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

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

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

PART 1

### Definition of Bull & Bear FRN

Definition of Bull & Bear 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
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 ?
Definition of Bull & Bear FRN

C = AR + B

• 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

PART 2

### Pricing on Bull & Bear FRNwithout Constraint

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)

Interest rate swap

Counterpart

No arbitrage!!

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

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

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%

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 !!

PART 3

### Bull Floating Rate Noteswith Constraint

Section 1

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%

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

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%

Problem

Solution

Sol Diagram

Pricing of Restricted Floater

CF

CF

9.75%

LIBOR

LIBOR

0.25%

0

19.5%

19.5%

Bull Floater + SWAP

CF

10%

LIBOR

0

Bull FRN with Constraint

Cost of CAP

( - )

( - )

( + )

( + )

( - )

( + )

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
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)

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 ) ]

### Bear Floating Rate Noteswith Constraint

Section 2

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%

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

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%)

Problem

Solution

Sol Diagram

Pricing of Restricted Floater

CF

CF

9%

1%

LIBOR

LIBOR

0

5.25%

5.25%

Bear FRN + 2SWAP

CF

10%

LIBOR

0

Bear FRN with Constraint

( - )

( - )

Cost of Floor

( + )

( + )

( - )

( + )

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
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)

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 ) ]

### Generalized Pricing ofBull & Bear Floater

Section 3

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

PART 4

### The Pricing Sensitivity of Bull & Bear Floaters

Pricing Sensitivity of Fixed Rate Notes

Price

115

110

103.24

105

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

Price

115

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

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

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

Implied Duration

NEGATIVE

Duration

Price

Bear floater at

2*LIBOR-10.5%

115

110

D = the time until the next reset date

LIBOR+0.25%

105

100

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

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

### PART 5

Market Condition

Trend of LIBOR

1 yr：7.5%

1 yr：3.7%

Redemption rage of bond fund

Recession

1 yr：1.0%

### PART 6

Example：

Far Eastern Textile Bull-Bear Notes

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%

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

Example: Far Eastern Textile
• Terms
• COF Diagram
• Historical coupon

Bull

Bear

Bull

Bear

Reference Rate

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