Given two benchmark bonds: a one-year zero-coupon bond trading at 100 and promising to pay 105 at maturity and a two-year 5.5% annual coupon bond with face value of 100 trading at 95.45, what must be the two-year spot rate implied by the two bond prices?

(Round to 4 decimal places.)

Question

Given two benchmark bonds: a one-year zero-coupon bond trading at 100 and promising to pay 105 at maturity and a two-year 5.5% annual coupon bond with face value of 100 trading at 95.45, what must be the two-year spot rate implied by the two bond prices?

(Round to 4 decimal places.)

BlackTom AI · Accepted Answer

We start by identifying the data provided and the relationship between bond prices and spot rates.
First, the 1-year zero-coupon bond: it costs 100 today and pays 105 at t=1. This determines the 1-year discount factor as DF1 = 100/105 = 20/21 ≈ 0.9523809524, which corresponds to a 1-year spot rate r1 = 1/DF1 − 1 = 105/100 − 1 = 0.05, i.e., 5.0000%.
Next, consider the 2-year coupon bond: face value 100, annual coupon 5.5 (since 5.5% of 100), maturing at 2 years with total payment 105.5 at t=2. Its price is 95.45. Using bootstrapping, the present value is:
Price = 5.5·DF1 + 105.5·DF2, where DF2 is the 2-year discount factor for cash flows at t=2.
Plugging in DF1 = 0.9523809524, we compute the portion of the price attributable to the first coupon:
5.5·DF1 = 5.5 × 0.9523809524 ≈ 5.238095238.
Subtract from the total price to solve for the t=2 cash flow’s present value:
95.45 − 5.238095238 ≈ 90.211904762.
Then DF2 = 90.211904762 / 105.5 ≈ 0.8550 (more precisely ≈ 0.855089). This is the 2-year discount factor.
Now relate DF2 to the 2-year spot rate s2 with annual compounding:
DF2 = 1 / (1 + s2)^2 ⇒ (1 + s2)^2 = 1 / DF2.
Compute 1/DF2 ≈ 1 / 0.855089 ≈ 1.1696. Taking the square root gives 1 + s2 ≈ √1.1696 ≈ 1.0815, so s2 ≈ 0.0815, i.e., 8.15% (approximately).
Carrying more precise arithmetic yields s2 ≈ 0.0814 (rounded to four decimal places).
In summary, the two-year spot rate implied by the given bond prices is about 8.14% per year. The key steps were to determine the 1-year discount factor from the zero-coupon price, use it to solve for the 2-year discount factor from the coupon bond price, and then convert that discount factor into a two-year annualized spot rate via (1 + s2)^2 = 1/DF2.

Given two benchmark bonds: a one-year zero-coupon bond trading at 100 and promising to pay 105 at maturity and a two-year 5.5% annual coupon bond with face value of 100 trading at 95.45, what must be the two-year spot rate implied by the two bond prices? (Round to 4 decimal places.)

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