The index was first introduced by Charles Dow and Edward Jones in 1986. The index was simply calculated as an average price of 12 companies. So, it was calculated by adding up the stock prices of 12 companies and dividing 12, the divisor then.
Since that time, many stocks were added and taken off the list of the Dow Jones. To ensure continuity of the index, i.e. the value of the index is exactly the same before and after any change, the divisor, d had to be altered. Therefore, it was changes based on the following formula, many times over the years.
Then divisor is changed the same exact way, as we are are replacing one stock with another that has a different (based on the split ratio) price.
That is really the gist of this article. Let us look in details of what happens with Apple (ticker: APPL) is going to do 1:4 split. That means the number of shares go up by 4 and the price of each share go down by 4.
Currently the DJIA is composed of 30 stocks. Let us calle the price of Apple stock, before spiting, p1.
\[ DIJA = \frac{P_{1} + \sum\limits_{i=2}^{30} P_{i}}{D_{1}} \]And fater the change, we can write:
\[ DIJA = \frac{\hat{P}_{1} + \sum\limits_{i=2}^{30} \hat{P}_{i}}{D_{2}} \]But we know that the prices of all stock, but Apple stayed the same. And that \[ \hat{P}_i = P_{i}, for \; i = 2 \; to \; 30 \] and \[ \hat{P}_1 = P_{1}/4 \]
If we put all the above together, and since we have only one unknown, D2, then we can write:
\[ D2= D_{1} * \frac{P_{1}/4 + \sum\limits_{n=2}^{30} P_{i}}{P_{1} + \sum\limits_{n=2}^{30} P_{i}} \]We can use the current value of the Dow Divisor was 0.14579812049809 as of April 6, 2020, to calculate D2.
The trick is that although the value of the index is continuous, the rate of the change of the index is now different. Let us look at what happen to the index when Apple stocks gains 10%, like it actually did on July 31, or loses 10%, under the old and the new divisor.
As shown in the figure above, a 10% change in Apple (like it did on July 31, 2019) is equal more than 1% change in the index. As of July 31, Apple was 11% of the index. But with the new multiplier, the index will move much less as function of Apple.
So let us say you were holding SDOW, which is 3x leveraged inverse ETF of the DJI, i.e. if the Dow Jones go up 10% SDOW is down 3%. The 10% climb of Apple added over 1% of gain to the index. With the new divisor, if apple goes down 10%, then it will contribute about 0.25 of loss to the Dow Jones and SDOW will only gain less than 1%. So, the "odds" of recovering that loss have been shifted.
The above argument could be yet another reason why one should not trade ETFs. And, also show how the index can be easily manipulated. So Apple prices can pushed up, the multiplier get reset, then Apple can go down but the index will be way less impacted.
On August 31/2020 the DOW was changed drastically. Apple was split, 1-4. Three stocks left the DOW, RTX, PFE, and XOM. Three stocks joined the DOW, AMGN, CRM and HON. Based on that the new Dow divisor is now ( 0.15198715). See Ref (3) and (2).
(1) What Were the Original Dow Jones Industrial Average (DJIA) Companies? (2) Dow Jones Industrial Average (3) dow-jones-industrial-average-changes