP is entered. iii. Thus, SWAP pricing schedule defines a set of fixed rate bonds worth par value - known as par yield bonds. iv. Bootstrap procedure can be used to determine the zero-coupon yield curve from these par yield bonds. v. This zero-coupon yield curve gives LIBOR Zero Rates: - defines appropriate discount rates for SWAP cash flows; - can be used to determine values of the bonds in a SWAP; - can thus be used to determine current value of a SWAP.

© Paul Koch 1-14 I.B. Valuation of Interest Rate Swaps e . Consider value of fixed rate bond, Bfix . Define: ti = time until ith payments are exchanged (1 < i < n); (discount rates) ri = LIBOR zero rate corresponding to maturity ti ; k = fixed payment made on each payment date; L = notional principal. n Then: Bfix = Σ k e-ri (ti) + Le-rn(tn) (= PV of future cash flows i=1 at time SWAP is entered).

© Paul Koch 1-15 I.B. Valuation of Interest Rate Swaps f . Consider next value of floating rate bond, Bfl . i. Immediately after a payment date, Bfl = L (d. above) ii. Between payment dates, can value floating rate bond by noting Bfl will = L immediately after the next payment date: In our notation, t1 = time to next payment date. Thus: Bfl = L e -r1 (t1) + (k*) e -r1 (t1) , where k* = floating rate payment (LIBOR) to be made at t1 . (This is known today - next LIBOR is paid in arrears.) g. Finally, VSWAP = Bfix - Bfl . i. Or, if you pay fixed & receiving floating, VSWAP = Bfl - Bfix . NOTE: VSWAP = 0 when SWAP is first negotiated. During life, VSWAP moves away from zero as interest rates move.

© Paul Koch 1-16 I.B. Valuation of Interest Rate Swaps h . Example (different from example in Hull). Financial institution has agreed to: - receive fixed: 8% p.a. (semiannual comp) and pay floating: 6-mo LIBOR; - notional principal of L = $100 million; - remaining life = 1.25 years (15 months), payments in 3, 9, & 15 months; - LIBOR zero (discount) rates ri, for i = 3 mo, 9 mo, and 15 mo: r1 = 10.0%, r2 = 10.5%, and r3 = 11.0%; - 6-month LIBOR on last payment date was 10.2% (with semiannual comp). Thus: k = $4.0 million (= ½ of 8.0% fixed coupon); k* = $5.1 million (= ½ of 10.2% last LIBOR). r1 x t1 r2 x t2 r3 x t3 Then Bfixed = 4 e -.10 x .25 + 4 e -.105 x .75 + 104 e -.11 x 1.25 = $98.24 MM; r1 x t1 r1 x t1 and Bfl = 5.1 e -.10 x .25 + 100 e -.10 x .25 = $102.51 MM. and VSWAP = Bfixed - Bfl = $ 98.24 - $ 102.51 = - $ 4.27 MM . [ Receive fixed & pay floating; Today, fixed rate = 8% < floating rate = 10.2%; losing money! ]

© Paul Koch 1-17 I.B. Valuation of Interest Rate Swaps 2. Valuing Interest Rate SWAP as sum of forward contracts (FRAs). a. First exchange is known at the time SWAP is entered: In 3 months, receive fixed rate = 8%; pay floating rate = LIBOR = 10.2%; Value of 1st exchange = [ 0.5 x 100 x (.08 - .102) ] e -0.10 x .25 = -$1.07 MM. b. Remaining two exchanges are forward contracts (FRA’s): To get 2nd exchange in 9 months, compute fwd rate between 3 & 9 months: r2 = (r2T2 - r1T1) / (T2 - T1 ) = (.105 x .75 - .10 x .25) / (.75 - .25) = .1075  or 10.750% with continuous compounding; or 11.044% with semiannual compounding ( from formula in Ch.4 ). Value of FRA - 2nd exchange = [0.5 x 100 x (.08 - .11044)] e -.105 x .75 = -$1.41 MM. To get 3rd exchange in 15 months, compute fwd rate between 9 & 15 months: r3 = (r3T3 - r2T2) / (T3-T2) = (.11 x 1.25 - .105 x .75) / (1.25 - .75) = .1175  or 11.75% continuous, or 12.102% semiannual. Value of FRA - 3rd exchange = [0.5 x 100 x (.08 - .12102)] e -.11 x 1.25 = -$1.79 MM. c. Together, VSWAP = ( -1.07 - 1.41 - 1.79) = -$4.27 MM . (same answer)

