This paper considers a class of leases with indexed rents subject to a floor. Such leases have an upward-only flavour to them, although not in exactly the same sense as in the traditional upward-only institutional lease. After deriving a general result, three empirically important cases are considered: a modified upward-only lease, the Swedish standard contract for commercial leases and the percentage lease. Analytical results and numerical examples are used to illustrate how the leases relate to other contract types.