Assignment
Zero rates and arbitrage pricing, and introduction to duration
Please turn in one copy per group. Please do not submit this printout along with your homework. Please show your work. Please write all of your names and your section number on your submission. It is recommended that you use Excel to solve this homework.
Bootstrapping
The following annual coupon bond prices with face values of $100 from a "risk-free" institution have been observed in the market as of 11/01/2015:
|
Maturity Date
|
Annual coupon rate
|
Price in decimals
|
|
11/01/2016
|
3.5%
|
100.00000
|
|
11/01/2017
|
3.5%
|
100.47299
|
|
11/01/2018
|
3.5%
|
101.39991
|
|
11/01/2019
|
3.5%
|
102.54795
|
|
11/01/2020
|
3.5%
|
103.97606
|
|
11/01/2021
|
3.5%
|
105.22412
|
1. Using bootstrapping, calculate the zero rates implied by the market prices above. Round answers to near the nearest basis point.
2. Using the zero rates above, calculate the discount factors in question 1. above
3. Using the zero rates, calculate the arbitrage-free price of a 3-year, 4% annual coupon bond with a par value of $100 maturing on 11/01/2018
4. Using the zero rates, calculate the coupon rate the "risk-free" institution would need to offer on a three-year annual coupon bond to result in an issuance price of par or $100. Round solution of c to nearest basis point. Here, given a set of zero rates, you'll need to guess a coupon rate ‘c' that sets the price of the bond equal to par.
Duration
5. Consider the following bonds:
|
Bond
|
YTM (%)
|
Coupon Rate (%)
|
Years to maturity
|
|
A
|
5
|
5
|
5
|
|
B
|
5
|
5
|
10
|
|
C
|
2
|
5
|
5
|
|
D
|
5
|
8
|
5
|
a. Calculate the duration for each of these $100 par, annual coupon bonds
b. How does time to maturity affect the duration of a bond? Why?
c. How does YTM affect the duration of a bond? Why?
d. How does the coupon rate affect the duration of a bond? Why?
Empirical
For this exercise we're going to analyze some historical data on zero-coupon bonds over the last 60 or so years. Included in this homework is a spreadsheet called "Zeros.xlsx". After the "Date" column, each column lists the price of a zero-coupon bond with a par value of $100 of a particular maturity, in years. For example on December 31, 1952, the price of a 1-year zero maturing on December 31, 1953 is $98.022 (Cell C3); the price of a 2-year zero maturing on December 31, 1954 is $95.713 (Cell D3); and so on. Notice that after a year the price of the 1-year zero is exactly $100, as the bond matures (Cell B4); analogously, after a year, the 2-year zero is now priced as a one-year zero at $98.438 (Cell C4); and so on.
1. Calculate the zero rates for each maturity from years 1 to 5 for each year. Plot them on a chart.
2. There is significant time variation in these rates. The National Bureau of Economic Research (NBER) "dates" business cycles: https://www.nber.org/cycles.html. What happens to the spread between the one-year zero rate and the five-year zero rates during recessions; that is, which rate falls relatively more, the one-year rate or the five-year rate? Why do you think that is?
Now we're going to look at some properties of the annual returns of these zero-coupon bonds. These aren't the "approximate" returns in the lecture slides because we'll be explicitly accounting for the one-year lapse in maturity. To calculate the annual return on zero-coupon bond you simply need to use this formula
Annual Return_today=(P_today-P_(last year))/P_(last year)
For example, the three-year zero as of December 31, 1952 was priced $93.587 (Cell E3). In one year's time the bond was now a two-year zero priced at $96.47 (Cell D4). Its annual return over the year, then, was
Annual Return_19531231=(P_19531231-P_19521231)/P_19521231 =(($96.47-$93.587))/$93.587=0.0308 or 3.08%
Loosely speaking, similar shifts of "one-cell-left-and-one-cell-down" (Cell E3 becomes D4, and so on) results hold for every other annual return. Note that you'll lose one time series observation when calculate returns for an asset; that is, if you have T price observations, you'll only have T-1 return observations.
3. Calculate the entire annual return series for each zero-coupon bond. In the end you'll have five annual return series, one for each maturity. Each annual return series should be 60 observations.Don't submit these time series!
4. Using Excel, calculate the average return for each of the zeros. Also calculate the standard deviations. To calculate the average return, use Excel's "=AVERAGE(...)" function. To calculate the standard deviation, use Excel's "STDEV(...)" function. For each maturity, report the zeros average return and standard deviation of those returns in table.
5. If we take standard deviation as a crude measure of risk, which bonds are riskier, the short-term bonds or the long-term bonds? Are you compensated for this risk in the form of a higher average return?