Non-parallel risk,duration bucketing and partial durations

Before discussing principal component analysis, we briefly review some more primitive
approaches to measuring non-parallel risk. These have by no means been
superseded: later in the chapter we will discuss precisely what role they continue to
play in risk management.
The easiest approach is to group securities into maturity buckets. This is a very
simple way of estimating exposure to movements at the short, medium and long
ends of the yield curve. But it is not very accurate: for example, it ignores the fact
that a bond with a higher coupon intuitively has more exposure to movements in the
short end of the curve than a lower coupon bond with the same maturity.
Next, one could group securities into duration buckets. This approach is somewhat
more accurate because, for example, it distinguishes properly between bonds with
different coupons. But it is still not entirely accurate because it does not recognize
that the different individual cash flows of a single security are affected in different
ways by a non-parallel yield curve shift.
Next, one could group security cash flows into duration buckets. That is, one uses
a finer-grained unit of analysis: the cash flow, rather than the security. This makes
the results much more precise. However, bucketed duration exposures have no direct
interpretation in terms of changes in some reference set of yields (i.e. a shift in some
reference yield curve), and can thus be tricky to interpret. More seriously, as
individual cash flows shorten they will move across bucket duration boundaries,
causing discontinuous changes in bucket exposures which can make risk management
awkward.
Alternatively, one could measure partial durations. That is, one directly measures
how the value of a portfolio changes when a single reference yield is shifted, leaving
the other reference yields unchanged; note that doing this at the security level and
at the cashflow level gives the same results. There are many different ways to define
partial durations: one can use different varieties of reference yield (e.g. par, zero
coupon, forward rate), one can choose different sets of reference maturities, one can
specify the size of the perturbation, and one can adopt different methods of interpolating
the perturbed yield curve between the reference maturities.
The most popular partial durations are the key rate durations defined in Ho (1992).
Fixing a set of reference maturities, these are defined as follows: for a given reference
maturity T, the T-year key rate duration of a portfolio is the percentage change in its
value when one shifts the T-year zero coupon yield by 100 bp, leaving the other
reference zero coupon yields fixed, and linearly interpolating the perturbed zero
coupon curve between adjacent reference maturities (often referred to as a ‘tent’
shift). Figure 5.1 shows some examples of key rate durations.
All the above approaches must be used with caution when dealing with optionembedded
securities such as callable bonds or mortgage pools, whose cashflow
timing will vary with the level of interest rates. Option-embedded bonds are discussed
in detail elsewhere in this book.