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Definition

Estimation

Motivation

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Core + Math

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Component Contributions

Shock Impact Comparison

Constraint Checklist

Implied Model Properties

Near-Boundary Warning

Controls how quickly correlations respond to new market shocks. Higher values mean faster adaptation to recent co-movements.

Determines how persistent correlation changes are. Higher values mean correlation shocks decay more slowly.

DCC models capture time-varying correlations crucial for risk management. The α parameter controls shock sensitivity while β controls persistence. Together, they determine how correlations evolve following market events. During crisis periods, correlations typically spike (high α effect) and remain elevated (high β effect), reducing diversification benefits when needed most.

Qt = (1-α-β)Q̄ + α(εt-1εt-1') + βQt-1

DCC Parameter Explorer

Educational Insights

Return Distribution Comparison

Histogram of 1008 returns simulated from a GARCH model, showing the characteristic fat-tailed distribution typical of financial time series.

GARCH Model Simulation Histogram

How GARCH Models Generate Fat Tails

Adjust parameters to see how coupon rate, maturity, and yield affect bond price and risk metrics.

Bond Pricing Calculator

Negative forward rates indicate pricing inconsistencies that could allow risk-free profit opportunities:

Build a zero-coupon yield curve from market instruments using the bootstrapping algorithm

Yield Curve Bootstrapping Simulator

Cash Flow Timeline

P&L attribution breaks down the total portfolio P&L to show how much each position contributed. Positions with larger market values and longer durations typically contribute more to total P&L risk.

represents the first-order price change from the yield shift, while the

captures the second-order curvature adjustment. For large yield shifts, convexity can significantly offset duration-based losses (or reduce gains).

Understanding P&L Attribution

Risk Concentration Detected

Shows each position''s contribution to total portfolio P&L under the {scenarioName} scenario

P&L Attribution by Position

Analyze dollar-based interest rate risk across multiple positions with parallel and curve scenarios

measures the dollar change in portfolio value for a 1 basis point change in yields. Your portfolio has a DV01 of

, meaning a 1bp parallel shift in yields would change your portfolio value by approximately this amount.

scenario shows how your portfolio would perform under this specific yield curve movement.{plDirection}

Understanding Portfolio DV01

Portfolio DV01 Scenario Analyzer

Compare different yield curve interpolation methods and their impact on curve shape and forward rates

Interpolation Methods Comparison

Key Rate Duration Profile

uses piecewise linear (continuous) weighting between adjacent key rates. Each cash flow is influenced by its two bracketing key rates with weights that sum to 1.0. This creates smooth transitions across the yield curve. Unlike step interpolation, there are no discontinuous jumps between buckets.

uses piecewise constant (discontinuous) bucketing. Each cash flow is assigned 100% weight to the nearest left key rate, creating distinct maturity buckets. Unlike linear interpolation, rate changes do not smoothly transition between keys—they jump discretely at bucket boundaries.

Visualize how bond price sensitivity decomposes across yield curve points

The sum of key rate durations ({total}) approximates modified duration ({modified})

Bonds with bullet payments show concentrated KRD at maturity

The {maturity}Y key rate accounts for {percentage}% of total duration

Match individual key rate exposures to hedge non-parallel curve risk

Key Observations:

Modified duration assumes parallel shifts—all rates move together. But during the 2013 "Taper Tantrum," 2-year yields rose 15bp while 30-year yields jumped 100bp. A portfolio with high 30-year key rate duration would have suffered far more than modified duration predicted. Key rate duration provides the granularity needed for precise risk management in non-parallel shift environments.

💡 Why This Matters: Beyond Modified Duration

Key Rate Duration Explorer

Use bootstrapping for discount curves, arbitrage-free short rate models (Vasicek, CIR, Hull-White) for derivatives, or spline methods for complex shapes.

