Nelson-Siegel and Svensson Curve Fitting

A central bank or bond trader rarely has yields for every single maturity they need — they might observe liquid bond prices at 2, 5, 10, and 30 years, but need a rate for 7 years or 15 years too. You could draw straight lines between the dots, but that produces kinks and doesn't smoothly extrapolate. The Nelson-Siegel model instead fits the whole curve with one smooth function controlled by just four parameters, each with a direct economic meaning.

The formula, piece by piece

$$y(\tau) = \beta_0 + \beta_1 \left(\frac{1 - e^{-\tau/\lambda}}{\tau/\lambda}\right) + \beta_2 \left(\frac{1 - e^{-\tau/\lambda}}{\tau/\lambda} - e^{-\tau/\lambda}\right)$$

In words: the yield at maturity $\tau$ is built from three pieces added together. $\beta_0$ is the level — the yield the curve approaches at very long maturities, since the other two terms decay to zero as $\tau$ grows. $\beta_1$ is the slope — it starts at full weight for very short maturities and fades to zero for long ones, so it controls how far the short end sits from the long-run level. $\beta_2$ is the curvature — a term that starts at zero, rises to a peak at intermediate maturities, then fades back to zero, controlling how humped the belly of the curve is. $\lambda$ sets where along the maturity axis that hump peaks.

The Svensson extension adds a second curvature term with its own $\lambda_2$, letting the curve have two humps instead of one — useful when real curves show more complex shapes than the four-parameter version can capture.

Worked example

Fitted parameters: $\beta_0 = 4.0\%$, $\beta_1 = -1.0\%$, $\beta_2 = 0.6\%$, $\lambda = 2$. Estimate the yield at $\tau = 2$ years.

1. Decay term: $e^{-2/2} = e^{-1} \approx 0.368$.
2. Slope factor: $(1 - 0.368)/(2/2) = 0.632$.
3. Curvature factor: $0.632 - 0.368 = 0.264$.
4. Assemble: $y(2) = 4.0\% + (-1.0\% \times 0.632) + (0.6\% \times 0.264) = 4.0\% - 0.632\% + 0.158\% \approx 3.53\%$.

Change only $\beta_1$ to $-0.3\%$ (a much flatter short end) and recompute the last two steps: $y(2) = 4.0\% - 0.190\% + 0.158\% \approx 3.97\%$ — the whole short-to-intermediate part of the curve shifts up, while the long end stays anchored near $\beta_0$, exactly matching what a pure slope parameter should do.

What this means in practice

Central banks (including the Fed and ECB) publish Nelson-Siegel or Svensson fits of the government curve because a four-to-six-parameter model is far easier to compare across time, forecast, and use for consistent interpolation than raw bond yields, which are noisy and unevenly spaced across maturities. Traders use it to spot bonds trading rich or cheap relative to the fitted curve, and researchers use the fitted $\beta$ parameters directly as the level/slope/curvature factors in macro-finance models.