Callable Bonds & Swaptions

Callable Bonds & Swaptions 1 Outline PART ONE Chapter 1: callable debt securities generally; intuitive approach to pricing embedded call Chapter 2: payer and receiver swaptions; intuitive pricing approach 2 Outline PART TWO Chapter 3: standard pricing model for European swaptions Chapter 4: standard pricing model for European and Bermudan callable bonds 3

Outline PART THREE Chapter 5: applications involving swaptions and callable bonds; combining swaptions with callable bonds to provide cost-efficient financing Chapter 6: Quiz 4 Chapter 1 Introduce callable debt securities Intuitive approach to pricing embedded call 5 Callable bonds: introduction Simplest form of callable bond is one that contains single option to prepay instrument on one specific future date prior to maturity Example: ABC is able today to issue 5-year debt at par, paying 6% s.a. ABC anticipates that rates will decline significantly in next couple of years, and would like to refinance fixed-rate debt at lower cost when this occurs ABC proposes to investor 25 bps yield pick-up, to 6.25%, in return for allowing ABC, on bond s second anniversary, to prepay it in whole at par 6

Callable bond: issuer benefits If issuer view is right, issuer will be able in 2 years time to issue 3-year debt at substantially lower rate than those prevailing on day one Issuer would prepay outstanding bond and reissue for remaining 3 years at lower rate Brings aggregate cost for entire 5-year period potentially lower than 6% originally available on 5-year straight debt 7 Callable bond: sample term sheet Term Sheet Issuer Face Value Issue price Tenor Status Coupon Prepayment Company ABC $100 $100 5 years Senior unsecured 6.25%, paid semi-annually The bond may be prepaid on its second anniversary at par (plus accrued and unpaid interest) 8 Scenario analysis On second anniversary, assume ABC can issue 3-year noncallable debt at 5% s.a. ABC would call original debt at par ABC would refinance it for 3 years at 5% ABC s all-in cost over entire 5-year period would fall between 6.25% and 5% 9

Calculating All-in Cost Over 5 Years Principal 100-100.000 Coupon yrs 1-2 6.25% 3.125 Coupon yrs 3-5 5.00% 3.125 3.125 3.125 2.500 2.500 2.500 2.500 2.500 102.500 Semi-annual IRR 2.77% Annualized IRR 5.54% 10 Scenario analysis By issuing callable, issuer is implicitly assuming that 3-yr rates have reasonable probability, on second anniversary, to have fallen below 5.81% from current level 3-year rate on second anniversary Scenario 1 Rate > 6.25% 2 6.25% > Rate > 5.81% 3 Rate < 5.81% Outcome Leave original bond outstanding Refinance Refinance All-in-cost AIC = 6.25% 6% < AIC < 6.25% AIC < 6% Not sufficient for rates generally to decline for issuer to save money They must decline enoughso issuer savings in years 3-5 > coupon premium incurred in years 1-2 on PV basis www.dnatrainingconsulting. net 11 Call provision: alternative styles Often call right is not European, but Bermudan ; so rather than arising only on a specific anniversary, right arises on any coupon payment date on or after that specific anniversary Call price may not be exactly par: common to set call price > par if bond is called early, then have call price decline gradually towards par the later the call date 12

Bermudan callable bond Term Sheet Issuer Face Value $100 Issue price $100 Tenor Status Coupon Prepayment Company ABC 5 years Senior unsecured 6.35%, paid semi-annually The bond may not be prepaid until the second anniversary. On or after the second anniversary, the bond may be prepaid as follows: If called on the second anniversary, call price is 101.60 If called on the third anniversary, call price is 100.80; and If called on the fourth anniversary, call price is 100 (par) 13 Bermudan callable investor perspective Call price premium compensates investor for early call, since investor undertook process of obtaining internal approvals and reviewing documentation, and incurred significant legal expenses Also investor may have to reinvest returned principal at unattractive levels, given decline in interest rates, impacting portfolio performance This is classic reinvestment risk Note 0.10% additional coupon under Bermudan alternative, and stepdown in call price from 101.80 to 100 over time Both features are consistent with greater issuer flexibility allowed by Bermudan call relative to European 14 Calculating Breakeven for Call on Third Anniversary -100.000 Principal 100 3.175 3.175 3.175 3.175 Coupon Call Price 3.175 Years 1-3 6.35% 100.80 3.975 Years 4-5 5.50% 2.750 2.750 2.750 102.750 Semi-annual IRR 3.10% Annualized IRR 6.20% 15

Breakevens for Bermudan callable Breakeven rate on call date for remaining tenor Called on Second Anniversary Third Anniversary Fourth Anniversary Breakeven Rate 5.14% 4.96% 4.37% The later issuer exercises call and refinances, the more rates for remaining maturity must decline to achieve breakeven 16 Volatility: intuitive impact on call price If interest rates change rapidly, and by large amounts, value to issuer of bond prepayment privilege > than if rates stable Chances that rates will fall below breakeven level are greater in more volatile markets. Manifests itself in larger upward adjustment for callable bond coupon versus non-callable alternative than in markets with stable rates Restatement of principle that options on high volatility assets are more valuable than options on low volatility assets 17 Effect of curve shape Intuitively would expect that flat curve implies higher chance of declining rates than steep curve, so prepayment option should be more expensive for issuer when curve is flat But offsetting this is fact that when curve is steep, shorter rates are lower than longer rates, so even absent any drop in rates it may make sense to refinance, just to take advantage of the curve rolldown effect 18

