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The Pricing of Bull and Bear Floating Rate Notes: An Application of Financial Engineering Donald J. Smith

The Pricing of Bull and Bear Floating Rate Notes: An Application of Financial Engineering Donald J. Smith. 財金所 碩一 蔡佩伶 林瑋莉 洪婉瑜. Agenda. Definition of bull & bear FRN Equilibrium pricing on bull & bear FRN without constraint

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The Pricing of Bull and Bear Floating Rate Notes: An Application of Financial Engineering Donald J. Smith

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  1. The Pricing of Bull and Bear Floating Rate Notes: An Application of Financial EngineeringDonald J. Smith 財金所 碩一 蔡佩伶 林瑋莉 洪婉瑜

  2. Agenda • Definition of bull & bear FRN • Equilibrium pricing on bull & bear FRN without constraint • Equilibrium pricing on bull & bear FRN with constraint • The pricing sensitivity of bull & bear floaters • Market condition • Example：Far Eastern Textile Bull-Bear notes

  3. PART 1 Definition of Bull & Bear FRN

  4. Definition of Bull & Bear FRN • Traditional FRN • C = LIBOR + 0.25% • Bull floaters(inverse floaters or yield curve notes) • C = 17.2% - LIBOR •  coupon rate as the market rate ( bond price ) •  attract investor who is “bullish” on bond price • Bear floaters • C = 2 LIBOR - 9.12% •  coupon rate as the market rate ( bond price ) •  attract investor who is “bearish” on bond price

  5. Definition of Bull & Bear FRN • General expression for the coupon reset formula on an FRN: • C = A R + B • ( C >= 0 non-negativity constraint ) • C: periodic coupon rate • R: variable reference rate (ex:LIBOR) • A is characteristic parameter which determines the type of the note. • How to determine B ?

  6. Definition of Bull & Bear FRN C = AR + B fixed rate note traditional FRN • 01 • bull floaterquasi-fixedbear floater • (17.2% - LIBOR) (0.5LIBOR+5%) (2LIBOR – 9.12%) • Fixed rate note: C=F A=0 , B=F • Traditional FRN: C=R+M A=1 , B=M A

  7. PART 2 Pricing on Bull & Bear FRNwithout Constraint

  8. Pricing on Bull & Bear FRN without Constraint --- Example 1 fixed rate note 10% (F) Firm Issuer Investor or LIBOR+0.25% (R+M) 9.75% (F-M) LIBOR (R) traditional FRN Interest rate swap Counterpart No arbitrage!!

  9. Pricing on Bull & Bear FRN without Constraint --- Example 1 • If the firm considers issuing a bull floater with C=X-LIBOR how to determine the break-even XB such that if X<XB , a cost saving is achieved? X-LIBOR Investor Firm Issuer 9.75% LIBOR COF = (XB - LIBOR) + (LIBOR - 9.75%) = 10% (F)  XB= 19.75% Counterpart

  10. Pricing on Bull & Bear FRN without Constraint --- Example 2 • If the firm considers issuing a bear floater with C=2*LIBOR-Y, how to determine the break-even YB such that if Y>YB, a cost saving is achieved? Firm Issuer 2 LIBOR - Y Investor LIBOR 9.75% COF = (2 LIBOR -YB) + 2(9.75% - LIBOR) = 10% (F)  YB= 9.5% Counterpart

  11. Pricing on Bull & Bear FRN without Constraint --- Generalized Floater coupon SWAP COF = AR + Bu+ A [ ( F – M ) － R] = Bu +A [ ( F – M )] Equilibrium condition : COF = F Bu = (1－A) F + AM for any A bull ex : A= -1Bu = 2 F - M = 2*10% - 0.25% = 19.75% bear ex : A= 2  Bu = (-1) F + 2M = (-10%) +2*0.25% = -9.5%

  12. Pricing on Bull & Bear FRN without Constraint Problem: • In example 1, the bull floater : C=19.75% - LIBOR What if LIBOR > 19.75% ?  COF = Max [0, (19.75% - LIBOR)] + (LIBOR - 9.75%) > 10% • In example 2, the bear floater : C= 2 LIBOR – 9.5% What if LIBOR < 4.75% ?  COF = Max [0, ( 2 LIBOR – 9.5%)] + 2 (9.75% - LIBOR) >10% C >= 0 --- the non-negativity constraint !!