© Paul Koch 1-18 I.B. Valuation of Interest Rate Swaps d. Since fixed rate in SWAP is chosen so that VSWAP = 0, the sum of values of FRAs underlying the SWAP must be zero when it is entered. However, the value of each individual forward contract ≠ 0; - Some FRAs will have positive values, others negative. i. Reconsider example from 2.a-c. Consider view of the counterparty to the bank: - pays fixed all future periods [ k = .5 x 8.0% ]; - will receive LIBOR for 1st exchange in 3 months [ k* is known today ]; - will receive LIBOR for i = 2,3; in 9 & 15 months [fwd rates ( ri ), unknown]; - Value of each fwd = (F - K) e -ri (ti) = (ri - 8.0) e -ri (ti) : ii. Value of fwd contract > 0 when fwd rate > 8.0% [when ri > 8.0; receive > 8.0]. Value of fwd contract = 0 when fwd rate = 8.0% [when ri = 8.0, receive = 8.0]. Value of fwd contract < 0 when fwd rate < 8.0% [when ri < 8.0, receive < 8.0]. iii. In this example, all 3 fwd rates (ri) > fixed = 8.0%; - Counterparty to bank is long these FRAs; expects to receive > it will pay; has positive value. - Bank is short these FRAs; expects to receive < it will pay; has negative value. (-$4.27 MM)

© Paul Koch 1-19 I.B. Valuation of Interest Rate Swaps d. (continued) iv. Whether the fwd rate for any given future payment date is > or < the fixed rate (k) in the SWAP, depends on slope of yield curve! - If yield curve slopes upward, then fwd rates  as maturity of fwd contract . - Since the sum of values of the fwd contracts = 0 when SWAP is entered, this means: fwd interest rate is < k for early payments, & fwd interest rate is > k for later payments; i.e., value of fwd contracts for early payments < 0, & value of fwd contracts for later payments > 0.  - If yield curve slopes downward, the reverse is true; i.e., value of fwd contracts for early payments > 0, & value of fwd contracts for later payments < 0.

© Paul Koch 1-20 II. Currency Swaps A. Mechanics of Currency SWAPS. 1. Example 1; Currency SWAP. Consider the following opportunities for A & B: $ £ . US Company A 8.0% 11.6% NOTE: rUK > rUS UK Company B 10.0% 12.0% since ∆PUK > ∆PUS Difference: 2.0% 0.4% . (US inflation lower) a. Comparative Advantage. Company B has lower credit rating. But risk premium is not same in both mkts: i. 2.0% greater in $; only 0.4% greater in £. (B is better known in UK.) ii. Difference represents a margin to be captured. iii. Can capture & share this margin with a SWAP. b. If A wants £ while B wants $, SWAP is in order. Principal amounts must be: i. Specified in both currencies; ii. Exchanged at beginning & end of SWAP; iii. Chosen so they are  in value at beginning. e.g., If £1 = $1.5, A may lend $15 million, while B lends £10 million. c. In reality, each company goes through a bank; Bank warehouses SWAPs.