Effect of curve shape Assume flat curve at 6%, and 5-year bond callable only on first anniversary. Pricing model has led you to 6.25% coupon for callable alternative In one year, 4-year rate, currently at 6%, needs to have fallen or remained same, or have risen no higher than 6.25%, i.e. by no more than 25 bps, for bond to be worth refinancing 19 Effect of curve shape Now assume a positive curve, whose 5-year point is still at 6%, but whose 4-year point is presumably lower, at 5.50% say. Again we suppose bond callable on first anniversary comes with 6.25% coupon True, curve shape implies generally higher probability of rate increase, which appears to diminish value of prepayment option; but offsetting this is fact that refinancing still makes economic sense if, in one year, 4-year rate has risen no more than 75 bps, versus only 25 bps with flat curve In fact option with flat curve still turns out generally more valuable than with steep curve, but simple intuition is not sufficient to lead to correct result 20 Chapter 2 Introduce payer and receiver swaptions Intuitive approach to pricing 21

Swaptions: introduction Borrower has 5-year debt at L + 75 bps, and is worried that rates may increase Borrower could simply swap from floating to fixed; if 5-year swap rate is 6%, all-in cost would come to 6.75% Borrower is contemplating currently divestiture of a division, expected to close in one year, which if implemented would enable him to prepay loan and eliminate interest rate worries Should divestiture not close, borrower would like to switch into fixed 22 Swaption as flexible hedge Rather than enter immediately into 5-year swap, borrower pays upfront premium of 80 bps for right to enter 4-year swap at 6% (against Libor) in one year s time, where borrower pays fixed 4-year swap < 6% 4-year swap > 6% Failure No exercise: Either retain floating debt at L+75bps, or Swap at prevailing market rate Exercise the swaption and fix debt at 6% Divestiture Success Debt prepaid Swaption is worthless so no exercise Debt prepaid Exercise swaption Assign swap to bank (unwind) and collect FMV 23 Swaption term sheet Term Sheet Notional Premium Swaption tenor Underlying swap tenor Party paying fixed Party paying floating Swap floating rate Fixed rate convention Floating rate convention Swap fixed rate $100 million 0.80%, payable upfront 1 year 4 years Borrower Bank 6m Libor 30/360 Act/360 6%, payable semi-annually www.dnatrainingconsulting. net 24

Swaption terminology Borrower has purchased a payer swaption; in reference to fact that he is party acquiring option, and option gives him right to pay fixed rate of 6% under underlying swap We always link position to swap s fixed leg Bank is described as having sold a payer swaption Note very importantly: verb purchase which applied to borrower becomes sell from bank s perspective; but underlying swap is still a payer and does not become a receiver 25 Swaption terminology Now assume investor owns FRN paying Libor flat, and is worried that rates may diminish, so is considering entering a receive-fixed swap to create a synthetic fixed-rate investment Investor however is still uncertain about direction of rates nearterm, so rather than enter into swap immediately, he pays 80 bps upfront to bank, in return for right, in one year s time, to receive fixed at 6% and pay Libor for another 4 years We say in this case investor has bought receiver swaption, and that bank has sold receiver swaption 26 Swaption notation We refer to swaption with which we began chapter as a 1 x 4 First figure ( 1 ) indicates tenor in years until option expiration, and second figure ( 4 ) indicates tenor of underlying swap from date of option expiration, i.e. from when swap comes alive 27

Swaption risk factors Receiver Payer Long Short Long Short Curve Up + + Curve Down + + Vol Up + + Vol Down + + 28 Short vol swaption trade What should trader do if she has no strong directional view on curve shifts, but believes vols are unsustainably high and very likely to decline? Trader sells swaption straddle, i.e. sells both receiver and payer with same strike, and earns two upfront premiums Assuming rates do not move too much (in either direction), she enjoys large gain from diminution in vols Any loss on either position from curve movement, if it is not too large, will be offset to some degree by gain on other position, but probably not of same magnitude 29 Swaption variations Swaptions may be European or Bermudan or even American Bermudan version may be exercised on more than one single date, but is still limited to specific dates only American swaption may be exercised on any date prior to expiration 30

Swaption variations Additional issue (relevant to Bermudan and American alternatives only): is tenor of underlying swap fixed upfront ( constant maturity swaption ), or does it depend on date of exercise, diminishing in tenor the later the exercise date ( remaining maturity swaption ) Most common version of remaining maturity swaption fixes maturity date of underlying swap irrespective of option exercise date. So if a 5x5 Bermudan is exercised after 3 years, underlying swap would have 7-year tenor; but if exercise happens one year later, underlying swap s tenor would be 6 years 31 Constant v. remaining maturity Notional Premium Swaption tenor Underlying swap tenor Party paying fixed Alternative 1 ( constant maturity swap ) $100 million 1.00%, payable upfront Option may be exercised on any of 1 st, 2 nd, 3 rd or 4 th anniversary after trade date 4 years Borrower Alternative 2 ( remaining maturity swap ) $100 million 0.90%, payable upfront Option may be exercised on any of 1 st, 2 nd, 3 rd, or 4 th anniversary after trade date (5-n) years,where n is number of years elapsed since trade date Borrower Party paying floating (6- mo. Libor, A/360) Swap strike Bank 6%,s.a. (30/360) Bank 6%,s.a. (30/360) www.dnatrainingconsulting. net 32 Constant v. remaining maturity Under Alternative 1, underlying swap tenor is always 4 years, irrespective of exercise date Under Alternative 2, tenor diminishes the later the exercise date and always equals remaining number of years since inception, assuming a total of 5 years 33

