As an adjective, it means able to be changed into a different form or use. As a noun, it often refers to a car with a roof that can be folded down or removed.
From Latin *convertibilis* 'able to be turned around', from *convertere* 'to turn, transform'. English applied the idea to many things that can change form, including cars and financial instruments.
A convertible car is literally 'turnable'—from closed to open. The same word is used in finance for 'convertible bonds' that can turn into shares. It’s a good example of how one physical image (a roof flipping back) helps explain more abstract 'shape-shifting' in money or ideas.
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