Volatility shocks and curvature risk

We have seen that, while principal component analysis seems to identify curvature
shifts as a source of non-parallel risk, on closer inspection the results are somewhat
inconsistent. That is, unlike parallel and slope shifts, curvature shifts do not seem
to take a consistent form, making it difficult to design a corresponding risk measure.
The main reason for this is that ‘curvature shifts’ can occur for a variety of quite
different reasons. A change in mid-range yields can occur because (a) market volatility
expectations have changed, (b) the ‘term premium’ for interest rate risk has changed,
(c) market segmentation has caused a temporary supply/demand imbalance at
specific maturities, or (d) a change in the structure of the economy has caused a
change in the value d above. We briefly discuss each of these reasons, but readers
will need to consult the References for further details.
Regarding (a): The yield curve is determined by forward short-term interest rates,
but these are not completely determined by expected future short-term interest
rates; forward rates have two additional components. First, forward rates display a
downward ‘convexity bias’, which varies with the square of maturity. Second, forward
rates display an upward ‘term premium’, or risk premium for interest rate risk, which
(empirically) rises at most linearly with maturity. The size of both components
obviously depends on expected volatility as well as maturity.
A change in the market’s expectations about future interest rate volatility causes a
curvature shift for the following reason. A rise in expected volatility will not affect short
maturity yields since both the convexity bias and the term premium are negligible.
Yields at intermediate maturities will rise, since the term premium dominates the
convexity bias at these maturities; but yields at sufficiently long maturities will fall,
since the convexity bias eventually dominates. The situation is illustrated in Figure
5A.2. The precise form taken by the curvature shift will depend on the empirical forms
of the convexity bias and the term premium, neither of which are especially stable.
Regarding (b): The term premium itself, as a function of maturity, may change. In
theory, if market participants expect interest rates to follow a random walk, the term
premium should be a linear function of maturity; if they expect interest rates to
range trade, or mean revert, the term premium should be sub-linear (this seems to
be observed in practice). Thus, curvature shifts might occur when market participants
revise their expectations about the nature of the dynamics of interest rates, perhaps because of a shift in the monetary policy regime. Unfortunately, effects like
this are nearly impossible to measure precisely.
Regarding (c): Such manifestations of market ineffiency do occur, even in the US
market. They do not assume a consistent form, but can occur anywhere on the yield
curve. Note that, while a yield curve distortion caused by a short-term supply/
demand imbalance may have a big impact on a leveraged trading book, it might not
matter so much to a typical mutual fund or asset/liability manager.
Regarding (d): It is highly unlikely that short-term changes in d occur, although it
is plausible that this parameter may drift over a secular time scale. There is little
justification for using ‘sensitivity to changes in d’ as a measure of curvature risk.
Curvature risk is clearly a complex issue, and it may be dangerous to attempt to
summarize it using a single stylized ‘curvature shift’. It is more appropriate to use
detailed risk measures such as key rate durations to manage exposure to specific
sections of the yield curve.