In this paper, we construct a new variance bound on any stochastic discount factor (SDF) of the form m = m(x), where x is a vector of random state variables. In contrast to the well known Hansen-Jagannathan bound that places a lower bound on the variance of m(x), our bound tightens it by a ratio of 1=½2x;m0 where ½x;m0 is the multiple correlation coefficient between x and the standard minimum variance SDF, m0. In many applications, the correlation is small, and hence our bound can be substantially tighter than Hansen-Jagannathan’s. For example, when x is the growth rate of consumption, based on Cochrane’s (2001) estimates of market volatility and ½x;m0 , the new bound is 25 times greater than the Hansen-Jagannathan bound, making it much more difficult to explain the equity-premium puzzle based on existing asset pricing models.