Question

1. a) How do bond analysts use bootstrapping?

b) Calculate the missing spot rates for the following Treasury yields. (All yields are shown on a bond equivalent basis). All the securities maturing from 1.5 years on are selling at par. The 0.5 and 1.0-year securities are zero-coupon instruments.

Year (Period)	Yield to Maturity (%)	Spot Rate (%)
0.5 (1)	4.00	4.00
1.0 (2)	4.25	4.25
1.5 (3)	4.50	4.51
2.0 (4)	4.75	4.78
2.5 (5)	5.00	?
3.0 (6)	5.25	5.32
3.5 (7)	5.50	5.58
4.0 (8)	5.75	5.85
4.5 (9)	6.00	6.12
5.0 (10)	6.25	?

Answer 1

Since the Bonds are trading at par, the coupopon rate = yield to maturity

For 2.5 year bond

Coupon per period = (Coupon rate / No of coupon payments per year) * Face value

Coupon per period = (5% / 2) * $100

Coupon per period = $2.5

Bond Price = Coupon / (1 + 0.5 year spot rate / 2)¹ + Coupon / (1 + 1 year spot rate / 2)² + Coupon / (1 + 1.5 year spot rate / 2)³ + Coupon / (1 + 2 year spot rate / 2)⁴ + Coupon / (1 + 2.5 year spot rate / 2)⁵ + Face value / (1 + 2.5 year spot rate / 2)⁵

$100 = $2.5 / (1 + 4% / 2)¹ + $2.5 / (1 + 4.25% / 2)² + $2.5 / (1 + 4.51% / 2)³ + $2.5 / (1 + 4.78% / 2)⁴ + $2.5 / (1 + 2.5 year spot rate / 2)⁵ + $100 / (1 + 2.5 year spot rate / 2)⁵

90.5391 = 102.5 / (1 + 2.5 year spot rate / 2)⁵

(1 + 2.5 year spot rate / 2)⁵ = 102.5 / 90.5391

(1 + 2.5 year spot rate / 2)⁵ = 1.1321

(1 + 2.5 year spot rate / 2) = (1.1321)^(1/5)

(1 + 2.5 year spot rate / 2) = 1.02512

2.5 year spot rate / 2 = 0.02512

2.5 year spot rate = 5.025 or 5.03%

For 5 year bond

Coupon per period = (Coupon rate / No of coupon payments per year) * Face value

Coupon per period = (6.25% / 2) * $100

Coupon per period = $3.125  

Bond Price = Coupon / (1 + 0.5 year spot rate / 2)¹ + Coupon / (1 + 1 year spot rate / 2)² + Coupon / (1 + 1.5 year spot rate / 2)³ + Coupon / (1 + 2 year spot rate / 2)⁴ + Coupon / (1 + 2.5 year spot rate / 2)⁵ + Coupon / (1 + 3 year spot rate / 2)⁶ + Coupon / (1 + 3.5 year spot rate / 2)⁷ + Coupon / (1 + 4 year spot rate / 2)⁸ + Coupon / (1 + 4.5 year spot rate / 2)⁹ + Coupon / (1 + 5 year spot rate / 2)¹⁰ + Face value / (1 + 5 year spot rate / 2)¹⁰

$100 = $3.125 / (1 + 4% / 2)¹ + $3.125 / (1 + 4.25% / 2)² + $3.125 / (1 + 4.51% / 2)³ + $3.125 / (1 + 4.78% / 2)⁴ + $3.125 / (1 + 5.03% / 2)⁵ + $3.125 / (1 + 5.32% / 2)⁶ + $3.125 / (1 + 5.58% / 2)⁷ + $3.125 / (1 + 5.85% / 2)⁸ + $3.125 / (1 + 6.12% / 2)⁹ + $3.125 / (1 + 5 year spot rate / 2)¹⁰ + $100 / (1 + 5 year spot rate / 2)¹⁰

75.3030 = 103.125 / (1 + 5 year spot rate / 2)¹⁰

(1 + 5 year spot rate / 2)¹⁰ = 102.5 / 75.3030

(1 + 5 year spot rate / 2)¹⁰ = 1.3695

(1 + 5 year spot rate / 2) = (1.3695)^(1/10)

(1 + 5 year spot rate / 2) = 1.03194

5 year spot rate / 2 = 0.03194

5 year spot rate = 6.388 or 6.39%