Constant v. remaining maturity Both versions are usually more expensive than European alternative, since they provide more flexibility to owner regarding exercise privilege Alternative 1 is more expensive than Alternative 2, since it carries same number of exercise rights as Alternative 2, but involves underlying whose duration (price volatility) is higher than Alternative 2 s for all but first exercise date 34 Swaption alternative: cross-currency swaption Buyer has right, but not obligation, on (or sometimes before) option expiration date, to enter cross-currency swap at pre-determined notionals, coupons and tenor International borrower issues 7-year USD debt, but purchases option to swap last 5 years of debt s coupons, as well as principal repayment, into EUR if he sees risk of USD appreciating against EUR 35 Cross-currency swaption Notional in CCY1 Notional in CCY1 Swaption Buyer Notional CCY2 x index 2 Notional CCY1 x index 1 Notional in CCY2 Notional in CCY2 Premium Bank 1) At inception 2) At swaption expiration, if swaption exercised 3) Periodic payments under CCS 4) At CCS maturity Index can be floating or fixed with different frequencies and day-counts www.dnatrainingconsulting. net 36

Swaption alternative: cross-currency swaption International borrower has substantial cash flows in both EUR and USD, so indifferent whether to issue in one currency or the other Borrower can subsidize his funding cost by borrowing in EUR but granting to lender option to convert principal and coupons into USD at pre-determined rates and on pre-determined dates Effectively embedding a cross-currency swaption into loan, whose premium may be paid upfront or, more typically, embedded in interest rate under loan facility 37 Swaption alternative: basis swaption Option to enter basis swap on defined date in future and for defined term and notional amount Example: bank pays counterparty upfront premium, in return for right to enter 5-year swap in 2 years time on $100 notional, under which bank pays 3m Libor and receives 6m Libor minus 25 bps, or pays 3m Libor and receives Fed Funds plus 50 bps 38 Basis swaption Swaption Buyer Basis Swaption Notional x (Index2 + Spr2) Notional x (Index1 + Spr1) Bank Premium 1) At inception 2) At swaption expiration, if swaption is exercised 3) Exchange of periodic payments 39

Callable Bonds & Swaptions (Part II) 40 Outline PART TWO Chapter 3: standard pricing model for swaptions Chapter 4: standard pricing model for callable bonds 41 Chapter 3 Standard pricing model for European swaptions 42

Pricing Swaption in Flat Curve Environment Strike 6.00% Fwd Vol 30% Option Expiration 2 Swap Tenor 5 Forward Swap 6.0000% Payer Swaption 3.8197% Receiver Swaption 3.8197% Period FRA/Spot DFs Fixed Payments PV of Fixed Payments Floating Payments PV of Floating Payments Principal at Begin 1 6.00% 0.97 3.00 2.91 3.00 2.91 100 2 6.00% 0.94 3.00 2.83 3.00 2.83 100 3 6.00% 0.92 3.00 2.75 3.00 2.75 100 8 6.00% 0.79 3.00 2.37 3.00 2.37 100 9 6.00% 0.77 3.00 2.30 3.00 2.30 100 10 6.00% 0.74 3.00 2.23 3.00 2.23 100 11 6.00% 0.72 3.00 2.17 3.00 2.17 100 18 6.00% 0.59 3.00 1.76 3.00 1.76 100 19 6.00% 0.57 3.00 1.71 3.00 1.71 100 20 6.00% 0.55 3.00 1.66 3.00 1.66 100 43 Pricing Swaption in Steep Curve Environment Strike 6.00% Forward Swap 7.0520% Fwd Vol 30% Payer Swaption 6.4005% Option Expiration 2 Receiver Swaption 2.4385% Swap Tenor 5 Period FRA/ Spot DFs Fixed Payments PV of Fixed Payments Floating Payments PV of Principal Floating at Begin Payments 1 5.00% 0.9756 3.0000 2.9268 2.5000 2.4390 100 2 5.25% 0.9507 3.0000 2.8520 2.6250 2.4955 100 3 5.50% 0.9252 3.0000 2.7756 2.7500 2.5443 100 8 6.75% 0.7933 3.0000 2.3798 3.3750 2.6773 100 9 7.00% 0.7665 3.0000 2.2994 3.5000 2.6826 100 10 7.25% 0.7396 3.0000 2.2189 3.6250 2.6812 100 11 7.50% 0.7129 3.0000 2.1387 3.7500 2.6734 100 18 9.25% 0.5327 3.0000 1.5982 4.6250 2.4639 100 19 9.50% 0.5086 3.0000 1.5257 4.7500 2.4157 100 20 9.75% 0.4849 3.0000 1.4548 4.8750 2.3640 100 44 Arbitrage strategy if payer/receiver parity does not apply Illustration of potential arbitrage if ATMF 2x5 payer is worth 4% while ATMF 2x5 receiver is worth 3.75%. Libor fixed @ strike Bank 1 Sell 2x5 payer Upfront premium of 4% fixed @ strike Arbitrageur: net profit of 25bps annuity fixed @ strike Libor Buy 2x5 receiver Upfront premium of 3.75% Libor Bank 2 2X5 forward-start swap If swap rate in 2 years < strike If swap rate in 2 years > strike Bank 3 45

