1. 1 Convertible Bonds with Call Notice Periods SONAD 2003 Friday, May 2

2. 2 Andreas Grau (agrau@uwaterloo.ca) Peter Forsyth (paforsyth@elora.uwaterloo.ca) Kenneth Vetzal (kvetzal@uwaterloo.ca) Convertible Bonds with Notice Periods

3. 3 Goals Build a precise model of convertible bonds with notice periods Evaluate the effect of a notice period for –the optimal call strategy –the value of the convertible bond Evaluate the effect of suboptimal call strategies

4. 4 Goals Build a precise model of convertible bonds with notice periods Evaluate the effect of a notice period for –the optimal call strategy –the value of the convertible bond Evaluate the effect of suboptimal call strategies

5. 5 Goals Build a precise model of convertible bonds with notice periods Evaluate the effect of a notice period for –the optimal call strategy –the value of the convertible bond Evaluate the effect of suboptimal call strategies

6. 6 Goals Build a precise model of convertible bonds with notice periods Evaluate the effect of a notice period for –the optimal call strategy –the value of the convertible bond Evaluate the effect of suboptimal call strategies

7. 7 CB Background CB is a bond with the option to convert it into shares Interesting for issuers with poor credit rating (start-up company) Same return, but lower risk as stocks

8. 8 CB Background CB is a bond with the option to convert it into shares Interesting for issuers with poor credit rating (start-up company) Same return, but lower risk as stocks

9. 9 CB Background CB is a bond with the option to convert it into shares Interesting for issuers with poor credit rating (start-up company) Same return, but lower risk as stocks

10. 10 CB Background CB is a bond with the option to convert it into shares Interesting for issuers with poor credit rating (start-up company) Same return, but lower risk as stocks

11. 11 CB features Company xy - Convertible bond - Amount issued Face value F Conversion ratio  Coupon payments c i Maturity T Options Put price B p Call period starting at time t s Call price B cl Trigger price, Notice period T n, Company xy - Convertible bond - Amount issued Face value F Conversion ratio  Coupon payments c i Maturity T Options Put price B p Call period starting at time t s Call price B cl Trigger price, Notice period T n,

12. 12 Company xy - Convertible bond - Amount issued Face value F Conversion ratio  Coupon payments c i Maturity T Options Put price B p Call period starting at time t s Call price B cl Trigger price, Notice period T n, Company xy - Convertible bond - Amount issued Face value F Conversion ratio  Coupon payments c i Maturity T Options Put price B p Call period starting at time t s Call price B cl Trigger price, Notice period T n, CB features Company xy -Stock- Volatility  Dividends D i Company xy -Stock- Volatility  Dividends D i

13. 13 Company xy - Convertible bond - Amount issued Face value F Conversion ratio  Coupon payments c i Maturity T Options Put price B p Call period starting at time t s Call price B cl Trigger price, Notice period T n, Company xy - Convertible bond - Amount issued Face value F Conversion ratio  Coupon payments c i Maturity T Options Put price B p Call period starting at time t s Call price B cl Trigger price, Notice period T n, CB features Company xy -Stock- Volatility  Dividends D i Company xy -Stock- Volatility  Dividends D i Risk free rate r

14. 14 CB models – no default

15. 15 CB models – no default S V Time =T

16. 16 CB models – no default V S

17. 17 CB models – T&F Kostas Tsiveriotis and Chris Fernandes 1998: Valuing Convertible Bonds with Credit Risk

18. 18 CB models – T&F V S

19. 19 CB models – AFV Elie Ayache, Peter A. Forsyth, and Kenneth R. Vetzal 2002: Next Generation Models for Convertible Bonds with Credit Risk

20. 20 CB models – AFV V S

21. 21 Notice periods t i-2 continue call stop t i-1 call continue stop call continue titi stop t i+1

22. 22 Notice periods continue t i-2 call continue t i-1 call continue titi t i+1 stop

23. 23 Notice periods continue t i-2 call continue t i+1 call continue titi t i+1 stop titi continue call

24. 24 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue call titi

25. 25 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue call titi

26. 26 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue call titi S V Time = t i+1

27. 27 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue call titi Time = t i+1 S V

28. 28 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue call titi Time = t i+1 S V

29. 29 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue call titi Time = t i+1 S V S V Time = t i +T n

30. 30 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue call titi S V Time = t i+1 S V Time = t i +T n

31. 31 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue S V call titi Time = t i+1 S V

