Gallant, Hansen and Tauchen (1990) show how to use conditioning information optimally to construct a sharper unconditional variance bound on pricing kernels. The literature predominantly resorts to a simple, sub-optimal procedure that scales returns with predictive instruments and computes standard bounds using the original and scaled returns. This article provides a formal bridge between the two approaches. We propose a optimally scaled bound, which, when the first and second conditional moments are known, coincides with the bound derived by Gallant, Hansen and Tauchen (GHT bound. When these moments are mis-specified, our optimally scaled bound still yields a valid lower bound for the standard deviation of pricing kernels, unlike the GHT bound. Moreover, the optimally scaled bound can be used as a diagnostic for the specification of the first two conditional moments of asset returns because it only achieves the maximum when conditional mean and conditional variance are correctly specified. Other potential applications include testing dynamic asset pricing models, studying the predictability of asset returns, dynamic asset allocation, and mutual fund performance measurement. The illustration in this article starts with the familiar Hansen-Singleton (1983) set-up of an autoregressive model for consumption gowth and bond and stock returns but adds time-varying volatility to the model. Both an unconstrained version and a version with the restrictions of the standard consumption-based asset pricing model imposed, serve as the data-generating processes to illustrate the behavior of the bounds. In the process, we explore an interesting empirical phenomenon: asymmetric volatility in consumption growth.