Using total return calculations is critical in developing security selection models.

Essentially, capital distributions/dividends to investors are deducted from the historical close price series. When calculating the return over the holding period using this series: $r(t) = \frac{P(t) - P(t-1)}{P(t-1)}$, you are calculating the actual profit/loss return including dividend distributions.

However, it's possible that the dividend adjustment will cause some adjusted historical prices to go below zero. For example, consider Avis adjusted close prices in 2003. The adjusted close price in May 2003 is about -$\$1.5$. The adjusted close price a year later is$\$80$. This is a considerable return. However, when calculating the return $\frac{85 - (-1.5)}{-1.5}$ a negative return is produced because of the denominator.

One approach would be to "forward adjust" dividend adjustments as opposed to deducting dividends paid on the prior series. But then the forward-adjusted close prices would not match prices traded on the exchange (complicating trade execution, etc.). Another approach is to shift the -$\$1.5$and$\$85$ above the zero line some arbitrary amount. This diminishes the actual return the farther up the number line the two price points are shifted.

Yahoo calculated dividends adjustments on a percentage basis, not on a absolo

Any suggestions on how to solve this problem?

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Where did you get the data for Avis' stock price in 2003/2004? Can you provide a link? Was this ownership of private or publicly traded stock? –  bill_080 Apr 23 '11 at 18:32
Publicly traded stock. Dataset is from CSI. www.csidata.com –  Quant Guy Apr 25 '11 at 0:23

Add-back the dividends as if they had not been paid out. That will ensure that you have a positive price when deriving the returns.

For example, MSFT paid a 0.16 dividend on 2011-15-02. Here are the raw prices, according to Yahoo:

date       close
----------------
2011.02.01 27.99
2011.02.02 27.94
2011.02.03 27.65
2011.02.04 27.77
2011.02.07 28.2
2011.02.08 28.28
2011.02.09 27.97
2011.02.10 27.5
2011.02.11 27.25
2011.02.14 27.23
----------------    <- dividend was paid here
2011.02.15 26.96
2011.02.16 27.02
2011.02.17 27.21
2011.02.18 27.06
2011.02.22 26.59
2011.02.23 26.59
2011.02.24 26.77
2011.02.25 26.55
2011.02.28 26.58


Yahoo's own adjustment is to subtract the dividend for all dates prior to the payment, exactly as your question states. But if all you want is the return, then add-back the dividend after the payment. Thus, we have:

date       close
----------------
2011.02.01 27.99
2011.02.02 27.94
2011.02.03 27.65
2011.02.04 27.77
2011.02.07 28.2
2011.02.08 28.28
2011.02.09 27.97
2011.02.10 27.5
2011.02.11 27.25
2011.02.14 27.23
2011.02.15 27.12
2011.02.16 27.18
2011.02.17 27.37
2011.02.18 27.22
2011.02.22 26.75
2011.02.23 26.75
2011.02.24 26.93
2011.02.25 26.71
2011.02.28 26.74


The return calculations won't match Yahoo's exactly. Yahoo's adjusted first and last price are 27.83 and 26.58, for a return of -4.49%, whereas the return from my adjusted prices above is -4.47%. But my returns are guaranteed to be more sensible simply because I know there will never be a negative price.

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Yes this is an approach that resolves the paradox. It does open a second technical issue in that my current "forward-adjusted" close prices will not match quotes observed on exchanges. This makes debugging and data quality control more difficult when you want to validate return figures. Also, to place trades, I will need to maintain an un-adjusted price series. –  Quant Guy Apr 24 '11 at 5:15
@Quant Guy Only persist the raw price table; the adjusted table should be materialized only when it will be used. (I'm a big believer in the immutability of historical data, which is why I advocate bi-temporal or point-in-time storage with log-only amendments.) Your "second issue" is only an illusion since you'll have the raw prices to compare against the exchange---adjusting is only used for assessing returns and never for comparing prices. As far as validating your computed returns, I'm not sure the Yahoo numbers are any more real than what you'll come up with. –  chrisaycock Apr 24 '11 at 6:06

The problem is, you're calculating the "return on investment" or "return on original investment". Anytime you are given back ALL or MORE THAN your original investment, you are no longer "invested". As a result, calculating "return on investment" no longer applies.

For example with a 100% return of capital, the "original investment" drops to zero. So, the return on "investment" is infinity.

Another example is a merger/restructuring, where you get back more than all of your original investment. Your "investment" is negative (someone gave you the money to invest rather than you giving them the money to invest), so your return on "investment" is now negative.

Notice that it doesn't matter whether you calculate the series forward or backward, a 100% return of capital situation provides an infinite return. And, when you are given back more than you invested, your return is negative.

One solution would be to check to see if the original investment is less than or equal to zero. If so, then flag the result accordingly.

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Just in case the question comes up....en.wikipedia.org/wiki/Return_of_capital –  bill_080 Apr 22 '11 at 20:58
I just ran across the following. It helps explain the restructuring of Avis/Cendant/Wyndham/etc in 2006. @Quant, was your 2003 negative number the result of owning stock in one of these companies? .... avisbudgetgroup.com/investor_relations/… ....and another link.... avisbudgetgroup.com/investor_relations/faqs.cfm#how_many_shares –  bill_080 Apr 23 '11 at 18:50
Hi Bill - flagging the company is a decent way to go. Then I can impose/impute some kind of max return. I'll have to scale/trim calculated returns anyway to avoid outliers. AVIS was just an example -- there are dozens of companies that exhibit this phenomenon. –  Quant Guy Apr 24 '11 at 5:19
@Quant, Most restructuring/return-of-capital situations mean that the original organization no longer exists. As with the link in my above comment, those that owned the original company get their "negative ROI" numbers as private tax information in their brokerage statement. As far as I know, the "negative ROI" doesn't show up much in publicly available data. Can you supply some links to public "-ROI" examples? –  bill_080 Apr 24 '11 at 22:56