Homework: Zero Coupon Bond

Topics: Bond, Zero-coupon bond, Yield curve Pages: 9 (1493 words) Published: February 25, 2014
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%

Year(n)
0.5
1
1.5
2

1. Price zero coupon bonds for each maturity (be sure to write down the general formula)
1

E4721

Professor Dastidar

Assignment #1

2. Price a 5% (annualized coupon rate) coupon bonds for each maturity. 3. Suppose the two-year 5% coupon bond is trading for 103.414 dollars. Is there an arbitrage? If so, explicitly construct a trade that delivers a risk-less profit of \$50,000. This means, write down the entire portfolio with initial cost and final payoffs at each time.

Question 3: Bloomberg
The objective of this exercise is to get you familiar with the Bloomberg screen and some feel for the market. Go to one of Bloomberg terminals in the Library. You do not want to wait to do this assignment until Monday evening, as you will see lots of your classmates sitting there with you! The point of this exercise is to remind of the institutional details we sometimes glaze over in class.

At any point in time, to return to the previous screen, hit the menu key. Follow these steps: Type in the letter t, hit the GOVT key and then hit GO. In front of you will see the complete list of U.S. Treasury bills, notes and bonds outstanding. I want you to find the 4% Treasury note expiring on 2/15/14. To scroll up or down at any point in time, hit page back or forward. The maturity date is the column with dates in it. To select the bond, click on it with the mouse. From now on we will refer to this screen as the Main Screen.

1. What is its price? When was it issued and how large was the issuance? To figure this out, click on item Security Valuation/Cashflow. After clicking on this, click on Description. A screen with a lot of information will pop-up. The amount issued will be under Security Information. Click the MENU key twice.

2. What is the yield to maturity of the bond? From the Main Screen, click on Yield and OAS Analysis and then click on YA Yield Analysis. Report the street convention yield.
3. Find the appropriate spot rates for discounting the...

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