Given two benchmark bonds: a one-year zero-coupon bond trading at 100 and promising to pay 105 at maturity and a two-year 4.0% 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 4.0% 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 are given two bonds and asked to infer the two-year spot rate implied by their prices.

First, interpret the data for the one-year zero-coupon bond: it costs 100 today and pays 105 in one year. The one-year discount factor is therefore DF1 = 100/105 = 0.95238095, which corresponds to a one-year spot rate of s1 = 5.0% (since 1 + s1 = 105/100 = 1.05).

Next, use the two-year coupon bond to extract the two-year discount factor. The two-year bond has a 4.0% annual coupon on a face value of 100, so it pays 4 at year 1 and 104 at year 2 (coupon 4 plus principal 100). Its price is 95.45. Using the spot-rate framework, its price equals the present value of its cash flows discounted at the appropriate spot rates:
  95.45 = 4/(1 + s1) + 104/(1 + s2)^2
Where s1 is the known one-year spot rate (5.0%) and s2 is the unknown two-year spot rate.

Compute the first cash flow’s present value using s1 = 5%: 4/(1 + 0.05) = 4/1.05 ≈ 3.80952381. Subtract this from the total price to isolate the second cash flow’s present value: 95.45 − 3.80952381 ≈ 91.64047619.

This remaining amount equals the present value of 104 paid at year 2 discounted by the two-year discount factor: 91.64047619 = 104/(1 + s2)^2.

Solve for (1 + s2)^2: (1 + s2)^2 = 104 / 91.64047619 ≈ 1.1355. Take the square root to get 1 + s2 ≈ sqrt(1.1355) ≈ 1.0653–1.0660. Subtract 1 to obtain the two-year spot rate: s2 ≈ 0.0653 (approximately 6.53%).

Summary of key steps:
- Determine DF1 from the zero-coupon bond: DF1 = 100/105 = 0.95238095, so s1 = 5%. 
- Use P = 4/(1+s1) + 104/(1+s2)^2 with P = 95.45 to solve for (1+s2)^2.
- Compute (1+s2)^2 = 104 / (95.45 − 4/(1.05)) ≈ 1.1355, hence s2 ≈ sqrt(1.1355) − 1 ≈ 0.0653 (6.53%).

Thus, the two-year spot rate implied by the two bond prices is approximately 6.53% per year.

Given two benchmark bonds: a one-year zero-coupon bond trading at 100 and promising to pay 105 at maturity and a two-year 4.0% 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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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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