Arbitrage strategies: payer/receiver parity Strategy: Payer Receiver Forward-start swap ATMF Payer > ATMF Receiver Short Long Pay fixed ATMF Payer < ATMF Receiver Long Short Receive fixed Outcome: Profit Risk Payer premium Receiver premium 0 Receiver premium Payer premium 0 46 Bermudian swaption: pricing Closed-form solution (available for pricing European swaption) not applicable to Bermudan alternative, nor American one True of both constant-maturity swaption and remaining maturity swaption Numerical solutions are required, typically involving binomial or trinomial trees, similar to ones used in FX Options module Will illustrate binomial technique in next chapter, on callable bond pricing 47 Chapter 4 Pricing model for European and Bermudan callable bonds using binomial trees 48

Callable Bond Pricing (Low vol) Period 0 1 2 3 Vol 10% 3.5000% 5.4289% 7.0053% 9.1986% 4.4448% 5.7354% 7.5312% 4.6958% 6.1660% 5.0483% 2-Year 4.2% Bond Coupon 4.20 100.000 103.034 104.200 103.966 104.200 104.200 3-Year 4.7% Bond Coupon 4.70 100.000 102.523 102.546 104.700 104.477 103.721 104.700 104.704 104.700 104.700 49 4-Year 5.2% Bond Coupon 5.20 100.000 101.961 100.789 101.538 105.200 105.039 103.238 103.032 105.200 105.316 104.290 105.200 105.344 105.200 105.200 4-Year 6.5% Bond Coupon 6.50 104.643 106.730 104.425 104.029 106.500 109.881 106.918 105.541 106.500 109.034 106.815 106.500 107.882 106.500 106.500 4-Year 6.5% Bond Callable Every year at Par Coupon 6.50 Call Price 100 102.899 106.500 104.425 104.029 106.500 106.500 106.500 105.541 106.500 106.500 106.500 106.500 106.500 106.500 106.500 50 4-Year 6.5% Bond Callable at 102 in Year 1, 101 in Year 2, and 100 in Year 3 Coupon 6.50 Call Price EOY 1 102 Call Price EOY 2 101 Call Price EOY 3 100 103.942 106.660 104.425 104.029 106.500 108.500 106.770 105.541 106.500 107.500 106.500 106.500 106.500 106.500 106.500 4-Year Step-Up Note, Coupons 5.50% Years 1-2, 9.5% Years 3-4, Callable Every Year at Par Coupon Yrs 1-2 5.50 Coupon Yrs 3-4 9.50 Call Price 100 102.453 105.567 105.500 109.500 109.500 106.510 105.500 109.500 109.500 105.500 109.500 109.500 109.500 109.500 109.500 51

Binomial tree: time steps Real tree would have many more time steps of substantially shorter duration Precision of model increases as we increase number of time steps 52 Callable bond: pricing Investor has granted issuer option to prepay on any anniversary, but coupon has not risen to compensate for this privilege So value of callable must be less than 104.643, price of noncallable 53 Deconstruction of callable bond Non-callable bond Callable bond Call option Call option can be European, Bermudan or American Callable bond = Non-callable bond call option Value of call option being positive in all case, callable bond is worth < non-callable version, in all cases 54

Deconstruction of puttable bond Non-puttable bond Puttable bond + Put option Puttable bond = Non-puttable bond + put option 55 Amending the tree Principal new feature in tree starting on Row 54, which reflects issuer s right to call at par, is stipulation that if price in that cell obtained under normal formula exceeds call price plus accrued interest which would give issuer immediate arbitrage since he could call the bond by paying 106.5 and then sell it for this higher price market would preempt this arbitrage by refusing to price this instrument above 106.50 Stated differently, no rational investor would agree to pay > 106.5 for this instrument on any call date, no matter what formula says, since investor would face then immediate loss when bond is called away at 106.5 exactly 56 Amending the tree We know that Callable Bond = Non-callable Bond Call Option Also value of call option increases when vol rises Therefore increase in vol will reduce bond value further, pulling it closer to 100 57

Callable Bond Pricing (High vol) Period 0 1 2 3 Vol 20% 3.5000% 5.9194% 8.3440% 12.0003% 3.9679% 5.5931% 8.0441% 3.7492% 5.3921% 3.6144% 2-Year 4.2% Bond Coupon 4.20 100.000 102.577 104.200 104.423 104.200 104.200 3-Year 4.7% Bond Coupon 4.70 100.000 101.562 101.337 104.700 105.438 103.854 104.700 105.616 104.700 104.700 58 4-Year 5.2% Bond Coupon 5.20 100.000 100.450 98.281 99.128 105.200 106.550 103.495 102.568 105.200 107.248 105.018 105.200 106.730 105.200 105.200 4-Year 6.5% Bond Coupon 6.50 104.643 105.185 101.872 101.589 106.500 111.427 107.180 105.071 106.500 111.000 107.551 106.500 109.285 106.500 106.500 4-Year 6.5% Bond Callable Every year at Par Coupon 6.50 Call Price 100 102.108 104.864 101.872 101.589 106.500 106.500 106.500 105.071 106.500 106.500 106.500 106.500 106.500 106.500 106.500 59 4-Year 6.5% Bond Callable at 102 in Year 1, 101 in Year 2, and 100 in Year 3 Coupon 6.50 Call Price EOY 1 102 Call Price EOY 2 101 Call Price EOY 3 100 103.116 104.950 101.872 101.589 106.500 108.500 106.682 105.071 106.500 107.500 106.500 106.500 106.500 106.500 106.500 4-Year Step-Up Note, Coupons 5.50% Years 1-2, 9.5% Years 3-4, Callable Every Year at Par Coupon Yrs 1-2 5.50 Coupon Yrs 3-4 9.5 Call Price 100 102.453 105.104 105.500 107.268 109.500 106.974 105.500 109.500 109.500 105.500 109.500 109.500 109.500 109.500 109.500 60

