CHAPEL HILL, N.C. (MarketWatch) — How many of you are willing to bet that Bank of America stock will outperform Apple’s in coming months?

Hardly any of you, I am sure.

After all, Apple AAPL, +1.50% is on an unbelievable tear — close on the heels of Exxon Mobil XOM, +0.14% to become the most valuable company in the world. Bank of America BAC, +1.34% , in contrast, is off nearly 50% from its 52-week high, which was set in early January — including a stunning 17% drop on Monday of this week.

What's the point of market caps?

But your aversion to betting on Bank of America over Apple just goes to show how hard it is to be a genuine contrarian.

One who nevertheless has those contrarian instincts is Rob Arnott, founder of Research Affiliates, an investment advisory firm that champions fundamental indexes — benchmarks in which a stock’s weight is not a function of its market cap but, instead, any of a number of fundamental variables like sales, earnings, book value, and so forth.

In an interview Tuesday afternoon, soon after Apple briefly surpassed Exxon Mobil to reach the peak of the market-cap rankings, Arnott told me that he believes Bank of America will outperform Apple in coming months. He said that the virtues of an alternate weighting scheme besides market cap are particularly evident in the case of Apple. “The company is currently being priced as the biggest future source of profits on the planet. That seems a tad aggressive.”

Where would Apple rank according to fundamental criteria rather than market cap? One answer is provided by the FTSE RAFI All-Caps US 1000 index. In that index, as of Aug. 8, according to Arnott, Apple was the 21st largest company. In the FTSE RAFI All World Index, Apple is the 48th largest company.

The difference between a first-place and a 21st or 48th place ranking is one measure of Apple’s aggressive pricing.

In contrast, Bank of America is an example of a company so feared and loathed that its market-cap rank has sunk markedly below its fundamental rank. As a result, Arnott said, even if Apple deserves to have a larger market cap than Bank of America, it still would underperform if its price were to decline far enough to bring its market-cap rank down to match its fundamental rank — and Bank of America’s price were to rise enough to bring its market cap-rank up.

Is there historical support for Arnott’s bet? Yes.

Consider a recent Hulbert Financial Digest study that started with a list from Standard & Poor’s showing, for each of the last 30 years, the stock that on Jan. 1 had the largest market cap of any in the S&P 500 index SPX, +0.82% . The study then calculated, for each of these stocks, its dividend-adjusted return over the subsequent 12 months, and compared that to the return of the overall stock market, as judged by the Wilshire 5000 Total Return index (W5000)

These market cap leaders lagged the overall market by an average of 4.9 percentage points per year.

Upon slicing and dicing the data in a different way, Arnott’s own research reached a similar conclusion. He focused on market sectors between 1952 and 2009, comparing the largest cap company in each sector to the performance of that entire category. On average over the 12 months after a company rose to become the “top dog,” it lagged its sector by 3.5%.

Note carefully that these statistics are based on long-term historical averages. Even if the future is like the past, therefore, it’s still possible that Apple will be the better bet over the coming months than Bank of America.

Nevertheless, those statistics underline just how much the odds are against it doing so.