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They differ because of
The first big assumption that the DDM makes is that dividends are steady, or grow at a constant rate indefinitely. This is assumption does not hold in reality
The dividend discount model generally understates the intrinsic value of the firm. Important considerations such as the value of patents, brand name, and other intangible assets should be used in conjunction with the DDM to assess the value of a firm's equity. These intangibles should be added to the result of a DDM calculation to arrive at a more appropriate valuation.
The dividend discount model does not account for the volatility in stock prices. The prices change everyday, but the model results will not change.
The Dividend Discount Model works best for firms with stable growth rates, firms which pay out dividends that are high and approximate FCFE and firms with stable leverage.
First the acronyms - DDM is Dividend Discount Model,FCFE is Free Cash Flow to Equity
The reasons that you do get the same price from DDM is as I have explained earler due to - volatility is stock prices, they chnage everyday but DDM values do not. DDM works best if the payout ratio is high and DDM undertstaes the price since it does not take into account the brand name an dother intrinsic factors.
When you use P/E, it is forward looking, the P/E captures the expecte dgrowth in the future but DDM is backward looking, you see the dividends paid and not what the diivdends could be due to growth of the firm.
I hope this helps