The statistical properties of various measures of risk were investigated with a view to explaining the reasons for lack of use in finance of risk measures other than the variance, and to see if there is a sensible measure to use for cross-European comparisons. As examples, the semi-variance, the lower partial moment, the Gini, and the absolute deviation were considered. A Monte Carlo study was conducted to study the behaviour of these measures of risk. Bootstrap methods were used by drawing random observations from real financial returns data, for the same purpose. Specifically, the equity indices from the following stock markets were used: Germany, United Kingdom, France, Italy, Holland and Belgium. It is concluded that there is little reason to reject measures of risk other than the variance based on a view that they are too volatile. Indeed the evidence shows that whilst the standard deviation may be the appropriate measure of risk for high volume markets (Germany, UK and France), it is not the most reliable risk measurement (in terms of volatility) when considering lower volume markets (Italy, Holland and Belgium). That is, the more non-normal the returns, the more likely is the standard deviation to be volatile compared to the Gini or the absolute deviation. Finally, optimized portfolios were investigated using different risk measures and substantial differences were found; this suggests these findings have practical consequences.