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The S&P 500 has recovered almost 33% from its late March low,but it may have gone too far too fast.

"Reopenings don't mean recovery,"says Jeff Kleintop, chief global investment strategist at theSchwab Center for Investment Research. It's also not clear that thecountries with the biggest lockdowns will have the strongestreopenings, according to Kleintop.

He notes that defensive stockshave been leading the rebound in global stock markets, which isvery unusual, suggesting that either global stock markets haven'thit their lows yet or the economic recoveries will belong and slow.

How stock markets behave willdepend heavily on how strong economies rebound from lockdowns and whether new cases of COVID-19, which are likely to followreopenings, will remain contained.

To understand those developments,Kleintop is following alternative data such as data on rush-hourcommuting, air pollution, retail foot traffic and electricity usagein addition to the usual economic indicators.

His colleague, Liz Ann Sonders,chief investment strategist at the Schwab Center for FinancialResearch, says the U.S. stock market is not paying enough attentionto "second-order economic effects" of the COVID-19pandemic. 

U.S. stocks, for example, roseslightly the day the government reported that 20.5 million peoplehad lost their jobs in April because it also reported that 80% ofthose job losses were temporary. But with a growing number ofcorporate defaults and bankruptcies at small and large companies,those temporary losses may become permanent, says Sonders, addingthat such factors "need to be included in marketexpectations."

The University of Chicago'sBecker Friedman Institute predicts that 42% of U.S. jobs lostthrough April 25 — or 11.6 million — due to the coronavirus willbecome permanent.

Sonders also notes that recentstock buying sprees are being propelled not only by quantitativeand algorithmic buyers but also by small traders and individualinvestors, including sports bettors who are now turning to makingstock bets, which adds another risk factor for the U.S. stockmarket.

Also underpinning the stockmarket recovery from the March lows — and the corporate bond market— is the Federal Reserve. It has been supporting the U.S. economywith near zero interest rates, asset purchases and multipleliquidity programs to backstop various bond market segments as wellas Treasury programs to aid businesses, such as the PayrollProtection Program and Main Street Lending program.

Most of those programs haven'teven started yet, and those that have begun have used just afraction of the money available, leaving the Fed with "lots offirepower" to do more, says Kathy Jones, chief fixed incomestrategist at the Schwab Center. "The Fed has even had success justby signaling what it can do," says Jones.

Among bond markets, the municipalmarket especially faces challenges, says Jones. "State revenues aredeclining while costs are going up, much of it for health care" dueto the COVID-19 pandemic and the economic shutdowns thatfollowed.

The Fed has announced a municipalliquidity fund, which it recently extended to three years, toprovide muni issuers bridge loans they desperately need, butultimately more federal support will be needed, says Jones. "Statescan't file for bankruptcy."

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