1. Consider three risk-free securities:

• a zero-coupon bond with a face value of $1,000 and 3 years to maturity, trading at $711.78,
• an annuity that pays $1,000 in year 1 and year 2. The annuity costs $1,809.72,
• a bond that pays off $1,000 in one year, $2,000 in two years, and $3,000 in three
• years. This bond trades at $4,802.40.
• (a) What is the term structure of spot interest rates implied by these instruments? What is the term structure of forward rates?
• (b) What are the YTMs (yields to maturity) of these three instruments? Write down the equation that you need to solve to compute YTMs. Solve the equations for the three-year zero-coupon bond and the annuity. Solve the third equation numerically, that is, compute the YTM using the functionIRR(or using Goal Seek or Solver Tool) in Excel.
• Suppose now that there is also a fourth instrument: a zero-coupon bond with the face value of $1,000 and two years to maturity. This bond trades at $857.34.

(c) Create a portfolio of the original three instruments that replicates the cash flows of the two-year zero-coupon bond.

(d) What is the YTM of the two-year zero-coupon bond? What is the YTM of the replicating portfolio you found in part (c)? (Try to answer this question without doing any calculations. However, if you wish, use whichever method you like to compute the YTM.)

(e) Suppose now that the two-year zero-coupon bond trades at $900.00 (all other prices are unchanged). Is there a free lunch available here? Describe a trading strategy that will allow you to get it.