Interest Rate Forwards and Swaps

Interest Rate Forwards and Swaps 1 Outline PART ONE Chapter 1: interest rate forward contracts and their pricing and mechanics 2 Outline PART TWO Chapter 2: basic and customized swaps and their pricing and mechanics 3

Outline PART THREE Chapter 3: Cross-currency swaps Chapter 4: Swap revaluation and trading strategies 4 Outline PART FOUR Chapter 5: Quiz 5 Chapter 1: Forward Rate Agreements Use of interest rate forwards Determination of forward rate Mechanics of FRA Arbitraging incorrect FRA pricing 6

Quoting a Forward Rate Assume your bank on Jan 1, 2009 posts zero-coupon rates from 3 months to 5 years equal to the $ Libor rates 3-mo Libor is 3%; 6-mo Libor is 3.5% Depositor wants to know rate you would offer on 3-month deposit starting in 3 months, i.e. on April 1, 2009 Usually denoted as 3x6 forward rate agreement, or FRA 7 Calculating FRA rate Done via no-arbitrage exercise First you borrow for 3 months and invest for 6 months No-arbitrage pricing exercise 3.50% 0 3 MO 6 MO 3.00%? 8 Calculating FRA rate Investment grows to (1+3.5%x181/360) = 1.0176 after 6 months Borrowing grows to (1 + 3% x 90/360) = 1.0075 after 3 months To break even you would need your borrowing in the next 3 months (91 days) to grow by 1.0176/1.0075 = 1.0100 This implies a borrowing rate for the 3x6 period of 3.9648%, calculated as (1.0100-1) x (360/91) Can think of FRA rate of 3.9648% as the rate that makes an investor who invests for 6 months achieve same return as one who invests for 3 months and then again for 3 more months at this forward rate 9

General FRA Equation RL DaysL RS Dayss F ( 1 ) ( 1 ) ( 1 360 360 360 F RL DaysL [ 1 ] 360 360 { 1} ( ) RS DaysS DaysF [ 1 ] 360 Where RL = spot Libor for the long period RS = spot Libor for the short period F = forward Libor DAYSL = number of days in the long period DAYSS = number of days in the short period DAYSF = number of days covered by the forward period or Days F ) 10 FRA Mechanics (1) Customer does not actually need to place funds with bank quoting the FRA. Customer places funds anywhere she likes, and collects or makes payment under the FRA which brings her yield back to the FRA quoted rate all-in Example: if Libor resets 1% below FRA rate, customer collects annualized 1% from bank, and vice-versa if Libor resets 1% above FRA rate 11 FRA Settlement Diagram Customer 3.9648% Libor Bank Libor 12

FRA Settlement Example Libor resets at 3.75%, for period in question Client deposit is for $100 (so this is FRA notional amount) Interest earned on deposit is $100 x (3.75% x 91/360) = $0.9479 Under the FRA, bank pays the depositor $100 x (3.9648% - 3.75%) x 91/360 = $0.0543 Sum of these two amounts gives customer $1.0022, which is exactly equivalent to an annualized 3.9648% for a 91 day period 13 FRA Mechanics (2) Amount of net settlement will be known as soon as Libor setting takes place, at the beginning of the period in question. Question is whether to wait until end of this period to pay it, or pay some discounted amount upfront. FRA convention is to settle net payment as soon as it is possible to calculate it; this reduces counterparty and operational risk in the system; net payment due is discounted at current Libor, known as the settlement rate, i.e. $0.0543/(1 + 3.75% x 91/360) = $0.0538 We can therefore write the formula for a FRA's net settlement amount as follows: Notional x (Contract rate - Settlement Rate) x (Days/360) (1 + Settlement Rate x Days/360) 14 3-month Libor FRA Table FRA 3x6 6x9 9x12 Period Covered starts in 3 months, ends in 6 months starts in 6 months, ends in 9 months starts in 9 months, ends in 12 months 21x24 starts in 21 months, ends in 24 months 15

6-month Libor FRA Table FRA 6x12 12x18 18x24 Period Covered starts in 6 months, ends in 12 months starts in 12 months, ends in 18 months starts in 18 months, ends in 24 months 54x60 starts in 54 months, ends in 60 months 16 Pricing 6x12 FRA We know from before that 6-mo. Libor is 3.50%; 12-mo. Libor is 4.00% As before, we can set (1+ 4% x 365/360) = (1 + 3.5% x 181/360) x (1 + F x 184/360); This leads to 6x12 FRA rate = 4.41% You should confirm this on Worksheet FRA 17 FRA Arbitrage Assume dealer is quoting 3x6 FRA at 4.50% You borrow at 3.50% for 6 months, invest at 3% for 3 months and lock in 4.5% investment rate for next 3 months; at maturity your profit per dollar is $0.00136, calculated as (1 + 3% x 90/360) x (1 + 4.50% x 91/360) (1 + 3.5% x 181/360) This may seem small, but remember you can do this on large notional amounts. On $100MM, you would earn $136,309 Conversely if dealer quotes FRA at 3.6%, you invert the trade: lend for 6 months at 3.5%, borrow for 3 months at 3%, and lock in the 3.6% rate at which you borrow again in 3 months time for 3 more months; at maturity your profit per dollar is $0.00093. You can check this calculation. 18

