This exercise uses US Treasury STRIP price quotes from www.bondsonline.com as of November 30, 2000. You can access the discount factors and the implied zero-coupon rates by clicking here. You can also look at a plot of the discount factors across maturities. The first column in this file is the time to maturity of the STRIP in years, the second column is the discount factor (computed as price divided by 100) and the last column is the zero-coupon rate that is implied by the discount factor and time to maturity per the formula from the slides in Lecture 2.
(A) Derive the zero-coupon and the par coupon yield curve as well as the implied forward rates for all the maturities for which there are quotes. Do you believe that all that jaggedness in the implied forward rates is really out there? What might potentially be causing this behavior?
Here is the picture that I obtained.
(B) Using the following functional form from the slides:
z(0,t) = a + b*exp(-t) + c*exp(-3*t) + d*exp(-9*t) + e*exp(-27*t) + error
to run a regression of the zero coupon rates on the "artificial regressors" above. You would need to construct x1 = exp(-t), x2 = exp(-3*t), x3 = exp(-9*t), etc., etc. Obtain the parameter estimates for a, b, c, d and e above. Do you get a reasonable fit? Are the estimated coefficients statistically significant?
Here are my estimates of a, b, c, d and e:
a = 0.058096, b = -0.028737, c = 0.140322, d = -1.197551, e = 34.493474
with the corresponding t-statistics:
ta = 41.31212, tb = -0.75761, tc = 0.58463, td = -0.44806, te = 0.38581.
Here is a plot of the actual zero coupon rates versus the fitted values of the zeros from the regression above.
(C) Now compute the zero coupon rates implied by the regression (!), i.e. using the values for the parameters, for t = 0, 1/2, 1, 3/2, ..., 25. These values are commonly referred to as "the fitted values" in econometrics lingo. Based on the fitted values of the zero coupon rates derive the par coupon yield curve and the implied forward rates.
Here is the plot of the smoothed zeros, par coupons and implied forward that I obtained.
(D) Thought question: If you do not believe the smoothed version of the yield curves and the implied forward rates try to play around with the functional form of the regression above and try to come up with another smoothed version of the zeros that you are comfortable with.
No unique answer to this one :-) Given that the regression coefficients are most significant for the first term, it seems sensible to add more terms with small coefficients for t in the exponential.
