From the chaos of the sovereign debt crisis to the acrimony over an influx of refugees, European authorities have proved something less than an exemplar of coordinated government action. The European Union has become a focus of populist anger, further constraining options. And Germany has opposed bailouts for lenders in other lands, making a Deutsche rescue politically radioactive.  All of which adds to worries that Deutsche amounts to a fire burning, one that might yet become an inferno, while the fire department is consumed with existential arguments over its purpose. If the alarm sounds, no one can be sure what, if anything, will happen.  In the worst case — now highly unlikely — the bank could collapse, inciting a scramble to pull money from markets around the globe. Institutions that trade with Deutsche would feel an urge to collect their cash immediately. Given the scale of the bank’s balance sheet — 1.8 trillion euros, or more than $2 trillion — that inclination is likely to spread to every crevice of finance. Economies would grind to a halt. Jobs and fortunes would disappear.  Despite murmurings in pundit quarters that this sort of situation may be unfolding, provoking comparisons with the catastrophic bankruptcy of the American investment banking giant Lehman Brothers eight years ago, most economists dismiss such talk as overwrought and overblown.  Deutsche is sitting on cash reserves worth €240 billion, or about $269 billion. It has sold bonds that can be converted to equity should the need arise. The Justice Department’s proposed fine of $14 billion is viewed as the opening of a negotiation that could cost Deutsche a fraction of that amount — thinking that sent the stock surging on Friday.  Not least, Deutsche Bank is a classic example of the species of financial animal known as Too Big To Fail.  “We saw what happened with Lehman,” said Nicola Borri, a finance professor at LUISS, a university in Rome. “It’s impossible that the authorities would let something like that happen again. It has ties with all the banks in the world. It is highly leveraged. A disorderly default would be very, very difficult for the entire financial system.”  On both sides of the Atlantic, the financial crisis prompted the construction of new regulatory authorities and requirements that banks set aside more funds in reserve against troubles.  “The system is much more robust and resilient because of the buffers,” said Nicolas Véron, a senior fellow at Bruegel, a research institution in Brussels. “There are pockets of fragility, but broadly speaking, the system is better prepared.”  But the markets do not appear to fully buy that the defenses are secure.  Deutsche is heavily involved in the trading of derivatives, the exotic financial instruments that were at the center of the 2008 crisis. Derivatives can be so mind-bendingly complex that no one fully grasps who owes what to whom until someone big enough to rattle markets suddenly cannot pay.  Then, fear takes over, and investors dump holdings indiscriminately. This lowers the value of even solid assets on bank balance sheets, giving rise to further cause for concern.  Because Deutsche has been dominated by its investment banking operations — meaning it is not sitting on a large pile of plain deposits as a cushion — it is especially vulnerable to such volatility.  Fear, in other words, is not just a symptom of trouble but also a cause.  This makes Deutsche’s problems the world’s problems. Not for nothing did the International Monetary Fund in June declare Deutsche to be “the most important net contributor to systemic risks” on earth.  A collapse may be exceedingly unlikely. Yet the beginning would probably feel something like recent days.  Thursday brought reports that hedge funds were quietly extracting their money from Deutsche’s coffers. The bank’s shares plummeted to a new low.  No lender in the world has more potential to create financial mayhem than Deutsche Bank.  Friday morning, Deutsche’s chief executive officer, John Cryan, released a letter to his staff offering assurances that the bank boasted “strong fundamentals.” The stock recovered slightly on those comments, but the sense remained that the need for reassurance attested to concerns.  The biggest form of insurance against panic is confidence that larger players — in this case, European authorities — stand at the ready to mount a rescue, should one be required.  But confidence is not something Europe has proved terribly skilled at instilling. Its abilities to marshal a bailout are dubious. New rules introduced to discourage reckless investments by large financial institutions bar taxpayer-financed bailouts.  Germany has been adamant that these strictures be applied, rebuffing a recent attempt by the Italian prime minister, Matteo Renzi, to secure an exemption allowing him to inject taxpayer money into the Italian banking system. The optics of Germany seeking a way around the rules for its largest lender would be especially problematic.  The Deutsche chief and the German government both shot down a report that the bank had asked that a bailout be prepared.  More broadly, Germany has been the most fervent voice that reckless economic pursuits should be punished, no matter the human toll.  As Athens has negotiated with European authorities and the International Monetary Fund for a series of bailouts, Germany has demanded deep cuts to Greek public spending, sharply cutting pension payments to retirees. The Greek government used much of the bailout money to pay back debts to German banks.  Against this backdrop, a German bailout of its largest bank would reinvigorate accusations that it uses the European Union as a cover to pursue its own national interests.  This dynamic has force in the markets, presenting another factor that investors must absorb as the evaluate they risks of holding Deutsche’s debts and shares.  “The fact that we don’t know the reaction of the authorities is a factor of uncertainty,” said Mr. Veron of Bruegel.  Here is a feedback loop that amplifies the risks. The likelihood that Deutsche needs a rescue appears small, yet the possibility that a rescue could be forged seems close to nil. That tightens the pressure on Deutsche.  And yet Deutsche’s stature may provide the decisive form of insurance. In event of emergency, the authorities might have to act, whatever the politics.  “Deutsche Bank is so big and so systemically important that the rules will be bent,” Mr. Borri said. “If I were an investor, I would assume that the rules would be bent.”  Follow Peter S. Goodman on Twitter: @petersgoodman  A version of this article appears in print on October 1, 2016, on page A1 of the New York edition with the headline: Too Big to Fail, but Is Bailout on the Table?. Order Reprints| Today's Paper|Subscribe
As investors’ concerns of another crisis are stoked, the arithmetic is less important than fears about the impotence of European authorities.