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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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slide1

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

slide2

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
definition of bull bear frn4
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
definition of bull bear frn5
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 frn6
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

pricing on bull bear frn without constraint example 1
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!!

pricing on bull bear frn without constraint example 19
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
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

pricing on bull bear frn without constraint generalized
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%

pricing on bull bear frn without constraint12
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 !!

bull frn with constraint
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%

bull frn with constraint16
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

bull frn with constraint17
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%

bull frn with constraint18
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

( - )

( - )

( + )

( + )

( - )

( + )

bull frn with constraint19
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
bull frn with constraint20
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)

bull frn with constraint21
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 frn with constraint
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%

bear frn with constraint24
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

bear frn with constraint25
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%)

bear frn with constraint26
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

( + )

( + )

( - )

( + )

bear frn with constraint27
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
bear frn with constraint28
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)

bull frn with constraint29
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 equilibrium pricing
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

pricing sensitivity of fixed rate notes
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

pricing sensitivity of traditional frn
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

slide35

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

slide36

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

slide37

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

slide38

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
part 5

PART 5

Market Condition

slide40

Trend of LIBOR

1 yr:7.5%

1 yr:3.7%

Redemption rage of bond fund

Recession

1 yr:1.0%

part 6

PART 6

Example:

Far Eastern Textile Bull-Bear Notes

example far eastern textile
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 textile43
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 textile44
Example: Far Eastern Textile
  • Terms
  • COF Diagram
  • Historical coupon

Bull

Bear

Bull

Bear

Reference Rate

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

Conclusions

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