How interest rates across different maturities are connected — and why it matters
A parametric framework for yield curve estimation used by central banks and financial institutions worldwide
When a government borrows money by issuing bonds, it pays different interest rates depending on how long it borrows. A one-year bond might pay 4%, while a 10-year bond might pay 4.5%. A yield curve is simply a line that plots these interest rates across all maturities.
The challenge is that governments don't issue bonds for every possible maturity. You might find bonds at 1, 2, 5, 10, and 30 years — but what's the rate for 7 years, or 12? A yield curve fills in those gaps by fitting a smooth line through the available data points, giving us an estimated rate for any maturity.
The Nelson-Siegel-Svensson method is one of the most widely used approaches for constructing these curves. It is the standard at many of the world's central banks.
The Nelson-Siegel-Svensson (NSS) model is a parametric approach to yield curve estimation that offers a practical balance between theoretical interpretability and empirical fit. By extending the original Nelson-Siegel three-factor structure with an additional curvature term, it can accommodate the range of yield curve shapes typically observed in sovereign bond markets.
The Nelson-Siegel-Svensson model constructs a yield curve using six parameters. Each one controls a different aspect of the curve's shape:
By adjusting these six values, the model can reproduce virtually any yield curve shape observed in practice.
β₀ (Level): The interest rate that very long-term bonds converge toward
β₁ (Slope): Negative values produce the typical upward-sloping curve; positive values produce a downward slope
β₂ (Curvature): Positive values create a hump; negative values create a trough
λ₁: Positions the first curvature effect along the maturity spectrum
λ₂: Positions the second curvature effect, typically at longer maturities
The estimation goal: Find the combination of parameters that best fits observed market yields
The NSS model specifies the zero-coupon yield at maturity τ as the sum of a constant plus three exponentially decaying components:
Where $\tau$ is time to maturity and $\{\beta_0, \beta_1, \beta_2, \beta_3, \lambda_1, \lambda_2\}$ are estimated via nonlinear least squares
Asymptotic yield
The yield to which the curve converges as maturity increases without bound. Reflects long-run expectations for real rates and inflation compensation.
Slope factor
Determines the spread between short- and long-term yields. Negative values produce the upward-sloping curve observed under normal market conditions.
First curvature factor
Governs curvature in the intermediate segment. Its hump-shaped loading function allows the model to capture humped or U-shaped yield curves.
Svensson extension
Provides a second curvature degree of freedom, enabling the model to fit double humps, S-curves, and other complex shapes often observed during policy transitions.
Medium-term decay rate
Controls the exponential decay speed of the first curvature component. Lower values concentrate the effect at shorter maturities.
Long-term decay rate
Controls the decay speed of the Svensson curvature term. Generally set larger than λ₁ to affect longer maturities.
Use the sliders below to adjust each parameter and see how the yield curve responds in real time. Start with small changes to one parameter at a time to build intuition for what each one does.
Adjust the parameters below to observe how each component of the NSS model affects yield curve shape. The visualization illustrates the sensitivity of the functional form to individual parameter changes.
The yield curve doesn't always look the same. Its shape changes as market expectations shift, and each configuration carries a distinct economic signal.
Yield curve morphology encodes market expectations about monetary policy, growth, and inflation. The NSS framework's parameter structure maps directly to the standard shape taxonomy, with level, slope, and curvature factors corresponding to distinct economic drivers.
Shape: Slopes upward from left to right
Meaning: Investors demand higher returns for locking up money longer. This is the default pattern in stable, growing economies.
Typical conditions: Steady economic expansion
Parameters: β₁ < 0, β₂ ≈ 0
Reflects a positive term premium that compensates investors for duration risk. Consistent with expectations of continued economic expansion and stable or tightening monetary policy.
Shape: Slopes downward — short-term rates exceed long-term rates
Meaning: Markets expect the economy to weaken and interest rates to fall. Historically, inversions have preceded most U.S. recessions.
Typical conditions: Late in the business cycle, before a downturn
Parameters: β₁ > 0, β₂ < 0
Indicates market expectations of monetary easing ahead, typically driven by anticipated economic contraction. An established leading indicator of recessions, reflecting both policy expectations and flight-to-quality dynamics.
Shape: Roughly the same rate across all maturities
Meaning: Markets see roughly equal risk at all horizons, often because the economic outlook is unclear
Typical conditions: Transition periods between expansion and contraction, or between policy regimes
Parameters: β₁ ≈ 0, β₂ ≈ 0
A near-zero term premium, typically arising during transitions between monetary policy regimes. Short- and long-term factors are approximately offsetting.
Shape: Medium-term rates are higher than both short- and long-term rates
Meaning: Markets may expect near-term tightening followed by eventual easing, creating a peak in the middle of the curve
Typical conditions: Policy uncertainty, mixed economic data
Parameters: β₂ > 0 (positive curvature)
Typically reflects expectations of a monetary policy tightening cycle followed by subsequent easing, or supply-demand imbalances concentrated in specific maturity segments.
