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.

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads2 8.1 Describing a Corporate Bond Terms and conditions Bond Type Coupon Description Issue size Bond Call Provision Bond Sinking Fund Provision

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads3 Outstanding bond market debt as of March 31, 2002

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads4 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 Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads5 The double-up option remaining principal at maturity 1st year2nd year( 3rd year ) no double up ( - 10 $ ) double up ( -20 $ ) double up ( -20 $ ) case4 no double up ( - 10 $ ) no double up ( - 10 $ ) $ $ case1 case2 case3 $60 $70 double up ( -20 $ )

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads6 Option adjusted spread 1

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads7 Option adjusted spread 2

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads8 Option adjusted spread 3

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads9 Option adjusted spread 4

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads10 Option adjusted spread 5

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads Bond Model Valuation of a Bond with No Embedded Options - principal : $100, annual coupon payment: $7 maturity: 6 years, spot yield: 6.5%

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads 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 P(T)= e T : discount function

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads13 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)

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads14 A coupon bond price Today1 yr2 yr3 yr4 yr5 yr6 yr

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads15 A callable bond price Today1 yr2 yr3 yr4 yr5 yr6 yr

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads16 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

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads17 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 Today1 yr2 yr3 yr4 yr5 yr6 yr N/A N/A Today1 yr2 yr3 yr4 yr5 yr6 yr

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads18 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 Principal to be retired each year Outstanding amount at the beginning of each year Remaining principal At the end of each year Interests 100* *0.0780*0.07

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads19 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) - $ at year 5 =

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads20 Sinking fund bond Today1 yr2 yr3 yr4 yr5 yr6 yr

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads21 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 case4th year5th year remaining principal at maturity no double up double up no double up double up no double up $80 $60 $70

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads Numerical Example(9) Valuation of a Double-up Sinking Fund (II) - The binomial lattices for each case Case Today1 yr2 yr3 yr4 yr5 yr6 yr

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads23 Case Today1 yr2 yr3 yr4 yr5 yr6 yr

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads24 Case Today1 yr2 yr3 yr4 yr 5 yr 6 yr

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads25 Case Today1 yr2 yr3 yr4 yr5 yr6 yr

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads26 Double Up Option Model Valuation of a Double-up Sinking Fund (II) - Denote: B j (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

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads27 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!! Today1 yr2 yr3 yr4 yr

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

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads Credit Scoring Approaches A scoring system to determine the credit risk of a bond Altman’s Z score

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

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads Bond Analysis (2) Key rate duration –Exposure to the yield curve risks

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

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads 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% Issuing day: February 21, Maturity day: February 21, 2005

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads A 2-factor model to 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 1yr2yr3yr4yr5yr20yr30yr σ σ 2 0.1

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

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads Valuation of the Bond: rolling back from the call date A binomial lattice of a Eurobond yr2 yr3 yr

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

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads Explaining the Concept of the Arbitrage-free Condition on a Solemn Occasion

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads39 Appendix: Callable Bond and Sinking Fund Bond Pricing Risk-neutral Pricing Proposition

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads40 Callable bond

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads41 Sinking Fund Bond

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads42 Sinking Fund Bond (2)