There is a fairly simple formula for banking profitability. Return-on-assets, ROA, given by

ROA = (1-T) * [ (1 - ER + NII) * NIM - L]

where

T = Tax rate

revenue = Net Interest Income + Non-interest income

ER = efficiency ratio = Non-interest expense divided by revenue

NII = Non-interest income divided by revenue

NIM = Net interest margin = net interest income divided by total earning assets

L = provisions for loan losses divided by total earning assets

So the general strategy for banking is always to try to increase non-interest income if it doesn't increase non-interest expense too much. You want to keep your non-interest expenses as low as possible. Then you want to have a high NIM which usually means low deposit costs. Then you want good underwriting i.e. low loan losses. You usually can't do much about taxes. That is really about all there is to banking. The devil of course is in the details of how you actually do this better than your competitors.

Return-on-equity, ROE, is just

ROE = ROA*(assets/equity)

where (assets/equity) is the leverage factor. This leverage factor is limited by banking regulators. It must be less than 25 and is usually required to be less than 20 to be consider "Well capitalized". Most banks try to get it around 12-15. If a bank is profitable, i.e. ROA > 0, then they usually benefit from more leverage. However this can be more dangerous because if things turn bad, the bank may become unprofitable and this leverage will magnify losses to equity. If the bank pays no dividend and retains earnings it can grow at a growth rate equal to ROE, if all of these ratios above stay fixed. It is pays out a fraction PR of earnings as a dividend, then it can grow at (1-PR)*ROE while continuing to pay the dividend at the same payout ratio.

It is also possible to leverage up the shareholders equity by raising other forms of equity such as preferred shares. This requires a fixed dividend payout to preferred shareholders which comes after tax. This involved another leverage factor

EQ/CEQ = (equity/common-equity).

The preferred equity, PE, is the difference between equity and common-equity, PE =EQ-CEQ. If the bank goes bust, the preferred share holders are paid back this preferred equity before common shareholders get anything.

The return-on-common-equity ROCE is given by

ROCE = ROE * (EQ/CEQ) - BC (EQ/CEQ-1)

where BC = the borrowing cost or the dividend yield of the preferred shares.

If you set this to zero you you can solve for the condition where the bank's total equity stays the same size (before payment of any common dividend). That is

ROE = BC (PE/EQ). For example of PE/EQ is 1/2, then the bank's common equity and total equity stay the same when the ROE is half the borrowing cost. When this is true the bank stays the same every year. If you want it to stay the same size after paying a dividend, you replace ROE with (1-PR)*ROE.

When can the common shareholders grow their common-equity faster by issuing preferred shared? Just set ROCE equal to ROE and solve and you find this is equal when ROE =BC. So your ROE had better be higher than your borrowing cost or issuing preferred shares is not worthwhile.

If you want to know the growth rate of common equity, this is equal to ROCE. If there is a common dividend, just replace the ROE with (1-PR)*ROE. That tells you the growth rate of retained common equity. If you keep issuing preferred shares to keep the ratio EQ/CEQ the same, and the other ratios remain the same, you can grow the whole bank at this faster rate. Like the usual leverage ratio, banking regulators put limits on this leverage factor. Usually they want at least half the equity to be common equity.

Ok, lets do an example. Lets assume:

NIM = 3%

ER= 50%

NII=10%

T=40%

L=0.5%

This results in ROA = (1-T) * [ (1 - ER + NII) * NIM - L] = 0.78%. Now lets assume assets/equity =15. This results in ROE = 11.7%. Now lets suppose the bank pays a dividend at a payout ratio of PR=25% of earnings. Then it can grow at G=(1-0.25)*11.7 = 8.78%. Mid to high single digit growth rates are fairly typical for banks. Now, what about issuing preferred shares? Lets suppose it can sell preferred shares at a yield of 6%. Since ROE > 6%, this sounds promising. Lets say that it raises total equity to twice common equity EQ/CEQ=2

ROCE = ROE * (EQ/CEQ) - BC (EQ/CEQ-1) = 17%

The growth rate after paying preferred and common dividends will be

G = (1-PR)*ROE * (EQ/CEQ) - BC (EQ/CEQ-1) = 11.6%. So as long as they continue to issue preferred shares to keep EQ/CEQ=2, and everything else stays the same, then they can grow at this faster rate. In reality, of course, nothing stays the same but that is a different matter.

Now what if the economy goes into recession and the loss ratio, L, goes to 2%? Now ROA=-0.12%. It is now slightly unprofitable. ROE=-1.8%. Equity of the bank has contracted by -1.8% before payment of preferred and common dividends. ROCE=-9.6%. The leverage due to the preferred shares has magnified this loss. Common equity has dropped by -9.6%. That is a pretty big hit for just 2% loan losses. The bank will likely cancel the common dividend payment and if this goes on for another couple of years, it will have to raise common capital and dilute the interest of current shareholders in order to keep the leverage ratio below the regulatory limit.