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Lecture 9

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2 3 10 8.04 6.00 4.84

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Maturity (years)YTM 13.0% 53.5% 103.8% 154.1% 204.3% 304.5% 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_3 020-tstrips.html?mod=topnav_2_3000

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01230123 Rates f 3-1 R n = spot rates f n = forward rates year

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R2R3R2R3 f 3 f 3-2 f2f2 0 1 2 3 year

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example 1000=1000 (1+R 3 ) 3 (1+f 1 )(1+f 2 )(1+f 3 )

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Forward Rate Computations (1+ R n ) n = (1+R 1 )(1+f 2 )(1+f 3 )....(1+f n )

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

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Bond Value = C 1 + C 2 + C 3 (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

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Bond Value Bond Value = C 1 + C 2 + C 3 e r e r2 e r3 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

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

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

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Example What is the 3rd year forward rate? 2 year zero treasury YTM = 8.995% 3 year zero treasury YTM = 9.660%

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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 yr1000 x (1.08995) 2 = 1187.99 3 yr1000 x (1.09660) 3 = 1318.70 IRR of ( FV= 1318.70 & PV= -1187.99) = 11%

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example (using previous example ) f 3 = 11% Q: What is the 2 year forward price on a 1 yr bond? A: 1 / (1+.11) =.9009

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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%

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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

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coupons paying bonds to derive rates Bond Value = C 1 + C 2 (1+r)(1+r) 2 Bond Value = C 1 + C 2 (1+R 1 )(1+f 1 )(1+f 2 ) d1 = 1 d2 = 1 (1+R 1 )(1+f 1 )(1+f 2 )

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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

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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 f 1 & f 2..9150 = 1/(1+f 1 ).8350 = 1 / (1.0929)(1+f 2 ) f 1 = 9.29% f 2 = 9.58% PROOF

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

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Example What is the 3rd year forward rate? 2 year zero treasury YTM = 8.995% 3 year zero treasury YTM = 9.660%

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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

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example (using previous example ) f 3 = 10.99% Q: What is the 2 year forward price on a 1 yr bond? A:

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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%

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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:

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coupons paying bonds to derive rates

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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

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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 f 1 & f 2. f 1 = 8.89% f 2 = 9.15% PROOF

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