  13. PART 3 Pricing on Bull & Bear FRNwith Constraint

  14. Bull Floating Rate Noteswith Constraint Section 1

  15. Problem Solution Sol Diagram Pricing of Restricted Floater Bull FRN with Constraint CF ( - ) 10% LIBOR 9.75% 19.75% ( + ) Problem: causing from SWAP If LIBOR>19.75%, the issuer can’t lock the cost of fund at 10%

  16. Problem Solution Sol Diagram Pricing of Restricted Floater • CAPCALL option on the RATE • PUT option on the PRICE • Cost of CAP(S, X, T, r, σ) 5-year semiannual settlement on 6-mon LIBOR X = 19.5% Premium = 96 b.p. 25 b.p.(per year) Bull FRN with Constraint SOLUTION To buy a INTEREST CAP

  17. Problem Solution Sol Diagram Pricing of Restricted Floater Bull FRN with Constraint COF FRN Max(0,19.5% - LIBOR) Swap LIBOR - 9.75% F = 10% CAP & its cost - Max(0,LIBOR-19.5%) + 0.25%

  18. Problem Solution Sol Diagram Pricing of Restricted Floater CF CF 9.75% LIBOR LIBOR 0.25% 0 19.5% 19.5% Buying a CAP Bull Floater + SWAP CF 10% LIBOR 0 Bull FRN with Constraint Cost of CAP ( - ) ( - ) ( + ) ( + ) ( - ) ( + )

  19. Problem Solution Sol Diagram Pricing of Restricted Floater • C = AR + Br • Bull Floater A < 0 • Cap • Payoff on Cap = Max(0, R-X) • C = AR + Br > 0 X = - Br/A • Zcap(- Br/A): amortized costs of a cap • ( ~premium of a call option ) Bull FRN with Constraint

  20. Problem Solution Sol Diagram Pricing of Restricted Floater • COF • = Max(0, AR + Br) Floater(1) + A [ ( F -M ) - R] SWAP(2) - A [ Z cap ( - Br/ A ) ] Cost of CAP(3) + A Max [ 0 , R - ( - Br/ A) ] CAP(4) • Simplification Procedure • R  - Br/ A(4) is 0 and (1) is AR + Br • R > - Br/ A(4) is AR + Br and (1) is 0 Bull FRN with Constraint COF = AR + Br + A [ ( F - M ) - R]- A [ Z cap ( - Br/ A ) ] (1) + (4) (2) (3)

  21. Problem Solution Sol Diagram Pricing of Restricted Floater Bull FRN with Constraint • COF • = AR + Br + A [(F -M) - R] - A [Z cap ( - Br/ A )] • = F if A< 0, Br = (1 - A) F + AM + A [ Z cap (- Br/ A ) ]

  22. Bear Floating Rate Noteswith Constraint Section 2

  23. Problem Solution Sol Diagram Pricing of Restricted Floater Bear FRN with Constraint ( - ) CF 10% LIBOR 9.75% 4.75% ( + ) Problem: causing from SWAP If LIBOR < 4.75%, the issuer can’t lock the cost of fund at 10%

  24. Problem Solution Sol Diagram Pricing of Restricted Floater • FLOORPUT option on the RATE • CALL option on the PRICE • Cost of FLOOR(S, X, T, r, σ) 5-year semiannual settlement on 6-mon LIBOR X = 5.25% Premium = 193 b.p. 50 b.p.(per year) Bear FRN with Constraint SOLUTION To buy a INTEREST FLOOR

  25. Problem Solution Sol Diagram Pricing of Restricted Floater Bear FRN with Constraint COF FRN Max(0, 2*LIBOR-10.5%) 2 Swaps 2*(9.75% - LIBOR) F = 10% 2 (FLOOR & its cost) - 2*Max(0, 5.25%-LIBOR) + 2*(0.5%)

  26. Problem Solution Sol Diagram Pricing of Restricted Floater CF CF 9% 1% LIBOR LIBOR 0 5.25% 5.25% Bear FRN + 2SWAP Buying two Floor CF 10% LIBOR 0 Bear FRN with Constraint ( - ) ( - ) Cost of Floor ( + ) ( + ) ( - ) ( + )

  27. Problem Solution Sol Diagram Pricing of Restricted Floater • C = AR + Br • Bear Floater A > 1 • Floor • Payoff on Floor = Max(0, X-R) • C = AR + Br > 0 X = - Br/A • Zfloor(- Br/A): amortized costs of a floor • ( ~premium of a put option ) Bear FRN with Constraint

  28. Problem Solution Sol Diagram Pricing of Restricted Floater • COF • = Max(0, AR + Br) Floater(1) + A [ ( F -M ) - R] SWAP(2) + A [ Z floor ( - Br/ A ) ] Cost of Flo(3) - A Max [ 0 , ( - Br/ A) - R ] FLOOR(4) • Simplification Procedure • R  - Br/ A(4) is AR + Br and (1) is 0 • R > - Br/ A(4) is 0 and (1) is AR + Br Bear FRN with Constraint COF = AR + Br + A [ ( F – M ) - R]+ A [ Z floor ( - Br/ A ) ] (2) (3) (1) + (4)

