The Pricing of Bull and Bear Floating Rate Notes - An Application of Financial Engineering

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

 _{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

Definition of Bull & Bear FRN

Definition of Bull & Bear FRN



_{Traditional FRN}

C = LIBOR + 0.25%



Bull floaters

 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

Definition of Bull & Bear FRN



C = A R + B

C: periodic coupon rate

R: variable reference rate (ex:LIBOR)



Definition of Bull & Bear FRN

fixed rate note traditional FRN

bull floater

quasi-fixed

bear 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

Pricing on Bull & Bear FRN

without Constraint

Pricing on Bull & Bear FRN

without Constraint --- Example 1

Firm Issuer Counterpart Investor 10%

(F)

LIBOR

(R)

9.75%

(F-M)

Interest

rate swap

traditional FRN

fixed rate note

Pricing on Bull & Bear FRN

without Constraint --- Example 1



how to determine the break-even X_B such that if X<X_B , a cost saving is achieved?

Firm Issuer Investor

Counterpart COF = (X_B - LIBOR) + (LIBOR - 9.75%) = 10% (F)  X_B= 19.75% LIBOR 9.75% X-LIBOR

Pricing on Bull & Bear FRN

without Constraint --- Example 2



C=2*LIBOR-Y, how to determine the break-even Y_B such that if Y>Y_B, a cost saving is achieved?

Firm Issuer _Investor

Counterpart COF = (2 LIBOR -Y_B) + 2(9.75% - LIBOR) = 10% (F)  Y_B= 9.5% LIBOR 2 LIBOR - Y 9.75%

Pricing on Bull & Bear FRN

without Constraint --- Generalized

COF =

AR + B

+

A [ ( F – M )

－ R]

=

B

+A [ ( F – M )]

SWAP Floater coupon

Equilibrium condition :

COF = F

B

= (1

－ A) F + AM for any A

bull ex : A= -1 Bu = 2 F - M = 2*10% - 0.25% = 19.75%

Pricing on Bull & Bear FRN

without Constraint

Problem



 COF = Max [0, (19.75% - LIBOR)] + (LIBOR - 9.75%) > 10%



 COF = Max [0, ( 2 LIBOR – 9.5%)] + 2 (9.75% - LIBOR) >10%

_{C >= 0}

_{the non-negativity constraint !!}

Pricing on Bull & Bear FRN

with Constraint

Bull Floating Rate Notes

with Constraint

Bull FRN with Constraint

Problem: causing from SWAP If LIBOR>19.75%,

the issuer can’t lock the cost of fund at 10%

 _Problem  _Solution  _{Sol Diagram}  _{Pricing of} Restricted Floater ( - ) ( + )

Bull FRN with Constraint



_CAP

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) To buy a INTEREST CAP

Bull FRN with Constraint

CAP & its cost

- Max(0,LIBOR-19.5%) + 0.25%

CF LIBOR Buying a CAP 19.5% 0.25% LIBOR CF 10% 0

Bull FRN with Constraint

 _Problem  _Solution  _{Sol Diagram}  _{Pricing of} Restricted Floater 9.75% CF 19.5% LIBOR 0

Bull Floater + SWAP

Cost of CAP ( - ) ( + ) ( + ) ( + ) ( - ) ( - )

Bull FRN with Constraint

C = AR + B

_Cap

Payoff on Cap = Max(0, R-X) C = AR + Br > 0 X = - Br/A

Z_cap(- Br/A): amortized costs of a cap

( ~premium of a call option )

Bull FRN with Constraint

_COF

- A [ Z _cap ( - Br / A ) ] _{Cost of CAP(3)}

+ A Max [ 0 , R - ( - Br / A) ] _CAP(4) 

_{Simplification Procedure}

Bull FRN with Constraint

_COF

Bear Floating Rate Notes

with Constraint

Bear FRN with Constraint

Problem: causing from SWAP If LIBOR < 4.75%,

the issuer can’t lock the cost of fund at 10%

 _Problem  _Solution  _{Sol Diagram}  _{Pricing of} Restricted Floater 4.75% LIBOR 10% 9.75% CF ( - ) ( + )

Bear FRN with Constraint



_FLOOR

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) To buy a INTEREST FLOOR

Bear FRN with Constraint

F = 10%

2 (FLOOR & its cost)

- 2*Max(0, 5.25%-LIBOR) + 2*(0.5%)