The chart displays {count} market yield observations as orange diamond markers

. The root mean squared error is {rmse} basis points.

 and the fitted {model} curve as a blue line

Yield Curve Visualization

Fit theoretical yield curve models to market yield data

Nelson-Siegel-Svensson Curve Fitter

These models are not arbitrage-free and can produce incorrect forward rates

NSS can fit at most two humps; complex curves need spline methods

Many regulations require arbitrage-free models (e.g., Hull-White, HJM)

With fewer than 6-8 points, bootstrapping may be more reliable

⚠️ When NOT to Use Parametric Models

Design liability-immunized portfolios and test their effectiveness under yield curve scenarios

Single-payment liabilities (e.g., zero-coupon bonds) or multi-period cash flows (e.g., pension obligations). The liability's duration and convexity determine the immunization target.

Define your liability stream:

Liability Metrics

Immunization Match Quality

💡 Why This Matters

Automatically construct a portfolio that matches liability duration and PV

Manually select assets or use auto-immunize to match liability duration. Redington conditions: PV match, duration match, and convexity cushion.

Build an immunized portfolio:

Auto-immunize uses exact optimization for 2-asset portfolios. For 3+ assets, it uses a heuristic equal-weighting approach. Manual fine-tuning of weights may be needed for optimal immunization with multiple assets.

Note:

Apply yield curve scenarios to see how well your portfolio protects against rate changes. Perfect immunization shows near-zero tracking error.

Test immunization effectiveness:

Testing parallel yield curve shifts where all maturities move by the same amount. This tests the effectiveness of duration matching under uniform interest rate changes across the entire curve.

Portfolio Immunization Explorer

Price-Yield Curve

Explore the inverse relationship between bond yields and prices

Yield-Price Relationship

Forecast Loss by Horizon

GJR-GARCH's superior performance is strongest at short horizons (1-5 days) where recent shock sign matters most. The γ parameter directly enters the 1-step-ahead forecast σ²ₜ₊₁ = ω + (α + γI₍ₜ₎)ε²ₜ + βσ²ₜ, but its influence decays exponentially: at h-step-ahead, the effective γ contribution is scaled by (α + β)ʰ⁻¹

Both models converge toward their unconditional variance σ̄² = ω/(1 - α - β - γ/2) as h → ∞. Beyond 10-day horizon, forecast differences become negligible because the autoregressive structure dominates: σ²ₜ₊ₕ ≈ σ̄² + (α + β + γ/2)ʰ(σ²ₜ - σ̄²)

The DM test answers "are forecasts statistically different?" (p < 0.05), but practical decisions require economic significance: is the loss reduction > 5%? A model can be statistically superior yet economically irrelevant, or vice versa in small samples

💡 Key Insights:

Diebold-Mariano Test Results (Selected Horizons)

Response to a 1% shock over time

Impulse Response Function

GARCH vs Historical Measures (252 days)

Volatility Comparison

Adjust GARCH model parameters and observe their effects on volatility dynamics

Parameter Interpretation

Model Properties

GARCH(1,1) Parameter Explorer

Showing 3.0x volatility shock convergence to long-run average

Volatility Forecast - Mean Reversion Dynamics

Volatility Convergence Analysis

High volatility with moderate persistence (35% annualized volatility)

Crisis Regime

Volatility shocks persist for very long periods (22% annualized volatility)

High Persistence

Quick return to normal volatility levels (22% annualized volatility)

Low Persistence

Standard parameters estimated from equity indices (16% annualized volatility)

Typical Equity Market

GARCH Volatility Forecasting Tool

Additional volatility impact from negative shocks (min = -α to ensure non-negative response)

Base level of volatility scaled by 10,000

Crisis Period Behavior

GARCH Adaptation Speed

Mean Reversion Properties

Model Performance Differences

Volatility Clustering Effect

Volatility Evolution Over Time (S&P 500 GARCH Model)

This chart shows actual GARCH(1,1) conditional volatility fitted to real S&P 500 returns using estimated parameters (ω=0.0186, α=0.1026, β=0.8821). Notice how volatility exhibits mean reversion: after spikes, it gradually declines back toward its long-run average (shown as the blue dashed line representing unconditional volatility).

With S&P 500 estimated parameters α = 0.1026 and β = 0.8821, the persistence parameter α + β = 0.9847 indicates very strong persistence. This means:

Understanding Persistence and Half-Life

Indicates whether the model allows volatility to respond differently to positive vs. negative shocks.

