Homework: Zero Coupon Bond
E4721
Professor Dastidar
Assignment #1
Question 1
Consider the following data. The column marked n gives the price today of one dollar delivered in half-year n, i.e., of a zero coupon bond which pays $1 in half-year n. In the next two columns there are the cash flows of two bonds, A and B. Essentially, bond A pays a 20% semi-annual coupon and bond B pays a 10% semi-annual coupon. Both bonds mature in 2.5 years, when each also pays its principal of 100. Assume semi-annual compounding.
Half
Year
1
2
3
4
5
n
Bond A Bond B
.95
.91
.87
.80
.70
10
10
10
10
110
5
5
5
5
105
A. Calculate the price of each bond assuming there are no arbitrage opportunities in the market. (That is, calculate the present value of each of the bonds.)
B. Now suppose that in fact bond A is traded at $111.97 while bond B is traded at
$91.41. Are there arbitrage opportunities in the market? If yes, how would you take advantage of them? [Assume that zero coupon bonds are traded.]
C. Calculate the yields to maturity (or the interest/discount rates rn) from the discount factors n given above (i.e., from the prices of the above zero coupon bonds). Plot these against the time to maturity t of the bonds. This is the term structure of interest rates or the yield curve based on zero coupon bonds.
D. Calculate the yield to maturity (i.e., the IRR) of bonds A and B assuming they trade at the prices quoted in part B. Which bond has a higher yield to maturity at these prices?
Compare these to the yields of the various zero coupon bonds.
E. Based on your answers to parts B and D above, does yield to maturity give you a guide as to possible mispricing in the market? Can you think about why this might be the case? (Hint: how would your answer to all the parts above change if the term structure or yield curve was flat at, say, 13%?)
Question 2: Bond arbitrage
Suppose that the current term structure is given by: rn 2.45%
2.70%
2.79%
2.98%
Professor Dastidar
Assignment #1
Question 1
Consider the following data. The column marked n gives the price today of one dollar delivered in half-year n, i.e., of a zero coupon bond which pays $1 in half-year n. In the next two columns there are the cash flows of two bonds, A and B. Essentially, bond A pays a 20% semi-annual coupon and bond B pays a 10% semi-annual coupon. Both bonds mature in 2.5 years, when each also pays its principal of 100. Assume semi-annual compounding.
Half
Year
1
2
3
4
5
n
Bond A Bond B
.95
.91
.87
.80
.70
10
10
10
10
110
5
5
5
5
105
A. Calculate the price of each bond assuming there are no arbitrage opportunities in the market. (That is, calculate the present value of each of the bonds.)
B. Now suppose that in fact bond A is traded at $111.97 while bond B is traded at
$91.41. Are there arbitrage opportunities in the market? If yes, how would you take advantage of them? [Assume that zero coupon bonds are traded.]
C. Calculate the yields to maturity (or the interest/discount rates rn) from the discount factors n given above (i.e., from the prices of the above zero coupon bonds). Plot these against the time to maturity t of the bonds. This is the term structure of interest rates or the yield curve based on zero coupon bonds.
D. Calculate the yield to maturity (i.e., the IRR) of bonds A and B assuming they trade at the prices quoted in part B. Which bond has a higher yield to maturity at these prices?
Compare these to the yields of the various zero coupon bonds.
E. Based on your answers to parts B and D above, does yield to maturity give you a guide as to possible mispricing in the market? Can you think about why this might be the case? (Hint: how would your answer to all the parts above change if the term structure or yield curve was flat at, say, 13%?)
Question 2: Bond arbitrage
Suppose that the current term structure is given by: rn 2.45%
2.70%
2.79%
2.98%