An index groups multiple companies' share prices together. An index is usually made up of that are listed together on the same exchange. They are often mentioned in the news media to measure how a certain market or exchange is performing.
Investors trade indices for the same reasons as they trade shares, hoping to profit from their growth but with the additional advantage of the diversity they add to a portfolio. When you trade an index you are trading multiple companies across multiple industries. For example there are 30 companies that make up the German DAX index. Including, at the time of writing, the carmaker Volkswagen, the sporting goods company Adidas, the airline Deutsche Lufthansa, and the pharmaceutical company Bayer, and the insurance and financial company Allianz. The companies in DAX are the 30 most valuable companies listed on the German Deutsche Boerse stock exchange. Trading the DAX gives an investor more diversity than investing in a single company, e.g. Adidas, which specializes exclusively in in manufacturing sporting goods.
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