12) (worth 13 pts): Compute today\'s arbitrage-free price of a three-year, $1000
ID: 2804876 • Letter: 1
Question
12) (worth 13 pts): Compute today's arbitrage-free price of a three-year, $1000 par-valued Bond that pays a 10% annual coupon. There is NO CREDIT RISK associated with this Bond. Use the following three default-free, zero-coupon Bonds to compute the price: 1-year zero: YTM = 3.50%, Face Value = $100.00 (i.e., the zero matures one-year from today) 2-year zero: YTM = 4.00%, Face Value = $100.00 3-year zero: YTM = 4.50%, Face Value-S 100.00 · · · B) Briefly explain why the price you have computed is the "arbitrage free" price. Recall, that "arbitrage free" means that an investor who buys (or sells) the coupon bond at your price CANNOT make a riskless profit.Explanation / Answer
By Law of One Price, we can compute the value of coupon bond using zero-coupon bonds.
Lets calculate cost of each zero coupon bond using YTM and Par value
Value of 1-year zero-coupon bond = $100/(1+YTM1) = $100/1.035 = $96.62
Value of 2-year zero-coupon bond = $100/(1+YTM2)2 = $100/1.042 = $92.46
Value of 3-year zero-coupon bond = $100/(1+YTM3)3 = $100/1.0453 = $87.63
Cashflows of 10% coupon bond must be matched with zero-coupon cashflows for each term
Therefore Face value required would be $100, $100 and $1100 for 1-year, 2-year and 3-year terms respectively.
Now to match the 3-year Face value the three-year coupon bond with par value $100 will have to be increased elevenfold ie 11 * 87.63 = 963.93
Add all the prices of zero-coupon bonds to obtain the price of coupon bond because Law of one price states that value of coupon bond is equal to the price of portfolio of zero-coupon bonds.
Price of coupon bond = $96.62 +$92.46 + $963.93 = $1153.01
Answer : $1153
B)
By Law of One Price, the three- year coupon bond price is $1153. If the price was any higher then an investor could easily buy zero-coupon bonds and sell three-year coupon bond to make a profit. It would then not be an arbitrage-free price. Or even if the three-year coupon bond price was lower, an investor would buy coupon bond and sell zero-coupon bonds to make a profit. Even this price would not be arbitrage free.
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