Stock market live updates: Stocks close at session low, Dow drops nearly 400, Walmart rolls over

4:07 pm: Fed's Rosengren warns of opening economy too soon

Boston Fed President Eric Rosengren, whose district covers one of the coronavirus national hotspots, warned Tuesday that opening businesses too soon could be a mistake. Even if places like restaurants and retail stores open, people may not yet feel safe enough to go there, the central bank official told CNBC's "Closing Bell." "It's really important to get the public health right. If we don't get the public health right, it's going to be impossible for us to get the economy right," he said. Boston has begun the first phases of its reopening as the state has been plagued by a high coronavirus fatality count, which most recently was near 5,800. – Cox

3:18 pm: Stocks fall to session lows on reported doubts over Moderna drug, before recovering losses

Stocks dropped do their lowest levels of the day following a report from Stat News that the Moderna vaccine for Covid-19 might not be as effective as some think. The report, which cited several vaccine experts, said that based on information that Moderna has released there's no way to determine the efficacy of the drug. The Dow briefly traded more than 200 points lower before paring losses. Moderna was about 9% lower after earlier falling more than 12%. The market's rally has in some part been tied to the latest developments surrounding a coronavirus vaccine. Stocks jumped on Monday due to hopes from the Moderna trial. – Stevens

2:57 pm: Royal Caribbean needs to raise an additional $2.5B-$3B to survive a 'no sail' scenario through 2021, analyst says

Ahead of Royal Caribbean's first quarter earnings results, Wells Fargo said it believes the company will have to raise significant capital. "We expect the raise will likely occur over the next 2-3 weeks and lean toward convertible debt and equity," the firm said in a note to clients. Wells Fargo expects "limited incremental information" from Royal Caribbean's results on Thursday as the company already preliminary reported back May 13. "We will be looking for potential fleet reactivation plans/timeline, update on customer deposit balance and refund/cancellation rates post March 31, details on booking/pricing trends over the past week including rebookings vs. new bookings for 2021, remaining unencumbered assets, incremental cost or cash burn reduction opportunities." Shares have gained 16% this week, but are down 67% this year. - Bloom

2:27 pm: Bank of America says market's rally can continue

2:14 pm: Citi sees 7% upside for stocks over next 12 months

2:10 pm: Watch out for the 'chart battle,' UBS Art Cashin says

12:00 pm: Markets at midday: Stocks mostly flat after Monday's massive rally

11:55 am: UBS: Outperformance in high-beta stocks signals more optimism about the economy

11:50 am: Warren presses Mnuchin about mandating companies to keep workers on payrolls

11:39 am: Mnuchin on new 20-year bonds: 'Lock in' interest rates at historic lows

Treasury Secretary Steven Mnuchin said with the launch of a new 20-year bond, the department is aiming to stretch the duration while locking in ultra-low interest rates. "Prior to this, we spent a lot of time looking at 50- and 100-year bonds and determined that there just wasn't enough demand to make it worth it given our borrowing sizes," Mnuchin said Tuesday during a Senate hearing. A 20-year bond "gives us the ability to extend the duration to raise significant amount of funds. It is my intention to borrow a lot of money in the short term to have the funding, but then to expand our financing in 10-, 20- and 30- year bonds. What I'd like to do is to lock in a significant amount of very low interest rates so that the money we are borrowing can be paid back and dealt with over a long period of time." The Treasury Department is launching a 20-year bond later this month in a $20 billion auction to fund a record level of borrowing the government will need to do this year to support the economy through the coronavirus pandemic. –Li

11:15 am: Monday's rally allowed more than 80% of S&P 500 stocks to close above their 50-day moving average

11:00 am: Here are Tuesday's biggest analyst calls of the day: Disney, Peloton, Beyond Meat & more

10:45 am: Mnuchin: 'We are fully prepared to take losses'

Stay-at-home stocks resumed their outperformance in the opening minutes of trading on Tuesday, with Amazon, Netflix and Facebook helping the Nasdaq gain 0.3%. Walmart also rose about 2% following its quarterly earnings report. The Dow meanwhile fell more than 100 points in the opening minutes of trading on Tuesday following Monday's 900-point rally. Declining stocks outnumbered gainers 24 to 6 on the 30-stock index. The KBW Bank Index was also down 2.3% after bank stocks surged on Monday. —Pound

9:30 am: Stocks open little changed ahead of Congressional testimonies

8:58 am: Southwest says it's seeing some demand improvement

8:56 am: Fed chiefs at May testimonies have a history - remember the 'taper tantrum'

Market pros are watching every nuance in the testimony of Fed Chairman Jerome Powell and Treasury Secretary Steven Mnuchin before the Senate Banking Committee. They are set to discuss the policy response to the coronavirus emergency. BMO rate strategist Jon Hill reminds us that former Fed Chairman Ben Bernanke was appearing before the Congressional Joint Economic Committee on May 22, 2013, when he started the 'taper tantrum' market sell-off. In response to a question, Bernanke said: "If we see continued improvement and we have confidence that that's going to be sustained then we could in the next few meetings...take a step down in our pace of purchases." Hill doesn't expect the same from Powell. "I don't expect anything like that to come out, but it's theoretically a live possibility. These are big names -Powell and Mnuchin. They do have the opportunity to move markets," he said. The 10-year yield was at 0.73% ahead of the testimony, and the next level to watch is 0.75%, the intraday technical peak going back to April 15. —Domm

8:23 am: Single stocks see record outflows last week, says Bank of America

8:21 am: Moderna falls after new stock offering

8:10 am: Kohl's net sales drop by more than 40%

7:58 am: Mnuchin, Powell to face Senate over emergency lending programs

The U.S. Treasury Secretary and the Chairman of the Federal Reserve will appear via video-conference before the Senate Banking Committee on Tuesday at 10 a.m. ET to answer questions on how they are appropriating $500 billion in crisis lending programs. Congress earlier this year made those funds available through its $2 trillion economic relief package known as the CARES Act. The Treasury received $46 billion to direct assistance to airlines and other bruised industries with another $450 billion earmarked to cover losses in the Fed's lending program. Senators will likely grill Treasury Secretary Steven Mnuchin on how much risk the department is willing to accept on its investment in the Fed's lending apparatus and the implications of losing a significant portion of those funds allocated by Congress. Fed chief Jerome Powell, meanwhile, could face questions about when the Fed's lending facilities will be functional and whether lawmakers need to provide additional funding to mitigate the fallout from the current economic downturn. In prepared remarks that Powell will deliver to lawmakers, Powell says the Fed is "committed to using our full range of tools to support the economy in this challenging time even as we recognize that these actions are only a part of a broader public-sector response." — Franck

7:50 am: Walmart climbs after reporting rapid rise for e-commerce

7:46 am: Home Depot misses profit expectations in the first quarter

7:45 am: Jamie Dimon: Coronavirus is a 'wake-up call' for more inclusive economy

7:25 am: Stock futures flat after 900-point Dow rally in previous session

U.S. stock futures pointed to a pause on Tuesday following sharp gains during Monday's session. Dow Jones Industrial Average futures traded just 11 points higher, or 0.04%. Dow futures were initially dragged down by a 2.1% drop in Home Depot shares during premarket trading. However, those losses were offset by a 4% gain in Walmart shares. S&P 500 futures dipped 0.1% while Nasdaq 100 futures gained 0.2%. On Monday, the Dow and S&P 500 logged in their best day since early April, with the Dow rallying more than 900 points. The S&P 500 gained 3.2%. —Imbert Subscribe to CNBC PRO for exclusive insights and analysis, and live business day programming from around the world.