Appendix: Economic factors driving the curve

Macroeconomic explanation of parallel and slope risk
This appendix presents some theoretical explanations for why (a) parallel and slope
shifts are the dominant kinds of yield curve shift that occur, (b) curvature shifts are
observed but tend to be both transitory and inconsistent in form, and (c) the behavior
of the short end of the yield curve is quite idiosyncratic. The theoretical analysis
helps to ascertain which empirical findings are really robust and can be relied upon:
that is, an empirical result is regarded as reliable if it has a reasonable theoretical
explanation. For reasons of space, the arguments are merely sketched.
We first explain why parallel and slope shifts emerge naturally from a macroeconomic
analysis of interest rate expectations. For simplicity, we use an entirely
standard linear macroeconomic model, shown in Table 5A.1; see Frankel (1995) for
details.

The model is used in the following way. Bond yields are determined by market
participants’ expectations about future short-term interest rates. These in turn are
determined by their expectations about the future path of the economy: output,
prices and the money supply. It is assumed that market participants form these
expectations in a manner consistent with the macroeconomic model. Now, the model
implies that the short-term interest rate must evolve in a certain fixed way; thus,
market expectations must, ‘in equilibrium’, take a very simple form.
To be precise, it follows from the theorem stated in Table 5A.1 that if i0 is the
current short-term nominal interest rate, it is the currently expected future interest
rate at some future time t, and iê is the long-term expected future interest rate, then
rational interest rate expectations must take the following form in equilibrium:
itóiêò(i0ñiê)eñdt
In this context, a slope shift corresponds to a change in either iê or i0, while a
parallel shift corresponds to a simultaneous change in both. Figure 5A.1 shows,
schematically, the structure of interest rate expectations as determined by the model.
The expected future interest rate at some future time is equal to the expected future
rate of inflation, plus the expected future real rate. (At the short end, some distortion
is possible, of which more below.)
Figure 5A.1 Schematic breakdown of interest rate expectations.
In this setting, yield curve shifts occur when market participants revise their
expectations about future interest rates – that is, about future inflation and output
growth. A parallel shift occurs when both short- and long-term expectations change
at once, by the same amount. A slope shift occurs when short-term expectations
change but long-term expectations remain the same, or vice versa.
Why are parallel shifts so dominant? The model allows us to formalize the following
simple explanation: in financial markets, changes in long-term expectations are primarily driven by short-term events, which, of course, also drive changes in shortterm
expectations. For a detailed discussion of this point, see Keynes (1936).
Why is the form of a slope shift relatively stable over time, but somewhat different
in different countries? In this setting, the shape taken by a slope shift is determined
by d, and thus by the elasticity parameters c,{ j, o of the model. These parameters
depend in turn on the flexibility of the economy and its institutional framework –
which may vary from country to country – but not on the economic cycle, or on the
current values of economic variables. So d should be reasonably stable.
Finally, observe that there is nothing in the model which ensures that parallel and
slope shifts should be uncorrelated. In fact, using the most natural definition of
‘slope shift’, there will almost certainly be a correlation – but the value of the
correlation coefficient is determined by how short-term events affect market estimates
of the different model variables, not by anything in the underlying model itself. So
the model does not give us much insight into correlation risk.