A computational device receives indications of a minimum retention time and a maximum retention time in cache for a first plurality of tracks, wherein no indications of a minimum retention time or a maximum retention time in the cache are received for a second plurality of tracks. A cache management application demotes a track of the first plurality of tracks from the cache, in response to determining that the track is a least recently used (LRU) track in a LRU list of tracks in the cache and the track has been in the cache for a time that exceeds the minimum retention time. The cache management application demotes the track of the first plurality of tracks, in response to determining that the track has been in the cache for a time that exceeds the maximum retention time.