The variance of a truncated random variable and the riskiness of the underlying variables

We start from a variable x̃, which has an unspecified (and possibly even infinite-variance) distribution, and we truncate x̃ from above and below with bounds that may linearly depend on a second variable, ỹ~. We investigate how the variance of this truncated variable is affected by a binomial version of the Rotschild-Stiglitz measure of increased riskiness of x̃ or ỹ. We find that, for most unimodel distributions of x̃, such an increase in the riskiness of x̃ increases the variance of the truncated variable. The effect of changed riskiness in ỹ is ambiguous.