Are there any cases where an average value can be maximized only at the cost of increasing variance?

Let’s say that there is some property whose value we would like to be as high as possible. We also want to keep the value from fluctuating too much. In particular, there may be some minimum threshold, which would be very undesirable to fall below. Are there ever cases where it is only possible to reduce the variance and, in particular, the chances of falling below the threshold, by reducing the average?