It is now tempting to concentrate entirely on parallel and slope shifts. This approach
forms the basis of most useful two factor interest rate models: see Brown and
Schaefer (1995). However, it is important to understand what is being lost when one
focuses only on two kinds of yield curve shift.
First, there is the question of whether empirical correlations are respected. Figure
5.7(a) shows, graphically, the empirical correlations between daily Treasury yield
shifts at different maturity points. It indicates that, as one moves to adjacent
maturities, the correlations fall away rather sharply. In other words, even adjacent
yields quite often shift in uncorrelated ways.
Figure 5.7(b) shows the correlations which would have been observed if only
parallel and slope shifts had taken place. These slope away much more gently as one
moves to adjacent maturities: uncorrelated shifts in adjacent yields do not occur.
This observation is due to Rebonato and Cooper (1996), who prove that the correlation
structure implied by a two-factor model must always take this form.
What this shows is that, even though the weights attached to the ‘other’ eigenvectors
seemed very small, discarding these other eigenvectors radically changes the
correlation structure. Whether or not this matters in practice will depend on the
specific application.
Second, there is the related question of the time horizon of risk. Unexplained yield
shifts at specific maturities may be unimportant if they quickly ‘correct’; but this will
clearly depend on the investor’s time horizon. If some idiosyncratic yield shift occurs,
which has not been anticipated by one’s risk methodology, this may be disastrous
for a hedge fund running a highly leveraged trading book with a time horizon of hours or days; but an investment manager with a time horizon of months or quarters,
who is confident that the phenomenon is transitory and who can afford to wait for it
to reverse itself, might not care as much.
This is illustrated in Figure 5.8. It compares the observed 10-year Treasury yield
from 1953 to 1996 to the yield which would have been predicted by a model in which
parallel and slope risk fully determine (via arbitrage pricing theory) the yields of all
Treasury bonds. The actual yield often deviates significantly from the theoretical
yield, as yield changes unrelated to parallel and slope shifts frequently occurred. But
deviations appear to mean revert to zero over periods of around a few months to a
year; this can be justified more rigorously by an analysis of autocorrelations. Thus,
these deviations matter over short time frames, but perhaps not over long time
frames. See Phoa (1998a,b) for further details.
Third, there is the question of effects due to market inhomogeneity. In identifying
patterns of yield shifts by maturity, principal component analysis implicitly assumes
that the only relevant difference between different reference yields is maturity, and
that the market is homogeneous in every other way. If it is not – for example, if
there are differences in liquidity between different instruments which, in some
circumstances, lead to fluctuations in relative yields – then this assumption may not
be sound.
The US Treasury market in 1998 provided a very vivid example. Yields of onthe-
run Treasuries exhibited sharp fluctuations relative to off-the-run yields, with
‘liquidity spreads’ varying from 5 bp to 25 bp. Furthermore, different on-the-run
issues were affected in different ways in different times. A principal component
analysis based on constant maturity Treasury yields would have missed this source
of risk entirely; and in fact, even given yield data on the entire population of Treasury
bonds, it would have been extremely difficult to design a similar analysis which
would have been capable of identifying and measuring some systematic ‘liquidity spread shift’. In this case, risk management for a Treasury book based on principal
component analysis needs to be supplemented with other methods.
Fourth, there is the possibility that an important risk factor has been ignored. For
example, suppose there is an additional kind of fundamental yield curve shift, in
which 30- to 100-year bond yields move relative to shorter bond yields. This would
not be identified by a principal component analysis, for the simple reason that this
maturity range is represented by only one point in the set of reference maturities.
Even if the 30-year yield displayed idiosyncratic movements – which it arguably does
– the analysis would not identify these as statistically significant. The conjectured
‘long end’ risk factor would only emerge if data on other longer maturities were
included; but no such data exists for Treasury bonds.
An additional kind of ‘yield curve risk’, which could not be detected at all by an
analysis of CMT yields, is the varying yield spread between liquid and illiquid issues
as mentioned above. This was a major factor in the US Treasury market in 1998; in
fact, from an empirical point of view, fluctuations at the long end of the curve and
fluctuations in the spread between on- and off-the-run Treasuries were, in that
market, more important sources of risk than curvature shifts – and different methods
were required to measure and control the risk arising from these sources.
To summarize, a great deal more care is required when using principal component
analysis in a financial, rather than physical, setting. One should always remember
that the rigorous justifications provided by the differential equations of physics are
missing in financial markets, and that seemingly analogous arguments such as
those presented in the Appendix are much more heuristic. The proper comparison is
with biology or social science rather than physics or engineering.
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