[Paper Review] Asset volatility forecasting:The optimal decay parameter in the EWMA model

Axel A. Araneda|arXiv (Cornell University)|May 29, 2021

Stochastic processes and financial applications4 citations

TL;DR

This paper investigates the optimal decay parameter (λ) in the Exponentially Weighted Moving Average (EWMA) model for forecasting asset volatility across multiple horizons. Using daily returns of 22 S&P 500 components (1994–2018), it finds that time-varying λ, optimized in-sample at each period, significantly improves out-of-sample forecasting accuracy over fixed full-sample λ, especially for weekly and monthly horizons.

ABSTRACT

The exponentially weighted moving average (EMWA) could be labeled as a competitive volatility estimator, where its main strength relies on computation simplicity, especially in a multi-asset scenario, due to dependency only on the decay parameter, $\\lambda$. But, what is the best election for $\\lambda$ in the EMWA volatility model? Through a large time-series data set of historical returns of the top US large-cap companies; we test empirically the forecasting performance of the EWMA approach, under different time horizons and varying the decay parameter. Using a rolling window scheme, the out-of-sample performance of the variance-covariance matrix is computed following two approaches. First, if we look for a fixed decay parameter for the full sample, the results are in agreement with the RiskMetrics suggestion for 1-month forecasting. In addition, we provide the full-sample optimal decay parameter for the weekly and bi-weekly forecasting horizon cases, confirming two facts: i) the optimal value is as a function of the forecasting horizon, and ii) for lower forecasting horizons the short-term memory gains importance. In a second way, we also evaluate the forecasting performance of EWMA, but this time using the optimal time-varying decay parameter which minimizes the in-sample variance-covariance estimator, arriving at better accuracy than the use of a fixed-full-sample optimal parameter.

Motivation & Objective

• To determine the optimal decay parameter λ in the EWMA model for forecasting asset volatility across different time horizons.
• To evaluate whether a fixed full-sample λ or a time-varying λ yields better out-of-sample forecasting performance for the variance-covariance matrix.
• To assess the impact of forecasting horizon on the optimal λ, particularly in relation to short-term memory and volatility clustering.
• To compare the predictive accuracy of the EWMA model using RiskMetrics-recommended λ (0.97) against empirically derived optimal values.
• To provide practical guidance for financial practitioners on selecting λ in multi-asset volatility forecasting under computational constraints.

Proposed method

• Empirical analysis using daily adjusted closing prices of 22 top US large-cap stocks (S&P 500 components) from 1994 to 2018.
• Rolling window scheme for out-of-sample forecasting, with variance-covariance matrices updated recursively using the EWMA formula: $ \sigma_t^2 = \lambda \sigma_{t-1}^2 + (1-\lambda) r_{t-1}^2 $.
• Estimation of full-sample optimal λ by minimizing the mean squared error (MSE) of the variance-covariance matrix over the entire sample period.
• Implementation of a time-varying λ strategy, where at each time t, the optimal λ is selected based on minimizing the in-sample MSE of the covariance matrix at t−1.
• Use of the Diebold-Mariano test to assess statistical significance of performance differences between fixed and time-varying λ approaches.
• Evaluation across three forecasting horizons: 1-day, 5-day (weekly), and 10-day (bi-weekly), with additional analysis for 21-day (monthly) horizon.

Experimental results

Research questions

• RQ1 What is the optimal fixed decay parameter λ for EWMA volatility forecasting across different time horizons (1-day, 5-day, 10-day, 21-day)?
• RQ2 How does the optimal λ vary with the forecasting horizon, and does it reflect changes in memory persistence?
• RQ3 Does using a time-varying λ, optimized in-sample at each period, improve out-of-sample forecasting accuracy compared to a fixed full-sample λ?
• RQ4 Is the improvement in forecasting accuracy from time-varying λ statistically significant across different horizons?
• RQ5 How does the performance of the EWMA model with empirically derived λ compare to the RiskMetrics-recommended λ = 0.97 for 1-month forecasting?

Key findings

• For monthly forecasting (21-day horizon), the optimal fixed full-sample λ is 0.98, closely aligning with the RiskMetrics recommendation of 0.97, which ranks second with minimal MSE difference.
• For weekly (5-day) and bi-weekly (10-day) horizons, the optimal fixed full-sample λ values are 0.92 and 0.95, respectively, indicating that shorter horizons require lower λ to emphasize recent volatility.
• The optimal λ is a function of the forecasting horizon, with lower values (higher weighting on recent returns) emerging for shorter horizons, reflecting increased importance of short-term memory.
• Using a time-varying λ that is optimized in-sample at each prior period (λ_{(t-1)T}^*) yields significantly lower MSE than using a fixed full-sample λ across all horizons except 1-day forecasting.
• The Diebold-Mariano test confirms statistical significance (p < 0.001) of the improvement in forecasting accuracy when using time-varying λ, with t-statistics of 5.895 (10-day), 5.174 (21-day), and 3.214 (5-day).
• For 1-day forecasting, the time-varying λ approach yields a slightly lower MSE (0.001160 vs. 0.001355) but the difference is not statistically significant, suggesting no clear advantage over fixed λ in this case.