Journal of Applied Finance & Banking

Downside Risk and Average Returns: A Condensed Cross-Sectional Analysis Using Semi-Continuous Complex Wavelet Frames

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• Abstract

   

  We study the cross-sectional pricing of systematic downside risk within a semi-continuous complex wavelet frame. Unlike conventional downside asset-pricing models built on the real-valued Discrete Wavelet Transform (DWT/MODWT which are tied to dyadic scales and cannot disentangle amplitude from phase we adopt a redundant, translation-invariant complex representation whose coefficients are complex-valued. This lets us extract, at every investment horizon, the instantaneous amplitude (synchronous co-movement) and the instantaneous phase (lead–lag, delayed transmission) of asset returns. Splitting each complex coefficient into its real and imaginary parts, we build multiscale downside beta and downside co-skewness measures and price them across 30 U.S. industry portfolios (1926–2025). Classical time-domain and MODWT specifications leave downside risk essentially unpriced (R² < 3%). The complex representation reverses this: a four-factor model combining synchronous and phase-shifted downside beta and co-skewness explains up to R² ≈ 60% of the cross-section at business-cycle horizons. The results show that amplitude and phase information must be considered jointly when assessing systematic downside risk.

   

  JEL classification numbers: G12, G17, C58.

  Keywords: Downside beta; Downside co-skewness; Continuous Wavelet Transform; Semi-continuous frame; Complex wavelet coefficients; Cross-sectional asset pricing; Time–frequency analysis.