Raising the strike Remind you that this feature is common, to discourage early redemption or at least compensate investor if it does happen for previous efforts by paying her extra amount for her troubles Increase in call strike for first two exercise dates diminishes call value So given equation Callable Bond = Non-callable Bond Call Option Anticipate increase in price of this bond versus previous version 61 Comparison Trade-off between this bond and 6.5% bond callable at par on each anniversary, priced at 102.899 Earn 1% less under this version in Years 1-2, then potentially 3% more in Years 3-4 if bond survives long enough Not enough to compare PV of 3%s to PV of foregone 1%s and deduce this instrument (under this simple PV analysis) offers better value 62 Comparison Likelihood of actually receiving high coupon in Years 3-4 is diminished by substantial probability bond will have been called by then Involves probability-weighted comparison; tree enables this comparison by solving for entire investment s fair value, revealing spot price of 102.453, so pointing to slight decline in value from fixed-coupon alternative Quiz question will ask you to price puttable bond 63

Callable Bonds & Swaptions (Part III) 64 Outline PART THREE Chapter 5: applications of swaptions and callable bonds Chapter 6: Quiz 65 Chapter 5 Applications of swaptions and callable bonds Combining swaptions and callable bonds to provide cost-efficient financing for issuers 66

Extendible interest rate swap Gives owner right at specific time in future to extend swap tenor for pre-determined period Example: Swap curve is flat at 6% Borrower contracted 7-year loan at L + 75 bps Borrower may prepay loan in two years from proceeds of asset disposal; otherwise, loan will remain outstanding for all 7 years Should borrower hedge debt with 2-year IRS, 7-year IRS, or something else? 67 Extendible interest rate swap Either 2-year or 7-year IRS brings all-incost to 6.75% fixed Could we construct 2-year IRS, extendible at borrower s option into 7-year IRS (to be available if Borrower doesn t prepay loan) Swap curve flat at 6% 7-year loan at L + 75 May be prepaid in 2 years, otherwise remains outstanding for 7 years What would be fixed rate under this instrument? 68 Extendible swap deconstruction 2-yr pay-fixed IRS extendible into 7-yr IRS at borrower s option = Normal 2-yr pay-fixed IRS @ K% + 2 x 5 bought payer swaption So borrower has fixed his cost for Years 1-2 at K%, and can extend protection again at K% if he does not prepay loan and if K% rate is still competitive under current market conditions 69

Extendible Swap Pricing Strike (K%) 6.00% Forward Swap 6.0000% Fwd Vol 30% Payer Swaption 3.8197% Option Expiration 2 Receiver Swaption 3.8197% Swap Tenor 5 Spot 2-year swap 6.0000% PV of K% swap 0.0000% Extendible Swap 3.8197% Period FRA/ Spot DFs Fixed Payments PV of Fixed Payments Floating Payments PV of Floating Payments Principal at Begin 1 6.00% 0.971 3.000 2.913 3.000 2.913 100 2 6.00% 0.943 3.000 2.828 3.000 2.828 100 3 6.00% 0.915 3.000 2.745 3.000 2.745 100 10 6.00% 0.744 3.000 2.232 3.000 2.232 100 11 6.00% 0.722 3.000 2.167 3.000 2.167 100 12 6.00% 0.701 3.000 2.104 3.000 2.104 100 13 6.00% 0.681 3.000 2.043 3.000 2.043 100 14 6.00% 0.661 3.000 1.983 3.000 1.983 100 70 Extendible swap valuation achieving zero NPV Solve for fixed rate K% that achieves zero upfront NPV as per equation below (which shows borrower perspective): 2-yr pay-fixed IRS, extendible into 7-yr IRS = Off-market 2-yr payfixed IRS, fixed leg at K% + 2x5 payer swaption on K% swap PV = ve PV = + ve PV = 0 71 Monetizing underlying option in callable bond Swaptions can be used in connection with callable bonds to generate cost savings for high-grade borrowers, such as FNMA and other US agencies before credit crisis Example: Assume AAA-rated FNMA can borrow for any maturity on fixedrate basis at swap rate flat Swap curve is perfectly flat at 6% and vols in swaption market are 30% for any combination of maturity and strike Retail bond market investors buy callable bonds at small coupon premium over non-callable bonds, say 50 bps, irrespective of other specifics 72

Monetizing underlying option in callable bond FNMA could issue either 7-year straight debt at 6%, or 7-year debt callable in 2 years (at par) at 6.50% FNMA CFO has expressed strong preference for 7-year fixed-rate funding, so is leaning away from callable instrument However CFO recognizes that 50 bps annual coupon premium appears modest relative to potential savings of refinancing at potentially much lower 5-year rate in 2 years 73 Monetizing underlying option in callable bond At issuance: FNMA issues 6.50% callable bond at par FNMA sells 2x5 receiver for 88 bps annuity Investors Callable Bond FNMA Receiver swaption Bank 88bps 74 Residual risks for FNMA Scenario 1: On second anniversary If 5-yr IRS 6.50% Swaption lapses FNMA still pays 5.62% net 0.88% Investors 6.50% annuity FNMA coupons Bank 75

