It is extremely tempting to conclude that (a) the analysis has determined that there
are exactly three important kinds of yield curve shift, (b) that it has identified them
precisely, and (c) that it has precisely quantified their relative importance.
But we should not draw these conclusions without looking more carefully at the
data. This means exploring datasets drawn from different historical time periods,
from different sets of maturities, and from different countries. Risk management
should only rely on those results which turn out to be robust.
Figure 5.3 shows a positive finding. Analyzing other 5-year historical periods, going
back to 1963, we see that the overall results are quite consistent. In each case the
major yield curve shifts turn out to be parallel, slope and curvature shifts; and
estimates of the relative importance of each kind of shift are reasonably stable over
time, although parallel shifts appear to have become more dominant since the late
1970s.
Figures 5.4(a) and (b) show that some of the results remain consistent when
examined in more detail: the estimated form of both the parallel shift and the slope
shift are very similar in different historical periods. Note that in illustrating each
kind of yield curve shift, we have carried out some normalization to make comparisons
easier: for example, estimated slope shifts are normalized so that the 10-year yield
moves 100 bp relative to the 1-year yield, which remains fixed. See below for further
discussion of this point.
However, Figure 5.4(c) does tentatively indicate that the form of the curvature shift
has varied over time – a first piece of evidence that results on the curvature shift
may be less robust than those on the parallel and slope shifts.
Figure 5.5 shows the effect of including 3- and 6-month Treasury bill yields in the
1993–8 dataset. The major yield curve shifts are still identified as parallel, slope and
curvature shifts. However, an analysis based on the dataset including T-bills attaches
somewhat less importance to parallel shifts, and somewhat more importance to
slope and curvature shifts. Thus, while the estimates of relative importance remain
qualitatively significant, they should not be regarded as quantitatively precise.
Figures 5.6(a) and (b) show that the inclusion of T-bill yields in the dataset makes
almost no difference to the estimated form of both the parallel and slope shifts.
However, Figure 5.6(c) shows that the form of the curvature shift is totally different.
Omitting T-bills, the change in curvature occurs at the 3–5-year part of the curve;
including T-bills, it occurs at the 1-year part of the curve. There seem to be some
additional dynamics associated with yields on short term instruments, which become
clear once parallel and slope shifts are factored out; this matter is discussed further
in Phoa (1998a,b).
The overall conclusions are that parallel and slope shifts are unambiguously the
most important kinds of yield curve shift that occur, with parallel shifts being
dominant; that the forms of these parallel and slope shifts can be estimated fairly
precisely and quite robustly; but that the existence and form of a third, ‘curvature’
shift are more problematic, with the results being very dependent on the dataset
used in the analysis. Since the very form of a curvature shift is uncertain, and
specifying it precisely requires making a subjective judgment about which dataset is
‘most relevant’, the curvature shift is of more limited use in risk management.
The low weight attached to the curvature factor also suggests that it may be less
important than other (conjectural) phenomena which might somehow have been
missed by the analysis. The possibility that the analysis has failed to detect some
important yield curve risk factors, which potentially outweigh curvature risk, is
discussed further below.
International bond yield data are analyzed in the next section. The results are broadly consistent, but also provide further grounds for caution. The Appendix
provides some theoretical corroboration for the positive findings.
We have glossed over one slightly awkward point. The fundamental yield curve
shifts estimated by a principal component analysis – in particular, the first two
principal components representing parallel and slope shifts – are, by definition,
uncorrelated. But normalizing a ‘slope shift’ so that the 1-year yield remains fixed
introduces a possible correlation. This kind of normalization is convenient both for
data analysis, as above, and for practical applications; but it does mean that one
then has to estimate the correlation between parallel shifts and normalized slope
shifts. This is not difficult in principle, but, as shown in Phoa (1998a,b), this
correlation is time-varying and indeed exhibits secular drift. This corresponds to the
fact that, while the estimated (non-normalized) slope shifts for different historical
periods have almost identical shapes, they have different ‘pivot points’. The issue of
correlation risk is discussed further below.
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