The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads The Oxford Guide to Financial Modeling

The Oxford Guide to Financial Modeling by Ho & Lee

Chapter 8. Investment Grade Corporate Bonds: Option Adjusted

Spreads

The Oxford Guide to

Financial Modeling

Thomas S. Y. Ho and Sang Bin Lee

Copyright © 2004 by Thomas Ho and Sang Bin Lee. All rights reserved.

Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads 2


8.1 Describing a Corporate Bond

• Terms and conditions

• Bond Type

• Coupon Description

• Issue size

• Bond Call Provision

• Bond Sinking Fund Provision



Outstanding bond market debt as of March 31, 2002



8.2 Valuation of a Bond

• Price Quote in Terms of Yield– Yield to maturity– Yield to worst– Yield spread

• Callability: Callable bond price = Non-callable bond price - Call option value

• Sinking Fund• Option Adjusted Spread (OAS)



The double-up optionremaining principal

at maturity1st year 2nd year ( 3rd year )

no double up( - 10 $ )

double up( -20 $ )

double up( -20 $ )

case4



$80

100 $

100 $

case1

case2

case3

$60

$70

$70

double up( -20 $ )



Option adjusted spread 1

Corporate Bond Spreads for Industrial

0

200

400

600

800

1000

1200

1400

1600

1 2 3 5 7 10 30

Time to Maturity

Basi

s Poin

t

Aaa/ AAAAa1/ AA+Aa2/ AAAa3/ AA-A1/ A+A2/ AA3/ A-Baa1/ BBB+Baa2/ BBBBaa3/ BBB-Ba1/ BB+Ba2/ BBBa3/ BB-B1/ B+B2/ BB3/ B-Caa/ CCC




Corporate Bond Spreads for Financials

0

100

200

300

400

500

600

700

800

900

1000

1 2 3 5 7 10 30

Time to Maturity

Basi

s Poin

ts





Corporate Bond Spreads for Bank

0

100

200

300

400

500

600

700

800

900

1000

1 2 3 5 7 10 30

Time to Maturity

Basi

s Poin

t





Corporate Bond Spreads for Transportation

0

200

400

600

800

1000

1200

1 2 3 5 7 10 30

Time to Maturity

Basi

s Poin

t





Corporate Bond Spreads for Utility

0

200

400

600

800

1000

1200

1 2 3 5 7 10 30

Time to Maturity

Basi

s Poin

t




8.3 Bond Model

• Valuation of a Bond with No Embedded Options

- principal : $100, annual coupon payment: $7

maturity: 6 years, spot yield: 6.5%

2 3 4 5 6

Bond Price

7 7 7 7 7 107

1.065 (1.065) (1.065) (1.065) (1.065) (1.065)

102.421



8.3 Callable Bond Pricing• Valuation of a Callable (I)- The call price schedule (linearly declining)

$106(0year), $105(1year), · · · , $100(maturity)- Using Ho-Lee one-factor model- Assumption

volatility: 15%, yield: 6.5%,

nominal volatility(σ): 0.15 x 0.065

P(T)= e-0.065T: discount function



Arbitrage-free Interest Rate Movement• Use backward substitution to determine the bond

price• Binomial annual discount rate [P(n, i, 1) ]:

- P(3, 3, 1)

2 ( 1)( , ,1)

( ) 1

exp 2 0.15 0.065 0.98068

n i

n

P nP n i

P n

4 3 3

3 3

(1.065) 0.9806882 0.96447

(1.065) (1 0.980688 )



A coupon bond price

107.000

102.382

100.045 107.000

99.633 104.260

100.933 103.619 107.000

103.845 104.837 106.176

101.366 107.801 107.332 107.000

112.502 110.344 108.128

115.201 111.191 107.000

116.174 110.120

115.200 107.000

112.150

107.000

Today 1 yr 2 yr 3 yr 4 yr 5 yr 6 yr



A callable bond price

107.000

102.382

100.045 107.000

99.633 104.260

100.920 103.619 107.000

103.602 104.809 106.176

100.208 107.289 107.272 107.000

110.273 109.280 108.000

111.000 109.000 107.000

110.000 108.000

109.000 107.000

108.000

107.000




Option Adjusted Spread (OAS)

• Option Adjusted Spread (OAS) is the constant spread added to the one period short rate such that the fair value of the bond equals the observed bond price

• OAS incorporates the credit risk and marketability

- Assumption The callable bond has credit risk and marketability. bond price: $99.5



Static Spread and the OAS• Static Spread: spread determined from the expected

payments based on the forward yield curve• Option Adjusted Spread: based on the binomial lattice

model The promised cash flow from the callable bond The option-adjusted cash flows from the callable bond

