A true alpha male is someone who does not need to do much exercise and dieting and still looks lean and mean and is popular with women. Money is actually irrelevant here.
Look at nature, look at the male lion - the perfect model of what an alpha male is: most of his time is spent eating and sleeping...he doesn't hunt (female lions do that for him) and his physical activities are limited to occasional short (although admittedly intense) fights for territory and females with other male lions, and mating.
Someone who kind of resembles this life style in human life is, to me, an alpha male proper.