FRA Arbitrage with Bid-Offer Bank is quoting the following rates: 3-months 6-months 3x6 FRA Bid 2.9% 3.4% 4.00% Offer 3.1% 3.6% 4.02% Bank is quoting a FRA rate that appears higher than the correct one, while quoting 3- and 6-month interest rates that straddle the rates we had seen before Again, you try to arbitrage the unusually high FRA rate by implementing the first version of the arbitrage we showed previously: you borrow for 6 months, lend for 3 months, and lock in the FRA rate to lend again for 3 months in 3 months time 19 FRA Arbitrage with Bid-Offer Given the bid offer spread, the outcome of this trade per dollar is (1 + 2.9% x 90/360) x (1 + 4% x 91/360) (1 + 3.6% x 181/360), which this time comes to a loss of $0.00067, or $66,558 if the notional is $100MM. If conversely you tried to lend for 6 months, borrow for 3 months, and lock in the FRA rate to borrow again for 3 months in 3 months time, the outcome per dollar would be (1 + 3.4% x 181/360) (1 + 3.1% x 90/360) x (1 + 4.02% x 91/360), which again generates a loss, this time of $0.00090 per dollar, or $89,597 on a notional of $100MM Apparent error in the FRA rate was not large enough to enable you to overcome the cost disadvantages of paying the bid-offer spread on 3 separate instruments 20 Final Remarks on Arbitrage No arbitrage is ever completely risk-free Main residual risks of arbitrage we illustrated are: Risk of default by the entity or entities to whom you lend the money Risk of default by the FRA counterparty when the FRA is in your favor at settlement Risk that the settlement payment under the FRA, since it is discounted and paid upfront, cannot be reinvested for the forward period at a rate at least as high as the one used for the discounting 21

Interest Rate Forwards and Swaps (Part II) 22 Chapter 2: Interest Rate Swaps IRS construction and pricing Basic interest swap pricing and use for hedging interest rate risk Use and pricing of customized swaps for more complex situations 23 Interest Rate Exposure (1) Assume on Jan 1, 2009 a borrower has borrowed $100MM for 3 years at 3-mo. Libor flat. With Libor at 3%, first interest payment will be $0.75MM. Borrower has exposure to Libor for all 11 subsequent interest periods. A rising Libor will increase interest expense and reduce reported income. 24

Interest Rate Exposure (2) Assume FRA rates you are quoting are as shown in this table: Period Number Contract Designation Period Start Date Number of Days in period Rate for client 1 Spot Jan 1,2009 90 3.00% 2 3x6 April 1, 2009 91 4.00%* 3 6x9 July 1, 2009 92 4.50% 4 9x12 Oct 1,2009 92 4.75% 5 12x15 Jan 1, 2010 90 5.00% 6 15x18 Apr 1, 2010 91 5.25% 7 18x21 July 1,2010 92 5.50% 8 21x24 Oct 1,2010 92 5.75% 9 24x27 Jan 1,2011 90 6.00% 10 27x30 Apr 1, 2011 91 6.25% 11 30x33 July 1,2011 92 6.50% 12 33x36 Oct 1,2011 92 6.75% */ This 4% rate for the 3x6 FRA is our old 3.9684%, adjusted to reflect the bank s bid-offer. Similarly, all other rates include the bank s profit margin 25 Interest Rate Exposure (3) Borrower could hedge each reset using a series of FRAs, but would have unequal and rising interest payments in each quarter. What the client needs is a smoother pattern of cash outflows this is where the interest rate swap comes in, as described in Worksheet Simplified IRS Pricing Swap fixed leg is single rate that makes all cash flows borrower pays have a PV equal to PV of cash flows he would have paid under series of FRAs 26 Simplified IRS Pricing (1) Period Libor/FRA Net payment 1 3.00% $750,000 2 4.00% $1,000,000 3 4.50% $1,125,000 4 4.75% $1,187,500 5 5.00% $1,250,000 6 5.25% $1,312,500 7 5.50% $1,375,000 8 5.75% $1,437,500 9 6.00% $1,500,000 10 6.25% $1,562,500 11 6.50% $1,625,000 12 6.75% $1,687,500 Notional 100,000,000 27

