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Recoupment is the practice, common in the music industry, of claiming an advance provided to an artist back from that artist rather than (or, as well as) from related sources. Many record labels practice recoupment, and most new artists have little choice but to accept it since they lack the negotiating power to obtain a better contract.
Recoupment is also a common practice within the insurance industry. It means virtually the same thing, but generally occurs when an insurance company has paid the benefits to a hospital and then takes back that money when they have determined that those benefits should not have been given.
Suppose that a music label gives a band a $250,000 advance to record an album. The label agrees to do so in return for 90% of the sales. In addition, the label will specify certain standards for production of the album, for example, which studios the band will engage. The label may even hold the advance and make all disbursements on the band's behalf, ensuring the funds are used exactly as agreed. In other words, the $250,000 advance is not simply pocketed by the band -- it is to be spent on album production and the band's reasonable expenses during production. The album is recorded, and sells 200,000* copies at $10 each, yielding $2m. The record company takes 90% of this as agreed, leaving the band with $200,000 of their own. This is the situation without recoupment.
With recoupment, the label advances the band $250,000 as before. As before, the band (or the label, on the band's behalf) spends substantially all of the advance to produce an album meeting the label's specifications. The album again sells 200,000 copies at $10 each, yielding $2m. The record company takes 100% of the first $250,000 in sales (recouping the $250,000 advance to the band), and 90% of all further sales, as agreed. The band's net is effectively reduced to 0% of the first $250,000 in sales and 10% of the further $1.75m, leaving them with a total of $175,000.
*Recoupment deals primarily involve new artists, or those without a proven sales history, so selling 200,000 copies would be fairly unusual. More typically, 10,000 or fewer copies are sold by a new artist. In this more typical case, the label recoups only $100,000 of the $250,000 advance, along with other non-recoupable expenses incurred in getting the album out into the world.