# Ex post risk report

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The following table describes the ex-post risk statistics currently calculated by the FIA engine.

Statistic | Notes | Formula |
---|---|---|

mean | Average return over N samples | <math>\mu=\frac{1}{N}\sum_i{r_i}</math> |

sigma | Standard deviation of return over interval. Also referred to as volatility, sigma. | <math>\sigma=\sqrt{\frac{1}{N}\sum_i{(r_i-\mu)}^2}</math> |

population_sigma | Sample standard deviation of return. Includes Bessel's sample count correction N-1 instead of N. | <math>sd=\sqrt{\frac{1}{N-1}\sum_i{(r_i-\bar{x})}^2}</math> |

variance | Square of standard deviation of return over interval. | <math>\sigma^2</math> |

population_variance | Square of sample standard deviation of return over interval. | <math>{sd}^2</math> |

tracking_error | Standard deviation of active return (portfolio return minus benchmark return) | |

information_ratio | Active return (portfolio return minus benchmark return) divided by tracking error | |

covariance | Covariance of portfolio return against benchmark return | |

correlation | Correlation of portfolio return against benchmark return | |

correlation_squared | Square of correlation of portfolio return against benchmark return | |

beta | Beta of portfolio against benchmark | |

omega | Omega statistic | |

jensen_alpha | Jensen's alpha | |

Sharpe ratio | Sharpe ratio for portfolio return | |

Active Sharpe ratio | Sharpe ratio for portfolio return against benchmark return | <math>S = \frac{R_P-R_B}{sd}</math> |

Treynor ratio | <math>T = \frac{R_P-R_B}{\beta}</math> | |

upside_volatility | ||

downside_volatility | ||

Skewness | A measure of the asymmetry of the distribution of returns about their mean | <math>S=\frac{N}{(N-1)(N-2)}\sum_{i=1}^N\left[\frac{(r_i-\bar{r})}{sd}\right]^3</math> |

Kurtosis | A measure of the "peakedness" of the distribution of returns | <math>S=\frac{N(N+1)}{(N-1)(N-2)(N-3)}\sum_{i=1}^N\left[\frac{(r_i-\bar{r})}{sd}\right]^4</math>
<math>+\frac{3(N-1)(N-1)}{(N-2)(N-3)}</math> |