Dividend Reinvestment and the Total Return Process

Summary

The note poses a continuous-time question about a dividend-paying stock whose price follows a diffusion with drift reduced by a continuous dividend yield. It asks how the value process changes when dividends are continuously reinvested into the same stock, starting from one share. The proposed differential adds the dividend cash flow back to the stock’s price change, suggesting a process with the full drift parameter rather than the dividend-adjusted price drift.

This frames the distinction between a stock’s ex-dividend price process and the total return earned by an investor who reinvests distributions. The document contains the question and a tentative derivation, but no answer or supporting evidence, so it does not establish whether the proposed process is correct or explain the exact share-count dynamics. Its result also depends on the continuous-yield model and frictionless reinvestment assumption; discrete dividends, taxes, and transaction costs are outside its scope.

Key ideas

• The stock price process includes a continuous dividend yield that reduces its price drift.
• The investor’s value after reinvesting dividends includes both price changes and dividend cash flows.
• The proposed differential suggests that reinvestment restores the dividend-adjusted drift, but the document gives no answer validating it.
• The setup assumes continuous dividends and does not address frictions such as taxes or transaction costs.

Tags

# Reinvesting the dividends of a dividend paying stock


# Reinvesting the dividends of a dividend paying stock












Suppose we have a dividend paying stock that has the following dynamics: $$dS_t=S_t((\mu-q)dt+\sigma dW_t)$$ With a continuous dividend yield $q$. What is the portfolio $Y_t$ that results out of starting with one unit of the stock and continuously reinvesting the dividends in the stock? My intuition tells me that $$dY_t=dS_t+qS_tdt=S_t(\mu dt+\sigma dW_t)$$ So, is this just the same as a regular nondividend paying stock?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.