Question

if ROA at. a given bank remains fixed but both total assets and equity double, how dies ROE change?why?

Answer 1

ROA=Net Income/Total assets

ie. Return generated on the assets employed.

which can be further analysed by splitting into 2 ratios/components :

ROA=(Net Income/ Revenues)*(Revenues/Total assets)

ie.the ultimate quantum of income ,the bank is able to net ,from the revenues, & how efficient, the total assets at its disposal are utilised to generate the same revenues as aforesaid.

On the otherhand,

ROE= Net Income/Total Equity

ie. The above same Return expressed in relation to the total funds belonging to the stockholders of the bank.

which can be further analysed by splitting into 3 ratios/components :

ie. (Net Income/Revenues)*(Revenues/Total assets)*(Total assets/Total equity)

Here,

The first 2 ratios form the ROA

Now,

if ROA at a given bank remains fixed but both total assets and equity double, how dies ROE change?why?

mathematically, the only non-cancelling component that has doubled, in the above formula, is the

Total Equity in the denominator - in the 3rd component ,ie. Equity Multiplier

ROA remaining the same as a ratio,is possible only if,

Net income has doubled.

so, when net income & also the equity in the denominator double,

ROE also remains the same

as explained below with a numerical example

ROE=(15/2250)*(2250/22500)*(22500/225)=

6.67%

ROA=(15/2250)*(2250/22500)

0.07%

When the assets & Equity are doubled

ROE=(30/2250)*(2250/45000)*(45000/450)=

6.67%

ROA=(30/2250)*(2250/45000)

0.07%