Non-Market Capitalization Weighted Indices

Non-Market Capitalization Weighted Indices

A non-market capitalization weighted index (also referred to as a non-market cap or modified market cap index) is one where the weights of the index constituents are not determined by the market capitalization weighted methodology. For example, the weights are user-defined. Between index rebalancing, most corporate actions generally don't affect index weights, as they are fixed. As the constituents' price move, their weights will change accordingly. Therefore, a non-market capitalization weighted index must be rebalanced from time to time to reestablish proper weighting.

The overall approach to calculating non-market capitalization weighted indices is the same as the capitalization weighted indices. However, the constituents' market value are set to a value to achieve a specific weight at each rebalancing that is different from a purely circulating-adjusted market capitalization weighting.

The index level of a non-market capitalization weighted index is calculated by:

$${Index Level}=\frac{{Index Value}}{{Divisor}}$$

The IndexValue is calculated by:

$${Index Value}={\sum_{i} P_{i} * Q_{i}}$$

Where the IndexValue is the market value of an index with all constituents' circulating market capitalization; Pi is the price of each constituent in the index, Qi is the quantity of each constituent's circulating supply, and i denotes a constituent.

However, to calculate a non-market capitalization weighted index, the market value for each constituent in the index is redefined so that it has the appropriate weight in the index at the initial calculation date and also at each rebalancing date. The new adjustment factor used to establish the appropriate weighting is called Weight Adjustment Factor (WAF).

Therefore, the constituent market value is calculated by:

$${CV}_{i}={P_{i} * Q_{i} * WAF_{i}}$$

Where CVi is the market value of constituent i, Pi is the price of constituent i, Qi is the quantity of the circulating supply of constituent i, and WAFi is the adjustment factor of constituent i.

Therefore, the index level is calculated by:

$${Index Level}=\frac{\sum_{i} CV_{i}}{{Divisor}}$$

And, the WAF for each index constituent i on rebalancing date t is calculated by:

$${WAF}_{i,t}=\frac{{AW}_{i,t}}{{W}_{i,t}}$$

Where AWi,t is the assigned weight of constituent i on rebalancing date t as determined by the weighting rule of the index in question, and Wi,t is the original weight of constituent i on rebalancing date t based on the circulating-adjusted market capitalization.

The index divisor is defined based on the index level and market value. The index level stays unchanged at the rebalancing. But the divisor will change at the rebalancing, since prices and circulating supplies will have changed since the last rebalancing.

Therefore, the divisor is calculated by:

$${{Divisor}}_{after}=\frac{{Index Value}_{after}}{{Index Value}_{before}}$$

An example

Let's assume we are constructing a non-market capitalization weighted index named XYZ, composed of three constituents X, Y, and Z with the user-assigned weights of 25%, 35% and 40% for X, Y, and Z, respectively. The price and quantity of circulating supply for each constituent are shown in the table:

Constituent	Price (USD)	Quantity
X	100	2,000,000
Y	200	5,000,000
Z	300	8,000,000

The original weight of each constituent before adjustment is shown below:

Constituent	Original Weight
X	5.555%
Y	27.778%
Z	66.667%

The weights after adjustments are:

Constituent	Adjusted Weight	WAF
X	25%	4.50
Y	35%	1.26
Z	40%	0.60

We set the divisor to 36,000,000 USD. Therefore, the index level of XYZ is:

$${XYZ}=\frac{(100*2000000*4.50)+(200*5000000*1.26)+(300*8000000*0.60)}{36000000} = 100$$

Equal Weighted Indices

An equal weighted index is one where the index constituents have the same weight. As the constituents' price change, their weights will shift and break the exact equality. Therefore, an equal weighted index must be rebalanced from time to time to reestablish the equal weighting.

The overall approach to calculating equal weighted indices is the same as the capitalization weighted indices. However, the constituents' market value are set to a value to achieve a specific weight at each rebalancing that is different from a purely circulating-adjusted market capitalization weighting.

The index level of an equal weighted index is calculated by:

$${Index Level}=\frac{{Index Value}}{{Divisor}}$$

The IndexValue is calculated by:

$${Index Value}={\sum_{i} P_{i} * Q_{i}}$$

Where the IndexValue is the market value of an index with all constituents' circulating market capitalization; Pi is the price of each constituent in the index, Qi is the quantity of each constituent's circulating supply, and i denotes a constituent.

However, to calculate an equal weighted index, the market value for each constituent in the index is redefined so that it has the same weight in the index at the initial calculation date and also at each rebalancing date. The new adjustment factor used to establish the appropriate weighting is called Equal Weighted Adjustment Factor (EWAF).

Therefore, the constituent market value is calculated by:

$${CV}_{i}={P_{i} * Q_{i} * EWAF_{i}}$$

Where CVi is the market value of constituent i, Pi is the price of constituent i, Qi is the quantity of the circulating supply of constituent i, and EWAFi is the adjustment factor of constituent i.

Therefore, the index level is calculated by:

$${Index Level}=\frac{\sum_{i} CV_{i}}{{Divisor}}$$

And, the EWAF for each index constituent i on rebalancing date t is calculated by:

$${EWAF}_{i,t}=\frac{{1}}{{N}*{W}_{i,t}}$$

Where N is the total amount of the constituents, and Wi,t is the original weight of constituent i on rebalancing date t based on the circulating-adjusted market capitalization.

The index divisor is defined based on the index level and market value. The index level stays unchanged at the rebalancing. But the divisor will change at the rebalancing, since prices and circulating supplies will have changed since the last rebalancing.

Therefore, the divisor is calculated by:

$${{Divisor}}_{after}=\frac{{Index Value}_{after}}{{Index Value}_{before}}$$

An example

Let's assume we are constructing an equal weighted index named XYZ, composed of three constituents X, Y, and Z. The price and quantity of circulating supply for each constituent are shown in the table:

Constituent	Price (USD)	Quantity
X	100	2,000,000
Y	200	5,000,000
Z	300	8,000,000

The original weight of each constituent before adjustment is shown below:

Constituent	Original Weight
X	5.555%
Y	27.778%
Z	66.667%

The adjusted weights are:

Constituent	Adjusted Weight	EWAF
X	33.333%	6.00
Y	33.333%	1.20
Z	33.333%	0.50

We set the divisor to 36,000,000 USD. Therefore, the index level of XYZ is:

$${XYZ}=\frac{(100*2000000*6.00)+(200*5000000*1.60)+(300*8000000*0.50)}{36000000} = 100$$