Fama-Bliss forward rates into unsmoothed Fama-Bliss zero yields.
Although most of our analysis does not require the use of fixed maturities, doing so greatly
simplifies our subsequent forecasting exercises. Hence we pool the data into fixed maturities. Because
not every month has the same maturities available, we linearly interpolate nearby maturities to pool into
fixed maturities of 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108, and 120 months, where a
month is defined as 30.4375 days. Although there is no bond with exactly 30.4375 days to maturity, each
month there are many bonds with either 30, 31, 32, 33, or 34 days to maturity. Similarly we obtain data
for maturities of 3 months, 6 months, etc.7
The various yields, as well as the yield curve level, slope and curvature defined above, will play
a prominent role in the sequel. Hence we focus on them now in some detail. In Figure 2 we provide a
three-dimensional plot of our yield curve data. The large amount of temporal variation in the level is
visually apparent. The variation in slope and curvature is less strong, but nevertheless apparent. In Table
1, we present descriptive statistics for the yields. It is clear that the typical yield curve is upward sloping,
that the long rates are less volatile and more persistent than short rates, that the level (120-month yield) is
highly persistent but varies only moderately relative to its mean, that the slope is less persistent than any
individual yield but quite highly variable relative to its mean, and the curvature is the least persistent of
all factors and the most highly variable relative to its mean.8 It is also worth noting, because it will be
relevant for our future modeling choices, that level, slope and curvature are not highly correlated with
each other; all pairwise correlations are less than 0.40. In Figure 3 we display the median yield curve
together with pointwise interquartile ranges. The earlier-mentioned upward sloping pattern, with long
rates less volatile than short rates, is apparent. One can also see that the distributions of yields around
their medians tend to be asymmetric, with a long right tail.
Fitting Yield Curves
As discussed above, we fit the yield curve using the three-factor model,
У( ω = β11+ β21
1 λτ
1 -e t
λτ
∖ t
1 ^λtτ
1 -e t
--------e
λ(T
∖
-λ(T
)
7 We checked the derived dataset and verified that the difference between it and the original
dataset is only one or two basis points.
8 That is why affine models don’t fit the data well; they can’t generate such high variability and
quick mean reversion in curvature.