Good afternoon,

While you did not mention what kind of palm you have, the leaf spots on your palm are most likely caused by a fungus. Here is a link with extensive information about palms:

http://edis.ifas.ufl.edu/pdffiles/PP/PP14200.pdf

It is noted that for juvenile palms, leaf spots are more severe when the palms are grown in full sun rather than with some shade.

You can prune severely diseased leaves and dispose of them to prevent spores from further infecting the plant.  Leaf  injury due to water stress, sunburn, fertilizer burn, herbicide phytotoxicity, or cold temperatures should be avoided.

It is also noted that some fungal leaf pathogens are more severe on nutritionally-stressed palms. For example, Wodyetia bifurcata (foxtail palm) suffering from iron deficiency is more susceptible to leaf spots (Figure 5). Leaf spots due to potassium deficiency create injuries that allow leaf pathogens to become established.

Hope this helps with your palm.