One question often asked in forums and blogs is how to adjust stock prices in order to take into account dividend payments. There are several reasons why we may be interested in adjusting stock prices:

• Analysing total returns, taking into account dividend reinvestments for reporting (performance, historical volatility, etc.).
• Applying trading strategies based on the stock prices series (momentum, volatility, etc).

Whether or not we reinvest the dividends, it has a tremendous effect in a long-term investment. We just have to compare the S&P 500 Index Price (unadjusted) version and the Total Return (adjusted) version to realise it. The election of one of these types of indexes is capital whenever we want to use them as a benchmark. A magnificent equity portfolio against the unadjusted index can actually be very poor if it turned out to have the dividends reinvested. We would be comparing with the wrong version of the index.

So, let’s get to it! What can we do to adjust a price series with the dividends distributed along the time?

Calculations

Depending on the “direction” of the correction you can find two methods: Backwards or Forwards. Although the most common way of adjusting is backwards, I’m first going to explain forward adjustment, since its logic is simpler to understand.

But before that, I’m going to explain the concept of exDate.

We refer to exDate as the date prior to which the stock has to be owned in order to receive a scheduled dividend distribution. This means that:

• If you buy more stocks between exDate and the dividend payment date (normally after several days), you won’t receive more dividends for those new buys.
• If you sell your investment before (even the day before) the exDate, you will not receive any dividends.
• At exDate, the market discounts the dividend distribution from the price, so we will normally see a step at that date’s opening.

Forwards method

This method consists of modifying the prices on the appearance of the dividend distribution (the exDate) and on. In formulas:

Where P’x_Ford is any adjusted price on the date x, being x equal or after the exDate of the z^th dividend distribution and prior to the exDate of the z+1^th dividend distribution. Px are the prices without adjustment.

Consequently, before the first dividend distribution, the adjusted and unadjusted prices are equal.

Let’s explain it considering the number of stocks and their evolution to better understand where this formula comes from:

Imagine you buy stock A and several weeks later a dividend is distributed. At that moment you decide to reinvest this dividend in the same company, A, as soon as you get it.

Then, the number of owned stocks has evolved. Now you’ll have the stock you already had plus D/S new stocks, being D the dividend amount reinvested and S the price at which it’s reinvested. I’m sure you’ve already noticed that this is exactly the factor fj seen before. If we wanted to translate this number to money, we only had to multiply by the price S. This obviously gives us an amount of S+D (the stock price plus the dividend).

For new dividend reinvestments, we’d have to multiply the factors fj to accumulate them, as done in the previous formula.

Backwards method

This method corrects the price series backwards. That is, we’re going to adjust all prices before the exDate.

Let be, P’x_Back any adjusted price in the date x, being x prior to the z^th dividend distribution and posterior or equal to the z-1^th dividend exDate. Px are the prices without adjustment.

So, from the last dividend distribution exDate on, the adjusted and unadjusted prices coincide.

As you could already expect, the returns with both the backwards method and the forwards method are the same.

Other approches

Some data providers, like Yahoo, use the following formula to calculate the factors in the backward method:

With this formula we are assuming that the dividend reinvestment is at price p_j-1 – d_j, being p_j-1 the close price prior to the exDate. This is like expecting that the price at which you reinvest the dividends will be near the close of the previous date to exDate, minus the dividend distributed (which will be near the open price on exDate).

To obtain the forward-method factor, we would only have to invert the previous formula.

Example

Let’s calculate the adjusted prices of the stock 3M Co in 2015, and the performance achieved if someone kept purchasing this stock during 2015. That year there were 4 dividends distributed on February 11^th, May 20^th, August 19^th and November 18^th. Each distribution gave 1.025 dollars per stock.

Following the formulas seen before we obtain the following table. Although I have left only two decimals, it is important to keep them all when multiplying, in order to not lose accuracy. I’m not getting into the matter of lost of accuracy here, but it’s an issue of importance when the number of reinvested dividends grows.

I have marked in blue the dates where the adjusted prices are equal to the unadjusted ones (all the prices between those dates are equal too, of course).

If you look at the table carefully, you will notice that the prices calculated with the Yahoo formula are slightly different. This is reflected in the final returns calculated in the final row.

Conclusion

You must take into account that the return calculated for unadjusted prices is nothing but an entelechy. As soon as you buy the stock, you’re going to receive the dividends, even though you don’t reinvest them. So, actually, your final return should include the dividends obtained, without reinvesting them. The final return in 2015 for 3M Co would be:

In this case, quite the same as the return with the dividends reinvested.