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So PE = PB / (Return on Equity) as you observe.It doesn't seem to work that way in practice. Here are the P/E, P/B and RoE figures for Apple (AAPL), for example, as taken from Yahoo Finance.
finance.yahoo.com - Key statisticsApple (AAPL)P/E = 33.0
P/B = 43.9
RoE = 147.25%
Note that the units for P/E and RoE are completely different. The RoE is reported as a percentage, whereas the P/E is not.
If you convert the 147% RoE to the decimal number 1.47, and divide the 43.9 PB by 1.47 you get 29.9, which is not the same as 33.
I did the same exercise for META, and got PB/RoE = 25 whereas the actual PE is 29.
So, for 2 of the biggest companies in the US, the PB/RoE ratio is about 10% lower than the P/E ratio (at least, as reported by Yahoo Finance).
