It’s possible, though unlikely, that some of those tax-loss carry-forwards, as they’re known, expired unused. In 1995 they could be used for up to 18 years, which means any of the losses incurred before 1987 would have expired by 2005. But there’s no evidence Mr. Trump had losses of anywhere near that magnitude that early in his career. Most of the losses are presumed to date from the problems his casino operations ran into in the early 1990s.

It’s also possible that Mr. Trump earned close to $1 billion in taxable income during that decade that he sheltered from federal tax using his loss carry-forward.

The reason that’s unlikely is also evident on Mr. Trump’s 2005 return: the alternative minimum tax. The A.M.T. is a parallel system for calculating tax liability intended to ensure that high-income taxpayers pay a substantial amount in federal tax even if they have large deductions or other items to offset income.

The A.M.T. is much reviled by tax experts across the political spectrum for its unintended consequences and fiendish complexity. “It’s received wisdom among tax people that having two tax systems makes no sense,” said Douglas Holtz-Eakin, an economist who served as director of the Congressional Budget Office and is now president of the American Action Forum, a pro-growth advocacy group. “It’s a shadow tax system. Personally, I believe you should have one tax system you believe in and then live with it.” (I routinely pay the A.M.T., as do many residents of high-tax states like New York and California.)

But in the Trumps’ case, the A.M.T. seems to have worked as planned. Using the regular tax calculations, which allowed the Trumps to take the full amount of the loss carry-forward, they were able to reduce their taxable income to $31.6 million. The federal tax due on that amount was $5.3 million, or just over 3 percent of their total income.

Because of the A.M.T., the Trumps had to recalculate their tax liability by adding back some of the deductions and offsets, such as payments of state, local and real estate taxes and, it turns out, the loss carry-forward they used to offset ordinary income. That meant the Trumps had to add back the $103 million and then subtract the A.M.T.’s version of the loss (an extremely complicated calculation) to produce what is known as A.M.T. income, which is taxed at a flat 28 percent rate. Since that figure was larger than their taxable income using the regular system, they had to pay the A.M.T.— $31.3 million (which is 28 percent of about $110 million) — plus the regular tax of $5.3 million.

In light of that, it is hardly surprising that President Trump has called for abolishing the A.M.T.

But a question remains: If the Trumps were subject to the A.M.T. every year, and had to add back any loss carry-forward and pay the resulting tax liability, how did they use up so much of it — $813 million — by 2005?