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Derivatives

Derivatives. Lecture 9. Term Structure & Spots Rates. 8.04 6.00 4.84. 2 3 10. Pure Term Structure. Maturity (years) YTM 1 3.0% 5 3.5% 10 3.8% 15 4.1% 20 4.3% 30 4.5%

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Derivatives

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  1. Derivatives Lecture 9

  2. Term Structure & Spots Rates 8.04 6.00 4.84 2 3 10

  3. Pure Term Structure Maturity (years)YTM 1 3.0% 5 3.5% 10 3.8% 15 4.1% 20 4.3% 30 4.5% The “Pure Term Structure” or “Pure Yield Curve” are comprised of zero-coupon bonds These are often only found in the form of “US Treasury Strips.” http://online.wsj.com/mdc/public/page/2_3020-tstrips.html?mod=topnav_2_3000

  4. Forward rates Rates f3-1 year 0 1 2 3 Rn = spot rates fn = forward rates

  5. Spot/Forward rates R2 R3 0 1 2 3 year f2 f3 f3-2

  6. Spot/Forward rates example 1000 = 1000 (1+R3)3 (1+f1)(1+f2)(1+f3)

  7. Spot/Forward rates Forward Rate Computations (1+ Rn)n = (1+R1)(1+f2)(1+f3)....(1+fn)

  8. Continuous Compounding Warning: Answers in book will be slightly different than calculator.

  9. Bond Value Bond Value = C1 + C2 + C3 (1+r) (1+r)2 (1+r)3 Example $1,000 bond pays 8% per year for 3 years. What is the price at a YTM of 6% 1053.46 = 80 + 80 + 1080 (1+.06) (1+.06)2 (1+.06)3

  10. Bond Value Bond Value = C1 + C2 + C3 er er2 er3 Example $1,000 bond pays 8% per year for 3 years. What is the price at a YTM of 6% 1048.39 = 80 + 80 + 1080 e.06 e.06x2 e.06x3

  11. Yields YTM Example zero coupon 3 year bond with YTM = 6% and par value = 1,000 Price = 1000 / (1 +.06)3 = 839.62

  12. Yields YTM Example zero coupon 3 year bond with YTM = 6% and par value = 1,000

  13. Spot/Forward rates Example What is the 3rd year forward rate? 2 year zero treasury YTM = 8.995% 3 year zero treasury YTM = 9.660%

  14. Spot/Forward rates Example What is the 3rd year forward rate? 2 year zero treasury YTM = 8.995% 3 year zero treasury YTM = 9.660% Answer FV of principal @ YTM 2 yr 1000 x (1.08995)2 = 1187.99 3 yr 1000 x (1.09660)3 = 1318.70 IRR of ( FV= 1318.70 & PV= -1187.99) = 11%

  15. Forward rates & Prices example (using previous example ) f3 = 11% Q: What is the 2 year forward price on a 1 yr bond? A: 1 / (1+.11) = .9009

  16. Spot/Forward rates Example Two years from now, you intend to begin a project that will last for 5 years. What discount rate should be used when evaluating the project? 2 year spot rate = 5% 7 year spot rate = 7.05%

  17. Forward rates & Prices Example (previous example) 2 yr spot = 5% 7 yr spot = 7.05% 5 yr forward rate at year 2 = 7.88% Q: What is the price on a 2 year forward contract if the underlying asset is a 5year zero bond? A: 1 / (1 + .0788)5 = .6843

  18. Spot/Forward rates coupons paying bonds to derive rates Bond Value = C1 + C2 (1+r) (1+r)2 Bond Value = C1 + C2 (1+R1) (1+f1)(1+f2) d1 = 1 d2 = 1 (1+R1) (1+f1)(1+f2)

  19. Spot/Forward rates Example – How to create zero strips 8% 2 yr bond YTM = 9.43% 10% 2 yr bond YTM = 9.43% What is the forward rate? Step 1 value bonds 8% = 975 10%= 1010 Step 2 975 = 80d1 + 1080 d2 -------> solve for d1 1010 =100d1 + 1100d2 -------> insert d1 & solve for d2

  20. Spot/Forward rates example continued Step 3 solve algebraic equations d1 = [975-(1080)d2] / 80 insert d1 & solve = d2 = .8350 insert d2 and solve for d1 = d1 = .9150 Step 4 Insert d1 & d2 and Solve for f1 & f2. .9150 = 1/(1+f1) .8350 = 1 / (1.0929)(1+f2) f1 = 9.29% f2 = 9.58% PROOF

  21. Continuous Compounding Warning: Answers in book will be slightly different than calculator.

  22. Spot/Forward rates Example What is the 3rd year forward rate? 2 year zero treasury YTM = 8.995% 3 year zero treasury YTM = 9.660%

  23. Spot/Forward rates Example What is the 3rd year forward rate? 2 year zero treasury YTM = 8.995% 3 year zero treasury YTM = 9.660% Answer FV of principal @ YTM IRR of ( FV= 1336.16 & PV= -1197.10) = 10.99% Trick: Use 365 days to get a near continuous compounding rate. Then multiply by 365

  24. Forward rates & Prices example (using previous example ) f3 = 10.99% Q: What is the 2 year forward price on a 1 yr bond? A:

  25. Spot/Forward rates Example Two years from now, you intend to begin a project that will last for 5 years. What discount rate should be used when evaluating the project? 2 year spot rate = 5% 7 year spot rate = 7.05%

  26. Forward rates & Prices Example (previous example) 2 yr spot = 5% 7 yr spot = 7.05% 5 yr forward rate at year 2 = 7.87% Trick: Use 365 x 5 days to approximate continuous compounding when calculating IRR. Q: What is the price on a 2 year forward contract if the underlying asset is a 5year zero bond? A:

  27. Spot/Forward rates coupons paying bonds to derive rates

  28. Spot/Forward rates Example – How to create zero strips 8% 2 yr bond YTM = 9.43% 10% 2 yr bond YTM = 9.43% What is the forward rate? Step 1 value bonds 8% = 975 10%= 1010 Step 2 975 = 80d1 + 1080 d2 -------> solve for d1 1010 =100d1 + 1100d2 -------> insert d1 & solve for d2

  29. Spot/Forward rates example continued Step 3 solve algebraic equations d1 = [975-(1080)d2] / 80 insert d1 & solve = d2 = .8350 insert d2 and solve for d1 = d1 = .9150 Step 4 Insert d1 & d2 and Solve for f1 & f2. f1 = 8.89% f2 = 9.15% PROOF

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