Given the return on assets, the lower the bank capital, the higher the return for the owners of the bank.

How the amount of bank capital affects returns to equity holders?

Because owners of a bank must know whether their bank is being managed well, they need good measures of bank profitability. A basic measure of bank profitability is the return on assets (ROA), the net profit after taxes per dollar of assets:

ROA = (net profit after taxes) / (assets)

The return on assets provides information on how efficiently a bank is being run, because it indicates how much profits are generated on average by each dollar of assets.

However, what the bank’s owners (equity holders) care about most is how much the bank is earning on their equity investment. This information is provided by the other basic measure of bank profitability, the return on equity (ROE), the net profit after taxes per dollar of equity (bank) capital:

ROE = (net profit after taxes) / (equity capital)

There is a direct relationship between the return on assets (which measures how efficiently the bank is run) and the return on equity (which measures how well the owners are doing on their investment). This relationship is determined by the so-called equity multiplier (EM), which is the amount of assets per dollar of equity capital:

EM = (assets) / (equity capital)

To see this, we note that:

(net profit after taxes) / (equity capital) = ((net profit after taxes) / (assets)) x ((assets) / (equity capital))

which, using our definitions, yields:

ROE = ROA x EM

The formula in above equation tells us what happens to the return on equity when a bank holds a smaller amount of capital (equity) for a given amount of assets. As we have seen, the High Capital Bank initially has \$100 million of assets and \$10 million of equity, which gives it an equity multiplier of 10 (= \$100 million/\$10 million). The Low Capital Bank, by contrast, has only \$4 million of equity, so its equity multiplier is higher, equaling 25 (= \$100 million/\$4 million). Suppose that these banks have been equally well run so that they both have the same return on assets, 1%. The return on equity for the High Capital Bank equals 1% x 10 = 10%, while the return on equity for the Low Capital Bank equals 1% x 25 = 25%. The equity holders in the Low Capital Bank are clearly a lot happier than the equity holders in the High Capital Bank because they are earning more than twice as high a return. We now see why owners of a bank may not want it to hold too much capital. Given the return on assets, the lower the bank capital, the higher the return for the owners of the bank.