PE is PB/ROE Mathematically

The mathematical proof that Price-to-Book (P/B) ratio divided by Return on Equity (ROE) equals Price-to-Earnings (P/E) ratio is well established and can be explained as follows:

This shows how P/E ratio can be decomposed into market valuation relative to book value (P/B) combined with profitability relative to equity (ROE).

Supporting Explanation from Research:

Financial analysts often use this relationship to understand stock valuation from different perspectives by decomposing P/E into asset-based valuation (P/B) and profitability (ROE). This is supported in practical valuation and academic sources, such as the analysis on Footnotes Analyst explaining how P/B and ROE together reflect Price-to-Earnings dynamics and valuation multiples in the market1.