Contrary to the common wisdom that asset prices are hardly possible to forecast, we show that high and low prices of equity shares are largely predictable. We propose to model them using a simple implementation of a fractional vector autoregressive model with error correction (FVECM). This model captures two fundamental patterns of high and low prices: their cointegrating relationship and the long memory of their difference (i.e. the range), which is a measure of realized volatility. Investment strategies based on FVECM predictions of high/low US equity prices as exit/entry signals deliver a superior performance even on a risk-adjusted basis.