Commonwealth Bank of Australia (ASX: CBA) shares have long been popular among passive income investors.

That's thanks to the S&P/ASX 200 Index (ASX: XJO) bank stock's lengthy track record of making two fully franked dividend payments a year. Even in the pandemic-addled, stock-market-crashing year of 2020.

CBA also has bragging rights for delivering consecutively higher interim and final dividends every year since 2020.

As an example, CBA increased its FY 2024 final dividend to $2.50 a share, up from the $2.25 a share final dividend the bank paid out in FY 2023. And CBA's interim FY 2025 dividend of $2.25 was up from $2.15 the prior year.

All told then, the past 12 months of passive income payments come to $4.75 a share. That's up 4.4% from the two dividends CBA paid in the prior 12 months.

Which brings us back to our headline question.

Should I buy CBA shares for the passive income?

As you're likely aware, you can calculate the trailing dividend yield of any stock by dividing the total amount of dividends a company has paid over the past 12 months by its current share price.

On Friday, CBA closed the day trading for $172.87 a share.

That sees CBA trading on a fully franked 2.8% trailing dividend yield, which is significantly below the level of passive income CommBank has paid out over recent years.

Why?

The answer is simply that the CBA share price has gained far more rapidly than the matching increases in the bank's dividend payments.

As mentioned up top, CommBank's past year dividend payouts were up 4.4% from the prior year. However, the CBA share price has surged 30% over the same time.

And, despite slipping from the 25 June all-time closing high of $191.40, CBA shares are up more than 137% in five years.

While that's great news for longer-term shareholders, the bank's passive income growth just hasn't kept pace.

Let's turn back the clock

To illustrate what I mean, let's go back two years to 7 July 2023, when CBA shares closed the day trading for $98.73.

Over the previous 12 months, CommBank had paid out two fully franked dividends, totalling $4.20 a share. A period that saw CBA shares gain a much more modest 6.4%.

As for the passive income on offer, in July 2023, CBA stock was trading on a fully franked trailing dividend yield of 4.3%, significantly above the current 2.8%.

So, if you're buying CommBank stock at current levels, I think you should be doing so more with further share price gains in mind rather than the diminishing dividend yield.

How about the passive income from the other ASX 200 bank stocks?

In part because their share prices have gained far less than CBA's over the past year, the other big four ASX 200 bank stocks offer investors a better passive income return today (based on their trailing yields).

Going by Friday's closing prices:

• ANZ Group Holdings Ltd (ASX: ANZ) shares trade on a 5.5% dividend yield, franked at 70%
• National Australia Bank Ltd (ASX: NAB) shares trade on a fully franked 4.5% dividend yield
• Westpac Banking Corp (ASX: WBC) shares trade on a fully franked dividend yield of 4.6%