Equity portfolio managers traditionally form portfolios using forecasts of returns. They seek to identify and hold stocks that will have high returns and to avoid stocks that will have low returns. In contrast, a number of equity strategies that have received attention recently do not employ return forecasts at all. Instead, investors form portfolios based on stocks' risk characteristics. This paper reviews three non-returns-based (ie, risk-based) alternative equity strategies: minimum variance, equal risk contribution and maximum diversification. Minimum variance strategies build intuitive, low-risk portfolios. Equal risk contribution strategies are hybrids of risk minimizing and equal weighting strategies, and they, too, have intuitive characteristics and can have lower risk. In contrast, maximum diversification strategies are puzzling. Maximum diversification strategies prefer stocks with low correlations, even if they are risky, and since they do not target risk reductions, return increases or Sharpe ratio increases per se, maximum diversification strategies can produce portfolios with desirable characteristics only by accident.