Shape: Medium-term rates are lower than both short- and long-term rates
Meaning: Unusual configuration that may reflect specific central bank interventions, such as large-scale bond purchases focused on particular maturities
Typical conditions: Rare, usually linked to unconventional monetary policy
Parameters: β₂ < 0 (negative curvature)
A relatively uncommon configuration, typically associated with quantitative easing programs that target specific maturity sectors, or with pronounced market segmentation effects.
Explore the model hands-on. This ready-to-use spreadsheet walks through the Nelson-Siegel-Svensson method with real data.
What's included:
A working implementation template for NSS parameter estimation using Excel Solver with market data.
Features:
Ready-to-use spreadsheet with sample data Professional optimization template with Solver setup
Download Excel Template
Excel 2016+ required
Solver add-in must be enabled
Once we have the model's formula, we need to find the specific parameter values that produce a curve matching actual bond yields as closely as possible. This is done through an optimization process: a computer systematically adjusts the parameters, compares the resulting curve to market data, and repeats until it finds the best fit.
In practice, the estimation uses nonlinear least squares — a standard technique that minimizes the squared differences between the model's predicted yields and the yields observed in the market.
Gather current bond prices across a range of maturities (e.g., 1-year, 5-year, 10-year, 30-year government bonds)
Translate bond prices into their corresponding interest rates (yields to maturity)
Choose reasonable initial parameter estimates to give the optimization algorithm a starting point
Let the algorithm iteratively adjust parameters until the model curve matches observed yields as closely as possible
Accuracy: The fitted curve should closely track actual market yields
Smoothness: The curve should be free of erratic jumps or implausible shapes
Economic plausibility: Implied rates should be realistic (e.g., no negative long-term rates when none exist in the market)
Consistency: The method should produce stable, reproducible results from day to day
Parameter estimation involves a constrained nonlinear least squares problem. The objective is to minimize squared deviations between observed market yields and model-implied yields, subject to positivity constraints on the decay parameters and optional bounds that enforce economic plausibility.
Assemble a clean cross-section of government bond yields spanning the maturity spectrum, filtering for liquidity and representativeness
Convert bond prices to zero-coupon yields via bootstrapping or iterative methods, accounting for coupon structure and accrued interest
Set starting parameter values using economic priors or a grid search to avoid convergence to local minima
Apply Levenberg-Marquardt, trust-region, or similar algorithms with appropriate parameter bounds
Subject to: λ₁, λ₂ > 0 and economic plausibility constraints
Yield curves may sound abstract, but they influence the interest rates you encounter every day — on mortgages, car loans, savings accounts, and retirement funds. Here's how different institutions put them to use.
What they do: Banks use the yield curve to set rates on mortgages, savings accounts, and business loans
Why it matters to you: A well-estimated curve helps ensure fair pricing — you neither overpay on borrowing nor get shortchanged on savings
Example: The rate on a 30-year fixed mortgage is derived, in part, from the long end of the yield curve
What they do: Central banks monitor the curve to assess how their policy decisions are being transmitted to the broader economy
Why it matters to you: These decisions influence inflation, employment, and the cost of credit
Example: When the Federal Reserve considers changing interest rates, yield curve signals are a key input
What they do: Fund managers use yield curves to price bonds and manage the interest rate risk in pension funds and mutual funds
Why it matters to you: Accurate pricing leads to more reliable valuations of the bonds in your retirement account or 401(k)
Example: A pension fund uses the curve daily to value its bond portfolio and assess whether it can meet future obligations
The NSS model serves as foundational infrastructure for fixed-income markets. Its widespread adoption across central banks, financial institutions, and regulatory bodies reflects the demand for a yield curve framework that is transparent, reproducible, and economically interpretable.
The Nelson-Siegel-Svensson method is widely trusted, but like any model, it has boundaries. Understanding where it works well and where it falls short is essential for using it responsibly.
Extreme market conditions: During severe market dislocations, the model's smooth functional form may not adequately capture sharp distortions in the curve
Data quality dependence: Illiquid bonds or stale prices can distort the fitted curve
Not a forecasting tool: The model describes the curve as of today — it does not predict where rates will go tomorrow
Extrapolation risk: Estimates are less reliable for very short maturities (under 3 months) or very long maturities (beyond 30 years) where data is sparse
Quality controls: Systematic checks ensure the fitted curve is economically plausible
Data screening: Illiquid or anomalous bond prices are identified and excluded before estimation
Confidence indicators: Many implementations report how well the model fits different parts of the curve
Frequent re-estimation: The curve is recalculated regularly with the latest market data
The NSS framework offers substantial flexibility and economic interpretability, but several inherent limitations warrant consideration in both research and operational settings.
Operational implementations address these limitations through robust optimization techniques, rigorous data validation, cross-model comparison, and ongoing monitoring. Standard practice includes economic plausibility checks, residual diagnostics, and out-of-sample validation.