Asymmetry

Summarizes the empirical contexts where the model is most effective (e.g., equities with leverage, FX with fat tails).

Best Suited For

Rates how easy it is to understand and explain the model's parameters in economic or statistical terms (High = simple, Low = complex).

Interpretability

Captures the empirical tendency of negative returns to increase volatility more than positive returns of the same magnitude.

Leverage Effect

Specifies if the model requires parameter restrictions to ensure variances remain positive (some models, like EGARCH, avoid this).

Positivity Constraints

Shows whether the model explicitly incorporates threshold terms that change volatility dynamics once shocks cross certain levels.

Threshold Effects

Compare key characteristics of GARCH model variants. Select your market type for tailored recommendations, then click on any model name in the table below to view detailed specifications, parameters, and use cases.

Risk management across different time horizons

Conceptual Decomposition:

Long-term Component:

Short-term Component:

Conceptual Framework

Parameter estimation is more intensive, requiring robust optimization algorithms and careful initialization.

Computational Complexity:

Multi-component models typically require longer time series (5+ years) to reliably identify different persistence patterns.

Data Requirements:

Implementation Considerations

Models for Short and Long-Term Volatility Components

Applications:

Two-component GARCH specification with separate persistence parameters for short and long-term dynamics. Extends standard GARCH intuition by explicitly modeling different mean-reversion rates for each component.

MF2-GARCH:

Models the long-run variance as a smooth function of time using splines, allowing for gradual regime transitions. The short-term component follows standard GARCH dynamics around this time-varying baseline.

Spline-GARCH (SGARCH):

V-Lab Multi-Component Models

Enhanced spline specification with constraints preventing the long-run component from drifting without bound. Ensures the long-term variance approaches a stable level, suitable for risk forecasting applications.

Zero-Slope Spline-GARCH (SOGARCH):

Multiplicative Error Models (MEM) model positive-valued processes (realized volatility, trading volumes, durations, trading range) where the observed value equals the conditional mean times a positive innovation. Positivity is ensured by construction.

Key Characteristics:

Multiplicative Models

V-Lab Multiplicative Models:

Use Standard GARCH models for conditional variance modeling of returns with straightforward parameter interpretation. Choose MEM models when modeling positive-valued processes like realized volatility, where the multiplicative structure and guaranteed positivity provide natural advantages over additive specifications.

Practical Guidance:

Standard GARCH models specify conditional variance directly through additive functions of past squared innovations and lagged conditional variances. These models maintain positive variance through parameter constraints and provide intuitive economic interpretation.

Standard GARCH Models

V-Lab Standard Models:

Additive Variance Models (GARCH) vs. Multiplicative Error Models (MEMs)

{model} - Detailed Specification

GARCH Model Family Comparison

Correlation Models

Advanced Nelson-Siegel framework for modeling the complete Treasury yield curve

Term Structure Model

Fixed Income Models

Core volatility modeling approaches for standard analysis

Fundamental Models

GJR-GARCH-based models for long-horizon Value-at-Risk estimation

Long-Run VaR Models

Domain-specific models for advanced scenarios and alternative approaches

Specialized Applications

Systemic risk models measuring capital shortfall under market stress using various market factors and estimation approaches

SRISK Models

Systemic Risk Models

Volatility Models

Quick Reference: GARCH Family Comparison

Answer the questions below to receive a data-driven recommendation for choosing between GARCH, GJR-GARCH, EGARCH, and other asymmetric volatility models. This decision tree incorporates statistical test results, information criteria, and application context.

Test Performance (Illustrative Monte Carlo Evidence)

Guided GARCH Family Model Selection

Best Estimate (Highest Log-Likelihood)

Starting points converged to global maximum

Convergence Summary

Parameter initialization chart showing initial parameter values

Optimization Convergence Paths

Run optimization from 3-5 different starting points, select highest log-likelihood

If single optimization fails, try grid of

✓ Practical Recommendations:

Estimate GARCH(1,1) first, then use α̂, β̂ with

Constrained optimization ensures α̂ + β̂ + γ̂/2 < 1 throughout iterations

V-Lab

Historical ILLIQ

Definition

References