© Paul Koch 1-21 II.A. Mechanics of Currency Swaps $ £ . US Company A 8.0% 11.6% Margin = 1.6% UK Company B 10.0% 12.0% [ 2.0% - 0.4% ] Difference: 2.0% 0.4% .SWAP: Bank takes .4%; Both A and B get .6% (A will pay 11% for £; B: 9.4% for $)d. Mechanics: Company A Bank Company B A borrows $15m @; <8.0%> -- -- B borrows £10m @; -- -- <12.0%> Through the bank, A lends $15m to B; B lends £10m to A: A gets 8% for $; 8.0% <8.0%> -- A pays 11% for £; <11.0%> 11.0% -- B pays 9.4% for $; -- 9.4% <9.4%> B gets 12% for £; -- <12.0%> 12.0% . Margin Captured on $: 0.0% *** 1.4% *** 0.6% Margin Captured on £: 0.6% *** <1.0%> *** 0.0% Net Margin Captured: 0.6% 0.4% 0.6% (Bank gets 0.4% if exch rates stable.) [11.6 - 11] [1.4 - 1] [10 - 9.4] _________________________________________________________________________ $8% $8% $9.4% <───── <────── <────── ──────> ──────> ─────> £11% £12 % £12% Company A Financial Institution Company B

© Paul Koch 1-22 II.A. Mechanics of Currency Swaps $ £ . US Company A 8.0% 11.6% Margin = 1.6% UK Company B 10.0% 12.0% [ 2.0% - 0.4% ] Difference: 2.0% 0.4% .SWAP: Bank takes .4%; Both A and B get .6% (A will pay 11% for £; B: 9.4% for $)d. Mechanics: Company A Bank Company B A borrows $15m @; <8.0%> -- -- B borrows £10m @; -- -- <12.0%> Through the bank, A lends $15m to B; B lends £10m to A: A gets 8% for $; 8.0% <8.0%> -- A pays 11% for £; <11.0%> 11.0% -- B pays 9.4% for $; -- 9.4% <9.4%> B gets 12% for £; -- <12.0%> 12.0% . Margin Captured on $: 0.0% 1.4% 0.6% Margin Captured on £: 0.6% <1.0%> 0.0% Net Margin Captured: 0.6% 0.4% 0.6% (Bank gets 0.4% if exch rates stable.) [11.6 - 11] [1.4 - 1] [10 - 9.4] _________________________________________________________________________How do you determine the numbers, 11.0% & 9.4%? Bank gets 0.4%; A & B get 0.6%.Thus, A will pay (11.6% - 0.6%) = (11.0%); B will pay (10.0% - 0.6%) = 9.4%.

© Paul Koch 1-23 II.A. Mechanics of Currency Swaps 2. Economic Rationale for Currency Swaps. a. Tendency for firms to get better rates in their own country. i. Better known, better info, easier for bank to assess risk. ii. Valid rationale, but expect less benefit for global firms. iii. As global markets become more integrated, would expect cost of capital to equalize across markets; would expect these benefits to decline. b. Tax advantages for firm in own country. c. Swaps are useful for sourcing funds in other countries, to match sources and uses of funds. d. Swaps are useful for hedging currency risk.

© Paul Koch 1-24 II.B. Valuation of Currency Swaps 1. Relationship of Currency SWAP Value to Bond Prices. a. Consider position of Company B in above SWAP. - borrows $ @ 9.4%; lends £ @ 12%. - short a $ bond @ 9.4%; long a £ bond @ 12%. b. Let VSWAP = $ value of SWAP to Company B (Company B is party paying US interest). Then VSWAP = S BF - BD; where BF = value of FC- denominated bond, in FC; BD = value of $ - denominated bond, in $; S = spot exchange rate ( $ / FC ). c. Thus, VSWAP can be determined from: term structure in US$ rates, FC rates, & spot exchange rate.

© Paul Koch 1-25 II.B. Valuation of Currency Swaps Again: a. Consider position of Company B in above SWAP. - borrows $ @ 9.4%; lends £ @ 12%. - short a $ bond @ 9.4%; long a £ bond @ 12%. Now: d. This example. Assume term structure is flat in both U.S. and U.K.: - rUS = 5% p.a. (continuous compounding); - rUK = 10% p.a. ( " " ); - B has entered Currency SWAP where it: receives 12.0% on £10 MM; .12 (£10) = £1.20 MM; pays 9.4% on $15 MM; .094($15) = $1.41 MM. - SWAP will last another 3 years; payments once / year; - Spot exchange rate: $1.50 = £1 [ S = 1.5 ($ / £) ]. Then: BD = 1.41 e-.05 + 1.41 e-.05x2 + 16.41 e-.05x3 = 1.341 + 1.276 + 14.124 = $16.74 MM and: BF = 1.20 e-.10 + 1.20 e-.10x2 + 11.20 e-.10x3 = 1.086 + 0.982 + 8.297 = £10.37 MM Thus: VSWAP = S BF - B D = (1.5 $ / £) x ( £ 10.37 ) - ( $16.74 ) = -$1.19 MM .