Period Libor/ FRA Net payment DFs PVNP Adjusted net payment PVANP 1 3.00% 750,000 0.9926 744,417 1,250,000 1,240,695 2 4.00% 1,000,000 0.9827 982,729 1,250,000 1,228,411 3 4.50% 1,125,000 0.9718 1,093,270 1,250,000 1,214,745 4 4.75% 1,187,500 0.9604 1,140,465 1,250,000 1,200,489 5 5.00% 1,250,000 0.9485 1,185,668 1,250,000 1,185,668 6 5.25% 1,312,500 0.9362 1,228,823 1,250,000 1,170,308 7 5.50% 1,375,000 0.9235 1,269,878 1,250,000 1,154,434 8 5.75% 1,437,500 0.9105 1,308,786 1,250,000 1,138,075 9 6.00% 1,500,000 0.8970 1,345,507 1,250,000 1,121,256 10 6.25% 1,562,500 0.8832 1,380,007 1,250,000 1,104,006 11 6.50% 1,625,000 0.8691 1,412,258 1,250,000 1,086,352 12 6.75% 1,687,500 0.8547 1,442,238 1,250,000 1,068,324 Totals 14,534,045 13,912,762 Notional 100,000,000 Fixed rate 5.0000% Simplified IRS Pricing (2) (Difference) 621,283 28 3-year Swap 5.22% Borrower Bank Libor Libor But: (i) all settlements under IRS occur at end of period; and (ii) there are 12 settlements here, versus only one under FRA 29 Approximate Swap Formula Previous approach ignored exact daycount to keep matters simple Worksheet IRS Correct Pricing shows correct calculation with appropriate daycounts. The correct swap rate turns out to be 5.23%, only one basis point away from our approximation Under the simplified approach, we effectively solved the following equation: 12 F N DF 12 Libori N DFi i i 1 4 i 1 4 30

IRS Correct Pricing Period Libor/ FRA Net payment DFs PVNP Adjusted net payment PVANP 1 3.00% 750,000 0.9926 744,417 1,250,000 1,240,695 2 4.00% 1,011,111 0.9826 993,538 1,263,889 1,241,923 3 4.50% 1,150,000 0.9714 1,117,166 1,277,778 1,241,296 4 4.75% 1,213,889 0.9598 1,165,088 1,277,778 1,226,408 5 5.00% 1,250,000 0.9479 1,184,936 1,250,000 1,184,936 6 5.25% 1,327,083 0.9355 1,241,531 1,263,889 1,182,410 7 5.50% 1,405,556 0.9226 1,296,718 1,277,778 1,178,834 8 5.75% 1,469,444 0.9092 1,336,027 1,277,778 1,161,763 9 6.00% 1,500,000 0.8958 1,343,654 1,250,000 1,119,712 10 6.25% 1,579,861 0.8818 1,393,181 1,263,889 1,114,545 11 6.50% 1,661,111 0.8674 1,440,895 1,277,778 1,108,381 12 6.75% 1,725,000 0.8527 1,470,940 1,277,778 1,089,586 Totals 14,728,091 14,090,488 Notional 100,000,000 Fixed rate 5.0000% (Difference) 637,604 31 Approximate Swap Formula So F 12 i 1 12 Libori DFi i 1 DFi Equation explains why swap rate is often described as the time-weighted average of the relevant Libors 32 Customized Swaps We will customize swaps with the following features: A profit margin for the dealer A notional that changes over time A forward start date Different settlement frequencies between the floating and fixed leg, and/or also different daycounts A feature that fixes the Libor reset under the swap at the end rather than the beginning of the period in question ( Libor-in-arrears ). Each of these except the first is examined via its own worksheet 33

Amortizing Swap (1) Now assume loan amortizes in last year in four equal installments Swap needs to amortize in the same way, otherwise borrower will end up overhedged in year 3 Careful with Amortizations: first amortization here occurs at end of period 9, so notional at beginning of period 9 is still $100 MM. Similarly notional at beginning of period 12 is $25 MM, not zero. 34 Amortizing Swap Notional (at beg of period) Adjusted net payment Libor/ FRA Net payment PVNP PVANP 3.00% 100,000,000 750,000 744,417 1,250,000 1,240,695 4.00% 100,000,000 1,011,111 993,538 1,263,889 1,241,923 4.50% 100,000,000 1,150,000 1,117,166 1,277,778 1,241,296 4.75% 100,000,000 1,213,889 1,165,088 1,277,778 1,226,408 5.00% 100,000,000 1,250,000 1,184,936 1,250,000 1,184,936 5.25% 100,000,000 1,327,083 1,241,531 1,263,889 1,182,410 5.50% 100,000,000 1,405,556 1,296,718 1,277,778 1,178,834 5.75% 100,000,000 1,469,444 1,336,027 1,277,778 1,161,763 6.00% 100,000,000 1,500,000 1,343,654 1,250,000 1,119,712 6.25% 75,000,000 1,184,896 1,044,886 947,917 835,908 6.50% 50,000,000 830,556 720,448 638,889 554,190 6.75% 25,000,000 431,250 367,735 319,444 272,396 Totals 12,556,143 12,440,472 Fixed Rate 5.00% (Difference) 115,671 35 Amortizing Swap (2) Reducing notionals in year 3 puts lower weights on higher Libors when curve is upward sloping so swap rate decreases. Swap rate would increase if curve is inverted Swap whose notional increases over time ( accreting swap ) has a higher rate than a bullet swap in an upward sloping curve environment, and a lower rate in an inverted curve environment. Such a swap is often used to hedge a construction or project finance loan These four results are summarized in this table: 36

