long ends. It has problems of its own, however, because its estimation requires iterative nonlinear
optimization, and it can be hard to restrict the implied forward rates to be positive.
A third and very popular approach to yield construction is due to Fama and Bliss (1987), who
construct yields not via an estimated discount curve, but rather via estimated forward rates at the
observed maturities. Their method sequentially constructs the forward rates necessary to price
successively longer-maturity bonds, often called an “unsmoothed Fama-Bliss” forward rates, and then
constructs “unsmoothed Fama-Bliss yields” by averaging the appropriate unsmoothed Fama-Bliss
forward rates. The unsmoothed Fama-Bliss yields exactly price the included bonds. Throughout this
paper, we model and forecast the unsmoothed Fama-Bliss yields.
Modeling Yields: The Nelson-Siegel Yield Curve and its Interpretation
At any given time, we have a large set of (Fama-Bliss unsmoothed) yields, to which we fit a
parametric curve for purposes of modeling and forecasting. Throughout this paper, we use the Nelson-
Siegel (1987) functional form, which is a convenient and parsimonious three-component exponential
approximation. In particular, Nelson and Siegel (1987), as extended by Siegel and Nelson (1988), work
with the forward rate curve,
» = 311 ÷ β2 e ÷ β3 t∖^'τ.
The Nelson-Siegel forward rate curve can be viewed as a constant plus a Laguerre function, which is a
polynomial times an exponential decay term and is a popular mathematical approximating function.4 The
corresponding yield curve is
f 1 _ -ʌ,ɪ ʌ t 1 _ ~Kτ .
ylft) = Э1t ÷ ⅛t ; ÷ β31 ∖ ■' ∙
λ,τ λ,τ
к t / ∖ t 7
The Nelson-Siegel yield curve also corresponds to a discount curve that begins at one at zero maturity
and approaches zero at infinite maturity, as appropriate.
Let us now interpret the parameters in the Nelson-Siegel model. The parameter λt governs the
exponential decay rate; small values of λt produce slow decay and can better fit the curve at long
maturities, while large values of λt produce fast decay and can better fit the curve at short maturities. λt
4 See, for example, Courant and Hilbert (1953).