B. Estimate term structure of discount factors, spot rates and forward rates by
ID: 2788042 • Letter: B
Question
B. Estimate term structure of discount factors, spot rates and forward rates by using data on five semi-annual coupon paying bonds with $100 face value each: The bonds, respectively, have 1.25, 5.35, 10.4, 15.15 and 20.2 years to maturity; pay coupon at annual rates of B1, B2, B3, B4, and B5 percent of face value; and are trading at quoted spot market prices in dollars of B6, B7, B8, B9 and B10. Specify the discount factor function d(t) by a third degree polynomial with unknown parameters a, b, and c, as done in class. Using estimated d(t) function, determine spot rate and forward rate functions by assuming half-year compounding. Then write the values of the following based on your estimation.
1.Spot rate for term 2 year.
2.Spot rate for term 5 year.
3.Spot rate for term 10 year.
4.Spot rate for term 17 year.
5.Forward rate for half year period 2.5 to 3.0 years.
6.Forward rate for half year period 5.5 to 6.0 years.
7.Forward rate for half year period 10.5 to 11.0 years.
8.Forward rate for half year period 15.5 to 16.0 years.
B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 2.3 3.3 4.3 5.3 6.3 96.3 97.3 98.3 99.3 100.3Explanation / Answer
1. Coefficient of parameter a in first bond price equation.
1. Coefficient of parameter b in first bond price equation.
1. Coefficient of parameter c in first bond price equation.
Solution:
102.3 = 4.15 x d (1.25) + 4.15 x d (5.35) + 4.15 x (10.4) + 4.15 (15.15) + 104.15 x d (20.2)
102.3 = 4.15 x [1 + a (1.25) + b (1.25)^2 + c (1.25)^3] + 4.15 x [1 + a (5.35) + b (5.35)^2 + c (5.35)^3] + 4.15 x [1 + a (10.4) + b (10.4)^2 + c (10.4)^3] + 4.15 x [1 + a (15.15) + b (15.15)^2 + c (15.15)^3] + 104.15 x [1 + a (20.2) + b (20.2)^2 + c (20.2)^3]
102.3 = (4.15 + 4.15 + 4.15 + 4.15 + 104.15) + a (1.25 + 5.35 + 10.4 + 15.15 + 20.2) + b (1.25^2 + 5.35^2 + 10.4^2 + 15.15^2 + 20.2^2) + c (1.25^3 + 5.35^3 + 10.4^3 + 15.15^3 + 20.2^3)
102.3 = 120.75 + 217.2525a + 13363.07b + 929126.8 c
-217.2525a - 13363.07b - 929126.8 c = 18.45
2. Coefficient of parameter a in second bond price equation.
2. Coefficient of parameter b in second bond price equation.
2. Coefficient of parameter c in second bond price equation.
Solution:
103.3 = 4.65 x d (1.25) + 4.65 x d (5.35) + 4.65 x (10.4) + 4.65 (15.15) + 104.65 x d (20.2)
103.3 = 4.65 x [1 + a (1.25) + b (1.25)^2 + c (1.25)^3] + 4.65 x [1 + a (5.35) + b (5.35)^2 + c (5.35)^3] + 4.65 x [1 + a (10.4) + b (10.4)^2 + c (10.4)^3] + 4.65 x [1 + a (15.15) + b (15.15)^2 + c (15.15)^3] + 104.65 x [1 + a (20.2) + b (20.2)^2 + c (20.2)^3]
103.3 = (4.65 + 4.65 + 4.65 + 4.65 + 104.65) + a (1.25 + 5.35 + 10.4 + 15.15 + 20.2) + b (1.25^2 + 5.35^2 + 10.4^2 + 15.15^2 + 20.2^2) + c (1.25^3 + 5.35^3 + 10.4^3 + 15.15^3 + 20.2^3)
102.3 = 123.25 + 52.35a + 775.9075b + 12999.62 c
-243.4275a -16777.06b – 1307042c = 20.95
3. Coefficient of parameter a in third bond price equation.
3. Coefficient of parameter b in third bond price equation.
3. Coefficient of parameter c in third bond price equation.
Solution:
104.3 = 5.15 x d (1.25) + 5.15 x d (5.35) + 5.15 x (10.4) + 5.15 (15.15) + 105.15 x d (20.2)
104.3 = 5.15 x [1 + a (1.25) + b (1.25)^2 + c (1.25)^3] + 5.15 x [1 + a (5.35) + b (5.35)^2 + c (5.35)^3] + 5.15 x [1 + a (10.4) + b (10.4)^2 + c (10.4)^3] + 5.15 x [1 + a (15.15) + b (15.15)^2 + c (15.15)^3] + 105.15 x [1 + a (20.2) + b (20.2)^2 + c (20.2)^3]
104.3 = (5.15 + 5.15 + 5.15 + 5.15 + 105.15) + a (1.25 + 5.35 + 10.4 + 15.15 + 20.2) + b (1.25^2 + 5.35^2 + 10.4^2 + 15.15^2 + 20.2^2) + c (1.25^3 + 5.35^3 + 10.4^3 + 15.15^3 + 20.2^3)