Buying two Floor CF LIBOR 5.25% 1% LIBOR CF 10% 0

Bear FRN with Constraint

 _Problem  _Solution  _{Sol Diagram}  _{Pricing of} Restricted Floater 9% CF 5.25% LIBOR 0 Bear FRN + 2SWAP Cost of Floor ( - ) ( + ) ( + ) ( - ) ( - ) ( + )

Bear FRN with Constraint

C = AR + B

Floor

Payoff on Floor = Max(0, X-R) C = AR + Br > 0 X = - Br/A

Z_floor(- Br/A): amortized costs of a floor

( ~premium of a put option )

Bear FRN with Constraint

COF

+ A [ Z _floor ( - Br / A ) ] _{Cost of Flo(3)}

- A Max [ 0 , ( - Br / A) - R ] _FLOOR(4) 

Simplification Procedure

Bull FRN with Constraint

_COF

Generalized Pricing of

Bull & Bear Floater

Generalized Equilibrium Pricing

_{if A < 0  Bull Floater}

_{if A > 1  Bear Floater}

PART 4

The Pricing Sensitivity of

Bull & Bear Floaters

Traditional fixed

rate note at 10%

_：

 Market rate & price are NEGATIVELY related

Price

Market fixed rates for four years to maturity 8% 9% 10% 11% 12% 115 100 95 90 110 105

Pricing Sensitivity of Fixed Rate Notes

96.89 103.24

Pricing Sensitivity of Traditional FRN

Traditional FRN at

LIBOR+0.25%

_：

 Coupon rate in line with market yield  Price is near PAR

 LESS price sensitive

Price

Market fixed rates for four years to maturity 8% 9% 10% 11% 12% 115

100

Pricing sensitivity of Bull Floater

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

Price

Market fixed rates for four years to maturity 8% 9% 10%  11% 12% 115 100 95 90 110 105 93.67 96.89 100 formula for Br Fixed FRN Bull

Pricing sensitivity of Bear

Floater

Bear floater at

2*LIBOR-10.5%

_：

are POSITIVELY related

 New bear floater: 2*LIBOR - 11.5%

 Opportunity gain of about 100bp

Price

Market fixed rates for four years to maturity 8% 9% 10%  11% 12% 115 100 95 90 110 105 100 103.17 formula for Br Fixed FRN Bull Bear

Implied Duration

Price

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

LONGER duration than fixed rate notes



Coupon reset formula

：

C

= AR+B

=

A

[R+M+Z

]+

(1-A)

F

if A<0

A

[R+M]+

(1-A)

F

if 0<A<1

A

[R+M+Z

]+

(1-A)

F

if A>1

Duration of Replicated Portfolio

Bull

Quasi

Bear

 Cr ： portfolio of capped / floored floaters & fixed rate

notes

 Cr>0 ： Bull ： Max(Cr) = -F(1-A) / A

Bear _{： Min(C}r_{) = -F(1-A) / A}

 D

Market Condition

PART

5

Trend of LIBOR

Example

：

Far Eastern Textile Bull-Bear Notes

Example: Far Eastern Textile

 _{COF Diagram}

 _Historical

coupon

Bear Floater Bond

Max(0, 7.5% + (R - 6.9%) ) CAP R<14.4% Lock at 15% Floor R>14.4%

Bull Floater Bond

Max(0, 7.5%+ (6.9% - R)

Cater to investors’

different needs for

floaters

Cater to Far

Eastern’s need for

fixed rate debts

14.40% CF 14.40% LIBOR 0 Max(0, 14.40%-R)+Cap CF 0.6% LIBOR 0 Max(0, R+0.6%)+Floor LIBOR CF 15% 0 14.40%

Example: Far Eastern Textile

 _Terms

 _{COF Diagram}

 _Historical

 _Terms

 _{COF Diagram}

 _Historical

coupon

Example: Far Eastern Textile

4 5 6 7 8 9 10 11

Sep-94 Dec-94 Mar-95 Jun-95 Sep-95 Dec-95 Mar-96 Jun-96 Sep-96 Dec-96 Mar-97 Jun-97 Sep-97 Dec-97 Mar-98 Jun-98 Sep-98 Dec-98 Mar-99 Jun-99 Bull Bear

Reference Rate

Bull Bear

Conclusions



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

Notice the role bull & bear floaters play in interest rate risk management