Varrying Notional Libor spot/forward curve Upward Downward Amortizing notional lower higher Accreting notional higher lower 37 Forward-starting Swap Assume commitment for a 2-year loan has been signed but disbursement will not take place until one year from now Forward-starting swap is priced simply by setting the notional to zero until the start date. In upward sloping curve environment, this will increase swap rate versus bullet swap; and vice versa in inverted curve environment 38 Forward-Starting Swap Libor/F RA Notional (at beg of period) Net payment DFs PVNP Adjusted net payment PVANP 3.00% - - 0.9926 - - - 4.00% - - 0.9826 - - - 4.50% - - 0.9714 - - - 4.75% - - 0.9598 - - - 5.00% 100,000,000 1,250,000 0.9479 1,184,936 1,464,399 1,388,175 5.25% 100,000,000 1,327,083 0.9355 1,241,531 1,480,670 1,385,216 5.50% 100,000,000 1,405,556 0.9226 1,296,718 1,496,941 1,381,027 5.75% 100,000,000 1,469,444 0.9092 1,336,027 1,496,941 1,361,028 6.00% 100,000,000 1,500,000 0.8958 1,343,654 1,464,399 1,311,764 6.25% 100,000,000 1,579,861 0.8818 1,393,181 1,480,670 1,305,711 6.50% 100,000,000 1,661,111 0.8674 1,440,895 1,496,941 1,298,490 6.75% 100,000,000 1,725,000 0.8527 1,470,940 1,496,941 1,276,471 Totals 10,707,882 10,707,882 Fixed Rate 5.86% (Difference) - 39

Frequencies and Daycounts Often it is necessary to structure swap with each leg having a different frequency and/or daycount For example borrower may want to swap 3-mo Libor-based loan into semi-annual fixed with 30/360 daycount to netter compare all-in cost to an outstanding bond 40 Frequency Libor/F Notional (at Adjusted net RA beg of period) Net payment PVNP payment PVANP 3.00% 100,000,000 750,000 744,417-4.00% 100,000,000 1,011,111 993,538 2,667,304 2,620,948 4.50% 100,000,000 1,150,000 1,117,166-4.75% 100,000,000 1,213,889 1,165,088 2,667,304 2,560,073 5.00% 100,000,000 1,250,000 1,184,936-5.25% 100,000,000 1,327,083 1,241,531 2,667,304 2,495,352 5.50% 100,000,000 1,405,556 1,296,718-5.75% 100,000,000 1,469,444 1,336,027 2,667,304 2,425,128 6.00% 100,000,000 1,500,000 1,343,654-6.25% 100,000,000 1,579,861 1,393,181 2,667,304 2,352,129 6.50% 100,000,000 1,661,111 1,440,895-6.75% 100,000,000 1,725,000 1,470,940 2,667,304 2,274,461 Totals 14,728,091 14,728,091 Fixed Rate 5.33% (Difference) - 5.23% qtly 5.33% Effective 5.26% s.a. 5.34% Daycount adjustment 41 Libor-in-arrears (1) Assume borrower is swapping from 5.53% fixed to floating to benefit from anticipated decline in rates. 5.23% Borrower Bank Libor 5.53% 42

Libor-in-arrears (2) Regular swap puts him into synthetic floating at Libor + 30bps Borrower would like spread over Libor reduced or eliminated Often this is done by asking borrower to sell some optionally to the bank, e.g. option to extend swap, or to allow bank to increase notional on certain dates; but some borrowers do not want to sell optionality LIA swap works exactly like a regular swap in all respects, except that fixing of Libor for each period takes place two business days before the end of that period, instead of the beginning 43 Libor-in-arrears (3) If curve is upward sloping, this will increase the rate on the fixed leg of the swap Borrower saves money under LIA if Libor forwards turn out to have overstated outcomes Accurate pricing needs a convexity adjustment, which is too complicated to discuss here. Its impact is small but increases with long-dated swaps and in volatile rate environments 44 LIA Libor/FR A Notional (at beg of period) Net payment PVNP Adjusted net payment PVANP 3.00% 100,000,000 1,000,000 992,556 1,391,082 1,380,726 4.00% 100,000,000 1,137,500 1,117,731 1,406,538 1,382,093 4.50% 100,000,000 1,213,889 1,179,231 1,421,995 1,381,395 4.75% 100,000,000 1,277,778 1,226,408 1,421,995 1,364,827 5.00% 100,000,000 1,312,500 1,244,182 1,391,082 1,318,674 5.25% 100,000,000 1,390,278 1,300,651 1,406,538 1,315,863 5.50% 100,000,000 1,469,444 1,355,660 1,421,995 1,311,884 5.75% 100,000,000 1,533,333 1,394,116 1,421,995 1,292,886 6.00% 100,000,000 1,562,500 1,399,639 1,391,082 1,246,088 6.25% 100,000,000 1,643,056 1,448,908 1,406,538 1,240,338 6.50% 100,000,000 1,725,000 1,496,314 1,421,995 1,233,479 6.75% 100,000,000 1,788,889 1,525,420 1,421,995 1,212,562 7.00% - - - - Totals 15,680,816 15,680,816 Fixed Rate 5.56% (Difference) - 45

