Risk Parity Optimality

We show that the probability of risk parity beating any other portfolio is more than 50 percent. We also prove that if portfolio performance is measured by Sharpe ratio, risk parity is the only maximin portfolio when (1) all assets' future Sharpe ratios are greater than an unknown constant and all correlations are less than another constant, or (2) when the sum of all assets' future Sharpe ratios is greater than some constant. If portfolio performance is measured by expected return, risk parity is the only minimax portfolio when the sum of assets' Sharpe ratios is greater than a constant