When investors think of growth stocks, they often think of companies that tout a medical cure or knock-out technology.

They could profit handsomely by suspending their disbelief and thinking of giant, Paris insurance company AXA SA. Already the world’s third largest pure insurer with a market value of more than $78 billion, the company has set sales and profit goals that would be ambitious for a company one-tenth its size. Chief executive Henri de Castries has made a strong case for how he can achieve them.

Mr. de Castries last year told investors that by 2012 the company would double 2004 revenue and triple that year’s “underlying” earnings — net income excluding items such as net capital gains and changes to the value of intangibles such as goodwill. Those are big numbers considering the company reported more than 104 billion euros ($139 billion) in total revenue last year, according to US generally accepted accounting principles, or GAAP. That was tops among insurers and among the highest pocketed by public companies world-wide, according to researcher Capital IQ.

To reach those goals, the company has said it has to raise its earnings at about a 15 percent annual clip. In an 87-page presentation in October, Mr. de Castries, 52 years old, listed the specific growth and market returns that would be needed, without factoring in acquisitions.

The company’s assumptions — such as three percent to five percent annual growth in property-casualty revenue and five percent to ten percent annual growth in sales of life insurance and savings products — aren’t heroic. AXA declined to comment beyond its public disclosures.

“I’d say they probably will achieve their 2012 targets,” says Nicholas Byrne, an insurance-stock analyst with J.P. Morgan Securities Ltd. in London, which has done investment-banking work for AXA in the past year. Mr. Byrne, who doesn’t own AXA shares, rates the stock a “buy” and the company is his firm’s top pick for this year in insurance.

AXA, No. 3 among pure insurers by market valuation, after American International Group Inc. of the US and Germany’s Allianz SE, primarily sells investments and insurance products designed to increase assets and secure retirement income. It has an enviable position in ageing US and European markets.

Its shares, which have had a bumpy ride this year, closed down 1.9 percent at 28.07 euros ($37.43) on Friday on the Paris bourse, up about three percent this year — trading at a discount to many of its peers despite its growth focus. They fetch about 11 times European analysts’ projected earnings for 2007.

“The stock is showing the strongest earnings momentum in the sector, but isn’t getting an average multiple,” which is more than 12 times projected earnings, says Joseph Rohm, a financial-services-stock analyst with T. Rowe Price Group Inc. in London, which owns AXA in client portfolios.

Its American depositary shares closed on Friday on the New York Stock Exchange at $37.70, up some 19 percent year-to-date, helped by a falling dollar, which makes the company’s shares worth more in dollar terms.

Some might wonder why the stock doesn’t have a higher valuation. Perhaps investors are unwilling to bank on goals that are so far off. If Mr. de Castries, who took AXA’s reins in 2000, hits those targets, shareholders may be looking at ample gains in coming years.

A few rough calculations show that, if the company reaches its 2012 goal, it would report net income of about 5.20 euros a share under US GAAP. If investors keep paying the current multiple of those earnings, the stock would be valued at about 52 euros, 85 percent higher than Friday’s price.

That works out to a nearly 11 percent compound annual return, without dividends. If the company hits the earnings target and investors pay a 12.5 price-to-earnings multiple — in line with its peers today — that compound annual return rises to about 15 percent, again not including the dividend.

The stock’s dividend yield — its dividends per share in the past 12 months divided by the share price — is more than three percent. That is above the average 1.8 percent for stocks in the Standard & Poor’s 500-stock index.

AXA is on track to deliver. Underlying earnings rose 24 percent last year from 2004 and were up an additional 19 percent in the first half of 2006 compared with the same period a year earlier.

There are risks. AXA earns the majority of its revenue from pension, life-insurance and asset-management units.

The profitability of life and asset-protection products often depends on whether the life-span, medical-cost and financial-markets estimates that go into the prices AXA charges are correct.

Tiny errors spread across more than 50 million customers in dozens of countries — let alone terrorist attacks or a string of natural catastrophes — can hack away profit in a hurry.

The company’s results and share price also could be dinged by events outside its control. Investment returns are a significant driver of the company’s profit so rocky financial markets — as well as any big terrorist attack or a string of natural catastrophes — could toss a wrench into Mr. de Castries’ plan.

Some who are cautious on the stock say the goals aren’t necessarily a stretch.

“Many things could make it difficult to reach these goals, but they’ve made pretty normal assumptions in their plan,” says Dafina Dunmore, an analyst with Morningstar Inc. in Chicago. Her firm doesn’t do investment-banking work. While she believes the stock is about 15 percent above its fair value today, she acknowledges there is room for “tremendous upside” if the company reaches its goals.