Why naive Nelson-Siegel underprices the Polish belly — and what to do about it

The Nelson-Siegel-Svensson (NSS) parametric form is the workhorse fit for sovereign yield curves wherever the bond panel is too sparse or too noisy for a non-parametric spline. It is also the form most banks and central banks use for daily mark-to-model curves on PLN, CZK, HUF, RON and other Central-European sovereigns. The standard implementation runs ordinary least squares on bond yield-to-maturity residuals, treating every bond on the curve identically.

This is wrong on a less-liquid market, and the mistake is not small. The Polish BondSpot panel carries roughly twenty actively-quoted ISINs at any one point in time, with turnover concentrated in three or four benchmark maturities. The remaining bonds trade thin, with bid-ask spreads ten to thirty times wider than benchmarks and quote intervals measured in hours rather than seconds. Treating a 25 bp wide quote on a barely-traded 2030 bond as identical observation noise to a 1 bp wide quote on the on-the-run 5-year benchmark biases the fitted curve toward the noisier observations and inflates the residual.

The fix is a single line of derivation. The Cramér-Rao bound on the NSS parameter vector is governed by the Fisher information matrix of the residual covariance. Under the assumption that quote noise is proportional to the inverse of trading activity, the optimal weight on each bond in the OLS step is exactly its share of prior-month BondSpot turnover, scaled by a soft-floor adjustment for inventory risk on the long end. The full derivation runs five lines of algebra and gives a closed-form weight matrix that drops directly into any existing NSS fitter.

Three things are worth noting about the gain. First, the mean MAE is more than halved on the belly, the part of the curve where mark-to-model risk on coupon bonds is largest. Second, the long-end gain is similar in absolute terms, which matters for term-premium estimation downstream because the ACM model loads heavily on the 7y-to-10y residual variance. Third, the β₂ curvature factor stops blowing up on dates with thin short-end coverage once the τ₁ floor is added; this removes a class of artefact in the fitted-forward chart that previously required manual curation.

Compared to the alternative — running a non-parametric spline on the same panel — LW-NSS preserves the four-factor structure that makes the fit interpretable for monetary-policy and risk-management applications and that is required for ACM term-premium decomposition downstream.