Question

Prices of zero-coupon, default-free securities with face values of $1,000 are summarized in the following table: Maturity (years) Price (per $1,000 face value) $971.52 $939.15 $904.51 Suppose you observe that a three-year, default-free security with an annual coupon rate of 10% and a face value of $1,000 has a price today of $1,182.85. Is there an arbitrage opportunity? If so, show specifically how you would take advantage of this opportunity. If not, why not?

Buy how many coupon bonds, sell short how many one-year Zeros, two-year Zeros, and three-year Zeros? What would be the net profit in that case?

Answer 1

SUGGESTED ANSWER

• In the present case, there is a situation of arbitrage because of the price difference between the zero coupon and coupon , default free securities.
• The coupon bond is undervalued as the market price today (ie $ 1,182.85) is more than the price of the bond (approximately 1,000) computed if we discount the cash flows $100 (ie $1000*10%) for 3 years and $ 1000 in the third year at the rate of 10%. When the bond is undervalued, the best course of action in the situation of arbitrage is to buy/ long the under-priced security.
• Further, to determine the number of coupon bonds to buy, we have to take the lowest/ minimum of the integer which is divisible by both $ 1,000(par value) and $ 1,100 [cash flow in the third year ($ 1000+$100)]. This figure is 10. Thus, we buy 10 coupon bonds.
• Simultaneously to perform the action of arbitrage, we have to sell/ short the zero coupon securities. This implies that the zero coupon securities are over priced and hence should be sold.
• Further, the cash flows for year 1 and year 2 of zero coupon bond can be easily offset as we can take 1 bond each for year 1 & 2 to offset the cash flow of zero coupon bond. Year 3 cash flow can be offset by taking a long position of n=10 bonds and a short position of 11 bonds, 3 year zero coupon to earn profit in arbitrage,
• The table shown in the attached image explains the derivation of the net profit by taking the above discussed long position of coupon and short position of zero coupon default free securities::
  | suggested Answer : Calwlation 3 : Arbitrage Prefit Years | Particulars !! Cost 2 ul 3 m _ Long 10 Icoupon bonds _ -11,835 [lox 1183.5] +1000. +1000 +11000 [100 x10] [100x10] [1100x10 years... o 1 2 3 Short Hairste rond 1 year zero coupon bond +971.52 -1000 Short 12year zero coupon bond +939.15 How ABROOD - 1000 t11,000 Short Il +9949.61 1342wt zero [904.51 XII] coupon bond INET PROFIT $ 25.28 Slanned with pmScannert Sh
  

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