32. 32 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods titi continue call titi Time = t i Time = t i+1 S V S V Time = t i

33. 33 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods titi titi continue call Time = t i S V S V

34. 34 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods titi continue call titi Time = t i S V

35. 35 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue titi call Time = t i S V

36. 36 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue titi titi call stop Time = t i S V

37. 37 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods call continue titi S V Time = t i

38. 38 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue titi titi call continue titi call S*S* S V Time = t i V V c,t i

39. 39 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue titi titi call continue titi call S V Time = t i V

40. 40 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue titi titi call continue titi call V Time = t i S V

41. 41 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue titi titi call continue titi call

42. 42 continue t i-2 call continue t i+1 call continue titi t i+1 stop Notice periods continue titi titi call continue titi call

43. 43 Notice periods continue titi t i-2 call continue t i+1 call continue titi t i+1 stop call titi

44. 44 Notice periods continue titi t i-2 call continue t i+1 call continue titi t i+1 stop call titi

45. 45 Notice periods continue titi t i-2 call continue t i+1 call continue titi t i+1 stop call titi

46. 46 Notice periods continue titi t i-2 call continue t i+1 call continue titi t i+1 stop call titi

47. 47 Notice periods continue titi t i-2 call continue t i+1 call continue titi t i+1 stop call titi

48. 48 Mathematical model

49. 49 Example CB Company xy - Convertible bond – Amount issued 100M Face value F 100 Conversion ratio  1 Coupon payments c i 2, semi-annually Maturity T 5 years Options Call period starting at year 1 Call price B cl 140 Notice period T n 30 days Company xy - Convertible bond – Amount issued 100M Face value F 100 Conversion ratio  1 Coupon payments c i 2, semi-annually Maturity T 5 years Options Call period starting at year 1 Call price B cl 140 Notice period T n 30 days Company xy - Stock – Market capitalization 10,000M Implied volatility 20% Dividends 2 once a year, immediately after the coupon Company xy - Stock – Market capitalization 10,000M Implied volatility 20% Dividends 2 once a year, immediately after the coupon Capital market - Bonds – Term structure of risk free rate: Flat, 5% continuously compounded Capital market - Bonds – Term structure of risk free rate: Flat, 5% continuously compounded

50. 50 Example: Effect on value Notice period T n = 30 days S V

51. 51 Example: Effect on value T n = 10 days T n = 20 days T n = 30 days T n = 60 days T n = 90 days S VV

52. 52 Example: Optimal call 160 155 150 145 140 135 time t [years] S*S* No notice period (T n = 0) 1 2 3 4 5

53. 53 Example: Optimal call 160 155 150 145 140 135 1 2 3 4 5 time t [years] S*S* Notice period T n = 30 days no dividends

54. 54 160 155 150 145 140 135 time t [years] S*S* Notice period T n = 30 days dividends (D i =2) Example: Optimal call 1 2 3 4 5

55. 55 Approximations Ingersoll: –No notice period S*(t) = B cl (t) + A(t) Jonathan Ingersoll 1977: An examination of corporate call policies on convertible securities.

56. 56 Approximations Ingersoll: –No notice period S*(t) = B c (t) + A(t) 160 155 150 145 140 135 time t [years] S*S* 1 2 3 4 5

57. 57 Approximations Butler Alexander W. Butler 2002: Revisiting Optimal Call Policy for Convertible Bonds

58. 58 Approximations Butler 160 155 150 145 140 135 time t [years] S*S* 1 2 3 4 5

59. 59 Effect of Approximations S VV Ingersoll

60. 60 Effect of Approximations S VV Ingersoll Butler

61. 61 Conclusion Notice periods in CB can be done by solving a one dimensional problem each time step of the solution The result is the value of the CB as well as the optimal call strategy Previously published approximations for the optimal call strategy were not very precise

62. 62 Conclusion Notice periods in CB can be done by solving a one dimensional problem each time step of the solution The result is the value of the CB as well as the optimal call strategy Previously published approximations for the optimal call strategy were not very precise

63. 63 Conclusion Notice periods in CB can be done by solving a one dimensional problem each time step of the solution The result is the value of the CB as well as the optimal call strategy Previously published approximations for the optimal call strategy were not very precise

64. 64 Conclusion Notice periods in CB can be done by solving a one dimensional problem each time step of the solution The result is the value of the CB as well as the optimal call strategy Previously published approximations for the optimal call strategy were not very precise

65. 65 Questions?! ?