Chapter 27. Estimating the Yield Curve

7. Which interpolation scheme (linear, exponential, or logarithmic)
gives the highest inter-polated rates? Why? Can you explain the relative
ordering of the interpolation schemes in your answer to the previous question?

8. In practice, bond prices are never available at conveniently
spaced intervals. Some interpolation scheme is called for. However, by making
an assumption of constant forward rates between non-standard maturities, we can
develop a spot rate curve even for unequal time intervals. In this question,
you will undertake a simple exercise of this type.

You are given the following discount bond prices at times t:

t
Discount Bond Price

0.70
0.9754

1.32
0.9256

2.11
0.8777

All compounding and
discounting is continuous.

(a) Assuming that forward rates are constant between these dates, nd
these forward rates.

(b)
Price a two-year $100 face value bond that pays 10% semiannually.

=

9. In the previous question, what type of interpolation scheme is
being e ectively used: linear, exponential, or logarithmic?

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10. This question requires you to develop zero-coupon and forward rate
curves using real-world data from the US government debt markets. Proceed by
implementing the fol-lowing steps:

(a) Collect data for any one recent date on bond prices and yields.
There are many sources for such data, such as The Wall Street Journal,
Bloomberg screens, etc. The Wall Street Journal is the easiest. You will obtain
a set of maturities and yields. You need to get enough data for up to seven
years of maturity. Anything from 20 bonds or more would be appropriate. Arrange
them on a spreadsheet in two columns: (1) Maturity in Years (fractions allowed)
from Today and (2) Yield to Maturity (ytm)]. Call this Table 1.

(b) Plot the points with maturity on the x-axis and ytm on the y-axis
(a scatter plot). Call this Plot 1.

(c)
Fit a curve through the plot. We leave this to your imagination,
and you are free to choose some way to t a smooth line through this data.
Spreadsheets usually provide a tool to do this. Remember, that a straight line
is probably not the best way to do this. Call this curve Plot 2.

On
the same plot, t a curve through the coupon rates and through the bond prices.
Now you have three interpolated curves for yields, coupons, and prices.

(d)
With your tted line for yields, generate a new table of ytms,
coupon rates, and prices, each observation being six months apart. Hence, if
your last maturity bond is of seven years, you will have 14 periods of a half
year each. Call this Table 2. Remember that your tted line gives you yield as a
function of maturity. Hence, for maturities t = 0:5; 1; 1:5; 2; :::; 6:5; 7:0,
you will compute matching yields.

(e) Using this table of ytms, prices, and coupons, compute (a) the
zero-coupon rates and (b) forward rates for all maturities in the table. Call
this Table 3.

(f) Present plots of the (a) ytm curve, (b) zero-coupon rate curve,
and (c) forward rate curve on the same graph. Call this Plot 3.

(g)
Now, as an alternative, show how you can use Table 1 with a
regression method to derive discount factors and zero-coupon rates. Feel free
to make any simplifying assumptions here. Create a table of zero-coupon rates
spaced half a year apart. Call this Table 4.

(h) Plot the zero-coupon curve from Table 3 versus the one from Table
4, and com-ment. Call this Plot 4.

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Be creative!

11.
If you receive a cash ow of $100 at time 1.25 years, explain how
you would allocate this cash ow into two cash ows, A received at one year, and
cash ow B, received at 1.5 years. Assume that the zero-coupon rate for 1 year
is 6% and that for 1.5 years is 6.5%. Assume continuous compounding.

12. Using a cubic splines scheme, t the following discount factors
using just one knot point at t = 0:5 years.

t
d(t)

0.1
0.9934

0.2
0.9845

0.6 0.9456

0.8 0.9267

Find the function that
describes the entire discount function for any maturity t.