IRS Replication (1) You have been asked to quote on a 5-year swap in which you will pay 6-mo Libor and receive fixed semiannually You issue a 5-year bond and invest the proceeds in a 5-year floating rate note that pays 6-month Libor flat Note that net initial proceeds are zero, and repayment of your investment s principal in 5 years time would repay the bond you have issued, so again there will be no net cash flow for the principal 46 IRS Replication (2) To break even under the swap, you would set the fixed leg equal to the fixed coupon you are paying on your bond In fact any IRS can be viewed as two bond positions, one long and one short: if you are receiving fixed under the swap and paying floating, it is as if you have issued a Libor-based liability and invested the proceeds in a fixed rate bond; and vice-versa 6-month Libor 6-month Libor Dealer Coupons on fixed rate bond Fixed 47 Basis Swap (1) Now assume your bank wants to borrow 100MM for 5 years but has no access to the funding market. The /$ spot rate is 1.50 Your bank issues a $150MM 5-year FRN which pays 6-month $ Libor and converts the proceeds into 100MM in the spot FX market Assume also another bank, in the UK, wants to borrow $150MM for 5 years but has no access to the $ funding market. This bank issues a 100MM 5-year FRN which pays 6-month Libor and converts the proceeds into USD 150MM Now the two banks enter into the swap described in this diagram: 48

Basis Swap (2) 100 MM at maturity $150 MM bond at $ Libor Your Bank Libor on 100 MM semiannually $ Libor on $150 MM semi-annually UK Bank 100 MM bond at Libor $150 MM at maturity This kind of swap is called a Libor basis swap, and plays a critical role in funding operations of large banks, and also in pricing cross-currency swaps for customers Note very carefully that unlike an IRS, a basis swap includes an exchange of principal at maturity, without which neither party would have been properly hedged in our example 49 Libor Basis Swap Screen GBP/USD 6 month Libor Basis Swap Term Bid Offer 2-year 1 3 3-year 1 3 5-year 2 4 7-year 3 6 10-year 4 8 50 Interest Rate Forwards and Swaps (Part III) 51

Outline PART THREE Chapter 3: Cross-currency swaps Chapter 4: Swap revaluation and trading strategies 52 Chapter 3: Cross-currency swaps Pricing a basic cross-currency swap Alternative pricing methods 53 Cross-Currency Swap Suppose you have issued a $150MM 3-year 5.5% bond but really needed to fund a UK expansion You convert the $ proceeds of your issuance into, and seek to hedge your future $ coupon and principal obligations into by buying $ and selling under a series of derivative contracts Outright FX forwards would create liability in with uneven cash outflows since forward rates for successive dates will not be equal You would like your bank to make all payments required in $ under the bond, in return for you making to the bank, usually on the same dates, payments in whose profile is identical to that of a bond issue 54

Cross-Currency Swap $ Libor PV of Curve DFs $ Pmts Pmts Days Libor Curve DFs Pmts PV of Pmts 3.00% 0.99 2.06 2.05 91 5.00% 0.99 1.00 0.99 4.00% 0.98 2.06 2.03 90 4.90% 0.98 1.00 0.98 4.50% 0.97 2.06 2.00 92 4.80% 0.96 1.00 0.96 4.75% 0.96 2.06 1.98 92 4.70% 0.95 1.00 0.95 5.00% 0.95 2.06 1.95 91 4.60% 0.94 1.00 0.94 5.25% 0.94 2.06 1.93 90 4.50% 0.93 1.00 0.93 5.50% 0.92 2.06 1.90 92 4.40% 0.92 1.00 0.92 5.75% 0.91 2.06 1.88 92 4.30% 0.91 1.00 0.91 6.00% 0.90 2.06 1.85 91 4.20% 0.90 1.00 0.90 6.25% 0.88 2.06 1.82 91 4.10% 0.89 1.00 0.89 6.50% 0.87 2.06 1.79 92 4.00% 0.88 1.00 0.88 6.75% 0.85 152.06 129.65 92 3.90% 0.88 101.00 88.44 Totals 150.82 98.70 148.05 $ Fixed 5.50% Fixed 4.00% $ Notional 150 Notional 100 PV Diff 2.77 FX Spot 1.5 55 CCS Diagram 100 MM at maturity $150 MM bond at $5.5% Your Company Fixed% on 100 MM semi-annually $5.5% on $150 MM semi-annually Bank $150 MM at maturity 56 Basis Point Conversion You cannot simply add or subtract equal numbers of basis points to each leg after you have finished pricing the swap Suppose in other words that after you priced the previous swap, it turned out that the $ bond could not be issued at 5.50%, but rather at 5.75% Client would logically ask you now to reset the $ leg of the swap to this level, and would understand that the leg also would need to be revised upward An identical adjustment of 25 bps to the leg would be incorrect, because PVs must be equal, and these PVs are obtained by discounting off two different curves 57

Constructing/hedging CCS Most typically this is done using 3 swaps, two IRS and one Libor basis swap, as shown in this diagram: $Fixed on $150 MM $150 MM at maturity $L on $150 MM $ Bond at 5.50% Company $Fixed on $150 MM Fixed on 100 MM 100 MM at maturity Bank L+ x bps on 100 MM $L on $150 MM 100 MM at maturity $150 MM at maturity L on 100 MM Fixed on 100 MM 58 CCS Variations (1) Note that you can also keep one of the legs floating, by eliminating one of the interest rate swaps 59 CCS Variations (2) $ Bond at 5.50% $150 MM at maturity $Fixed on $150 MM $Fixed on $150 MM L+x bps on 100 MM Company Fixed on 100 MM Bank 100 MM at maturity $150 MM at maturity 100 MM at maturity 60

