Correlations: between markets,between yield and volatility

Recall that principal component analysis uses a single correlation matrix to identify
dominant patterns of yield shifts. The results imply something about the correlations
themselves: for instance, the existence of a global parallel shift that explains around
50% of variance in global bond yields suggests that correlations should, on average,
be positive.
However, in global markets, correlations are notoriously time-varying: see Figure
5.12. In fact, short-term correlations between 10-year bond yields in different countries are significantly less stable than correlations between yields at different maturities
within a single country. This means that, at least for short time horizons, one
must be especially cautious in using the results of principal component analysis to
manage a global bond position.
We now discuss a somewhat unrelated issue: the relationship between yield and
volatility, which has been missing from our analysis so far. Principal component
analysis estimates the form of the dominant yield curve shifts, namely parallel and
slope shifts. It says nothing useful about the size of these shifts, i.e. about parallel
and slope volatility. These can be estimated instantaneously, using historical or
implied volatilities. But for stress testing and scenario analysis, one needs an
additional piece of information: whether there is a relationship between volatility and
(say) the outright level of the yield curve. For example, when stress testing a trading
book under a ò100 bp scenario, should one also change one’s volatility assumption?
It is difficult to answer this question either theoretically or empirically. For example,
most common term structure models assume that basis point (parallel) volatility is
either independent of the yield level, or proportional to the yield level; but these
assumptions are made for technical convenience, rather that being driven by the
data.
Here are some empirical results. Figures 5.13(a)–(c) plot 12-month historical
volatilities, expressed as a percentage of the absolute yield level, against the average
yield level itself. If basis point volatility were always proportional to the yield level,
these graphs would be horizontal lines; if basis point volatility were constant, these
graphs would be hyperbolic.
Neither seems to be the case. The Japanese dataset suggests that when yields are
under around 6–7%, the graph is hyperbolic. All three datasets suggest that when
yields are in the 7–10% range, the graph is horizontal. And the US dataset suggests
that when yields are over 10%, the graph actually slopes upward: when yields rise,
volatility rises more than proportionately. But in every case, the results are confused
by the presence of volatility spikes.
The conclusion is that, when stress testing a portfolio, it is safest to assume that
when yields fall, basis point volatility need not fall; but when yields rise, basis point
volatility will also rise. Better yet, one should run different volatility scenarios as well
as interest rate scenarios.