107.000

7.000

7.000 107.000

7.000 7.000

7.000 7.000 107.000

7.000 7.000 7.000

99.500 7.000 7.000 107.000

7.000 7.000 7.000

7.000 7.000 107.000

7.000 7.000

7.000 107.000

7.000

107.000


107.000

102.101

99.562 107.000

99.004 104.021

100.199 103.209 107.000

102.839 104.306 105.981

99.500 106.759 107.002 107.000

109.877 109.007 107.980

111.000 109.000 N/A

110.000 N/A

N/A N/A

N/A

N/A




Sinking Fund Bond • Valuation of a Sinking Fund Bond with No Market

Purchase Option and No Call Provision

- The principal repayment and interest schedule

- Calculation

Year 0 1 2 3 4 5 6

Principal to be retired each year

0 0 　 0 　 0 　 10 10 80

Outstanding amount at the beginning of each year

100 　 100 　 100 　 100 100 90 80

Remaining principal At the end of each year

100 100 100 100 90 80 0

Interests 　 100*0.07 100*0.07 100*0.07 100*0.07 90*0.07 80*0.07

2 3 4 5 6

7 7 7 17 16.3 85.6102.315

1.065 (1.065) (1.065) (1.065) (1.065) (1.065)



Delivery Option of a Sinking Fund• Delivery Option: the issuer can satisfy the sinking fund

requirement by open market purchase or calling the bonds at par

- Terminal condition at maturity: bond value is 80 * (1.07)- $92.144 at year 5 =

985.6 0.89142 , 85.6 0.89142 10 90 0.07

8Min



Sinking fund bond

85.600

92.144

100.045 85.600

99.633 93.834

100.915 103.619 85.600

103.755 104.797 95.558

101.155 107.625 107.246 85.600

112.143 110.010 97.203

114.616 110.569 85.600

115.284 98.796

113.978 85.600

100.420

85.600




Double Up Sinking Fund• Double-up Sinking Fund provides issuers to retire

twice the sinking fund amount on each sinking fund date

- The double-up sinking fund principal process

case 4th year 5th yearremaining principal

at maturity

no double up

1

2

3

4

double up

double up

no double up

double up

no double up

$80

$60

$70

$70



8.3 Numerical Example(9)• Valuation of a Double-up Sinking Fund (II)

- The binomial lattices for each case

Case 164.200

81.906

100.045 64.200

99.633 83.408

100.896 103.619 64.200

103.665 104.757 84.940

100.945 107.449 107.160 64.200

111.783 109.675 86.277

114.031 109.947 64.200

114.395 87.472

112.755 64.200

88.690

64.200




Case 2

74.900

81.906

100.045 74.900

99.633 83.408

100.908 103.619 74.900

103.702 104.782 84.940

101.021 107.518 107.213 74.900

111.908 109.796 86.390

114.228 110.150 74.900

114.687 87.784113.158 74.900

89.205

74.900




Case 3

74.900

92.144

100.045 74.900

99.633 93.834

100.904 103.619 74.900

103.718 104.772 95.558

101.079 107.557 107.193 74.900

112.017 109.889 97.090

114.419 110.366 74.900

114.992 98.484

113.575 74.900

99.905

74.900




Case 4

85.600

92.144

100.045 85.600

99.633 93.834

100.915 103.619 85.600

103.755 104.797 95.558

101.155 107.625 107.246 85.600

112.143 110.010 97.203

114.616 110.569 85.600

115.284 98.796

113.978 85.600

100.420

85.600




Double Up Option Model• Valuation of a Double-up Sinking Fund (II)

- Denote: Bj(n, i)

where j denotes the jth case, n is the period, and i is the state.- The bond value for the end of year, using cases 1 and 2.

- Similarly , for cases 3 and 4 at the end of year 5

- The appropriate value under the optimal decision at the end of year 4

1,2 1 2B (5, ) (5, ), (5, ) for each i=0,1,2,3,4,5i Min B i B i

3,4 3 4B (5, ) (5, ), (5, ) for each i=0,1,2,3,4,5i Min B i B i

1,2 3,4B(4, ) (4, ), (4, ) for each i=0,1,2,3,4i Min B i B i



Valuation of a Double Up Sinking Fund Bond

• Lattice of the bond prices from the end of the fourth year

to the starting date: NOTE that this figure is wrong!!100.045

99.633

100.896 103.619

103.665 104.757

100.945 107.449 107.160

111.783 109.675

114.031 109.947

114.395

112.755

Today 1 yr 2 yr 3 yr 4 yr



8.4 Liquidity (Marketability) Spread

• Liquidity Spread is the addition return of the bond for the lack of marketability.

