Yes, the plan is needed for any sort of business. Many peoples making investment through online in different companies such as Forex , indices , stocks. You can also start a business immediately but before that you need a business plan. I am trading forex for last 2 years and i am making 30% profit in a month of my initial investment using my business plan .

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if you want financing or better chance of success yes. the business plan might show you that your business WON'T succeed due to unrealistic sales projections and save the agony of actually opening the business, failing and going broke

For a loan or to obtain an investor? Yes.

Always