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IntermediateOptions, Derivatives & FinancePython

Run this module

cd "Finance - Yield Curve"
python "yield_curve_tutorial.py"

View source on GitHub

Finance – Yield Curve

Overview

The yield curve is the most closely watched chart in global finance. It plots interest rates (yields) across different maturities for bonds of equal credit quality — most commonly US Treasury bonds. Its shape and movements drive pricing for virtually every financial asset, from mortgages to corporate bonds to equity discount rates.

This utility demonstrates how to construct, smooth, and interpret a yield curve using the Nelson-Siegel parametric model and implied forward rate extraction.

Key Concepts

Types of Rates

Rate Type Description
Par Yield Coupon rate making a bond price equal to face value ($100)
Zero / Spot Rate Yield on a zero-coupon bond — the pure rate for that maturity
Forward Rate Implied rate for a future period, derived from today's spot curve

Yield Curve Shapes

Shape Description Economic Signal
Normal Long rates > Short rates Growth expected
Inverted Short rates > Long rates Recession warning
Flat Short ≈ Long rates Transition / uncertainty
Humped Mid > both ends Complex rate expectations

The 2s10s spread (10yr minus 2yr yield) turning negative has preceded every US recession since 1955.

Nelson-Siegel Model

A 4-parameter model that fits a smooth curve through noisy market yields:

y(T) = beta0
 + beta1 × [(1 – e^(–T/τ)) / (T/τ)] ← slope loading
 + beta2 × [(1 – e^(–T/τ)) / (T/τ) – e^(–T/τ)] ← curvature loading
  • beta0 (Level): Long-run asymptotic yield
  • beta1 (Slope): Short-to-long rate difference
  • beta2 (Curvature): Hump or trough at intermediate maturities
  • tau (Decay): Controls where the slope/curvature effects peak

Logic Implemented

  1. Raw yield data — Hardcoded Treasury par yields representing a high-rate environment
  2. Curve classification — Categorises the shape (normal/inverted/flat/humped)
  3. Nelson-Siegel fitting — Minimises squared errors via scipy.optimize.minimize
  4. Forward rate extraction — Derives the implied path of future short rates

Files

  • yield_curve_tutorial.py: Sample data, Nelson-Siegel model, fitting engine, forward rate computation, and shape analysis.

How to Run

python yield_curve_tutorial.py

Financial Applications

1. Bond Pricing

  • Zero-coupon (spot) rates are used to discount each cash flow from a coupon bond.
  • The yield curve determines the "risk-free discount factors" for all maturities.

2. Interest Rate Derivatives

  • Swap rates, caps, floors, and swaptions are all priced off the forward rate curve.
  • The Nelson-Siegel model provides a smooth forward curve without oscillations.

3. Central Bank Policy Analysis

  • The short end of the curve is heavily influenced by the Fed Funds rate.
  • The long end reflects growth/inflation expectations and term premium.

4. Credit Spread Analysis

  • Corporate bond yields = Treasury yield (risk-free) + credit spread.
  • The yield curve provides the risk-free baseline for all credit analysis.

5. Mortgage-Backed Securities

  • Mortgage rates are typically benchmarked off the 10-year Treasury yield.

Best Practices

  • Use zero rates (not par rates) for discounting: Par rates mix maturity effects; spot rates are "pure" rates for each specific maturity.
  • Forward rates are volatile: Small changes in the spot curve create large swings in forward rates at long maturities.
  • Nelson-Siegel has limits: It cannot fit bimodal or heavily distorted curves — use cubic splines for more complex shapes.
  • Data sources: Free access to daily US Treasury par yields via FRED (St. Louis Fed).

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