CCS Variations (3) Also frequencies and daycount for the two legs do not have to match so we could have the $ leg based on quarterly under a 30/360 day convention, while the leg could be based on 6-month Libor and under an act/365 day basis 61 Chapter 4: Swap Revaluation & Trading Strategies IRS factor sensitivities Effect of parallel shift and pivot on swap value Use of IRS by traders 62 Sensitivities Period Original Curve New Curve Net payment DFs PVNP Adjusted net payment PVANP 1 4.00% 4.00% 1,000,000 0.9901 990,099 1,334,067 1,320,858 2 4.25% 4.25% 1,074,306 0.9796 1,052,363 1,348,890 1,321,339 3 4.50% 4.50% 1,150,000 0.9684 1,113,704 1,363,712 1,320,671 4 4.75% 4.75% 1,213,889 0.9568 1,161,477 1,363,712 1,304,832 5 5.00% 5.00% 1,250,000 0.9450 1,181,264 1,334,067 1,260,707 6 5.25% 5.25% 1,327,083 0.9326 1,237,683 1,348,890 1,258,020 7 5.50% 5.50% 1,405,556 0.9197 1,292,699 1,363,712 1,254,216 8 5.75% 5.75% 1,469,444 0.9064 1,331,887 1,363,712 1,236,053 9 6.00% 6.00% 1,500,000 0.8930 1,339,490 1,334,067 1,191,313 10 6.25% 6.25% 1,579,861 0.8791 1,388,863 1,348,890 1,185,815 11 6.50% 6.50% 1,661,111 0.8647 1,436,430 1,363,712 1,179,257 12 6.75% 6.75% 1,725,000 0.8501 1,466,382 1,363,712 1,159,260 Totals 14,992,343 14,992,343 Notional Fixed rate 100,000,000 5.34% Shift 0.00% Pivot 0.00% (Difference) - 63

Libor Curve Libors 0.07 0.065 0.06 0.055 0.05 0.045 0.04 0.035 0.03 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Period 64 Sensitivities (cont) Period Original Curve New Curve Net payment DFs PVNP Adjusted net payment PVANP 1 4.00% 5.02% 1,255,000 0.9876 1,239,445 1,334,067 1,317,532 2 4.25% 5.27% 1,332,139 0.9746 1,298,332 1,348,890 1,314,658 3 4.50% 5.52% 1,410,667 0.9611 1,355,742 1,363,712 1,310,616 4 4.75% 5.77% 1,474,556 0.9471 1,396,551 1,363,712 1,291,571 5 5.00% 6.02% 1,505,000 0.9331 1,404,250 1,334,067 1,244,760 6 5.25% 6.27% 1,584,917 0.9185 1,455,745 1,348,890 1,238,954 7 5.50% 6.52% 1,666,222 0.9034 1,505,342 1,363,712 1,232,041 8 5.75% 6.77% 1,730,111 0.8881 1,536,479 1,363,712 1,211,087 9 6.00% 7.02% 1,755,000 0.8728 1,531,701 1,334,067 1,164,325 10 6.25% 7.27% 1,837,694 0.8570 1,574,931 1,348,890 1,156,018 11 6.50% 7.52% 1,921,778 0.8409 1,615,937 1,363,712 1,146,685 12 6.75% 7.77% 1,985,667 0.8245 1,637,150 1,363,712 1,124,359 Totals 17,551,606 14,752,606 Notional Fixed rate 100,000,000 5.34% Shift 1.02% Pivot 0.00% (Difference) (2,799,000) DO NOT MODIFY 65 12.00% 10.00% Libor Curve Libors 8.00% 6.00% 4.00% 2.00% 0.00% 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Period 66

Swap Revaluation Fixed rate receiver has a gain when rates decline, since fixed payments are discounted at lower discount rates; it follows that receiving fixed under a swap is one way of taking a view on declining interest rates Can reach same result by remembering that fixed rate receiver can be viewed as owning a fixed rate bond which he is funding at Libor; so he would have gain on his asset when rates decline, while liability s value remains essentially the same Price gain for the Receiver of a 1 bp drop in rates should be roughly equal to the price gain of owning a 3-year, $100MM quarterly-pay bond whose coupon is 5.34% More generally DV01 of a swap is very nearly the same as DV01 of a fixed rate bond whose principal, coupon, maturity and daycount/frequency match those of the fixed leg of the swap Swap also has convexity like a bond which means changes in its value are not linear relative to changes in rates 67 DV01 of 3-year, quarterly-pay 5.34% bond Inputs Discount rate 5.34% Tenor in years 3.00 Coupon 5.34% Initial price of the bond $100,000,000 New price of the bond $100,027,555 68 Effect of Curve Shift on Swap P&L 20,000,000 15,000,000 10,000,000 5,000,000 PV (USD) Shift (%) - -6.00% -4.00% -2.00% 0.00% 2.00% 4.00% 6.00% (5,000,000) (10,000,000) (15,000,000) (20,000,000) 69