Residual risks for FNMA Scenario 2 On second anniversary If 5-yr IRS < 6.50%: Swaption is exercised, FNMA pays 6.50% fixed and receives Libor FNMA calls back old bonds FNMA issues FRN at Libor FNMA pays 5.62% net Old Investors New Investors Old bond Par Par FRN at Libor FNMA Libor 6.50% 88bps Bank 76 Conclusion FNMA is assured to pay fixed, directly or synthetically, for entire 7 years under either scenario But FNMA received on Day 1 a premium whose annualized value was 88 bps, bringing down effective all-in cost for 7 years to only 5.62% and saving it 38 bps net per annum 77 Monetizing underlying option in callable bond Is example realistic? To what do we attribute significant value generated for issuer? Borrower has taken advantage of serious mis-pricing of interest rate optionality in retail bond market that purchases FNMA paper We demonstrated that 7-year callable bond issued by FNMA can be decomposed into 7-year straight bond plus 2x5 receiver swaption, priced in interbank swaption market at 88 bps per annum Less sophisticated retail market, attracted to FNMA principally on account of triple-a rating, considers 50 bps annually under callable almost free money! 78

Monetizing underlying option in callable bond FNMA in effect is buying this option cheap in retail market, then reselling it in market which appreciates its value fully and is prepared to pay fair price Savings of this size are unrealistic in US market, but still occur in less liquid markets Illustration is excellent example of arbitrage funding strategies 79 Structured investment products Infinite variety of structured investment products, relating to numerous asset classes, from interest rates and FX to credit and commodities Consider investor who expects USD Libor to decline so asks to be shown variety of inverse floaters: 1. You show her first simple vanilla alternative paying (10% Libor) when curve lies at 5% 2. Then you show her leveraged version paying 20% (3 x L) 3. Then you improve this to 22% (3 x L) but place a cap on her coupon at 13% 4. Next you make instrument callable on its first anniversary only, which enables you to lift coupon to 23% (3 x L) 5. Finally you make call Bermudan, and increase coupon to 24% (3 x L) 80 Structured investment products Process is one in which different features are added gradually to structure, each introducing its own risks but enabling you to offer headline coupon, in return for asking investor to accept additional risks that may not trouble her Last two steps involved inclusion of call feature to enhance yield, initially European and then more expensive Bermudan version Numbers in this example were not confirmed via pricing model 81

Chapter 6 Quiz 82 Question 1 Using a Libor curve that lies completely flat at 6%, calculate first the rate on the fixed leg of a 7-year, semiannual spot-starting swap. Then assuming the same vol input of 30% we used for swaptions throughout the module, combine a 7-year offmarket rate spot-starting swap with a 2x5 receiver swaption to engineer a 7-year IRS cancelable on its second anniversary at the option of the fixed-rate payer, so that the NPV of the aggregate package is zero. What is the strike of this cancelable swap, accurate to two decimal places? a) 6.83% b) 7.03% c) 7.23% d) 7.43% 83 Solution to Question 1 Libor curve being flat at 6% means that all swap rates, whether spot-starting or forward-starting, lie at 6% 84

Solution to Question 1 Strike (K%) 6.00% Forward Swap 6.0000% Fwd Vol 30% Payer Swaption 3.8197% Option Expiration 2 Receiver Swaption 3.8197% Swap Tenor 5 Spot 2-year swap 6.0000% PV of K% swap 0.0000% Extendible Swap 3.8197% Period FRA/ Spot DFs Fixed Payments PV of Fixed Payments Floating Payments PV of Floating Payments Principal at Begin 1 6.00% 0.971 3.000 2.913 3.000 2.913 100 2 6.00% 0.943 3.000 2.828 3.000 2.828 100 3 6.00% 0.915 3.000 2.745 3.000 2.745 100 10 6.00% 0.744 3.000 2.232 3.000 2.232 100 11 6.00% 0.722 3.000 2.167 3.000 2.167 100 12 6.00% 0.701 3.000 2.104 3.000 2.104 100 13 6.00% 0.681 3.000 2.043 3.000 2.043 100 14 6.00% 0.661 3.000 1.983 3.000 1.983 100 85 Solution to Question 1 Initial position: pay fixed under 7-yr swap 1y 2y 3y 4y 5y 6y 7y Receives L L L L L L L Pays 7.23% 7.23% 7.23% 7.23% 7.23% 7.23% 7.23% + 3y 4y 5y 6y 7y Pays L L L L L Receives 7.23% 7.23% 7.23% 7.23% 7.23% If holder exercises swaption = 1y 2y Receives L L Pays 7.23% 7.23% 86 Solution to Question 1 Initial position: pay fixed under 7-yr swap 1y 2y 3y 4y 5y 6y 7y Receives L L L L L L L Pays 7.23% 7.23% 7.23% 7.23% 7.23% 7.23% 7.23% If holder does not exercise = + Do nothing 1y 2y 3y 4y 5y 6y 7y Receives L L L L L L L Pays 7.23% 7.23% 7.23% 7.23% 7.23% 7.23% 7.23% 87

