Sharpe, Sortino, Calmar:three ratios, one set of returns.

Sharpe ratio = (mean return − risk-free rate per period) ÷ standard deviation of returns × √(periods per year). Risk is all variation, up or down. The standard deviation here is the sample one, dividing by n − 1.

Sortino ratio = (mean return − target) ÷ downside deviation × √(periods per year). Downside deviation is √(Σ min(0, return − target)² ÷ n): only shortfalls below the target count, but every period is in the denominator. The target here is the risk-free rate per period.

Calmar ratio = compound annual return ÷ maximum drawdown. The annual return is (Π(1 + return))^(periods per year ÷ n) − 1; the maximum drawdown is the largest peak-to-trough fall of the compounded equity curve. Traditionally it is measured over 36 months.

Worked example. Four monthly returns of 2%, −1%, 3% and 0%, risk-free rate 0. The mean is 1.00%. The deviations from it are 1, −2, 2 and −1, whose squares sum to 10, so the sample standard deviation is √(10 ÷ 3) = 1.83%. Sharpe = 1 ÷ 1.826 × √12 = 1.90. Only −1% falls below zero, so downside deviation is √(1 ÷ 4) = 0.50%, and Sortino = 1 ÷ 0.5 × √12 = 6.93. The equity curve peaks at 1.02 and falls to 1.0098, a 1.00% drawdown; the four months compound to 12.52% a year, so Calmar = 12.52.

The example shows why a short record flatters every ratio: four months with one small loss annualise into numbers no strategy sustains. These are descriptions of past returns, not predictions of future ones.