All bonds considered below have a face value of $1000. The two-year zero-coupon bond is trading at $906.86 while the two-year bond with annual coupon rate 3% is trading at $962.96. In the absence of arbitrage, what is the fair price of the one-year zero-coupon bond?

Question

BlackTom AI · Accepted Answer

We are given two bonds with the same face value of 1000 and asked to find the fair price of a one-year zero-coupon bond under no-arbitrage.

First restate the data: the two-year zero-coupon bond with face 1000 trades at 906.86. This means the present value of 1000 due in 2 years is 906.86, so the discount factor for 2 years is 906.86/1000 = 0.90686 per 1000 of face. Equivalently, the price of receiving 1000 in 2 years is 906.86, and the per-dollar discount factor for 2 years is 0.90686/1000 for a unit cash flow at t=2, but we’ll work with cash flows in their actual amounts.

Next, consider the two-year bond with a 3% annual coupon. It pays 30 at year 1 and 1030 at year 2 (since it also repays the face value at maturity). The price of this bond is 962.96. By no-arbitrage, its present value equals the sum of each cash flow discounted by the appropriate discount factors:

PV = 30 × d1 + 1030 × d2

where d1 is the 1-year discount factor (for a cash flow at year 1) and d2 is the 2-year discount factor for a cash flow at year 2.

We know d2 from the given 2-year zero price: the PV of 1000 at t=2 is 906.86, so the PV of 1030 at t=2 is 1030 × (906.86/1000) = 1030 × 0.90686 = 934.0658.

Plugging into the bond price equation:
962.96 = 30 × d1 + 934.0658
So 30 × d1 = 962.96 − 934.0658 = 28.8942
Hence d1 = 28.8942 / 30 = 0.96314 (approximately).

This d1 ≈ 0.96314 is the price today of a 1-dollar payoff one year from now (the 1-year discount factor). Therefore, the fair price of a one-year zero-coupon bond with face value 1000 is 1000 × d1 = 1000 × 0.96314 = 963.14.

In summary, the no-arbitrage price for the one-year zero-coupon bond is about 963.14, derived from using the 2-year zero-coupon price to back out the 2-year discount factor and then solving for the 1-year discount factor from the coupon bond.

All bonds considered below have a face value of $1000. The two-year zero-coupon bond is trading at $906.86 while the two-year bond with annual coupon rate 3% is trading at $962.96. In the absence of arbitrage, what is the fair price of the one-year zero-coupon bond? Single choice

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