Solution to Question 1 Put-call parity brings about this result More generally, swap with a maturity M years, extendible at one party s option for N years at same fixed rate, is exactly equivalent in cash flows terms to swap with maturity (M+N) years (with same fixed rate as before), but cancelable in that party s option after M years Shape of the curve does not affect this outcome, which is true whether curve is flat, positive, inverted or humped Correct answer is (c) 88 Question 2 Please answer this question assuming the same market data as in worksheet Callable Bonds. You are offered a 6.5% 4-year bond paying annual coupons and puttable on each anniversary by the investor to the issuer, at 98 on the first anniversary, 99 on the second, and 100 on the third, in each case plus accrued and unpaid interest. What is the fair value of this bond? You will need to adapt the binomial models provided for callable bonds so they can be used instead for puttable bonds a) 99.87 b) 101.22 c) 101.85 d) 105.22 89 Solution to Question 2 Period 0 1 2 3 Vol 10% 2-Year 4.2% Bond Coupon 4.20 3.5000% 5.4289% 7.0053% 9.1986% 4.4448% 5.7354% 7.5312% 4.6958% 6.1660% 5.0483% 100.000 103.034 104.200 103.966 104.200 104.200 3-Year 4.7% Bond Coupon 4.70 100.000 102.523 102.546 104.700 104.477 103.721 104.700 104.704 104.700 104.700 4-Year 5.2% Bond Coupon 5.20 100.000 101.961 100.789 101.538 105.200 105.039 103.238 103.032 105.200 105.316 104.290 105.200 105.344 105.200 105.200 90

4-Year 6.5% Bond Puttable at 98 in Year 1, 99 in Year 2, and 100 in Year 3 Coupon 6.50 Put Price EOY 1 98 Put Price EOY 2 99 Put Price EOY 3 100 105.219 107.706 106.028 106.500 106.500 110.098 107.372 106.500 106.500 109.034 106.815 106.500 107.882 106.500 106.500 91 Question 3 The 5-year swap rate is 5% while the 10-year swap rate is 5.50%. A borrower with floating rate debt enters into a 5- year swap with HSBC, but which can be extended, in HSBC s sole discretion, for 5 additional years at the same fixed rate as for the first 5 years. Which of the following is true? You may not use any spreadsheets to answer this question a) The rate the borrower will pay on the fixed leg of the swap will be below 5% b) The rate the borrower will pay on the fixed leg of the swap will be below 5.50% but not necessarily below 5% c) The rate the borrower will pay on the fixed leg of the swap will be below 5.50% but above 5% d) The rate the borrower will pay on the fixed leg of the swap will be above 5.50% 92 Solution to Question 3 Problem leaves significant majority of people confused Many are inclined to assume that positive shape of yield curve pushes fixed leg of extendible swap above 5% level of spot-starting 5-year swap Simple observation dispels this illusion: extendible is sum of spot-start swap plus (or minus) swaption Here borrower has entered spot-starting 5-year swap at K%, but has also sold to bank 5x5 European receiver swaption, with strike K% as well 93

Solution to Question 3 Fixed rate K% must achieve zero upfront PV as per equation below (which shows bank s perspective) 5-yr receive-fixed IRS, extendible at bank s option into 10-yr IRS = Off-market 5-yr receive-fixed IRS, fixed leg at K% PV = ve + 5x5 receiver swaption on K% swap PV = + ve Thus the correct answer is (a) PV = 0 94 Question 4 Using a Libor curve that is completely flat at 5% and a 30% vol for all interest rates, determine the rate on the fixed leg of a 5-year, semi-annual, interest rate swap for a borrower who wishes to pay fixed, and is eager to earn a subsidy by granting to the bank the right to double the swap s notional amount on its first anniversary for its remaining life: a) 4.58% b) 4.66% c) 4.88% d) 4.98% 95 Solution to Question 4 Structure referred to sometimes as expandable swap was not discussed in module Can intuit that bank s right to double notional is long position in 1x4 swaption struck at same fixed rate as original swap If bank exercises option, swap notional for remaining life doubles, since swap underlying swaption and original swap have identical terms from that point until maturity 96

Solution to Question 4 Solve for fixed rate K% that achieves zero upfront NPV as per equation below (which shows bank perspective) 5-yr $100 receivefixed IRS, expandable to $200 at bank s option on first anniversary = Off-market 5-yr $100 receive-fixed IRS, fixed leg at K% PV = ve + 1x4 $100 receiver swaption on K% swap PV = + ve PV = 0 97 Solution to Question 4 Strike (K%) 5.00% Forward Swap 5.0000% Fwd Vol 30% Payer Swaption 2.0343% Option Expiration 1 Receiver Swaption 2.0343% Swap Tenor 4 Spot 2-year swap 5.0000% PV of K% swap 0.0000% Extendible Swap -2.0343% Period FRA/ Spot DFs Fixed Payments PV of Fixed Payments Floating Payments PV of Floating Payments Principal at Begin 1 5.00% 0.976 2.500 2.439 2.500 2.439 100 2 5.00% 0.952 2.500 2.380 2.500 2.380 100 3 5.00% 0.929 2.500 2.321 2.500 2.321 100 10 5.00% 0.781 2.500 1.953 2.500 1.953 100 11 5.00% 0.762 2.500 1.905 2.500 1.905 100 12 5.00% 0.744 2.500 1.859 2.500 1.859 100 13 5.00% 0.725 2.500 1.814 2.500 1.814 100 14 5.00% 0.708 2.500 1.769 2.500 1.769 100 98 Solution to Question 4 Correct answer is (b) Expandable version of IRS appeals to borrowers who have no strong view on what maximum percentage of floating-rate debt should be swapped into fixed Consider for example borrower with $1BN of debt in aggregate, all of it currently floating, determined to swap at least $300MM but prepared to reach $600MM if right incentive is available 99

