Dividends are the multifunction pocket tool of the investor. You can use them in various ways, including as a source of income to pay your bills, or you can supercharge your investment returns by reinvesting them into more shares. It takes patience and commitment to a dividend strategy to get the ball rolling, but once it gets going, the results can be impressive.

Telecommunications giant AT&T (T 0.58%) is a great dividend stock. You'll have difficulty finding yields higher than the 7% it offers at today's share price. And it's not hard to generate $1,000 a year in passive income by investing in it.

As an initial investment

There are multiple paths to get to that destination. Of course, a more significant initial investment would be the most straightforward. AT&T's current dividend pays quarterly. Hold the stock through four quarters, and you'll get a total of $1.11 per share. To generate $1,000 in dividends in year one, you'd need to buy 901 shares.

2022 has been brutal for the market broadly, and AT&T has been battered. It's down more than 25% over the past 12 months to less than $16 per share, its lowest price in more than a decade. But if generating passive income is your primary focus, that could be a good thing. A lower share price creates a higher dividend yield because companies declare the payout amount, and the share price at which you bought your shares determines how much yield you get. Of course, a company must be financially stable enough to pay its dividend consistently. Sometimes yields increase because the market doesn't believe a payout is safe, and bids down the shares accordingly.

At today's prices, an investment of $14,416 will get you enough shares to generate $1,000 in passive income each year. You could also see payout increases from AT&T, though investors should know that the telecom giant isn't going to wow anyone with its bottom-line growth. Analysts are looking for earnings-per-share growth averaging 3% annually over the next three to five years. You're probably buying the stock for the dividend. Understandably, $14,416 isn't pocket change, especially if you're buying AT&T as part of a diversified portfolio (which is the right way to invest). But for retirees or older investors, that amount could be doable.

The snowball method

Don't assume that dividend investing can't work for you if you're starting with smaller amounts of money, especially if you're young. That just means you have more time for your investments to experience compound growth and do more of the work for you. You can still get to that $1,000 a year in passive income if you don't have $14,000 laying around to put to work this way. It will just take some time and commitment.

For example, let's say you buy 100 shares of AT&T. That would cost you just under $1,600 at today's share price. Assume that AT&T's profits and dividend payouts grow by an average of 3% annually. If you reinvest all your dividends, but don't otherwise add another penny to your initial investment, your position will grow into one that produces $1,000 in annual dividends after 16 years. Too long? Invest another $600 each year ($50 a month) in AT&T, and you'll cut that time down to nine years.

Different factors impact this. A higher stock price will lower the dividend yield, which means you get less income for your investment. Still, the point is that a small investment can produce significant results if you give it enough time.

Why you can trust that fabulous dividend yield

Making AT&T or any dividend stock part of a diversified long-term portfolio means you want to be able to count on its payouts. That's why it's wise to check a company's dividend payout ratio to see how well it can afford those distributions. In AT&T's case, dividends consume 67% of the company's profits. Management cut the dividend earlier this year as it frees up money and absorbs the Time Warner spinoff.

T Payout Ratio Chart

T Payout Ratio data by YCharts

The payout ratio is now more manageable, and AT&T's core business has been doing well in recent quarters, so investors should feel confident that the dividend checks will keep coming. But if at some point that ratio starts to go beyond 80%, investors will have cause for concern. Regardless, though, buying and holding an array of strong dividend payers like AT&T can eventually turn your portfolio into a passive income juggernaut.