|Loss Standard Deviation|
When thinking about the concept of risk, investors usually think about losses. Most often people think about risk as the standard deviation or volatility of all returns. In contrast, loss standard deviation measures the variability of returns below the target return. All positive returns are treated as zeros in the calculation as below:
T in the above equation can be thought of as a target rate where outperformance is measured. For example, a pension fund may have a target funding assumption that it must earn to be able to fund its pensioners.
Any return lower than this can be considered a loss (even if the absolute return is positive) because the result falls short of what must be earned by the fund to meet its liabilities. In this case if the funding assumption is a 0.5% monthly return, anything less than 0.5% is taken into consideration in the calculation. Alternative target rates are typically the risk-free rate or zero.
The loss standard deviation was proposed because some investors do not believe that positive returns should be included in measurement of risk and therefore look to only consider negative returns.
As such some investors replace the concept of standard deviation with loss standard deviation in various statistics as well as look at loss standard deviation as a stand-alone metric. A prominent example of the use of this concept is the Sortino Ratio, which replaces the standard deviation in the Sharpe Ratio with loss standard deviation in the denominator.
However, loss standard deviation numbers should be viewed with caution due to the limited data points involved with its calculation. Since all positive observations are ignored, the number of data points that are available may not be sufficient to make a valid statistical argument.