© Paul Koch 1-26 II.B. Valuation of Currency Swaps 2. Decomp . of Currency SWAP Value into Forward Contracts.Again: a. Consider position of Company B. - borrow $ @ 9.4%; lend £ @ 12%. - short a $-bond @ 9.4%; long a £-bond @ 12%. i. One payment / year; On each payment date, B will exchange: an inflow of £1.20MM (= 12% of £10MM) for an outflow of $1.41MM (= 9.4% of $15MM). iii. On final payment date, B will exchange: an inflow of £10MM for an outflow of $15MM. b. Each such exchange represents a forward contract. Let: ti = time of ith settlement date; ri = US$ discount rate for time ti; Fi = fwd exchange rate for time ti ($ / FC). The value of a long forward contract is the PV of the amount by which forward price exceeds delivery price: fi = (Fi - K) e -ri (ti) ; where Fi = S e (r-rf) ti .

© Paul Koch 1-27 II.B. Valuation of Currency Swaps c . Thus, the value to Company B of the forward contract (in $) for the exchange of interest payments at time ti is: Value (exchange of interest at ti ) = (1.2 Fi - 1.41) e -ri ti ; 1  i  n. where F1 = S e (r-rf) t1 = (1.5) e (.05 - .10) 1 = 1.4268 $ / £ ; F2 = = (1.5) e (.05 - .10) 2 = 1.3573 $ / £ ; F3 = = (1.5) e (.05 - .10) 3 = 1.2911 $ / £ . so f1 = [ (1.2 x 1.4268) - 1.41 ] e -.05 x 1 = $.29 ; ( > 0 )(receiving 12% on £10MM) f2 = [ (1.2 x 1.3573) - 1.41 ] e -.05 x 2 = $.20 ; ( > 0 )( paying 9.4% on $15MM) f3 = [ (1.2 x 1.2911) - 1.41 ] e -.05 x 3 = $.12 . ( > 0 ) d. Likewise, the value to Company B of the forward contract for the exchange of principal at time tn is: Value (exchange of principal at tn ) = ( 10 F n - 15 ) e -r n tn . = [ ( 10 x 1.2911) - 15 ] e -. 05 x 3 = -$ 1.80 . ( < 0 ) e . Thus , V SWAP = $. 29 + $.20 + $.12 - $1.80 = -$1.19 MM . (same)

© Paul Koch 1-28 II.B. Valuation of Currency Swaps 3. Another Example. Assume: - Term structure is flat in both Japan and U.S.; rJ = 4% p.a. ; rUS = 9% p.a. (continuous compounding); - NOTE: Interest differential = ( rUS - rJ ) = 5%. (US inflation higher) - Bank has entered Currency SWAP where it : receives 5% p.a. in ¥ ; pays 8% p.a. in $ (once / yr); - Principal amounts: $10 million & 1,200 mllion ¥ ; - SWAP will last another 3 years ; - Current exchange rate: 110¥ = $1 [S = 1 / 110 = .009091 ($ / ¥)]. a. Valuing SWAP as portfolio of US$ & Yen Bonds.(.08)*( 10MM $) Then: BD = .8 e -.09 + .8 e -.09 x 2 + 10.8 e -.09 x 3 = $9.64 MM ;(.05)*(1200MM ¥) and: BF = 60 e -.04 + 60 e -.04 x 2 + 1260 e -.04 x 3 = 1,230.55 MM ¥. Thus: VSWAP = S BF - BD = ( 1$ / 110¥ ) x ( 1,230.55 MM ¥ ) - ( $9.64 MM ) = $1.55 MM.

