# Ex post risk report

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

Statistic Notes Formula
mean Average return over N samples $\mu=\frac{1}{N}\sum_i{r_i}$
sigma Standard deviation of return over interval. Also referred to as volatility, sigma. $\sigma=\sqrt{\frac{1}{N}\sum_i{(r_i-\mu)}^2}$
population_sigma Sample standard deviation of return. Includes Bessel's sample count correction N-1 instead of N. $sd=\sqrt{\frac{1}{N-1}\sum_i{(r_i-\bar{x})}^2}$
variance Square of standard deviation of return over interval. $\sigma^2$
population_variance Square of sample standard deviation of return over interval. ${sd}^2$
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 $S = \frac{R_P-R_B}{sd}$
Treynor ratio $T = \frac{R_P-R_B}{\beta}$
upside_volatility
downside_volatility
Skewness A measure of the asymmetry of the distribution of returns about their mean $S=\frac{N}{(N-1)(N-2)}\sum_{i=1}^N\left[\frac{(r_i-\bar{r})}{sd}\right]^3$
Kurtosis A measure of the "peakedness" of the distribution of returns $S=\frac{N(N+1)}{(N-1)(N-2)(N-3)}\sum_{i=1}^N\left[\frac{(r_i-\bar{r})}{sd}\right]^4$

$+\frac{3(N-1)(N-1)}{(N-2)(N-3)}$