Exponentially Weighted Moving Average (EWMA)

The single parameter λ (lambda) controls the persistence of the model. A value close to 1 (e.g., 0.97) creates a slow decay, retaining memory of distant events for stable long-term forecasts. A lower value (e.g., 0.90) creates a fast decay, making the model highly reactive to immediate market jumps.

• RiskMetrics Standard: The industry benchmark sets λ = 0.94 for daily volatility and λ = 0.97 for monthly.
• Effective Window: The half-life of a shock is calculated as ln(0.5) / ln(λ).

EWMA avoids storing long historical windows by using a recursive formula. Today's variance estimate depends only on yesterday's variance and today's squared return. This makes it computationally lightweight for high-frequency systems.

• Formula: σ²_t = λ * σ²_{t-1} + (1 - λ) * r²_t
• Memory Efficiency: Only the previous period's variance needs to be cached, enabling real-time updates on streaming tick data.

EWMA naturally captures volatility clustering—the empirical phenomenon where large price changes tend to follow large changes. Because recent large returns immediately inflate the variance forecast, the model automatically adjusts risk budgets upward during turbulent regimes.

• Regime Responsiveness: Risk parity weights shift rapidly to de-leverage assets with spiking EWMA volatility.
• Limitation: EWMA does not model mean-reversion in volatility (unlike GARCH), so forecasts remain elevated until the shock decays out of the window.

EWMA extends naturally to multivariate covariance for risk parity. The covariance between asset i and j is updated using the cross-product of their returns, weighted by the same decay factor.

• Formula: σ_{ij,t} = λ * σ_{ij,t-1} + (1 - λ) * r_{i,t} * r_{j,t}
• Consistency: Using a single λ across all pairs ensures the resulting matrix is positive semi-definite, a critical requirement for convex optimization solvers.

By updating the covariance matrix with EWMA, the implied correlation between assets also becomes dynamic. A sudden co-movement spike increases the covariance term, raising the correlation estimate and signaling a breakdown in diversification precisely when risk parity needs to rebalance.

• Correlation Breakdown: During crises, correlations tend toward 1. EWMA captures this drift faster than simple historical averages.
• Risk Budget Reallocation: Higher correlations reduce the Effective Number of Bets (ENB), prompting a reduction in leverage.

Unlike a simple rolling window that applies equal weight to all observations and then drops them abruptly, EWMA applies a smooth decay. This eliminates the ghosting effect where a large shock suddenly drops out of the window, causing an artificial volatility cliff.

• Smooth Transition: The geometric decay ensures no single day causes a discontinuous jump in the risk forecast.
• Parameter Selection: The half-life (e.g., 11 days for λ=0.94) effectively defines the 'memory' of the system without hard cutoffs.

A mathematical theorem that perfectly decomposes total portfolio risk into additive contributions from each constituent. This is the mechanism that makes risk parity possible.

• Applies to homogeneous risk functions like volatility and Value-at-Risk
• The Marginal Risk Contribution (MRC) multiplied by weight equals the component risk
• Guarantees that the sum of all risk contributions equals 100% of total portfolio risk

A time-series model that estimates how correlations between assets evolve over time. While EWMA captures volatility clustering, DCC captures correlation clustering.

• Models both volatility dynamics and correlation dynamics simultaneously
• Two-stage estimation: univariate GARCH for volatilities, then multivariate for correlations
• Critical for risk parity when diversification benefits shift rapidly during crises

A covariance estimation model that assumes markets shift between distinct states, such as bull, bear, or crisis regimes. EWMA can be embedded within each regime for state-conditional responsiveness.

• Uses a hidden Markov model to infer the current regime probability
• Each regime has its own decay factor, allowing faster adaptation in crisis states
• Prevents the portfolio from being anchored to stale correlations from a previous market environment

A forward-looking forecast of portfolio risk based on current weights and a predicted covariance matrix. EWMA is the most common engine for generating this forecast in risk parity implementations.

• Contrasts with ex-post volatility, which measures historical realized risk
• The decay factor directly controls how aggressively the forecast responds to recent shocks
• Used to set leverage targets in volatility targeting overlays

An optimization boundary that limits the maximum percentage of total portfolio risk any single asset can contribute. EWMA-based risk estimates feed directly into these constraints.

• Typical constraint: no asset contributes more than 25-35% of total risk
• Enforces diversification even when a single asset's volatility spikes
• Works with convex optimization solvers to find the risk-balanced solution