1. 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.

2. 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

3. 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

4. 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)

5. 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 $ ) $80 100 $ case1 case2 case3 $60 $70 double up ( -20 $ )

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

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

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

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

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

11. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads11 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%

12. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads12 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

13. 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)

14. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads14 A coupon bond price 107.000 102.382 100.045107.000 99.633104.260 100.933103.619107.000 103.845104.837106.176 101.366107.801107.332107.000 112.502110.344108.128 115.201111.191107.000 116.174110.120 115.200107.000 112.150 107.000 Today1 yr2 yr3 yr4 yr5 yr6 yr

15. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads15 A callable bond price 107.000 102.382 100.045107.000 99.633104.260 100.920103.619107.000 103.602104.809106.176 100.208107.289107.272107.000 110.273109.280108.000 111.000109.000107.000 110.000108.000 109.000107.000 108.000 107.000 Today1 yr2 yr3 yr4 yr5 yr6 yr

16. 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

17. 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 107.000 7.000 107.000 7.000 107.000 7.000 99.5007.000 107.000 7.000 107.000 7.000 107.000 7.000 107.000 Today1 yr2 yr3 yr4 yr5 yr6 yr 107.000 102.101 99.562107.000 99.004104.021 100.199103.209107.000 102.839104.306105.981 99.500106.759107.002107.000 109.877109.007107.980 111.000109.000N/A 110.000N/A Today1 yr2 yr3 yr4 yr5 yr6 yr

18. 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 Year0123456 Principal to be retired each year 00 0 0 10 80 Outstanding amount at the beginning of each year 100 9080 Remaining principal At the end of each year 100 90800 Interests 100*0.07 90*0.0780*0.07

19. 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) - $92.144 at year 5 =

20. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads20 Sinking fund bond 85.600 92.144 100.04585.600 99.63393.834 100.915103.61985.600 103.755104.79795.558 101.155107.625107.24685.600 112.143110.01097.203 114.616110.56985.600 115.28498.796 113.97885.600 100.420 85.600 Today1 yr2 yr3 yr4 yr5 yr6 yr

21. 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 1 2 3 4 double up no double up double up no double up $80 $60 $70

22. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads22 8.3 Numerical Example(9) Valuation of a Double-up Sinking Fund (II) - The binomial lattices for each case Case 1 64.200 81.906 100.04564.200 99.63383.408 100.896103.61964.200 103.665104.75784.940 100.945107.449107.16064.200 111.783109.67586.277 114.031109.94764.200 114.39587.472 112.75564.200 88.690 64.200 Today1 yr2 yr3 yr4 yr5 yr6 yr

23. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads23 Case 2 74.900 81.906 100.04574.900 99.63383.408 100.908103.61974.900 103.702104.78284.940 101.021107.518107.21374.900 111.908109.79686.390 114.228110.15074.900 114.68787.784 113.15874.900 89.205 74.900 Today1 yr2 yr3 yr4 yr5 yr6 yr

24. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads24 Case 3 74.900 92.144 100.04574.900 99.63393.834 100.904103.61974.900 103.718104.77295.558 101.079107.557107.19374.900 112.017109.88997.090 114.419110.36674.900 114.99298.484 113.57574.900 99.905 74.900 Today1 yr2 yr3 yr4 yr 5 yr 6 yr

25. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads25 Case 4 85.600 92.144 100.04585.600 99.63393.834 100.915103.61985.600 103.755104.79795.558 101.155107.625107.24685.600 112.143110.01097.203 114.616110.56985.600 115.28498.796 113.97885.600 100.420 85.600 Today1 yr2 yr3 yr4 yr5 yr6 yr

26. 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

27. 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!! 100.045 99.633 100.896103.619 103.665104.757 100.945107.449107.160 111.783109.675 114.031109.947 114.395 112.755 Today1 yr2 yr3 yr4 yr

28. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads28 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

29. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads29 8.5 Credit Scoring Approaches A scoring system to determine the credit risk of a bond Altman’s Z score

30. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads30 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

31. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads31 8.6 Bond Analysis (2) Key rate duration –Exposure to the yield curve risks

32. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads32 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

33. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads33 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

34. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads34 8.7 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 1 0.150.140.130.120.1 σ 2 0.1

35. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads35 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

36. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads36 8.7 Valuation of the Bond: rolling back from the call date A binomial lattice of a Eurobond 104.6000 100.4075104.6000 101.5641 102.7347104.6000 94.3863104.6000 96.6269104.6000 101.9138104.6000 91.4491103.0886 104.2776104.6000 97.5433104.6000 99.8595104.6000 103.4438104.6000 104.6370 105.8447104.6000 01 yr2 yr3 yr

37. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads37 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

38. The Oxford Guide to Financial Modeling by Ho & Lee Chapter 8. Investment Grade Corporate Bonds: Option Adjusted Spreads38 8.9 Explaining the Concept of the Arbitrage-free Condition on a Solemn Occasion

39. 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

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

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

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