We derive new entropy and moment bounds for the stochastic discount factor (SDF). Our results generalize existing bounds which exploit risk-adjusted measures of investment opportunities---such as Sharpe ratios or expected log returns---that are maximized in the cross-section, across assets. By contrast, we can fix a single asset and optimally exploit information in its true and risk-neutral return distributions. Applying the framework to the S&P 500 index, we find that the $\theta$th SDF moment grows extremely rapidly when $\theta > 1$, and appears to diverge to infinity before $\theta=2$. But entropy measures and the $\theta$th moments with $\theta \in (0,1)$ are well-behaved theoretically and empirically, and can be related to measures of market risk aversion and of the attractiveness of investment opportunities.