[c#] Create a date time with month and day only, no year

There is no such thing like a DateTime without a year!

From what I gather your design is a bit strange:

I would recommend storing a "start" (DateTime including year for the FIRST occurence) and a value which designates how to calculate the next event... this could be for example a TimeSpan or some custom structure esp. since "every year" can mean that the event occurs on a specific date and would not automatically be the same as saysing that it occurs in +365 days.

After the event occurs you calculate the next and store that etc.