The efficiency of a single band-gap solar cell is specified by the Shockley-Queisser limit, which defines the maximal output power as a function of the solar cell's band-gap. One way to overcome this limit is by using a down-conversion process whereupon a high energy photon is split into two lower energy photons, thereby increasing the current of the cell. Here, we provide a full analysis of the possible efficiency increase when placing a down-converting material on top of a pre-existing solar cell. We show that a total 7 efficiency improvement is possible for a perfectly efficient down-converting material. Our analysis covers both lossless and lossy theoretical limits, as well as a thermodynamic evaluation. Finally, we describe the advantages of nanoparticles as a possible choice for a down-converting material.