Many investors only own a handful of different stocks, so they can individually track the performance of each stock they own. However, it's not sufficient to just keep your eyes on your own basket. Investors and traders also need information about the overall market sentiment.
That is what incides are for. They provide a single measurable and traceable number, which aims to represent the overall market or a selected set of stocks or sector and its movement. Indexes also serve as benchmark for investment comparisons – say your individual portfolio of stocks (or your mutual fund) returned 15%, but the market index returned 20% during same period. Hence, your performance (or your fund manager’s performance) is lagging behind the market.
What is a stock market index, e.g. the DJIA (Dow Jones Industrial Average)?
The Dow is an indicator of how 30 large, listed US companies have traded during a standard trading session.
A stock market index is a mathematical construct, which provides a single number for measurement of the overall stock market (or a selected portion of it). The index is calculated by tracking prices of selected stocks (e.g. the top 30, as measured by prices of the largest companies, or top 50 oil-sector stocks) and based on pre-defined weighted average criteria (e.g. price-weighted, market-cap weighted, etc.)
The calculation behind Dow Jones:
To better understand how the Dow changes value, let’s start at its beginnings. When Dow Jones & Co. first introduced the index in the 1890s, it was a “simple average” of the prices of all constituents. For example, let's say there were 12 stocks in the Dow index; in that instance, the Dow's value would have been calculated by simply taking the sum of closing prices of all 12 stocks and dividing it by 12 (the no. of companies or “constituents of the Dow index”). Hence, the Dow started as a simple price average index.
where n is the no. of constituents in the index (n=12 in our above example) and Sigma represents the summation of prices.
To explain the concept better with other scenarios and twists, let’s build our own simple hypothetical index along the lines of the Dow.
To keep it simple, assume that there is a stock market in a country that has only two stocks trading (Ally Inc. and Belly Inc. – A & B). How do we measure the performance of this overall stock market on a daily basis, as the stock prices are changing each moment and with every price tick? Instead of tracking each stock separately, it would be much easier to get and track a single number representing the overall market constituting both the stocks. The changes in that single number (let’s call it “AB index”) will reflect how the overall market is performing.
Let’s assume that the exchange constructs a mathematical number represented by “AB Index”, which is being measured on the performance of the two stocks (A and B). Assume that stock A is trading at $20 per share and stock B is trading at $80 per share on day 1.
Applying the initial concept of Dow to our hypothetical example of AB index:
 At the start, AB index =
= ($20 + $80) / 2 = 50.
Now suppose the next day, the price of A moves up from $20 to $25 and that of B moves down from $80 to $75.
 The new AB index = ($25 + $75)/2 = 50.
i.e. the positive price movement in one stock has canceled the equal value but negative price movement of another stock. Therefore, the index value remains unchanged.
Suppose on the third day, stock A moves to $30, while stock B moves to $85.
 The new AB index = ($30 + $85)/2 = $115/2 = 57.5
In the case of (2), the net sum price change was ZERO (Stock A had +5 change, while B has -5 change making the net sum change zero).
In the case of (3), the net sum price change was 15 (+5 for Stock A [25 to 30] while +10 for Stock B [75 to 85]). This net price sum change of 15 divided by n=2 gives the change as +7.5 taking the new changed index value on day 3 at 57.5.
Even though Stock A had a higher percentage price change of 20% ($30 from $25), and Stock B had a lower percentage change of 13.33% ($85 from $75), the impact of Stock B's $10 change contributed to a bigger change in the overall index value. This indicates that price-weighted indices (like Dow Jones and Nikkei 225) depend on the absolute values of prices rather than relative percentage changes. This has also been one of the criticizing factors of price weighted indexes, as they don’t take into account the industry size or market capitalization value of the constituents.
Now assume that another company C lists on the stock exchange at the price of $10 per share on the fourth day. AB index wants to expand and increase the number of constituents from two to three, to include the newly listed C company stock in addition to the existing A and B stocks.
From the perspective of the AB index, a new stock's coming onboard should not lead to a sudden jump or drop in its value. If it continues with its usual formula of [Sum(Prices)/number of constituents], then:
[4 - Incorrect] The new AB index = ($30 + $85 + $10) /3 = $125/3 = 41.67
This is a sudden dip in index value from previous 57.5 to 41.67, just because a new constituent is getting added to it. (Assuming that stock A & B maintain their earlier day prices of $30 and $85). This would not be a very useful reflection of the overall health of the market.
To overcome this calculation anomaly problem, the concept of a divisor is introduced.
