**Unformatted text preview: **Backus&Zin B40.3176.33 Spring 1999 Assignment 1: Answers
March 24, 1999 1. Calibration to DM rates. a Properties of DM rates: Maturity Mean Std Dev
1
3.503 0.437
24
4.154 0.433
36
4.609 0.454
60
5.377 0.519
120
6.388 0.663
b You might choose parameters like this: i ' = 0:92. In the data, the autocorrelations are a little less than this. ii = 3:503=1200 = 0:00292 the mean
short rate. iii = 1 , '2 1=20:437=1200 the standard deviation of the
short rate. This is a lot smaller than we saw for the US. The big issue here is
the short sample. With a highly persistent series, you need a lot of data to get
reliable estimates, and this isn't enough. iv = ,1:50 hits the 10-year rate.
See graph below.
c The long rates here are swap rates, which are par yields. They solve for a
seminannual swap and a monthly model
6
12 +
n
Swap Rate = 200 b + b 1 , bn + b ;
where n is the maturity of the swap in months. For the model, we could compute
discount factors bj for the mean value of the state variable namely, and
compute the value of the swap rate associated with it. The result is shown in the
gure o's, where we see that they're not much di erent from the spot rates.
What we have here is two o -setting errors: semiannual compounding raises the
rates and swap rates which mix short maturities with long ones lower them.
The result would have little e ect on our calibration one way or the other.
2. Vasicek for long rates. a This follows similar principles, but is a little harder since we can't nd the parameters one at a time. i ' = 0:978, the autocorrelation of the 5-year rate.
ii The variance of the 5-year rate in the data is 2:212=12002. In the model,
it's
!
1 , '60 2 2=1 , '2 :
2 2=1 , '2 =
B60=60
1,' 2
Equating the two gives us = 0:0159. iii,iv We need to pick and together.
= ,0:055 and = 0:00623 comes close. But as before, there is too little
curvature, so short rates are too high.
b This is a little terse, but here it goes: volatility of spot rates in this model declines
with maturity see below. With the given choices of ', the rate of decline is
greater than we see in the data. For this reason and others, many experts think a
model should have a second factor that is very persistent. Even many one-factor
models build in the presumption that ' is one, Ho-Lee and Black-Derman-Toy
being prominent examples.
Figures follow:
6.5 Mean Yield (Annual Percentage) 6 5.5 5 4.5 4 3.5
0 20 40 60
Maturity in Months 80 100 120 20 40 60
Maturity in Months 80 100 120 4 Standard Deviation of Spot Rate 3.5 3 2.5 2 1.5 1 0.5
0 ...

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