Dividend Adjustments and Drift Estimation in Stock Models
Summary
The document distinguishes price-return drift from total-return drift when estimating a stock model that includes dividends. In a continuous-dividend Black–Scholes-style setup, the stock’s price drift equals its total-return drift minus the dividend yield. A price series that excludes dividend adjustments therefore estimates price appreciation, while a dividend-adjusted series that adds paid distributions back estimates total return.
This distinction helps determine whether historical data should be adjusted before estimating drift for a model with future dividends. The answer suggests that both price and total-return drifts, as well as a continuous yield, can be estimated when the relevant data series are available. It also cautions that discrete dividend modeling is more involved. The explanation assumes clear conventions for adjusted and unadjusted prices; data-provider adjustment methods can vary, and the discussion does not provide a specific estimation procedure or address changing yields in detail.
Key ideas
- A continuous-dividend stock model separates total-return drift from price-return drift by the dividend yield.
- Unadjusted price data estimate price-return drift, while dividend-adjusted prices can represent total-return drift.
- Consistent data conventions are needed when estimating drift for a model that separately accounts for dividends.
- Discrete dividend modeling is more involved than the continuous-yield setup described.
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Full text
# Should I adjust historical data for dividends when estimating drift?
# Should I adjust historical data for dividends when estimating drift?
I'm building a Geometric Brownian Motion model which incorporates future dividends which vary over time. Since these should reduce stock price when paid, I can incorporate that into the model, however, I just realized that when I estimated drift from historical data, that data already incorporates dividends. Should I adjust the historical to remove the impact of the dividends?
## Answer by Daneel Olivaw (score 1)
https://quant.stackexchange.com/a/34697
In a Black-Scholes framework, the Geometric Brownian Motion specifying the dynamics of a stock continuously paying dividends is given by:
$$ dS_t = (\mu_{\text{Total}}-q)S_tdt+\sigma S_tdW_t $$
Under the model above, the stock price $S_t$ is continuously paying a dividend equal to $qS_tdt$, and as a consequence the stock price depreciates continuously at the same rate $-$ the same way in stock markets, stock prices plunge immediately after a dividend payment by an amount equal to the dividend paid.
Let us define $\mu_{\text{Price}} = \mu_{\text{Total}}-q$. Now, depending on what you mean by:
> "[...] data already incorporates dividends."
We have:
- If your stock data is not adjusted for dividends $-$ "not adjusted" meaning that stock prices do not incorporate paid dividends $-$ then if you estimate the drift on that data series you will be estimating $\mu_{\text{Price}}$ i.e. the price return drift which only captures returns from variations in the stock price independent of dividend payments.
- If your stock data is adjusted for dividends $-$ "adjusted" meaning that stock prices are corrected by adding back paid dividends $-$ then if you estimate the drift on that data series you will be estimating $\mu_{\text{Total}}$, i.e. the total return drift which incorporates dividend return.
You can manually adjust your data series to "remove the impact of the dividends" $-$ I am unsure whether you mean subtracting paid dividends from the stock price, or adding them back $-$ and you will be able to estimate both drifts $\mu_{\text{Total}}$ and $\mu_{\text{Price}}$ from each data series, adjusted and unadjusted. From there you can also estimate the continuous dividend yield $q$.
For discrete dividend modelling, things get messier. You can take a look at Wikipedia's article on the Black-Scholes model, where they consider extensions with dividend-paying stocks.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.