Let's begin by stating the obvious: If you've got $10 million burning a hole in your pocket, and your biggest problem right now is figuring out where to invest it -- well, as problems go, that's a nice one to have.

It also has a number of very simple, prudent solutions.

Firstly, you could hand your money over to a financial advisor to invest for you. True, there's always the chance you'll end up with a Bernie Madoff kind of character. And even if you manage to avoid that landmine, many of these folks will take 1% to 3% of your assets -- annually -- as their fee to manage your money.

Still, as my Foolish colleague Selena Maranjian has pointed out, you might get lucky: "A good [financial advisor] can serve you quite well," and help you make back that fee many times over in outperformance of the S&P 500.

Little girl in glasses fanning a wad of $100 bills

Image source: Getty Images.

K-I-S-S: Keep It Simple, Stockholder

Prefer to manage your money yourself? Me too. The good news is that there are plenty of easy ways to invest $10 million on your own.

The simplest way to invest would be to take your $10 million and invest it in a low-cost S&P 500 index fund such as the Vanguard S&P 500 Index (NASDAQMUTFUND:VFINX). (This also works for folks with less than $10 million to invest, by the way).

Sure, with a big investment like $10 million, one of your biggest concerns is ensuring you don't lose it all in a single bad investment. But that's just a great reason to go the index fund route.

The S&P 500 includes investments in 500 separate companies (that's about 6% of all stocks traded globally), so an investment in this one index gives you instant diversification. It also gives you a very good chance of making a profit.

Historically, the S&P 500 has returned about 10% to investors over long periods of time. So long as you don't need your $10 million immediately (maybe hold back just a million or two "for emergencies"), this should be a safe bet.  

What's more, if you assume that even professional money managers are a subset of the class of investors where half underperform the S&P and half outperform it (because that's how averages work), you have a 50-50 chance of doing as well in an S&P 500 index fund as you would handing your money over to a professional to make individual stock purchases. And by skipping the middleman, you'll save yourself $100,000 to $300,000 in fees annually.

First save, then earn

If you're willing to take on just a bit more risk, with the potential to not just save on fees, but profit from dividends as well, you might want to invest in a low-cost dividend-focused mutual fund -- or even in a select handful of dividend-paying stocks.

Depending on how you crunch the numbers, experts estimate that dividends paid by stocks and reinvested have been the source of anywhere from 30% to 90% of the stock market's gains over time. Thus, by weighting your investments toward dividend-paying stocks, it stands to reason you should be able to outperform the S&P 500 by capturing the greatest source of its gains, at minimal additional risk.  

Vanguard's low-cost High Dividend Yield Index ETF (NYSEMKT:VYM), for example, yields 3.2% right now -- about 72% better than the S&P 500's 1.9% yield. Even subtracting the fund's expense ratio (a mere 0.8%), that means you can get a 68% better dividend yield from investing in this dividend index fund, than if you bought the more plain vanilla Vanguard S&P 500 Index.

Though not quite as diverse as the S&P 500, the High Dividend Yield index also spreads your investment out among 400 different companies, which is more diversification than you would get from picking a handful of individual stocks, be they dividend payers or not.  

(Slightly) riskier investing

Of course, it's highly unlikely that each and every one of those 400 stocks will outperform the market. If you're comfortable taking on just a bit more risk, trying to pick and choose the very best investments offers a chance to juice your returns even further.

After all, while a 3.2% dividend is certainly better than 1.9%, you can find individual high quality dividend paying stocks that pay you even more.

In fact, there are more than ­200 stocks of mid-cap size or larger ($2 billion in market cap and up) that both (a) pay dividend yields twice the S&P 500 average and (b) sell for P/E ratios below the market average. That combination of high yield and low price should be hard to beat -- and a recipe for stock market outperformance. (A good free stock screener like this one from finviz.com can help you find them).  

Granted, you won't want to drop your entire $10 million into the first stock you see. No matter how attractive it might look, diversity is key to ensuring that one bad accident doesn't cost you a substantial portion of your nest egg.

In fact, academic studies suggest that you may need as many as 50 stocks to roughly duplicate the diversity inherent in owning just one broad-based index fund such as the Vanguard S&P 500 Index or High Dividend Yield Index -- so be prepared to cut your $10 million into piles of $200,000 or less.

You'll spend a bit more time on research and entering buy orders, true. But with the added assurance that your portfolio is properly diversified, you'll sleep more soundly because of it.