1) Complete columns V and W. (Note, I have made your life easier by using the dollar-sign in
appropriate places in cells V4 and W4.) Also, define date-t changes as the difference between
the values at date-(t+1) and date-t. Use this to fill in columns AB through AS, noting that there is
one less row when changes are introduced (hence, the red line in row 614). The eigenvector
associated with the highest eigenvalue are the weights for the level principal component, and
the eigenvector associated with the second highest eigenvalue are the weights for the slope
principal component. (I originally planned to ask you to derive these yourself, but it is too
difficult to do in Excel, so I decided to give them to you). With that said, if you are given a vector
and a covariance matrix, you can check whether that vector is an eigenvector of that matrix. If
so, you can identify the eigenvalue associated with this eigenvector. Here I ask you to do this for
the proposed level and slope eigenvectors. Basically, for the level and slope factors separately, I
am asking you to identify the matrix product Cψ, which is a vector, and then divide each term by
ψ. If Cψ = λψ, then each term in the vector Cψ divided by ψ should identify the same value λ.
2) Use yield changes in columns AB through AP to determine their covariance matrix. I have
already pasted all but the last row in, which you need to add in cells (AY17, BM17) and in cells
(AY41, BM41)
3) For both proposed eigenvectors, show that the condition Cψ = λψ holds (at least
approx imately). To do this for the level factor, highlight cells (BQ3, BQ17) and then write the
command =MMULT(AY3:BM17,BO3:BO17), and then press ctrl/shift/enter. If done cor rectly,
the cells (BQ3, BQ17) should now be populated. Moreover, if done correctly, if you divide cell
BQ3 by BO3, and BQ4 by BO4, etc, you should get almost the same value each time. The average
of these values determines the eigenvalue for the first PC (i.e., the level 2 factor), which you
should report. Repeat this process for the second PC, using the command
=MMULT(AY27:BM41,BO27:BO41). Then again check if the BQ27 divided by BO27 gen erates
approximately the same value as BQ28 divided by BO28, etc. If so, again take an average, and
define this as the eigenvalue for the slope PC.
4) Now that you have shown that the proposed level and slope weight vectors are in fact the first
and second Principal Components of the covariance matrix, we are ready to investigate how
well these two factors describe the entire yield curve. First, use the data in columns AR and AS
to determine the correlation between changes in the level and slope factors. One reason
Principal Component Analysis is so popular is that, by construction, these two factors are
uncorrelated. Thus, on average, when the level factor goes up or down, it does so independently
of the slope factor, and vice-versa. Report your correlation estimate. (It should be close to zero).
The implication is that, similar to associating to each fixed income portfolio a single statistic
“Duration,” Principal Component Analysis allows us to associate with any fixed income portfolio
the amount of exposure this portfolio has to changes in Level and Slope, and that changes to
these factors are uncorrelated with each other.
5) Finally, using a “scatter plot with lines”, plot the factor loadings on the Level factor as a function
of maturity (i.e., 1, 4, 7, 10, 15). You should find that all of the factor loadings are positive. The
implication is that when the level factor increases, the yields for all maturities tend to increase
(qualitatively consistent with the duration approximation assumption). Then on the same
scatter plot, plot the factor loadings for the Slope factor as a function of maturity. 3 (Look at
columns CX-CZ). In this case you should find that these coefficients flip sign. The implication is
that if the slope factor changes, its impact on short maturities will in general be opposite of its
impact on longer maturities, consistent with the slope of the yield curve changing.