The divisor allows the index values to maintain uniformity and continuity, without sudden high value fluctuations. The basic concept of a divisor goes as follows. Simply because a new constituent is getting added, this should not justify high value variations in the index. Hence just before the new constituent is introduced, a new “calculated” divisor value should be introduced. It should be such that the following condition should hold true:
New Price summation = $125 (3 stocks)
Last known good value of index = 57.5 (based on 2 stocks)
which leads to n-new = 125/57.5 = 2.1739
This new value becomes the new “divisor” of the AB index.
So on the day when the stock C in included in the AB index, it’s correct (and continuous value) becomes:
[4 - Correct] The new AB index = ($30 + $85 + $10) /2.1739 = $125/2.1739 = 57.5
This same value on the fourth day makes sense because we are assuming that the stock prices of A and B have not changed compared to the third day, and just because a new third stock is added, this should not lead to any variations.
On the fifth day, suppose the prices of stocks A, B, C are respectively $32, $90 and $9, then
 The new AB index = ($32 + $90 + $9) /2.1739 = $131/2.1739 = 60.26
Going forward, this new value of 2.1739 would continue to be the divisor (instead of the whole number of constituents). It will change only in the case of new constituents getting added (or deleted) or any corporate actions taking place in the constituents (example below).
Let’s continue further with calculation variations. Suppose that Stock B goes for a corporate action that changes the price of the stock, without changing the company valuation. Say it is trading at $90 and goes for a 1-to-3 stock-split, leading to tripling the number of available shares and reducing the price by a factor of three, i.e. from $90 to $30.
In essence, the company has not created (or reduced) any of its valuations because of this stock-split corporate action. This is justified by the number of shares tripling and the price coming down to a third of original. However, our index is solely price-weighted and does not account for share volume change. Taking the new $30 price into calculation will lead to another big variation as follows:
[6 - Incorrect] The new AB index = ($32 + $30 + $9) /2.1739 = $71/2.1739 = 32.66
This is way below the earlier index value of 60.26 (at step 5)
Here again, the divisor needs to change to accommodate for this change, using the same condition to hold true:
New Price summation = $71 (3 stocks)
Last known good value of index = 60.26 (step 5 above)
which leads to n-new or divisor value = 71/60.26 = 1.17822
Using this new divisor value,
[6 - correct] The new AB index = ($32 + $30 + $9) /1.17822 = $71/1.17822 = 60.26
(Assuming that stock A & C maintain their earlier day prices of $32 and $9)
Arriving at the same previous day value validates the correctness of our calculations. This new 1.17822 will become the new divisor going forward. The same calculation would apply for any corporate action affecting the stock price of any of the constituents.
Let’s conclude this with one last example:
Suppose Stock A goes for delisting and needs to be removed from the AB index, leaving only 2 stocks B & C.
 New Price summation = $30 + $9 = $39 (2 stocks)
Last known good value of index = 60.26 (step 6 above)
New divisor value = 39/60.26 = 0.64719
New index value = $39/0.64719 = 60.26
Dow calculations and value changes work in a similar way. The above cases cover all possible scenarios for changes for price-weighted indices like the Dow or the Nikkei. At the time of writing this article, the Dow Jones divisor value was 0.15571590501117.
The divisor value has its own significance. For every $ change in price of underlying constituent stocks, the index value moves by an inverse value. For e.g. if a constituent like VISA moves up $10, then it will lead to 10*(1/0.15571590501117) = 64.21952 change in value of DJIA.
Till there is any change in the number of constituents or any corporate actions in the same affecting the prices, the existing divisor value will hold.
Assessment of the Dow Jones calculation methodology:
No mathematical model is perfect –each comes with its merits and demerits. Price weighting with regular divisor adjustments does enable the Dow to reflect the market sentiments at a broader level, but it does come with a few criticisms. Sudden price increments/reductions in individual stocks can lead to big jumps/drops in DJIA. For a real-life example, an AIG stock price dip from around $22 to $1.5 within a month’s time led to a fall of almost 3000 points in the Dow in 2008. Certain corporate actions, like dividend going ex (i.e. becoming an ex-dividend, wherein the dividend goes to the seller rather than to the buyer), leads to sudden drop in DJIA on the ex-date. High correlation among multiple constituents also led to higher price swings in the index. As illustrated above, this index calculation may get complicated on adjustments and divisor calculations.
Despite being one of the most widely recognized and most followed index, critics of price-weighted DJIA index advocate using float-adjusted market-value weighted S&P 500 or the Wilshire 5000 index, although they too come with their own mathematical dependencies.
The Bottom Line
The second oldest index of the world since 1896, despite all of its known challenges and mathematical dependencies, the Dow still remains the most followed and recognized index of the world. Investors and traders looking at using DJIA as the benchmark should keep the mathematical dependencies in consideration. Additionally, indices based on other methodologies should also be worth considering for efficient index based investments.