Assignment
Problem 1:
Consider the following ask yields for T-bills that were quoted on 3/15/10, with settlement on 3/16/10. Fill in the third and fourth columns. Note: the quoted ask yield is a discount yield in percentage points. Thus .043 percent is .00043. Since yields are so low right now, carry around quite a few digits past the decimal point.
Compute BEY based on the semi-annual compounded rate as on slide #30 in the Teaching Note 1. (You could alternatively use the formula in the Teaching Note 2 (slide #18), but it will give you a slightly different answer because it's based on a simple, rather than semi-annually compounded rate.)
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Maturity date
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Ask Yield
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Bond Equivalent Yield
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Price (per $100 in FV)
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4/19/10
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.0430
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|
|
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4/27/10
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.0360
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|
|
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5/03/10
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.0430
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|
|
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5/10/10
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.0380
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|
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Problem 2:
There is a 4 percent coupon T-note trading in the market today. Its next semi-annual coupon payment is in 53 days. After this coupon payment, the T-note has 9 more semiannual coupon payments and a final face value payment of $1,000. The quoted asked yield is currently 2.3 percent (compounded semi-annually).
The settlement will occur in 2 days. The number of days in the current semi-annual period is 183 days. What is the full (or dirty) price and what is the clean price? Also, what accrued interest do you owe to the current owner at settlement?
Problem 3:
Consider the following three bonds and bond prices (0s denotes 0% annual coupon rate):
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Bond
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Price
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0s of 5/15/2012 (expiration date)
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96-12
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7.5s of 5/15/2012
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103-12
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15s of 5/15/2012
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106-02
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All bonds have identical face values and the same (but uncertain) settlement dates. Do these prices imply any arbitrage opportunity? If so, what are the weights of each security in an arbitrage portfolio (normalize the weight of the 15s bond to 1)? What is the arbitrage profit?
Problem 4: You are encouraged to use spreadsheet software to solve this problem. Feel free to modify the Google Sheets example presented in class. Solutions will be also provided in a form of a spreadsheet.
Q1: Use the following list of Treasury bond prices as of March 15, 2013 to derive (bootstrap) discount factors for 6 months, 1 year, 18 months, and 2 years:
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Maturity
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Coupon
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Price
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9/15/2013
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4.750%
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102-077
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3/15/2014
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4.125%
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103-21
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9/15/2014
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5.000%
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106-102
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3/15/2015
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5.000%
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107-276
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Q2: The Treasury is issuing a 2-year new note on March 15, 2013. The total offering is $15,000,000,000, and we observe the following demand schedule:
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Tendered
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Yield
|
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1,000,000,000
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0.999%
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3,000,000,000
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1.000%
|
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1,000,000,000
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1.001%
|
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4,000,000,000
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1.002%
|
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2,000,000,000
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1.003%
|
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3,000,000,000
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1.004%
|
|
5,000,000,000
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1.005%
|
|
1,000,000,000
|
1.006%
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- What yield clears the market?
- If the Treasury wishes to offer the issue at as close to par as possible, but not at a premium (i.e. price must be below par value), what is the coupon on the bond? The Treasury offers coupons in increments of 1/8%.
- What is the price of the bond given the discount factors above?
- What is the price of the bond given the market clearing yield (YTM)?
Q3: Create a portfolio to replicate the bond in Q2 using the bonds in Q1. Is there an arbitrage opportunity? Assume the bond in Q2 is priced according to its YTM (the market clearing yield you found in Q2). What are the weights of each of the four bonds in your replicating portfolio? What is the arbitrage profit?
Bonus question (Q4): Fit a discount rate curve to the bonds in Q1 using the Nelson-Siegel discount rate function. What are the implied parameter values and the total square error? Allow the solver to optimally pick all four parameters. Note: MS Excel and Google Sheets solvers often converge to local minima. Your results can vary widely as a consequence - that's OK.
Note: You might face issues with convergence in Excel when solving for the Nelson- Siegel discount rate function. I suggest trying the following things:
a. experiment with initial values of parameters; I suggest setting all of them to zero except h=0.1; see Nelson-Siegel tab in the Discount Factors.xlsx file;
b. make sure that "Make unconstrained variables non-negative" in the Solver dialog is unchecked;
c. you can try running optimization one more time after you find a solution; Excel's built-in solver is quite bad and this additional step may lead to a better solution.