# Treasury Yield-Curve Nelson-Siegel / Svensson Bootstrap

Bootstrap the discount-factor and spot curve from treasury quotes, fit Nelson-Siegel or Svensson parametrically, and chart spot plus instantaneous forward curves with basis-point residuals.

> Canonical page: https://elysiatools.com/en/tools/treasury-yield-curve-nelson-siegel-bootstrap-interpolator

- **Category:** Finance

- **Keywords:** yield curve, nelson-siegel, svensson, bootstrap, spot rate, forward curve

## Overview

The Treasury Yield-Curve Nelson-Siegel / Svensson Bootstrap tool extracts discount factors and zero-coupon spot rates from coupon bond prices, fits parametric yield-curve models, and generates continuous spot and instantaneous forward curves with basis-point error metrics.

## Inputs

- **Bonds (one per line: maturityYears price coupon%)** (textarea): 0.5 99.20 0 2 100.10 4.25 5 99.85 4.40 10 98.60 4.50 30 97.90 4.65
- **Parametric model** (select)
- **Spot-rate compounding** (select)
- **Show instantaneous forward curve** (checkbox)

## When to use

- Constructing zero-coupon spot rate and discount factor curves from raw Treasury bill and bond market prices.
- Fitting 4-parameter Nelson-Siegel or 6-parameter Svensson models to smooth bond yield curves and quantify slope or curvature factors.
- Deriving instantaneous forward rate curves and interpolating yields across non-standard maturity tenors.

## How it works

- Enter market quotes with maturity in years, bond price, and coupon percentage per row.
- Select the parametric model (Nelson-Siegel or Svensson) and compounding convention (semiannual bond-equivalent, annual, or continuous).
- Run the bootstrap algorithm to iteratively extract discount factors and spot rates from shortest to longest maturity.
- Review the resulting parameter estimates (level, slope, curvature, decay constants), RMSE residuals in basis points, fitted spot rates, and instantaneous forward curve chart.

## Use cases

- Fixed-income portfolio managers pricing illiquid bonds by discounting future cash flows against a parametrically fitted zero curve.
- Risk analysts and quantitative researchers modeling term structure movements and tracking level, slope, and curvature beta shifts.
- ALM teams needing precise discount factors across standard and non-standard tenors for balance sheet valuation.

## Frequently asked questions

### What input format is required for bond data?

Provide one bond per line containing three space-delimited values: maturity in years, clean or dirty market price, and annual coupon rate percentage.

### When should I choose Svensson over Nelson-Siegel?

Use Svensson (6 parameters) when your yield curve exhibits complex shapes with multiple humps or inflections that Nelson-Siegel (4 parameters) cannot capture.

### Which compounding conventions are supported?

You can output spot rates using semiannual bond-equivalent yield, annual compounding, or continuous compounding.

### How does the tool bootstrap discount factors?

It solves for zero rates sequentially from the shortest zero-coupon instruments outward, stripping intermediate coupon cash flows using previously determined discount factors.

### What does the instantaneous forward curve represent?

It represents the marginal implied interest rate at an exact future moment, derived analytically from the fitted parametric curve parameters.

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