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The ratio of net profit to shareholders' equity (also called book value, net assets or net worth), expressed as a percentage. A measure of how well a company uses shareholders' funds to generate a profit. 
Return on equity (ROE), is a financial ratio that measures the return generated on stockholders’/shareholders’ equity, the book or accounting value of stockholders’/shareholders’ equity which reflects the accumulation over time of amounts received by the company from stock/share issues plus the profits/earnings retained by the company, i.e., not yet distributed in dividends (accounting value of shareholders’ equity is also equal to a company’s net assets, i.e., assets minus liabilities). The typical formula can be expressed as follows:
profit for the year (or net income after taxes) / stockholders’ or shareholders’ equity
This is generally calculated over a year and expressed as a percentage, so a company that generated £100 worth of profit for the year for £1000 of equity has a ROE of 10%.
ROE is often said to be the ultimate ratio or ‘mother of all ratios’ that can be obtained from a company’s financial statement. A company can only create shareholder value, economic profits, if the ROE is greater than its cost of equity capital (the expected return shareholders require for investing in the company given the particular risk of the company).
Historically, the average ROE has been around 10% to 12%, at least in the US and UK.
Furthermore, the ROE can be decomposed to understand the fundamental drivers of value creation in a company. This is known as the DuPont decomposition and can be calculated as:
ROE = return on assets (ROA) X gearing (also called leverage)
ROE = (profit for the year ÷ assets) X (assets ÷ shareholders’ equity)
Note that since return on assets (ROA) is profit margin multiplied by asset turnover, the DuPont decomposition is sometimes represented as follows:
ROE = profit margin X asset turnover X gearing
ROE = (profit for the year ÷ sales) X (sales ÷ assets) X (assets ÷ shareholders’ equity)
The profit margin, asset turnover and gearing ratios can further be decomposed to complete the financial statement analysis or ratio analysis of a company.
The profit margin tells us how much profit a company makes on every dollar of sales. The asset turnover indicates how efficient a company is in using its assets and reflects how many dollars of sales a company generates from every pound/dollar of assets in the company.
Finally, the gearing ratio indicates how a company finances the assets it holds or more precisely the amount of assets per dollar of shareholder/stockholder equity investment in the company.
Assets being financed either by shareholders (equity) or by creditors such as banks and suppliers (called liabilities or debt), a higher ratio means a firm is getting more financing from outside creditors (since assets must equal equity plus liabilities, an all equity firm has a gearing ratio of one since all assets are financed by equity).
It would appear that greater gearing increases ROE, but this must be traded off against higher financing costs which reduces profit.
Unfortunately, there are variations in ratio definitions, a normal part of practical financial ratio analysis. There are issues regarding the numerator, e.g., which profit number should be used: before or after taxes; adjusted or not for non-recurring or one-time items; or operating type profit numbers, e.g., earnings before interest and taxes (EBIT), earnings before interest, taxes and depreciation & amortisation (EBITDA).
As for the denominator, a balance sheet number is for a specific point in time by contrast to the numerator which is for a period, so some use beginning-of-year equity, others use end-of-year equity or some average over the year.
Note: One of the challenges in giving a definition of accounting or finance terms is the differences in the terminology used. E.g., in the US and UK: shareholder/stockholder and profit for the year/net income.