Recently Convolutional Neural Networks (CNN) have been used to reconstruct hyperspectral information from RGB images. Moreover, this spectral reconstruction problem (SR) can often be solved with good (low) error. However, these methods are not physically plausible: that is when the recovered spectra are reintegrated with the underlying camera sensitivities, the resulting predicted RGB is not the same as the actual RGB, and sometimes this discrepancy can be large. The problem is further compounded by exposure change. Indeed, most learning-based SR models train for a fixed exposure setting and we show that this can result in poor performance when exposure varies. In this paper we show how CNN learning can be extended so that physical plausibility is enforced and the problem resulting from changing exposures is mitigated. Our SR solution improves the state-of-the-art spectral recovery performance under varying exposure conditions while simultaneously ensuring physical plausibility (the recovered spectra reintegrate to the input RGBs exactly).