Lecture 9. 2 3 10 8.04 6.00 4.84 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.

DerivativesLecture 9

Term Structure & Spots Rates

2 3 10

8.04

6.00

4.84

Pure Term StructureMaturity (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

Forward rates

0 1 2 3

Rates

f3-1

Rn = spot rates

fn = forward rates

year

Spot/Forward rates

R2

R3

f3

f3-2

f2

0 1 2 3 year

example

1000 = 1000 (1+R3)3 (1+f1)(1+f2)(1+f3)

Spot/Forward rates

Forward Rate Computations

(1+ Rn)n = (1+R1)(1+f2)(1+f3)....(1+fn)

Spot/Forward rates

Continuous Compounding

rtt

tt

e

C

r

C

)1(

Warning:

Answers in book will be slightly different than calculator.

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

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

YieldsYTM

Examplezero coupon 3 year bond with YTM = 6% andpar value = 1,000Price = 1000 / (1 +.06)3 = 839.62

YieldsYTM

Examplezero coupon 3 year bond with YTM = 6% andpar value = 1,000

27.835

1000Price

306.

e

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

Spot/Forward rates

ExampleWhat 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%

Spot/Forward rates

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

Forward rates & Prices

ExampleTwo 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%

Spot/Forward rates

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

Forward rates & Prices

coupons paying bonds to derive rates

Spot/Forward 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)

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 1value bonds 8% = 975 10%= 1010

Step 2 975 = 80d1 + 1080 d2 -------> solve for d11010 =100d1 + 1100d2 -------> insert d1 & solve for d2

Spot/Forward rates

example continuedStep 3 solve algebraic equationsd1 = [975-(1080)d2] / 80insert d1 & solve = d2 = .8350insert 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

Spot/Forward rates

Continuous Compounding

rtt

tt

e

C

r

C

)1(

Warning:

Answers in book will be slightly different than calculator.

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

Spot/Forward rates

ExampleWhat 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

Spot/Forward rates

16.133610003

10.119710002309660.

208995.

eyr

eyr

example (using previous example )f3 = 10.99%Q: What is the 2 year forward price on a 1 yr bond?

A:

Forward rates & Prices

8959.

1Price

11099.

e

ExampleTwo 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%

Spot/Forward rates

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:

Forward rates & Prices

6747.

1Price

50787.

e

coupons paying bonds to derive rates

Spot/Forward rates

221 Value Bondrr e

C

e

C

211

21 Value Bond fff ee

C

e

C

1

1 d1fe

21

12d

ff ee

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 1value bonds 8% = 975 10%= 1010

Step 2 975 = 80d1 + 1080 d2 -------> solve for d11010 =100d1 + 1100d2 -------> insert d1 & solve for d2

Spot/Forward rates

example continuedStep 3 solve algebraic equationsd1 = [975-(1080)d2] / 80insert d1 & solve = d2 = .8350insert d2 and solve for d1 = d1 = .9150

Step 4

Insert d1 & d2 and Solve for f1 & f2.

f1 = 8.89% f2 = 9.15%

PROOF

Spot/Forward rates

1

1 .9150 fe

20889.

18350. fee