Interest Rate Forwards and Swaps (Part IV) 70 Question 1 On January 1, 2009, a borrower whose interest cost is Libor + 75 bps under a $100MM loan signs up for the 12x18 FRA. The rate under the FRA is 5%. 6-month Libor resets at the beginning of January 2010 at 6%. What is the exact all-in amount paid in that period by the borrower, taking into account both the loan and the FRA, and assuming the FRA settles at the end of that period? a) $2,555,556 b) $2,890,972 c) $3,066,667 d) $3,450,000 71 Question 1 Solution Under the loan, the borrower pays 100MM x 6.75% x 181/360, while under the FRA he receives 100MM x (6% - 5%) x 181/360 So his all-in cost comes to 100MM x 5.75% x 181/360, which equals $2,890,972 72

Question 2 How much money could you make if you tried to arbitrage the following rates a dealer quotes you on Jan 1, 2009? Assume a $100MM notional for your arbitrage. Bid Offer 3-month Libor 3.96% 4.00% 6-month Libor 4.28% 4.32% 3x6 Libor FRA 4.40% 4.45% a) $15,779 b) $18,534 c) $19,621 d) $22,222 73 Question 2 Solution You borrow for 3 months at 4.00%, lock in a borrowing rate for the next 3 months of 4.45%, and invest the money for 6 months at 4.28% Per one dollar, your profit would be (1 + 4.28% x 181/360) (1 + 4.00% x 90/360) x (1 + 4.45% x 91/360) which amounts to $0.00016 per dollar, or $15,779 per $100MM 74 Question 3 You are asked on Jan 1, 2009 to price the fixed rate leg of an interest rate swap having the terms below. You (the dealer) will be receiving the floating payments, and will need to earn a profit. Please use the same Libor curve we have used in the early part of this module. Notional Initially $100,000,000, but amortizing in 2 equal repayments in the last 2 semesters Tenor Floating Rate Index Payment Frequency Daycount a) 4.98% b) 5.10% c) 5.29% d) 5.49% 2 years 3-month Libor Quarterly for floating leg Semi-Annual for fixed leg Act/360 for floating leg 30/360 for fixed leg Swap Start date July 1 2009 Dealer profit $125,000 75

Libor/ Days FRA Notional (at beg of period) Qn 3 Solution Net payment DFs PVNP Adjusted net payment PVANP 90 3.00% - 0.9926 - - - 91 4.00% - 0.9826 - - - 92 4.50% 100,000,000 1,150,000 0.9714 1,117,166-92 4.75% 100,000,000 1,213,889 0.9598 1,165,088 2,645,201 2,538,858 90 5.00% 100,000,000 1,250,000 0.9479 1,184,936-91 5.25% 100,000,000 1,327,083 0.9355 1,241,531 2,645,201 2,474,674 92 5.50% 100,000,000 1,405,556 0.9226 1,296,718-92 5.75% 100,000,000 1,469,444 0.9092 1,336,027 2,645,201 2,405,032 90 6.00% 50,000,000 750,000 0.8958 671,827-91 6.25% 50,000,000 789,931 0.8818 696,590 1,322,601 1,166,319 92 6.50% - 0.8674 - - - 92 6.75% - 0.8527 - - - 5.2904% 8,709,883 8,584,883 (Diff) 125,000 76 Question 4 You have entered into a 10-year interest rate swap in which you receive 6% fixed semi-annually and pay Libor semi-annually on a notional of $10MM. Right after you enter the swap, the whole Libor curve shift upwards by a parallel 100bps. Which of the following is most likely to be your approximate P&L? Do not use Excel to estimate your answer. a) A gain of $700,000 b) A loss of $700,000 c) A gain of $300,000 d) A loss of $300,000 77 Question 4 Solution Since you are receiving the fixed payments and rates have risen, you should expect a loss since the PV of the cash flows you are receiving will diminish, as we discussed before The duration of the swap should pretty obviously be much closer to 7 than to 3, since the swap we modeled earlier in the module had a 3-year maturity and a duration of around 2.8 A duration of 7 looks about right, meaning the swap would lose 7% of its notional given the 1% shift in rates The actual duration of this swap is in fact around 7.1 It would be a good exercise for you to confirm this 78

Question 5 Which of the following methods would not be a possible hedge for an interest rate swap you have entered into where you are paying fixed and receiving Libor? a) Issue a fixed rate note and invest the proceeds in a floating rate note b) Enter another interest rate swap with another counterparty under which you receive fixed and pay Libor c) Enter into a series of FRAs in which you are paying the settlement rate and receiving the contract rate d) Purchase a fixed rate bond and pledge it to a bank s repo desk for funding at the Libor rate 79 Question 5 Solution You are paying fixed under the swap so you need to receive fixed under your hedge and pay Libor The only hedge that fails to do this is the first one 80 Question 6 The Libor curve in USD is flat at 3%, while in GBP it is flat at 5%. On Jan 1, 2009, a borrower issues a GBP 200MM 3-year bond paying 7% quarterly, and wishes to swap it into a USD synthetic liability based on 3-month Libor. The FX spot rate is 1.75. The dealer would like to earn a profit margin of $500,000 on the swap. What is the all-in cost the dealer can quote the borrower? a) Libor -10 b) Libor +107 c) Libor + 179 d) Libor + 203 81

