In national accounting, debts are added according to those who are indebted. Household debt is the debt held by households. "National" or Public debt is the debt held by the various governmental institutions (federal government, states, cities ...). Business debt is the debt held by businesses. Financial debt is the debt held by the financial sector (from one financial institution to another). Total debt is the sum of all those debts, excluding financial debt to prevent double accounting. These various types of debt can be computed in debt/GDP ratios. Those ratios help to assess the speed of variations in the indebtness and the size of the debt due. For example the USA have a high consumer debt and a low public debt, while in eastern European countries, for example, the opposite tends to be true.
There are differences in the accounting of debt for private and public agents. If a private agent promises to pay something later, it has a debt, and this debt is enforceable by public agents. If a public body passes a law stating that it'll pay something later (a kind of promise), it keeps the right to change the law later (and not to pay). This is why, for instance, the money governments promised to pay for retirements does not show up in the public debt assessment, whereas the money private companies promised to pay for retirements do.
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