Why are examinees often not allowed to leave during the start and end of an exam?
Well the 10 minutes at the beginning is to allow some to arrive late without the possibility of collusion.
Where I work, it is 15 minutes at the beginning just for any who are late...
The 10 minutes at the end is probably to allow those rushing to finish a quieter time to concentrate - most students leaving manage to make noise - chairs scraping etc
When you forbid students from leaving early, this is usually a trade-off between the total disturbance caused by leaving and the discomfort caused to the student. To summarise:
Without that rule, far more students would leave early, because the vast majority of students actually uses most of the time. Most of the students who leave in the last ten minutes without the rule leave because the few remaining minutes do not allow to reasonably start a new task or revise anything – not because they finished everything. These students would not want to leave earlier with the rule in place.
A single student can easily leave in an orderly, quiet fashion. Multiple students can’t. They are going to be in each other’s way, will accidentally talk to each other, etc. In the last ten minutes you are very likely to have such an accumulation.
In most exams I attended or proctored, we used the following rule: If you want to leave early, you raise your hand, a proctor collects your exam, and then you leave quietly (to reduce the disturbance). However, if you stay until the end, everybody stays at their desk and the proctors go round to collect the exams and then everybody is free to leave (because the number of proctors is limited, and to avoid last-minute cheating). Obviously this only works with a forced separation of the two phases.
In the beginning, you will have some people arriving late, whom you do not want to confuse with the people leaving very early.
The first ten minutes establish the behaviour for the rest of the exam. Therefore you want to avoid disturbances as much as possible. (Thanks to O.R. Mapper for pointing this out).)
In some systems, there are always some students who just take a look at the exam and then immediately leave due to being discouraged or because they just want to know the tasks. If the students at least have to wait for ten minutes, they usually have nothing better to do than to actually look at the exam, and thus get an idea of what the tasks are like, be it to better prepare for the repeat exam or to think twice before impulsively making a bad decision. If they still want to leave after that, there is probably no helping it anyway. Moreover, those who do leave tend to spread their departures a bit (unless you publicly announce that the ten minutes have passed).
If something is horribly wrong with the exam, there is a good chance that it will be found in the first ten minutes; the professor or proctors can then take immediate action. In some systems, it may be helpful to ensure that all participants are present for this, to avoid somebody complaining of unfairness later (“If I had known about the correction, I would not have left early.”).
So, to summarise, the ten-minute rule avoids of the trouble caused by early leavers, without causing too much discomfort to those having to wait.
The 10 mins at the beginning is possibly to prevent cheating - an examinee sees the paper, leaves the room and phones it to friends who have arranged to arrive late. Or am I being cynical?