© Paul Koch 1-29 II.B. Valuation of Currency Swaps b. Valuing SWAP as portfolio of forward contracts. NOTE: The exchange of interest involves: Receiving ¥60 million (=.05 x 1,200MM ¥); Paying $.8 million (=.08 x $10MM). Forward foreign exchange rates are computed as follows: Fi = S e (r - rf) t ($ / FC), where (r - rf) = .09 -.04 = 5%. Thus, F1 = .009091 e.05 x1 = .0096 ($/ ¥); F2 = .009091 e.05 x2 = .0100 ($/ ¥); F3 = .009091 e.05 x3 = .0106 ($/ ¥). Value of forward contracts for exchange of interest:( receiving 5% on ¥ ) f1 = ( 60 x .0096 - .8 ) e -.09 x 1 = -$.21 MM ; ( < 0 )( paying 8% on $ ) f2 = ( 60 x .0100 - .8 ) e -.09 x 2 = -$.16 MM ; ( < 0 ) f3 = ( 60 x .0106 - .8 ) e -.09 x 3 = -$.13 MM . ( < 0 ) Value of forward contracts for exchange of principal: ( 1,200 x .0106 - 10 ) e -.09 x 3 = $ 2.04 MM . ( > 0 ) Thus, V SWAP = 2.04 - .21 - .16 - .13 = $1.54 MM . (same)

© Paul Koch 1-30 II.B. Valuation of Currency Swaps 4. The sign and magnitude of ( rUS - rFC ) is important. Assume principal amounts in the two currencies are equivalent at start. At this time, total value of Currency SWAP = 0. However, (like interest rate SWAPs), this doesn't mean each forward contract has zero value. When ( rUS - rFC ) is large: ( when $ inflation > ¥ inflation ) a. For the payer of low-interest currency , the early forward contracts (exchanging int.) have values > 0, & the final forward contract (exchanging princ.) has value < 0. Thus, there is tendency for SWAP to have value < 0 most of its life. * See first example with UK Sterling currency SWAP; Company B has entered Currency SWAP where it receives 12% in £, & pays 9.4% in $ (once / yr); forward contracts for exchange of interest > 0; forward contract for exchange of principal < 0. (VSWAP = -$1.19)

© Paul Koch 1-31 II.B. Valuation of Currency Swaps b. For the payer of high-interest currency, the early forward contracts (exchanging int.) have values < 0, & the final forward contract (exchanging princ.) has value > 0. Thus, there is tendency for SWAP to have value > 0 most of its life. * See second example with Yen currency SWAP; Bank has entered Currency SWAP where it receives 5% p.a. in ¥, & pays 8% p.a. in $ (once / yr); forward contracts for exchange of interest < 0; forward contract for exchange of principal > 0. (VSWAP = +$1.54) c. Thus, when bank enters a currency SWAP, it has a good idea whether SWAP will have value > 0 or < 0 as time passes, over its life. THIS IS IMPORTANT when credit risk is considered! Then you need to worry about which party owes money (SWAP < 0).

© Paul Koch 1-32 II.C. Other Swaps 1. Interest Rate SWAPs can be based on other floating rates. a. 6-month LIBOR (most common); 3-month LIBOR; 1-month Prime; 1-month commercial paper rate; T. Bill rate; Tax-exempt (muni-) rate. b. Choice depends on exposure. 2. Can construct SWAP to exchange one floating rate for another. a. e.g., LIBOR for Prime. b. Hedge exposure from assets & liabilities subject to diff. floating rates. 3. Principal amount can be varied throughout life of SWAP. a. Amortizing SWAP -- principal is reduced according to amortizing schedule on a loan. b. Step-Up SWAP -- principal is increased according to draw-downs on a loan agreement.

