Hey there!
Actually I had the same problem, at least I thought I did, as I also carry a Lumia 1020. The fact is, after working this out for HOOOURS, it turned out that the alarm starts at a low volume and then it gradually increases to a deafening level!
How I found this out is that I accidentally selected a short MP3 file (added it via USB connection into the 'Ringtones' folder) of say 5-12 sec, to be my alarm tone, and with that I could observe a difference after 1 minute in the alarm volume ... so it really works

!! though as far as I know, this is not mentioned anywhere :angry:
What I believe you should do is select a short sound file as your alarm tone, because each time it is iterated, it plays at a constant volume, but is louder than the preceding iteration. Meaning that if you select a short sound file, you will get a gradual loudness in your alarm, but with a 3-4 minute song you might not notice that, as this will take much longer.
Hope this helped
Cheers