B. Estimate term structure of discount factors, spot rates and forward rates by
ID: 2788039 • 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. Coefficient of parameter a in first bond price equation.
2. Coefficient of parameter b in first bond price equation.
3. Coefficient of parameter c in first bond price equation.
4. Coefficient of parameter a in second bond price equation.
5. Coefficient of parameter b in second bond price equation.
6. Coefficient of parameter c in second bond price equation.
7. Coefficient of parameter a in third bond price equation.
8. Coefficient of parameter b in third bond price equation.
9. Coefficient of parameter c in third bond price equation.
10. Coefficient of parameter a in fourth bond price equation.
11. Coefficient of parameter b in fourth bond price equation.
12. Coefficient of parameter c in fourth bond price equation.
13. Coefficient of parameter a in fifth bond price equation.
14. Coefficient of parameter b in fifth bond price equation.
15. Coefficient of parameter c in fifth bond price equation.
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.
2. Coefficient of parameter b in first bond price equation.
3. 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
4. Coefficient of parameter a in second bond price equation.
5. Coefficient of parameter b in second bond price equation.
6. 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
7. Coefficient of parameter a in third bond price equation.
8. Coefficient of parameter b in third bond price equation.
9. 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
10. Coefficient of parameter a in fourth bond price equation.
11. Coefficient of parameter b in fourth bond price equation.
12. 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
13. Coefficient of parameter a in fifth bond price equation.
14. Coefficient of parameter b in fifth bond price equation.
15. 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
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