© Paul Koch 1-33 II.C. Other Swaps 4. Deferred SWAP or Forward SWAP -- Parties do not begin to exchange interest payments until later. Can be created from two other SWAPS; e.g., Buy 6-year SWAP, Sell similar 3-year SWAP; (first 3 years cancel) effect is a deferred SWAP that goes from year 4 to year 6. 5. Can agree to exchange fixed interest rate in one currency for a floating interest rate in another currency. a. Combination of 'vanilla' Interest Rate SWAP & Currency SWAP. Can value with above procedures. (See problems at end of chapter.) 6. Can have embedded options to extend or shorten the life. a. Extendable SWAP -- one party has option to extend. b. Puttable SWAP -- one party has option to terminate.

© Paul Koch 1-34 II.C. Other Swaps 7. Options on SWAPs are traded on CBOT -- SWAPtions . a. Option on an Interest Rate SWAP is an option to exchange a fixed-rate bond for a floating-rate bond. i. Since the floating rate bond will remain close to its face value as interest rates change, the swaption can be considered as an option on the value of the fixed-rate bond. ii. See Hull, Ch. 18, 19. 8. "Constant Maturity" SWAP agreements. a. CMS SWAP -- exchange a LIBOR rate for a SWAP rate. [e.g., exchange 6-month LIBOR for 10-year SWAP rate every 6 months for next 5 years.] b. CMT SWAP -- exchange LIBOR for constant maturity Treasury rate. 

© Paul Koch 1-35 II.C. Other Swaps 9. Indexed Principal SWAP -- SWAP where principal reduces in a way that depends on interest rates. a. The lower the interest rate, the greater the reduction in principal. 10. Differential SWAP or Diff SWAP -- floating-interest rate in domestic currency is exchanged for floating-interest rate in a foreign currency, with both interest rates applied to same domestic principal. 11. SWAPs on commodities -- a. A company might consume  100,000 barrels of oil / year. Could agree to pay $8 million each year for 10 years, in exchange for $100,000 x S, where S = spot price of oil. i. Would effectively lock in cost of $80 / barrel. b. An oil producer might agree to opposite SWAP; i. Would effectively lock in revenue at $80 / barrel.

© Paul Koch 1-36 III.D. Credit Risk 1. SWAPs are OTC arrangements that have credit risk. 2. Consider position of bank that has offsetting SWAPs to 2 parties. a. If neither party defaults, bank is fully hedged. b. However, if one party defaults, Bank must honor the other contract. c. Suppose, after the start, contract with party A has negative value. i. If party B defaults, bank would lose the positive value in this contract. Bank would then be unhedged. ii. NOTE: Bank only has credit risk exposure when other party owes bank. iii. When bank owes other party (value of SWAP <0), & party A defaults, Bank should be able to realize a gain, since its liability is gone. However, in practice, party A will sell the SWAP or rearrange its affairs in some way, so A’s positive value is not lost. iv. Hence, if a counterparty goes bankrupt, must assume: there will be a loss to the bank if value to the bank > 0 (VSWAP > 0); there will be no gain to the bank if value to the bank < 0 (VSWAP < 0).

© Paul Koch 1-37 III.C. Credit Risk 3. Sometimes bank can predict which of two offsetting contracts will likely have positive or negative value. a. e.g., with Currency SWAPs, for the payer of low-interest currency, there is tendency for SWAP value to be < 0 most of its life; this would translate into a SWAP value > 0 for the bank. (See B.2.c.iii on previous page.) i. Thus, bank should be more concerned with creditworthiness of this counterparty! 4. In general, expected loss from Currency SWAP is > expected loss from Interest Rate SWAP. a. With Currency SWAP, principal amounts are also exchanged. (These have the greatest value, as maturity approaches.) b. For both Currency SWAPs and Interest Rate SWAPs, expected loss from default is much less than on a regular loan.

© Paul Koch 1-38 III.C. Credit Risk 5 . Distinguish between Credit Risk and Market Risk. Credit Risk arises from possible default of counterparty. Market Risk arises from possibility that market variables such as interest rates or exchange rates may change, so that value of contract could decline. Market Risk can be hedged by entering offsetting contracts. Credit Risk cannot be hedged in this way, but can be managed using Credit Derivatives. More later (see Glossary).