also governs where the loading on β3t achieves its maximum.5
We interpret β1t, β2t andβ3t as three latent dynamic factors. The loading on β1t is 1, a constant
that does not decay to zero in the limit; hence it may be viewed as a long-term factor. The loading on β2t
is (1 -e λtτ) / λtτ , a function that starts at 1 but decays monotonically and quickly to 0; hence it may be
viewed as a short-term factor. The loading on β3t is (( 1 -e λtτ) / λtτ) - e λtτ , which starts at 0 (and is thus
not short-term), increases, and then decays to zero (and thus is not long-term); hence it may be viewed as
a medium-term factor. We plot the three factor loadings in Figure 1. They are similar to those obtained
by Bliss (1997a), who estimated loadings via a statistical factor analysis.6
An important insight is that the three factors, which following the literature we have thus far
called long-term, short-term and medium-term, may also be interpreted in terms of level, slope and
curvature. The long-term factor β1t, for example, governs the yield curve level. In particular, one can
easily verify that y((∞)=β1t. Alternatively, note that an increase in β1t increases all yields equally, as the
loading is identical at all maturities, thereby changing the level of the yield curve.
The short-term factor β21 is closely related to the yield curve slope, which we define as the ten-
year yield minus the three-month yield. In particular, y( (120) -yt (3) = -.78β2t+.06β3t. Some authors such
as Frankel and Lown (1994), moreover, define the yield curve slope as yt (∞) -yt (0), which is exactly
equal to -β21. Alternatively, note that an increase in β21 increases short yields more than long yields,
because the short rates load on β21 more heavily, thereby changing the slope of the yield curve.
We have seen that β11 governs the level of the yield curve and β21 governs its slope. It is
interesting to note, moreover, that the instantaneous yield depends on both the level and slope factors,
because yt (0) = β11 + β21. Several other models have the same implication. In particular, Dai and
Singleton (2000) show that the three-factor models of Balduzzi, Das, Foresi and Sundaram (1996) and
Chen (1996) impose the restrictions that the instantaneous yield is an affine function of only two of the
three state variables, a property shared by the Andersen-Lund (1997) three-factor non-affine model.
Finally, the medium-term factor β31 is closely related to the yield curve curvature, which we
define as twice the two-year yield minus the sum of the ten-year and three-month yields. In particular,
5 Throughout this paper, and for reasons that will be discussed subsequently in detail, we set λt =
0.0609 for all t.
6 Factors are typically not uniquely identified in factor analysis. Bliss (1997a) rotates the first
factor so that its loading is a vector of ones. In our approach, the unit loading on the first factor is
imposed from the beginning, which potentially enables us to estimate the other factors more efficiently.