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HW 4
发布时间：2023-12-06
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HW 4
1) I took the monthly data from HW1 and kept data only from January. You will find this data in Moodle under the name “HW 4 Data.” The purpose of this HW problem is to investigate the Expectations Hypothesis, and more particularly, how useful forward rates are for predicting future spot rates.
1A) Determine the forward rates f0,1,2, that is, the one year forward rate one year from now.
1B) run a bivariate regression, with the dependent variable being the future one year spot rate z1,2, and the two regressors being 1) the current one year spot rate z0,1, and 2) the current forward rate f0,1,2. (It is easiest to run a multivariate regression in excel when the regressors are in columns next to each other.) As in previous cases, we will have one less row of data for the regression than in the raw data (hence the cells in red). What is the adjusted R2 of the regression? What are the coefficients and t-stats on the two independent variables? Interpret these results. In particular, how useful are the current spot rate and the forward rate for predicting future spot rates?
1C) One concern with the regression in 4B is that the interest rate process is very persistent. So instead, regress the change in spot rates [ z1,2 – z0,1 ] on [ f0,1,2 – z0,1 ].
(Intuitively, if you want to make money, you need to be able to predict how prices (or equivalently, yields) will change.) Once again, report the adjusted R2 and the coefficient and t-stat of the independent variable. How useful are forward rates for predicting changes in spot rates?
2) (For the following question, assume for simplicity that coupons are only paid annually. Further, assume all compounding is done annually). A one-year, zero-coupon bond with Maturity value = $1100 is selling for $095. A two-year coupon bond with Maturity value = $2100 and coupon rate 6% is selling for $099.50.
2A) Determine the 1-year (z01) and 2-year (z02) spot rates
2B) Determine f012, that is, the one-year forward rate one year from now.
2C) After careful analysis, you have estimated that the most likely one-year spot rate one-year from now, E0[z12], is 6.5%, which is higher than both the current one-year spot rate, z01, and the two-year spot rate, z02. You are considering two investments:
1) Invest in 2-year zero today, hold until maturity
2) Invest in 1-year zero today, and roll-over proceeds into another one-year zero one year from now.
Which of the two strategies should you choose and why?
3) The 6-month Treasury Bill has a YTM of 5.25%. The one-year Treasury Bill has a YTM = 5.5%. Both of these are zero coupon bonds, with the standard convention being used here. Thus, $100 today gets you an IOU from the Gov’t of $102.625 due in 6 months. Similarly, $100 today gets you an IOU from the Gov’t of $105.5756 due in one year. All bonds with maturity greater than one year are coupon bonds which are selling at par. The 1.5 year bond has a YTM = 5.75%. The 2-year bond has a YTM = 6.00%. The 2.5-year bond has a YTM = 6.25%. The 3-year bond has a YTM = 6.50%.
3A) Calculate D01, the price today for a piece of paper that pays $1 one period ( = 6 months) from now. Also calculate D02, D03, D04, D05, D06.
3B) Calculate the spot rates z01, z02, z03, z04, z05, z06, and then annualize them to obtain the YTM’s of zero-coupon bonds.
3C) Determine the Price of a Treasury bond with face value M = $100, Coupon = 4% and Maturity = 3 years.
