Private equity, generally speaking, is an equity (stock) investment in a privately-held company. Private = privately held company. Equity = stock. It's a little more complicated than that, but that's a useful starting point.
The term "private equity" can be used in different contexts and can mean different things. I group these in 4 categories:
Asset Class. First, the term "private equity" can refer to the private equity asset class. An asset class is basically a group of assets that have similar characteristics and behave similarly. Common asset classes include: public stocks; public bonds (also called fixed income), cash, real estate and private equity. There are also others, and different investors may define asset classes differently. This usage of the term "private equity" typically occurs when investors are discussing allocations to asset classes - for example "we are allocated 40% to public equities, 40% to public fixed income; 18% to private equity and 2% to cash." Within the private equity asset class, there are different strategies, including venture capital, growth equity and buyouts. For more on venture capital, growth equity and buyouts, please see my posts "LP Corner: What is Venture Capital? Here's My Definition", "LP Corner: Venture Capital vs Growth Equity - A Comparison" and "LP Corner: What are Buyouts? Here's my Definition."
Transaction. Second, the term "private equity" can refer to a type of transaction – which is primarily buyouts. However, it can also describe a growth equity investment. However, the term "private equity" isn't used to describe a venture capital investment. Venture capital is a unique strategy within the private equity asset class, and the term "venture capital" is used very specifically.
Firm. Third, the term “private equity” can be used to describe an investment firm. A “private equity” firm is one that primarily invests in buyout transactions, but the term can also apply to firms that invest in growth equity. One doesn’t use the term “Private Equity firm” to describe a venture capital firm - those are simply called "venture capital" firms.
Industry. Finally, the term "private equity" may refer to the private equity industry. The private equity industry starts with the private equity strategies of venture capital, growth equity and buyouts and also includes investors in the private equity asset class, such as public and private pensions, sovereign wealth funds, corporations, foundations, endowments, family offices and high net worth individuals. Also included are firms and funds - venture capital firms, growth equity firms and buyout firms, as well as corporate venture capital efforts. Add in service providers such as attorneys, accountants, valuation firms, database firms, back office support, etc. as well as industry groups (such as the NVCA and ILPA), conference providers and the press, and you end up with a very large industry.
So you need to understand the context in which the term is being used.
I hope this explanation is helpful.
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