A big dividend yield is one of the most alluring statistics in all of investing. It feels like free money. In fact, you often hear the related phrase, "this company is paying you to own its stock."

And when the market is also underestimating the value of a dividend stock, you have the opportunity to get paid while you watch the stock price appreciate.

That is absolute nirvana for a contrarian dividend investor -- getting large dividends and large capital gains. Today, there are four stock plays that could provide just such an opportunity.  

Contrarian dividend play No. 1: dogs of the S&P 500
My colleague Dan Dzombak made a list of the 10 S&P 500 stocks with the largest dividend yields -- ranging from 5.6% to 8.4%. Five telecom stocks dominated that list, with Frontier Communications (NYSE: FTR) sporting the largest yield.

The S&P 500 is made up of the largest, most venerable companies in the U.S. But what you have to watch out for is the dividend payers on the way down -- companies on this list that have run out of growth options and are just paying out their cash flow. For example, I worry that this may be the case for mail processor Pitney Bowes (NYSE: PBI). Hence, despite its 6.7% dividend yield, I'd require a very low normalized earnings multiple to buy into that stock.

Conclusion: This list is definitely worth investigating for beaten-down prospects.

Contrarian dividend play No. 2: mortgage REITs
I've warned in the past about the dangers of taking a huge dividend yield at face value, and no stocks have bigger face values than mortgage REITs. Annaly Capital (NYSE: NLY), American Capital Agency (Nasdaq: AGNC), and Hatteras Financial (NYSE: HTS) all have dividend yields between 15% and 20%. Yowza!

They have similar business models -- borrowing short-term at cheap interest rates and buying longer-term, higher-interest rate mortgage-backed securities that are guaranteed by government agencies or government-sponsored entities like Fannie Mae.

How is this a contrarian play? Because investors haven't bid up share prices to lower these dividend yields. That's because they know that this party ends at some point, when interest spreads will contract and these companies won't be able to support such high dividends.

But Ben Bernanke's announcement yesterday that the Fed will again start buying government bonds -- $600 billion worth -- shows that this interest rate party may last quite awhile. Even after the party's over, it's possible these companies will stick around with lower but still attractive dividend yields.   

Conclusion: This is something to consider for advanced investors who want to make a bet on the government's will to keep interest rates low. But read this warning before proceeding.

Contrarian dividend play No. 3: bank stocks
I've said it before and I'll say it again: The financial industry is a ripe area for contrarians these days. I love areas of the market that investors don't even consider considering. So many people got burned by the dividend cuts of the thought-to-be-safe banks that a large percentage of former bank investors are once bitten, twice shy.

Small banks provide great opportunities. But today, I want to highlight the opportunity in two big banks that have cut their dividends to basically nil. Because of the uproar over the foreclosure mess, Bank of America and JPMorgan Chase are trading at price-to-tangible book values of 0.9 and 1.3. That's impressive considering anything under 1.5 catches my eye.

Besides more clarity around foreclosures, dividend increases could be a share price catalyst for these two (and all the big banks).

Conclusion: If you believe the foreclosure headlines are overblown (I do) and if you're willing to roll the dice on banks with virtually no balance sheet visibility and virtually no current dividends, these two banks are quite cheap.

Contrarian dividend play No. 4: defense stocks
When the big kahuna of clients -- Secretary of Defense Robert Gates -- announces he wants to cut $100 billion over the next five years from a $700 billion annual U.S. defense budget, you can bet defense stocks of all stripes react negatively.

But here's the thing. Even though stock prices across the industry are depressed, individual players will have different outcomes. Sales at the big defense contractors are chunky. A big win at the expense of your competitors can negate a lot of the ill effects of newfound government thrift.

Picking out the winners is tricky, but the prices we're seeing make it worth the effort. Lockheed Martin (NYSE: LMT), General Dynamics (NYSE: GD), and Raytheon all have forward earnings multiples below 11 ; Lockheed also has a trailing multiple below 10 and pays the highest dividend yield of the group -- 4.2%.

Conclusion: It would be great if defense companies were rendered obsolete, but strife isn't going anywhere. I'll definitely be looking further into these three to determine if any will be less affected than the group.

The next step
These four dividend plays -- the dogs of the S&P 500, mortgage REITs, banks, and defense stocks -- are being given short shrift by the market. It's left to us contrarians to profit from their neglect.

If you'd like to read about some more dividend opportunities, we've put together a five-page free report: 13 High-Yielding Stocks to Buy Today. Just click here to take a look.

Anand Chokkavelu doesn't own shares of any company mentioned. General Dynamics is a Motley Fool Inside Value selection. The Fool owns shares of Annaly Capital Management, Bank of America, and JPMorgan. Try any of our Foolish newsletter services free for 30 days. We Fools may not all hold the same opinions, but we all believe that considering a diverse range of insights makes us better investors. The Motley Fool has a disclosure policy.