Qn 6 Solution (1) $ Libor PV of Curve DFs $ Pmts Pmts Days Libor Curve DFs Pmts PV of Pmts 3.00% 0.99 4.40 4.37 91 5.00% 0.99 3.50 3.46 3.00% 0.99 4.40 4.34 90 5.00% 0.98 3.50 3.41 3.00% 0.98 4.40 4.30 92 5.00% 0.96 3.50 3.37 3.00% 0.97 4.40 4.27 92 5.00% 0.95 3.50 3.33 3.00% 0.96 4.40 4.24 91 5.00% 0.94 3.50 3.29 3.00% 0.96 4.40 4.21 90 5.00% 0.93 3.50 3.25 3.00% 0.95 4.40 4.17 92 5.00% 0.92 3.50 3.21 3.00% 0.94 4.40 4.14 92 5.00% 0.91 3.50 3.17 3.00% 0.93 4.40 4.11 91 5.00% 0.89 3.50 3.13 3.00% 0.93 4.40 4.08 91 5.00% 0.88 3.50 3.09 3.00% 0.92 4.40 4.05 92 5.00% 0.87 3.50 3.05 3.00% 0.91 354.40 323.58 92 5.00% 0.86 203.50 175.29 Totals 369.85 211.06 in 369.35 in $ $ Fixed 5.03% Fixed 7.00% $ Notional 350 Notional 200 PV Diff 0.50 FX Spot 1.75 82 Qn 6 Solution (2) Period Libor/F RA Net payment DFs PVNP Adjusted net payment PVANP 1 3.00% 750,000 0.9926 744,417 750,000 744,417 2 3.00% 758,333 0.9851 747,023 758,333 747,023 3 3.00% 766,667 0.9776 749,486 766,667 749,486 4 3.00% 766,667 0.9702 743,784 766,667 743,784 5 3.00% 750,000 0.9629 722,198 750,000 722,198 6 3.00% 758,333 0.9557 724,727 758,333 724,727 7 3.00% 766,667 0.9484 727,116 766,667 727,116 8 3.00% 766,667 0.9412 721,584 766,667 721,584 9 3.00% 750,000 0.9342 700,643 750,000 700,643 10 3.00% 758,333 0.9272 703,096 758,333 703,096 11 3.00% 766,667 0.9201 705,414 766,667 705,414 12 3.00% 766,667 0.9131 700,047 766,667 700,047 Totals 8,689,535 8,689,535 Notional 100,000,000 Fixed rate 3.0000% (Difference) - 83 Question 6 Solution 200 MM at maturity $Libor on $350 MM 7% on 200 MM Borrower $3% on $350 MM $5.03% on $350 MM $350 MM at maturity 200 MM Bond at 7% 84

Question 7 Today is January 1, 2009. The USD Libor curve is monotonically downward sloping, with 6-month Libor spot at 6% and the 5-year forward 6-month Libor at 4%. You have just priced a spot-starting bullet interest rate swap, and are now asked to price a swap which starts on January 1, 2010 and amortizes by 25% of its original notional amount on each anniversary. The fixed leg of the swap will be increased as a result of: a) Both the delayed-start date and the amortization schedule b) The delayed-start date but not the amortization schedule c) The amortization schedule but not the delayed-start date d) Neither the delayed-start date nor the amortization schedule 85 Question 7 Solution Since the curve is downward sloping, the highest values for Libor are the earlier ones, and the lowest values are the last ones Postponing the start date of the swap removes the highest Libors from the calculation of the swap rate, so reduces the swap rate Amortizing the swap notional towards the end puts lower weights on the last few Libors, so increases the swap rate Therefore only the second effect increases the swap rate 86 Question 8 A company has just issued $100 million of fixed rate 5-year bonds at a price of par with a coupon of 6.00% (annual) and up-front fees of.50. The company wishes to swap the debt into. The spot rate is 1.25. Annual swap rates for 5-years are: $ interest rate swaps: 6.12 / 6.15 interest rate swaps: 7.98 / 8.02 /$ LIBOR basis swaps: Libor + 1 / Libor + 4 Assuming the company pays full bid-offer spreads, what is the all-in cost of the dollar debt swapped into fixed? (Be sure to use the all-in cost of the $ debt by taking into account the up-front fees paid on the offering.) a) 8.06% b) 8.42% c) 8.44% d) 8.46% 87

Bond YTM Inputs Proceeds (99.50) Tenor in years 5.00 Coupon 6% YTM of the bond 6.12% 88 Question 8 Solution $Libor $6.12% $6.12% Bond Company $Libor Libor + 4bps $100 MM at maturity 80 MM at maturity 89 Libor 8.02%