Dear,
Appointment letter is the bridge between the employer and employee. Without appointment letter, the link between employer and employee does not take place.
Advantages:
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It is gives you the details of your remuneration,your job responsibility, your current place of posting, the date from which you are hired, conditions of employment, your age of retirement, etc.
This means, you are very very clear about your purpose of very existence in the organization at what level. This helps you to get job done from others and assist your peers without any hiccups.
Disadvantages:
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when there is no link is seen between employer and employee, you cannot prove your belonginess to the organization. You can be asked to push off, any time without any notice or something.
Secondly, you will miss all information with respect to conditions of your employment. You will not have role clarity which will affect your performance too.
Balaji