Some New Variance Bounds for Asset Prices

When equity prices are determined as the discounted sum of current and expected future dividends, Shiller (1981) and LeRoy and Porter (1981) derived a relationship between the variance of the price of equities, p[subscript t] and the variance of the ex post realized discounted sum of current and future dividends: p[superscript * subscript t]: Var(p[superscript * subscript t]) >= Var(p[subscript t]). The literature has long since recognized that this variance bound is valid only when dividends follow a stationary process. Others, notably West (1988), derive variance bounds that apply when dividends are nonstationary. West shows that the variance in innovations in p[subscript t] must be less than the variance of innovations in a forecast of the discounted sum of current and future dividends constructed by the econometrician, p-hat[subscript t]. Here we derive a new variance bound when dividends are stationary or have a unit root, that sheds light on the discussion in the 1980s of the Shiller variance bound: Var(p[subscript t] - p[subscript t - 1]) >= Var(p[superscript * subscript t] - p[superscript * subscript t - 1])! We also derive a variance bound related to the West bound: Var(p-hat[subscript t] - p-hat[subscript t - 1]) >= Var(p[subscript t] - p[subscript t - 1]).