adjustments - Total Return measurement paradox w/ Adjusted Close Prices

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

The standard way to measure total return is to develop a series of price-adjusted data. Investopedia describes the standard method here: http://www.investopedia.com/ask/answers/06/adjustedclosingprice.asp

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?

Answer

Don't subtract dividends; add them.

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
----------------    <- add-back dividend
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.