Sharpe ratios measure the excess return of an asset per unit of risk. The excess return is defined as the return of an asset minus the return of a benchmark asset. The risk is defined as the standard deviation of the returns. Note that this definition of risk also includes risk to the upside. The Sharpe ratio \( S_h \) is defined as \[ S_h = \frac{\mu}{\sigma},\]where \[\mu = \int_{-\infty}^{+\infty} (r_c-r_b)p(r_c)\text{d}r_c,\] and \[\sigma=\left(\int_{-\infty}^{+\infty} (r_c-r_b-\mu)^2 p(r_c) \text{d}r_c \right)^{1/2},\]and \( r_c \) and \( r_b \) denote the returns of the cryptocurrency and benchmark asset, respectively, and \( p(r_c) \) is the probability density function of the cryptocurrency returns. Examples of these probability density function are provided here. A similar metric, the Sortino ratio, uses another definition of risk. Shown above are annualized cryptocurrency Sharpe ratios based on the previous 90, 180, and 365 days of log returns calculated from volume weighted average daily prices. The benchmark asset \( r_b \) was the risk-free LIBOR.