Style Extremity and Idiosyncratic Risk: An Incremental, Not Categorical, Determinant of Volatility on Valuation Ranks

<div> This paper separates two claims conflated in the style-investing literature: that a firm's growth/value direction predicts risk, and that its extremity-distance from the ambiguous midpoint-predicts risk incrementally to direction. In a survivorship-free, point-in-time panel of 6,546 U.S. common stocks (market cap ≥ $300 million, 2004-2025, 196,438 firm-quarters), we run a horse race between the signed rank and its own extremity transform on three independent constructions. On price-to-sales, extremity is the whole story: direction is null (t =-0.25) while extremity survives fully intact (t = 9.7)-a genuine, if economically modest (0.050 SD), effect and our cleanest identification. On price, both terms survive jointly (direction t =-4.2, extremity t = 14.2), reported only as an upper bound given disclosed mechanical volatility content. On book-to-market the pattern inverts entirely: extremity collapses to zero (t = 0.43) once direction is included, while direction stays large (t = 17.2)-what looked like extremity is the familiar growth-options finding that low-book-to-market firms are more volatile (Cao, Simin, and Zhao 2008), not a distance-from-median effect.&nbsp; </div> <div> <br> </div> <div> A battery of six placebo extremity transforms from characteristics unrelated to valuation (size, beta, bookto-market, momentum, MAX, idiosyncratic skewness) shows this symmetric pattern is not unique to valuation-all six predict forward idiosyncratic volatility, two more strongly than price-to-sales extremity itself-yet price-to-sales extremity survives every pairwise and joint test against them (t = 8.2 jointly, a 15% attenuation from its solo t = 9.9): a genuine incremental fact, not a newly documented class of risk determinant.&nbsp; </div> <div> <br> </div> <div> Economic value concentrates in the price-based measure, which improves out-of-sample volatility forecasts against HAR-RV and EGARCH benchmarks (Clark-West t ≥ 3.4) and lowers realized portfolio volatility in 18 of 18 sample years, surviving a correlation-matched synthetic placebo. The identifying price-to-sales measure adds far less out-of-sample value and no portfolio benefit, and there is no return premium to either (t = 0.7): style extremity is a risk characteristic, not a return anomaly. </div>