Yield-Curve-Modeling

Overview

The yield curve represents a snapshot of government bond yields across all maturities at a given date. In practice, yields are only directly observable at a discrete set of maturities — the tenors at which benchmark bonds are actively traded. This leaves the curve discontinuous by nature, with no direct observation between traded maturities.

The objective of this project is

1. To construct a smooth, continuous curve from these discrete observations — enabling reliable yield estimation at any maturity on a given date.

2. To provide a detailed risk decomposition of the portfolio

3. Stress test P&L under different scenarios.

The project fits a continuous curve through those discrete points using two methods — Nelson-Siegel and Cubic Spline — and compares how each performs.

Furthermore, the project evaluates risk measures such as Modified Duration, DV01, Key Rate Duration and parallel and non parallel shocks to stress test.

Yield Curve Fitting Methods

To form a curve from discontinuous points we use two difeerent methodologies :

1. Nelson-Siegel factor model
2. Cubic Spline Interpolation

1. Nelson-Siegel Model

The Nelson-Siegel (1987) model represents the yield curve as a linear combination of three factor loadings:

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

Fitted Parameters (US Treasury Curve — 2024-09-13)

Parameter	Value	Interpretation
β₀	3.618%	The level - Long-run neutral rate
β₁	2.0137%	The Slope - Monetary policy stance
β₂	-0.7415%	The Curvature - Medium-term rate expectations
λ	3.2945	Decay rate — controls where hump peaks (τ* = 1/λ) = 0.3039

Fitting strategy: λ is found by grid search; β₀, β₁, β₂ are solved analytically via OLS at each candidate λ.

2. Cubic Spline Model

Two variants are implemented — an interpolating spline that passes exactly through every observed yield, and a smoothing spline that trades fit for smoothness via a penalty parameter.

Two variants are implemented:

Variant	Description	Use Case
Interpolating	Passes exactly through every observed yield	Exact pricing, CSA discounting
Smoothing	Penalised spline; trades fit for smoothness	Noisy data

Portfolio

The portfolio comprises four US Treasury bonds spanning the 2 to 30 year maturity spectrum, with positions specified in config.py. All risk metrics are computed as of the valuation date defined in that file.

Portfolio

Bond Name	Coupon Rate	Maturity(year)	Face Value($m)
UST 4.750% 2026	4.750	2.0	50
UST 3.875% 2029	3.875	5.0	120
UST 3.875% 2034	3.875	10.0	90
UST 4.250% 2054	4.250	30.0	40

Results

Nelson-Siegel fits the curve with ~24bp RMSE using only 4 parameters. The interpolating spline fits exactly by construction but produces unstable forward rates. The smoothing spline sits between the two.

Metric	Nelson-Siegel	Cubic Spline (Interp)	Cubic Spline (Smooth)
RMSE	~24 bp	~0 bp	~8.5 bp
MAE	~20.73 bp	~0 bp	~7.7 bp
Parameters	4	N (one per knot)	N + penalty
Extrapolation	Converges to β₀	Diverges beyond data	Moderate
Forward Rates	Smooth, analytic	Can oscillate	Smooth

Nelson-Siegel is used for all downstream analysis in this project given its stable forward rates and forecastable parameter structure.

Using Nelson Siegel curve on a valuation date of 24th September, 2024 we obtain the following risk metrics.

Risk Metrics

Metric	Portfolio Value	Interpretation
Market Value	~ $306.8m	Future cash flows discounted
Modified Duration	~ 7.05yr	Change in price for a 1% rise in yield
Portfolio DV01	~ $216382.60	Change in the market value of a position for a 1 basis point
+100bp P&L	~ $-20.68m	Parallel shock of 100 basis point
+200bp P&L	~ $-38.96m	Parallel shock of 200 basis point
Key Rate Duration	~ 10-30yr	Which part of the yield curve poses the greatest risk to my portfolio?

The results/ folder contains charts generated by running the two notebooks. These visualisations cover three areas:

Yield Curve Fitting — plots of the fitted Nelson-Siegel and Cubic Spline curves against observed Treasury yields, forward rate term structure, discount function, and residual fitting errors across maturities.

Risk Management — charts showing the portfolio's interest rate risk profile including modified duration and DV01 by bond, key rate duration decomposition revealing where risk is concentrated along the curve, stress test P&L across parallel and non-parallel yield shocks.

Data

US Treasury yields downloaded from FRED. See data/README.md for download instructions.

How to Run

pip install -r requirements.txt

jupyter notebook notebooks/yield_curve_fitting.ipynb

jupyter notebook notebooks/yield-curve-risk_management.ipynb