Question

If the company were to issue an annual zero-coupon bond with a maturity of 2 years and a par value of $1,000, what would be the arbitrage-free price of that bond?

Answer 1

Explanation:

`\text{here is the missing text:} \\ \\ \text{ Suppose a company issues an annual coupon bond with a maturity of 2 years,} \\ \\ \text{ price of $950 \ par \ value \ of $1,000, and coupon rate of 5\% They also issue an}`

`\text{ annual zero-coupon bond with a maturity of 1 year, price of $900, \ and \ par \ value \ of \ $1,000.}`

`\text{To determine the 1-year spot rate using the 1-year zero bond.} \\ \\ FV = PV * (1 + S_1) \\ \\ 1+ S_1 = (FV)/(PV) \\ \\ 1 + S_1 = (1000)/(900) \\ \\ S_1 = 11.11\%`

`\text{PV of the 2-year bond = 950} \\ \\ Annual coupon = 1000 * 5\% = \$50 \\ \\ \\ 950 = (50)/((1+S_1) )+ ((50+1000))/((1+S_2)^2) \\ \\ 950 = (50)/(1.1111)+ (1050)/((1+S_2)^2) \\ \\ (1050)/((1+S_2)^2)=950 -45\\ \\ (1050)/((1+S_2)^2)=905 \\ \\ (1+S_2)^2 = (1050)/(905) \\ \\ 1 + S_2= √((1.16022)) \\ \\ S_2 = 7.714\%`

`\text{the arbitrage-free price of the 2-year bond}= (1000)/((1+0.07714)^2) \\ \\ = \mathbf{\$861.90}`