This paper considers the possibility of extending the Arrow-Pratt results on risk aversion to cases in which initial wealth is random. Specifically, we consider a situation in which an individual's wealth is the sum of two independent random variables x and y. We define the risk premium @?(x, y) which represents the reduction in mean wealth an individual is willing to accept to eliminate the random variable x while retaining the random variable y. It is shown that if u"1 is uniformly more (Arrow-Pratt) risk averse than u"2 and if either u"1 or u"2 exhibit nonincreasing (Arrow-Pratt) risk aversion, then @?"2(x, y) is always smaller than @?"1(x, y). An example is given in which both u"1 and u"2 exhibit increasing risk aversion and in which this result fails.