Pandora (P) priced its shares at $16 in June 2010. On its first day of trading, the stock shot up to $26 before plunging below its offering price the next day. After dipping as low as $8 a share last year, Pandora has rallied to $11.73 at yesterday’s close. That’s still 27% below its IPO price and 55% down from its record high.

Groupon’s (GRPN) stock has followed a similar if more extreme trajectory. Offered at $20 a share, the stock surged to $31.14 in a matter of hours, then began a slow painful slide. The day Mason left Groupon, the company’s stock traded as low as $4.24 a share, a 79% discount from its offering price and 86% off its fleeting, first-day high.

Looking at these two stocks, the first apparent lesson is one well-known to experienced investors: Beware the first-day pop. Once upon a time, buying a stock on the first day of trading could possibly let you in on a rally that could last months, if not weeks. But recently bearish investors have been impatient to cash in on quick gains or short stocks they believe have a brief shelf-life.

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But with more hindsight, something else is becoming clear: It seems to be getting harder for many companies to make money from the web. The business models that have resonated with consumers on a large scale simply aren’t translating into businesses that can generate steady profit growth.

In the first wave of Internet startups, companies like Amazon (AMZN) and Yahoo (YHOO) posted early losses for years before moving into steady profitability. Newer companies like Google (GOOG) and Facebook (FB) have emerged capable of delivering profit margins of 20% or 30%.

But these are the exceptions. Many recent web IPOs have struggled with fleeting profits or regular quarterly losses. Compare the profit margins of Pandora and Groupon with that of Google in the chart below. Granted, Google is an older, more mature company, but when it went public it had net profit margin of 13% and a pretax margin of 20%.

Both Pandora and Groupon grew rapidly and gained early market share by offering discounted or ad-supported services that appealed to cost-conscious consumers in the recent recession. Both became emblematic of a new generation of startups that could entice skittish investors into new tech IPOs. Both IPOs showed that, in fact, investors still aren’t hungry for companies better at growing revenue than growing profit.

Groupon’s profits have been pressured by, first, high marketing costs made to fuel rapid expansion and, more recently, an aggressive move into lower-margin direct sales. Pandora’s profits are hurt by the onerous fees demanded by music labels for licensing their artists under contract.

The earnings report Pandora made alongside news of the management change underscores the trend. Pandora is still growing at rates most companies would envy. Total listener hours rose 53% to top 4 billion hours last quarter. Revenue grew 54% to $125 million from the year-ago quarter. Yet the company still reported a net loss of 4 cents a share.

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Pandora’s earnings and future guidance were in line with analyst expectations, although the stock rose 21% in afterhours trading Thursday to $14.16. That may reflect optimism that a new CEO could help Pandora improve its performance. Groupon’s stock saw a similar pop after Mason’s departure hit the wires.

But it’s not clear how a new CEO is going to be a sure-fire solution for either company. Groupon’s board indicated that a seasoned executive could be needed to steer the company forward. But Kennedy is just such a seasoned hand, with several years at e-Loan and Saturn Corp. before joining Pandora in 2004. If Pandora’s board feels a younger face is necessary, perhaps they can arrange a CEO swap with Groupon.

The real reason weighing down the stocks of Groupon, Pandora and many other recent IPOs is a truth investors may not want to hear. The business models themselves are troubled. Using the web to build or market a business can cut costs, but it’s also trained consumers to expect — and even shop around for — prices to be low. Or free if ad-supported. Or, ideally, free without any ads.

And so, under pressure from investors, boards and companies are blaming CEOs. When in fact it’s the business models that are problematic. The web just doesn’t generate profits the way it used to.