Gross Or Net Debt?
The argument for excluding intragovernmental debt may seem convincing at first glance: why should you include debt which is an asset for another government agency, thus being no net liability for the government as a whole?
Assuming that it doesn't represent some external liability for the government agency that holds it, this would indeed be a valid argument for excluding it. But at least in the case of the U.S. government the vast majority of it does in fact represent external liabilities.
When a corporation partially pays it employees in the form of future pensions, the rule in corporate accounting is that the present value of those future pensions should count as a liability in the balance sheet. It is certainly not permissible for the company to count the assets in the pension fund that in the future will make those payments as equity.
The same rule should apply to the assets of the Social Security trust fund. Since these assets are supposed to cover future payments of Social Security to senior citizens, it is not valid to subtract the holdings of the Social Security trust fund, or any other intragovernmental holding that represents similar external liabilities, from estimate of government debt.
One problem is that unlike in the Swedish pension system there is no exact link between the level of assets in the Social Security trust fund and the level of payments to retirees. Payments are instead determined by the development of inflation, while the value of the fund is determined by income from Social Security taxes plus interest minus Social Security payments. This means that the present value of future payments could differ from the assets of the Social Security fund.
However, unless you have specific reasons for believing that there is a certain difference, the value of the trust fund is a good proxy for the value of future payments. After all, the trust fund could just as well underestimate the value of future payments as overestimate it. And clearly the value of the trust fund is closer to the truth than the zero value that advocates of excluding the holdings of the trust fund from government debt implicitly assumes.
By contrast, in some other countries assets of funds should be subtracted from gross debt as they are not linked to future external payments. One good example of this is Norway where the government has saved revenues from oil exports in a special oil fund that is not directly related to future payments of pensions and other benefits. Indeed, in Norway the value of that fund is greater than gross debt, so that the Norwegian government has no net debt at all, and instead has a large net worth.