No, he didn't beat the S&P, but holding some investment in the S&P and some in his fund beat the S&P in risk-adjusted returns (Sortino ratio, Shapre ratio, etc.

No, he didn't beat the S&P, but holding some investment in the S&P and some in his fund beat the S&P in risk-adjusted returns (Sortino ratio, Shapre ratio, etc.). As I mentioned, look up portfolio optimization in Post-Modern Portfolio Theory or Modern Portfolio Theory for details, but the gist is that Var(aX + bY) = a^2 * Var(X) + b^2 * Var(Y) + 2ab * Cov(X,Y) His fund was essentially uncorrelated with the S&P, so Cov(X,Y) was approximately zero. MPT uses Var(aX + bY + ... )^0.5 as its risk measure, and PMPT uses downside_variance^0.5.