Dividend Adjustments, Adjusted Closes, and Historical Returns
Summary
The document explores why a common historical-price adjustment for dividends may not match a shareholder’s simple holding-period return calculation. It sets out a hypothetical stock path with a dividend and compares adding the dividend to the sale proceeds with adjusting earlier prices by a factor based on the prior close. In the example, those calculations give different returns, prompting the author to ask whether the standard adjusted-close method is only an approximation or has another interpretation.
The text mentions a common data-provider explanation and a paper as references, then suggests that the adjustment can be viewed as accounting for a known upcoming dividend by treating the purchase price as reduced by the dividend amount. That interpretation is offered tentatively, not established with a derivation. The document contains no answer resolving the convention, and the numerical example alone does not establish which adjusted series is appropriate for every return or backtesting use. Readers should distinguish raw prices, dividend cash flows, and vendor-adjusted total-return series.
Key ideas
- The author compares a cash-inclusive holding-period return with a historical price adjustment based on a dividend factor.
- The example produces different returns under the two calculations.
- A tentative interpretation treats an anticipated dividend as reducing the effective purchase cost.
- The document does not establish a definitive formula or resolve which adjusted series suits each use.
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# Rigorous formula for adjusted close price
# Rigorous formula for adjusted close price
I'm a mathematician who is new to the stock market, and I'm hoping to shine a rigorous light on a simple example of adjusting close prices for, say, dividends.
Let time $t$ represent today. Say that the price of a stock two days ago (at time $t-2$) was $p_{t-2}=4$. Then, yesterday, the price climbs to $p_{t-1}=6$, and the stock pays a dividend of $d=1$. Today, let the stock price by $p_t=4$. If I'm correct, today would be called the 'ex-dividend date'.
If I bought the stock two days ago and sold it today, I would think that the appropriate calculation for my return (assuming no dividend reinvestment) is $\frac{p_t + d}{p_{t-2}} = 1.25,$ or $25\%$.
However, in most of the explanations I see online, they do the following: adjust the prices on $t-1$ and before by multiplying these prices by the quantity $(1-\frac{d}{p_{t-1}})$. If I use this adjusted price at time $t-2$, I would get: $p'_{t-2} = p_{t-2}(1-\frac{d}{p_{t-1}}) \approx 3.33$. If I then use the formula $\frac{p_t}{p'_{t-2}}$ for my return, I get $1.2,$ or $20\%$.
I would think that my return is $25\%$, but this popular formula does not agree. I notice that in order to derive an adjusted price $p'_{t-2}$ which gives the true return today, this adjusted price would need to be a function of $p_t$, implying that these adjusted prices would need to continually be adjusted every day. Perhaps this is cumbersome or inconvenient, so people settle for the popular formula as an approximation (which only needs to be adjusted when another dividend is paid or a split is made), but I'm not sure. Maybe I'm thinking about this all wrong. Any ideas of why these two formulae aren't rectifying/any good references are very much appreciated. Thanks!
An edit, since I can't comment:
nbbo2 - Thanks for the link, that's helpful. I suppose this version of adjusted price, in which the adjustment factor is fixed (until next dividend) and doesn't depend on today's price, is just a convenient approximation.
AlRacoon- Here's the sample calculation that yahoo finance gives: https://help.yahoo.com/kb/SLN28256.html. This paper: https://arxiv.org/abs/1105.2956 also explains the common formula very nicely.
After thinking about this some more, I suppose there's an interpretation that justifies the formula in the links above: if you knew a \$0.1 dividend was coming next week, and you bought the stock for \$1 today, you could think of having bought the stock for \$0.9 today when you come back to adjust. Anyway, thanks for your help everyone. Was a good exercise to think about this.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.