Solution to Question 4 Borrower could find very appealing immediate 34 bps of savings under first $300MM swap he initiates Downside is only that if rates have declined by first anniversary, notional of this swap doubles, but only at same fixed rate as original one Of course by that time 4.66% may not be that attractive relative to market rates for 4-year swaps 100 Question 5 Assume a completely flat Libor curve at 4% (s.a.), and an annualized volatility for interest rates of 40% Assume also, as we did in Chapter 5, that FNMA is rated triple-a and can issue non-callable debt for any maturity at the swap rate flat or Libor flat Assume finally that FNMA debt containing a European call option of any tenor requires a 30 bps coupon premium for successful placement 101 Question 5 (continued) How much would the issuer save, in basis points annually, if it issued a 5-year bond, paying coupons semi-annually and callable on its first anniversary, and immediately sold an appropriate swaption to lock in guaranteed 5-year fixed-rate funding under any interest rate scenario? You are encouraged to use the same approach we used in Chapter 5 to answer this question. a) 12 b) 23 c) 32 d) 41 102

Solution to Question 5 Strike 4.30% Forward Swap 4.0000% Fwd Vol 40% Payer Swaption 1.8251% Option Expiration 1 Receiver Swaption 2.8813% Swap Tenor 4 Annualized 0.6415% receiver premium Period FRA/Spot DFs Fixed Payments PV of Fixed Payments Floating Payments PV of Floating Payments Principal at Begin 1 4.00% 0.980 2.150 2.108 2.000 1.961 100 2 4.00% 0.961 2.150 2.067 2.000 1.922 100 3 4.00% 0.942 2.150 2.026 2.000 1.885 100 10 4.00% 0.820 2.150 1.764 2.000 1.641 100 11 4.00% 0.804 2.150 1.729 2.000 1.609 100 12 4.00% 0.788 2.150 1.695 2.000 1.577 100 13 4.00% 0.773 2.150 1.662 2.000 1.546 100 14 4.00% 0.758 2.150 1.629 2.000 1.516 100 15 4.00% 0.743 2.150 1.597 2.000 1.486 100 16 4.00% 0.728 2.150 1.566 2.000 1.457 100 17 4.00% 0.714 2.150 1.535 2.000 1.428 100 18 4.00% 0.700 2.150 1.505 2.000 1.400 100 19 4.00% 0.686 2.150 1.476 2.000 1.373 100 20 4.00% 0.673 2.150 1.447 2.000 1.346 100 103 Solution to Question 5 If 4-yr IRS 4.30% on first anniversary Swaption lapses FNMA still pays 3.66% net 4.30% 64bps Investors FNMA Bank coupons 104 Solution to Question 5 If 4-yr IRS < 4.30% on first anniversary Swaption is exercised, FNMA pays 4.30% fixed and receives Libor FNMA calls back old bonds FNMA issues FRN at Libor FNMA pays 3.66% net Old Investors Old bond Par Par New Investors FRN at Libor So correct answer is (c) FNMA Libor 4.30% 64bps Bank 105

Question 6 Turkey can borrow at 5% for 5 years in either Euros or US dollars. Spot and forward FX rates for EUR/USD are 1.50 for all maturities. Since the country has significant remittances in both of these currencies from a combination of exports (mostly in US dollars) and worker remittances (mostly in Euros), Deutsche Bank proposes to Turkey that it borrow EUR 100 MM for 5 years at 3%, provided that on the loan s first anniversary Deutsche can, in its sole discretion, convert half the loan into a 4-year USD 75MM loan at 3%, and on the second anniversary Deutsche can, in its sole discretion, convert the second half of the loan into a 3-year USD 75MM loan at 3%. 106 Question 6 (continued) How might you deconstruct the loan described above: a) Turkey has borrowed EUR 100MM at market rates and has also bought one or more European crosscurrency swaptions b) Turkey has borrowed EUR 100MM at market rates and has also bought one or more Bermudan crosscurrency swaptions c) Turkey has borrowed EUR 100MM at market rates and has also sold one or more European crosscurrency swaptions d) Turkey has borrowed EUR 100MM at market rates and has also sold one or more Bermudan crosscurrency swaptions 107 Solution to Question 6 To achieve 2% subsidy over normal 5-year rates Turkey must have sold optionality to Deutsche, not bought it And since each right to convert portion of loan into USD can be exercised on one specific date only, these are European options, not Bermudan 108

Solution to Question 6 In addition to borrowing EUR 100MM for 5 years, Turkey has sold to Deutsche i. 1x4 cross-currency swaption enabling Deutsche on loan s first anniversary to initiate cross-currency swap with Deutsche under which Deutsche will deliver EUR and receive USD, with notionals of EUR 50 and USD 75MM, and ii. 2x3 cross-currency swaption enabling Deutsche on loan s second anniversary to initiate a cross-currency swap with Deutsche under which Deutsche will deliver EUR and receive USD, with notionals of EUR 50 and USD 75MM 109 Solution to Question 6 Premium earned from selling these two crosscurrency swaptions is built into loan and reduces interest rate to 3% in either currency Therefore correct answer is (c) 110