Python 2.7 56 58 56 51 50
With the stolen Number One Dime from Scrooge McDuck, I made my fortune and now have more wealth than Scrooge.
y=lambda s:2*abs(sum((1j)**(ord(i)%15)for i in s))
Python 2.7 - 61 53 52 (case insensitive)
y=lambda s:2*abs(sum((1j)**(ord(i)%16%9)for i in s))
Implementation
>>> from random import sample
>>> y=lambda s:2*abs(sum((-1j)**(ord(i)%15)for i in s))
>>> path=sample('RLUD'*1000, 100)
>>> y(path)
20.0
>>> path=sample('RLUD'*1000, 100)
>>> y(path)
34.058772731852805