# Calculating index values

An index is a construct meant to monitor the changes in the prices of its constituents over time. But a collection of numbers is bulky and inefficient to use—hence the need for a single value easily comparable and trackable over time.

Differences in calculating index values exist depending on weighting scheme, but in the discussion that follows, we’ll focus exclusively on how to calculate the value of market cap weighted indexes (which are the dominant form of index).

## Index divisors and index values

Before you can understand how an index value is calculated, first you must understand how it’s normalized.

That is, index values can be tough to compare because the values of their constituents all start at different price points – not to mention they can be very large. That’s why indexers usually give their benchmarks a conceptually simple starting point by means of an index divisor. This arbitrary number, defined when an index is first established, allows us to divide the index to produce an initial value that’s easier to wrap your head around, such as 100.0.

Under most circumstances the index divisor remains constant, though it can change when securities are added or dropped from the index or when certain corporate actions occur, such as share adjustments, IPOs or M&As.

Once you know the index divisor, calculating the daily value of a cap-weighted index is easy:

Index value = Σ(market cap of securities)/index divisor

As an example, let’s revisit our hypothetical five-member index and set this as our index’s starting basket—that is, these are the stocks we’ll hold and the weights we’ll hold them in on Day 1:

 Stock Current Price Outstanding Shares Market cap Weighting A \$3 50 150 15% B \$1 50 50 5% C \$7 70 490 51% D \$9 20 180 19% E \$10 10 100 10% Total market cap 970 100%

Now let’s assume we want our index’s starting value to be a nice, round number—say, 100.0. That means we’d need to set our initial index divisor at:

Index divisor = total market value/ index value

= (970)/100.0

= 9.7

Our index divisor will stay at this value until something happens—a stock split, a security delisting, so forth—that would necessitate changing it.

Now let’s see how the value of our index changes when we look at it tomorrow:

 Stock Current Price Outstanding Shares Market cap Weighting A \$4 50 200 20% B \$1 50 50 5% C \$7 70 490 49% D \$9 20 180 19% E \$9 10 90 9% Total market cap 1010 100%

Index value = total market value/index divisor

= (1010)/ 9.7

=104.1

Our index value went up, from 100.0 to 104.1.

## Index divisors – not just for calculating value

Index divisors can be handy tools that allow us to gauge at-a-glance the effect a security price change will have on the total index. For example, in our example above, Security A gained \$1 on Day 2, while Security E lost \$1. That means we could expect a total change in index points of:

Total change in index points = (outstanding shares/index divisor)*change in share price * free float factor)

= ((50/9.7)*(\$1)*1.00) + ((10/9.7)*(-\$1)*1.00)

= 4.1

The index divisor can also be used to quickly calculate the change in the index’s total market cap:

Total change in market cap = change in index points * index divisor

= 4.1 *9.7

= \$40

## Time keeps on slippin’: Laspeyre vs. Paasche indices

As stated before, index values monitor price changes over time, which means that the interval of time observed can greatly impact your end value. It’s not always easy to know which time interval is most appropriate to measure. Should you calculate your index value based on some starting “base” basket and then see how the changes fluctuate from then on? Or should you set your “base” to what the index holds right now, and compare those prices to some earlier point in time?

The choice matters more than you think.

• The first example is known as a Laspeyre index (or base-weighted index),in which you calculate the change in prices of yesterday’s basket as they’ve changed from yesterday to today.
• The second is known as a Paasche index (or current-weighted index), in which you calculate the change in prices of today’s basket as they’ve changed from yesterday to today.

The distinction is easier seen than read:

For a Laspeyre index, any price changes in the underlying stocks are tracked on a daily basis—but changes in share quantity don’t factor in until the following day’s calculation.

For a Paasche index, however, those share quantity changes are baked right in; you start with a base basket that already incorporates the share quantity changes, then looks backward at what price changes have already occurred.

It seems like a fussy little distinction, but as you can see in the above simplified example, where your base is set does impact your index’s values.

Neither index is subjectively better than the other, and both types of indices are used in the industry; MSCI uses Laspeyre indexes, for example, while FTSE Russell prefers Paasche. Both have their advantages and tradeoffs. For example, a Paasche index may be more up-to-the-minute than a Laspeyre index, but it can tend to overestimate values as well (compared to a Laspeyres, which can underestimate).

In general, calculating a value for a Laspeyres index easier than for a Paasche one. If you already have price and share quantity data for your base Laspeyre basket, then all you need to calculate tomorrow’s value is new price data. For a Paasche index, however, you need new price and new share quantity data—which isn’t always as easy to collect.

## What happens when the number of shares in my index changes?

In the above examples, we’ve only looked at the effect of price changes, and held share quantity constant. But both Laspeyre and Paasche indexes will suffer discontinuities should the number of shares of their constituents change – say, if one security is dropped from the index and replaced by another, or if a security experiences a stock split or other corporate action that changes its quantity.

To avoid disruption, the index divisor (or price adjustment factor, as it’s called for some Laspeyre indexes) will need to be adjusted to ensure the numerator and denominator of their equations are still comparable.

For a Laspeyre index, which uses in its numerator yesterday’s quantities at today’s prices, the index must adjust today’s (closing) prices to make them comparable to yesterday’s.

For a Paasche index, however, whose denominator uses today’s quantities at yesterday’s prices, one must adjust yesterday’s (closing) prices to make them comparable to today’s.

How the divisor must be adjusted depends on action that takes place. To give you an idea how of this would work, let’s look at one fairly simple example, that of a stock split:

## A quick word about “chaining indexes”

Up until now, we’ve taken our index values relative to some fixed base period, but it’s usually more relevant to assume the base basket for your calculations is whatever the basket was in the time period that immediately precedes the current one. This is known as “chaining” an index. FTSE Russell, among others, chains their indexes.

Chaining can be done with either Laspeyre or Paasche indexes (or really any other index), and all that happens is that t=0 becomes t=n-1. Nothing else about the index changes.