The 'ideal' answer is definitely yes.
String matching against an indexed column will always be slower than matching a hashvalue stored in an index column. This is what hashvalues are designed for, because they take a large dataset (e.g. 3000 comparison points, one per character) and coalesce it into a smaller dataset, (e.g. 16 comparison points, one per byte).
So, the most optimized string comparison tool will be slower than the optimized hash value comparison.
However, as has been noted, implementing your own optimized hash function is dangerous and likely to not go well. (I've tried and failed miserably) Hash collisions are not particulrly a problem, because then you will just have to fall back on the string matching algorithm, which means that would be (at worst) exactly as fast as your string comparison method.
But, this is all assuming that your hashing is done in an optimal fashion, (which it probably won't be) and that there will not be any bugs in your hashing component (which there will be) and that the performance increase will be worth the effort (probably not). String comparison algorithms, especially in indexed columns are already pretty fast, and the hashing effort (programmer time) is likely to be much higher than your possible gain.
And if you want to know about performance, Just Measure It.