-269.6025a – 20579.01b – 1775630c = 21.45
4. Coefficient of parameter a in fourth bond price equation.
4. Coefficient of parameter b in fourth bond price equation.
4. Coefficient of parameter c in fourth bond price equation.
Solution:
105.3 = 5.65 x d (1.25) + 5.65 x d (5.35) + 5.65 x (10.4) + 5.65 (15.15) + 104.65 x d (20.2)
105.3 = 5.65 x [1 + a (1.25) + b (1.25)^2 + c (1.25)^3] + 5.65 x [1 + a (5.35) + b (5.35)^2 + c (5.35)^3] + 4.65 x [1 + a (10.4) + b (10.4)^2 + c (10.4)^3] + 4.65 x [1 + a (15.15) + b (15.15)^2 + c (15.15)^3] + 104.65 x [1 + a (20.2) + b (20.2)^2 + c (20.2)^3]
105.3 = (4.65 + 4.65 + 4.65 + 4.65 + 104.65) + a (1.25 + 5.35 + 10.4 + 15.15 + 20.2) + b (1.25^2 + 5.35^2 + 10.4^2 + 15.15^2 + 20.2^2) + c (1.25^3 + 5.35^3 + 10.4^3 + 15.15^3 + 20.2^3)
-295.7775a – 24768.91b – 2344639c = 22.95
5. Coefficient of parameter a in fifth bond price equation.
5. Coefficient of parameter b in fifth bond price equation.
5. Coefficient of parameter c in fifth bond price equation.
Solution:
106.3 = 6.15 x d (1.25) + 6.15 x d (5.35) + 6.15 x (10.4) + 6.15 (15.15) + 106.15 x d (20.2)
106.3 = 6.15 x [1 + a (1.25) + b (1.25)^2 + c (1.25)^3] + 6.15 x [1 + a (5.35) + b (5.35)^2 + c (5.35)^3] + 6.15 x [1 + a (10.4) + b (10.4)^2 + c (10.4)^3] + 6.15 x [1 + a (15.15) + b (15.15)^2 + c (15.15)^3] + 106.15 x [1 + a (20.2) + b (20.2)^2 + c (20.2)^3]
106.3 = (4.65 + 4.65 + 4.65 + 4.65 + 104.65) + a (1.25 + 5.35 + 10.4 + 15.15 + 20.2) + b (1.25^2 + 5.35^2 + 10.4^2 + 15.15^2 + 20.2^2) + c (1.25^3 + 5.35^3 + 10.4^3 + 15.15^3 + 20.2^3)
-321.9525a -29346.76b -3023821c = 24.45
6. Parameter a.
6. Parameter b.
6. Parameter c.
Solution:
A1
A2
A4
A5
1
-217.2525
-243.4275
-269.6025
-295.7775
-321.953
2
-13363.07
-16777.06
-20579.01
-24768.91
-29346.8
3
-929126.8
-1307042
-1775630
-2344639
-3023821
C1
C2
C3
1
370278.8463
29309659.71
2665757566
2
29309659.71
2358268260
217698517593.55
3
2665757566
217698517593.55
20365322777504.20
A1
A2
A3
1
0.0117789674
-0.0003077150
0.0000017475
2
-0.0003077150
0.0000080709
-0.0000000460
3
0.0000017475
-0.0000000460
0.0000000003
C1
C2
C3
C4
C5
1
-0.070642113
0.011172536
0.053917579
0.040555
-0.04596
2
0.001739684
-0.000375748
-0.001451419
-0.001039
0.00131
3
-4.36856E-05
-4.57574E-05
-5.71849E-05
-8.09E-05
-0.00012
a
-0.105661333
b
0.001295726
c
-0.007777017
A1
A2
A4
A5
1
-217.2525
-243.4275
-269.6025
-295.7775
-321.953
2
-13363.07
-16777.06
-20579.01
-24768.91
-29346.8
3
-929126.8
-1307042
-1775630
-2344639
-3023821
C1
C2
C3
1
370278.8463
29309659.71
2665757566
2
29309659.71
2358268260
217698517593.55
3
2665757566
217698517593.55
20365322777504.20
A1
A2
A3
1
0.0117789674
-0.0003077150
0.0000017475
2
-0.0003077150
0.0000080709
-0.0000000460
3
0.0000017475
-0.0000000460
0.0000000003
C1
C2
C3
C4
C5
1
-0.070642113
0.011172536
0.053917579
0.040555
-0.04596
2
0.001739684
-0.000375748
-0.001451419
-0.001039
0.00131
3
-4.36856E-05
-4.57574E-05
-5.71849E-05
-8.09E-05
-0.00012
a
-0.105661333
b
0.001295726
c
-0.007777017
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