  29. Problem Solution Sol Diagram Pricing of Restricted Floater Bull FRN with Constraint • COF • = AR + Br + A [(F -M) - R] + A [Z floor( - Br/ A )] • = F if A > 1, Br = (1 - A) F + AM - A [ Z floor(- Br/ A ) ]

  30. Generalized Pricing ofBull & Bear Floater Section 3

  31. Generalized Equilibrium Pricing • if A < 0 Bull Floater Br = ( 1- A ) F + AM + A [ Z cap ( - Br/ A ) ] • if A > 1 Bear Floater Br = ( 1 – A ) F + AM - A [ Z floor ( - Br/ A ) ] replicated portfolio

  32. PART 4 The Pricing Sensitivity of Bull & Bear Floaters

  33. Pricing Sensitivity of Fixed Rate Notes Price 115 110 103.24 105 Traditional fixed rate note at 10%： • Market rate & price are • NEGATIVELY related 100.00 100 95 96.89 90 8% 9%10%11% 12% Market fixed rates for four years to maturity

  34. Pricing Sensitivity of Traditional FRN Price 115 Traditional FRN at LIBOR+0.25%：  Coupon rate in line with market yield  Price is near PAR  LESS price sensitive 110 105 FRN 100 95 Fixed 90 8% 9% 10% 11% 12% Market fixed rates for four years to maturity

  35. Pricing sensitivity of Bull Floater Price Bull floater at 19.5-LIBOR： F to 11% • Future coupon , DR • MORE price sensitive • New bull floater: • 21.5% - LIBOR • Opportunity loss of about 200bp 115 110 105 FRN 100 96.89 100 Fixed 95 93.67 Bull 90 8% 9% 10%  11% 12% Market fixed rates for four years to maturity formula for Br

  36. Pricing sensitivity of BearFloater Bear floater at 2*LIBOR-10.5%： F to 11% • Future coupon , DR • ΔDR < Δfuture coupon • Market rate & price are POSITIVELY related • New bear floater: • 2*LIBOR - 11.5% • Opportunity gain of about 100bp Price Bear 115 103.17 110 105 FRN 100 100 95 Fixed Bull 90 8% 9% 10%  11% 12% Market fixed rates for four years to maturity formula for Br

  37. Implied Duration NEGATIVE Duration Price Bear floater at 2*LIBOR-10.5% 115 110 D = the time until the next reset date Traditional FRN at LIBOR+0.25% 105 100 Traditional fixed rate note at 10% 95 LONGER duration than fixed rate notes 90 Bull floater at 19.5-LIBOR 8% 9% 10% 11% 12% Market fixed rates for four years to maturity

  38. Duration of Replicated Portfolio • Coupon reset formula： Cr = AR+Br formula for Br = A [R+M+Zcap]+(1-A)Fif A<0 A [R+M]+(1-A)Fif 0<A<1 A [R+M+Zfloor]+(1-A)Fif A>1 Bull Quasi Bear • Cr ：portfolio of capped / floored floaters & fixed rate notes • Cr>0： Bull：Max(Cr) = -F(1-A) / A • Bear：Min(Cr) = -F(1-A) / A • Dbull/bear= ADcap/floor + (1-A)Dfixed

  39. PART 5 Market Condition

  40. Trend of LIBOR 1 yr：7.5% 1 yr：3.7% Redemption rage of bond fund Recession 1 yr：1.0%

  41. PART 6 Example： Far Eastern Textile Bull-Bear Notes

  42. Terms COF Diagram Historical coupon Example: Far Eastern Textile Bear Floater Bond Max(0, 7.5% + (R - 6.9%) ) Bull Floater Bond Max(0, 7.5%+ (6.9% - R) Cater to investors’ different needs for floaters CAP R<14.4% Floor R>14.4% Cater to Far Eastern’s need for fixed rate debts Lock at 15%

  43. Example: Far Eastern Textile CF • Terms • COF Diagram • Historical coupon CF 14.40% LIBOR 0.6% LIBOR 0 0 14.40% 14.40% Max(0, 14.40%-R)+Cap Max(0, R+0.6%)+Floor CF 15% LIBOR 0

  44. Example: Far Eastern Textile • Terms • COF Diagram • Historical coupon Bull Bear Bull Bear Reference Rate

  45. Equilibrium pricing condition of Bull & Bear floaters: Implicit rate on synthetic structure equals the explicit alternative Bull：more sensitive to market rate than fixed rate higher interest risk Bear：price positively related to market rates negative duration Conclusions Notice the role bull & bear floaters play in interest rate risk management

  46. Thanks for Your Attention

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