Three weeks into the new year, the Down Jones Industrial Average is already down 8%. That probably sounds like bad news to you. But here's an aspect of falling stock prices you may not have considered:

A stock's dividend yield equals the dollar value of the dividend it pays, divided into the dollar value of the stock that pays it. When the numerator on that equation (i.e., the stock price) gets smaller, the number on the other side of the equation gets bigger -- and the dividend yield increases.

Not to put too fine a point on it, but the more the stock market falls, the bigger the dividend yields get. And here are three of the dividends we like best.

Dan Caplinger: One dividend stock to keep an eye on in 2016 is Chevron (NYSE:CVX), and the news won't necessarily all be good. The energy industry continues to stagger as the New Year begins, and crude oil prices have fallen to their lowest levels in years while creating fears that further declines could follow. Although Chevron and its Big Oil peers are better insulated from price drops than some smaller independent oil and gas producers, even the largest players in the industry are feeling the pinch.

Chevron made the unusual move of not increasing its dividend during 2015, something that has been exceeding rare during the Dividend Aristocrat's 28-year history of consecutive annual dividend boosts. So far, the move doesn't jeopardize Chevron's Dividend Aristocrat status because its previous-year increase came in the middle of the year, causing its annual total payout to be higher. Yet if oil prices remain under pressure, it could be tough for Chevron to justify a boost from its current 5% yield. As Chevron's payout ratio rises due to falling earnings from lower oil and natural gas prices, the threat to the oil giant's dividend will only worsen, and investors need to hope that energy prices will rebound before that threat spurs action.

Sean Williams: We witnessed some ugliness in the coal industry in 2015, but I've got my eye on master limited partnership Alliance Resource Partners (NASDAQ:ARLP) in 2016.

Let's get the basics out of the way: Alliance Resource Partners struggled mightily last year. Shares of the company tumbled more than 65% as it modestly cut coal production amid a supply glut, forecast for weak demand, and ongoing weakness in coal prices. The company also broke a streak of raising its dividend for 29 consecutive quarters by choosing to keep its dividend flat from the prior quarter in October 2015. Its current yield of 20% is bound to make any dividend investor salivate, but chances are that a cut in some form is coming to help it preserve capital.

Now for the good news: Alliance Resource Partners has been such a coal standout throughout the years because it forges deals for its production years in advance. Doing so ensures that very little of its production is exposed to wholesale coal price volatility. Additionally, Alliance Resource has lower production costs than many of its peers, and it also has a considerably healthier balance sheet. Whereas many of its peers are billions of dollars underwater in debt, Alliance Resource has just $933 million in net debt, and cleared $322 million in levered free cash flow over the trailing-12-month period.

We also shouldn't cast aside the fact that coal is still the most important energy source in the United States. The Energy Information Administration's data shows that coal was still responsible for 39% of all electricity generation in 2014. It may not be as clean a fuel as natural gas, but it's not going to disappear overnight. Coal also has inroads in China, India, and other emerging markets, so there's exportation opportunities to be had, too.

Although Alliance Resource could be facing a dividend cut, I still envision its dividend yield surpassing 10% for years to come, making it an intriguing dividend (and rebound) candidate for 2016 and beyond.

Rich Smith: One of my personal favorite stocks for dividend investors today is Emerson Electric (NYSE:EMR). This widely diversified industrial manufacturer makes everything from valves, actuators, and regulators to entire alternators, motors, and even drives -- plus the software to run them all. In an economy that's rapidly going all-electric (even in our cars), this is a business that's very likely to thrive. And yet, with a stock down nearly 30% in the past 52 weeks, Emerson Electric's share price doesn't really reflect those prospects.

For anyone who loves dividends, that's a good thing.

Thanks to its falling stock price, Emerson Electric's annual dividend payment of $1.90 now yields a whopping 4.4% -- or about 13 times what your bank is probably paying you in interest today. That's a great bargain for dividend shoppers, and the stock itself doesn't look too shabby, either.

Emerson shares today cost a mere 10.8 times earnings. Add up the stock's 4.4% dividend yield and its projected 5.6% long-term earnings growth rate, and that means you're getting a total return yield of 11% on this stock, and a total return ratio of less than 1.0.

My one reservation about the stock, and the reason I haven't bought it myself just yet? Emerson generated only $1.8 billion in free cash flow over the past year, which falls pretty far short of the $2.7 billion in GAAP profits the company has reported for the trailing 12 months. If and when free cash flow recovers though, I'll be putting this stock on my shopping list.

Caveat investor
And there you have it, folks. However, before pressing any "buy" buttons, click on over to our Motley Fool discussion boards, and see what other investors think of these stocks. Two heads are better than one, and the tens of thousands of Fool readers populating our discussion boards are even better than that.

Dan Caplinger has no position in any stocks mentioned. Rich Smith has no position in any stocks mentioned. Sean Williams has no position in any stocks mentioned. The Motley Fool recommends Alliance Resource Partners, Chevron, and Emerson Electric. 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.