• Treasury STRIPS= default free liquid bonds

• U.S. Government-backed mortgage securities– Also has liquidity spread in addition to the spread of the prepayment risk

-

Default free illiquid bond Treasury equivalent bond

Knock in option on liquidity



8.5 Credit Scoring Approaches

• A scoring system to determine the credit risk of a bond

• Altman’s Z score

1 2 3 4 5

1

2

3

1.4 1.2 3.3 0.6 1.0 ,

= working capital/ total assets (%),

= retained earnings/total assets (%),

= earnings before interest and taxes/ total assets

Z x x x x x

x

x

x

4

5

(%),

= market value of equity / total liabilities (%),

= sales/ total assets(%).

x

x



8.6 Bond Analysis

• Cheap/Rich Analysis– Relative value a bond with other bonds via a bond model

• Effective duration– Exposure to parallel movement of the yield curve

effective duration spot yieldP

P



8.6 Bond Analysis (2)

• Key rate duration– Exposure to the yield curve risks

Key Rate Duration

- 1

0

1

2

3

4

5

6

7

8

9

0.25 1 2 3 5 7 10 15 20 25 30 dur

Time to maturity

(Key

Rat

e) D

urat

ion

9% coupon bond 8% coupon bond



8.6 Bond Analysis (3)

• OAS duration– Exposure to the change of the OAS

• Convexity– Exposure to a large yield curve movement, particularly for

the option embedded bonds

*OAS OAS

OAS duration = effective duration



8.7 Valuing a Eurobond

• Terms and conditions of the new issue- Face value: 50 million euros- Annual coupon rate: 4.2%(callable at par 2/03)- 4.6% 2003-2005- Issuing day: February 21, 2002- Maturity day: February 21, 2005



8.7 A 2-factor modelto Value the Callable Bond

• The steps to price the Eurobond- Step1. Specify the swap curve, which we assume to be 4% flat.

- Step2. Specify the volatility surface, applying the Ho-Lee two-factor model.

- Step 3. Construct the binomial lattice

- Step 4. Value by backward substitution

1yr 2yr 3yr 4yr 5yr 20yr 30yr

σ 1 0.15 0.14 0.13 0.12 0.1 0.1 0.1

σ 2 0.1 0.1 0.1 0.1 0.1 0.1 0.1



8.7 Optimal Call Condition on the Call Date

- Determining the value of each node point on 2003 call date

- X(1, i)* is the value of the bond after using the backward substitution, rolling back from the maturity of the bond

- Or X(1,i)* is the bond value at each node point with no option, applying the bond model

*1, 1, ,100X i Min X i



8.7 Valuation of the Bond: rolling back from the call date

• A binomial lattice of a Eurobond104.6000

104.6000

104.6000

100.4075 104.6000

101.5641

102.7347 104.6000

94.3863 104.6000

96.6269 104.6000

101.9138 104.6000

91.4491 103.0886

104.2776 104.6000

97.5433 104.6000

99.8595 104.6000

103.4438 104.6000

104.6370

105.8447 104.6000

104.6000

104.6000

104.6000

0 1 yr 2 yr 3 yr



8.8 Applications of Bond Analytics

• Total Return Approach– The valuation model can simulate the bond returns under

different market scenarios

• Managing Interest Rate Risk and Basis Risks– Use key rate durations, duration/convexity, OAS duration

to control each risk exposure

• Index Enhancement Strategy and Asset/Liability Management– Use the index or liability as benchmark to target the risk

exposures



8.9 Explaining the Concept of the Arbitrage-free Condition on a Solemn

Occasion

-0.02 -0.01 0 0.01 0.02

Yield Curve Shift

96

97

98

99

100

101

102

elballaCdnoB

Callable bondpricing by HoLee 2 factor model



Appendix: Callable Bond and Sinking Fund Bond Pricing

• Risk-neutral Pricing Proposition

, 0.5 1, 1 1, 1niB n i B n i B n i P

1 21

1

n in

i n

P nP

P n



Callable bond

11, 1 1Tc iX T i c P

*11, 1, ,c c TB T i Min X T i k

*1, 1,c cB T i B T i c

22, 0.5 1, 1 1, 1Tc c c iX T i B T i B T i P *

22, 2, ,c c TB T i Min X T i k *2, 2,c cB T i B T i c

, 0.5 1, 1 1, 1nc c c iX n i B n i B n i P , , ,c c nB n i Min X n i k c



Sinking Fund Bond

11, 1 1Ts T iX T i F c P

11, 1 1Ts T iX T i F c P

* 111, 1, , 1,T

s s T T sT

FB T i Min X T i F F X T i

F

*11, 1,s s TB T i B T i cF

, 0.5 1, 1 1, 1ns s s iX n i B n i B n i P

*

1

, ,ns s

n

FX n i X n i

F



Sinking Fund Bond (2)

*1

1

, , , ,ns s n n s

n

FB n i Min X n i F F X n i

F

*, ,s s nB n i B n i cF

,) 0.5 1, 1 1, 1ns s s iX n B n i B n i P

1

1

1 1, , , 1 ,n

s s sn n n

FB n i Min X n i X n i c

F F F

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The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads The Oxford Guide to Financial Modeling