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...
TRANSCRIPT
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