These three concepts are often used as synonyms but they are not. Though the common use of these words may lead to confusion, these are financial and legal terms and thus, they should be used properly and with caution. There are various related concepts needed to explain the above three: personal financial failure, business financial failure, cash flow, and several more.
When a business or a person can no longer cope with ordinary expenses and exactable debt is higher than liquid assets, then this state is known as financial failure. But this is a financial concept. There are also legal concepts related to this financial concept. Bankruptcy for example is a legal concept that reflects the state of financial failure and rules its consequences. In certain countries, the term bankruptcy is only reserved for individuals while other terms are used for businesses (insolvency, liquidation, etc.). In any case, bankruptcy implies a financial failure where the debtor cannot afford to repay debt any longer.
The cash flow is the movement of money, the transactions that a company or an individual make every day, month and year. Cash flow implies income and spending. Under a normal cash flow income is expected to provide the necessary funds to cope with the company's or individual's obligations (payment of services, debt payments, etc.). But, most importantly, this has to be done in a timely fashion. If for any reason the company or individual get behind on payments, it is imperative to catch up. This can be done by increasing income or using savings (selling assets, renting, additional jobs, etc.) or by reducing spending (closing accounts, cancelling services, reducing staff, consolidating debt, etc.). If for some reason, none of these solutions can be achieved, the unavoidable resolution would be a bankruptcy.
Insolvency, Bankruptcy, Liquidation
These concepts are used often both on the financial field and on the legal field. Truth is that bankruptcy is a legal concept well defined by the law and requires no clarification. Bankruptcy is the financial failure of an individual (or company in most countries). Under bankruptcy, debts are discharged by selling the debtor's assets (certain assets may be left aside) and transferring the amounts produced with those sales to the creditors proportionally to the amounts owed (as required by law certain debts are privileged).
Insolvency is the inability of an individual or company to cope with debt payments with current liquid assets. This can be solved by selling non-liquid assets, by borrowing money, by negotiating new terms with creditors, etc. If no solution is achieved, insolvency will probably lead to bankruptcy but these concepts are most certainly not synonyms. Truth is that insolvency is commonly applied to businesses and seldom to individuals (due to a British heritage). As regards to liquidation, it refers to the sell of the debtor's assets to cover the debts. It is also mainly used with companies but sometimes it can also be applied to the process within a bankruptcy that consists on selling in public auctions the debtor's assets.