Equal-weighted Indices

We consider introducing a market Index which, rather than being Market-cap Weighted, like the TSE or S&P, is Equal-weighted, so that we ...

>Remind me, please, of the difference.

Okay. We start with a bunch of money, namely \$A, and consider an N-stock Index.
We'll consider the following two scenarios:

1. We divide our money into N equal piles and buy equal dollar amounts of each stock in the Index. Now A/N dollars divided by the stock price, Pk, gives (for the kth stock purchase)
(1) ....       Sk = A/(NPk)     EQUAL weighted
shares of each stock in the Index (where the Price, Pk, varies from stock to stock, of course, so we buy a variable number of shares). We now have equal dollars invested in each stock.

2. A different tack is to buy, with our \$A, a number of shares which is proportional to the total number of outstanding shares of each stock in the Index. That is, if stock #1 has n1 outstanding shares and stock #2 has n2 outstanding shares etc. then we buy C n1 shares of stock #1 and C n2 shares of stock #2 etc. where "C" is a constant (which we'll determine momentarily). Our total cost is then
C n1P1+C n2P2+...+C nNPN = \$A
so the constant can be determined as:
C = A/ΣniPi
and the number of shares of each stock is:
(2) .....       Sk = Ank/ΣniPi     MARKET CAP weighted, like the TSE or S&P
where the number of shares, nk, varies from stock to stock, of course, so we buy a variable number of shares.

For sanitary reasons, we'll write simply ΣnP to mean n1P1 + n2P2 + ... + nNPN

>Wait! You call C a constant, yet it depends upon the stock prices which change ....
Uh ... yes. This constant is only constant long enough to generate our formula (2). Notice, however, that the amount of money we're investing in each stock is proportional to the market capitalization of that stock. We can see that clearly from:

C n1P1+C n2P2+...+C nNPN = \$A
where, for example, C n1P1 is invested in stock #1 and n1P1 is the market cap of that stock, and C n2P2 is invested in stock #2 and ..

>... and you rebalance the Index every minute, with every stock trade?
Pay attention. This is not our stock portfolio. This is a brand new market Index and, like an Index, its value changes minute by minute.

>Okay, I get it. Please continue.

For scenario (1), the EQUAL-WEIGHT Index, our Index begins with shares in each of N stocks according to:
Stock-Shares: Sk(0) = (A/N)/Pk(0)
Stock-Value: Vk(0) = A/N
Index-Value: V(0) = Σ Vk(0) = A*
where P1(0), P2(0), ..., PN(0), denote the N stock prices at time = 0.

* Of course, we may want to start our Index with a value, like 1000, so we'd choose A = 1000.

If the prices of the N stocks change from month to month (meaning time = 0 to time = 1 etc.) ...
>Or minute to minute ...
... from P(0) to P(1), for example, the value of each component changes too, according to:

Stock-Value: Vk(1) = (A/N)Pk(1)/Pk(0)     multiplying the number of shares by the new stock price
and the total Index-Value changes to:
Index-Value: V(1) = Σ Vk(1) = (A/N) ΣPk(1)/Pk(0)

The GainFactor (LossFactor?) in our Index is:

 V(1) / V(0) = (1/N) ΣPk(1)/Pk(0)

See? It's the Average GainFactor of all stock prices in the Index.

>zzzZZZ
Pay attention. Now we'll do the same for a MARKET-CAP WEIGHT Index.

Our Index begins with shares in each of N stocks according to:

Stock-Shares: Sk(0) = Ank/ΣniPi(0)
Stock-Value: Vk(0) = AnkPk(0)/Σ niPi(0)
Index-Value: V(0) = Σ Vk(0) = A
At the next time period, time = 1, we have:
Stock-Value: Vk(1) = AnkPk(1)/ΣniPi(0)     multiplying the number of shares by the new stock price
and the total Index-Value changes to:
Index-Value: V(1) = Σ Vk(1) = AΣnkPk(1)/ ΣniPi(0)

The GainFactor (LossFactor?) in our Index is:

 V(1) / V(0) = ΣnkPk(1)/ ΣniPi(0)

It's the Gain (or Loss) in the Total Market Capitalization of all stocks in our Index.

>Don't you have a single picture? A picture is worth a thousand ...
Okay, here's a picture of a hypothetical Nasdaq 10 Index where ...
>Nasdaq 10?
... where we consider the top 10 Nasdaq stocks (currently) and assume the number of outstanding shares haven't changed for the past ten years ...
>You're kidding, right?
... and calculate the average monthly gains in stock prices (for the EQUAL-WEIGHT Index) and the monthly gains in total market cap (for the MKT-CAP-WEIGHT Index) and rebalance each month and plot the evolution of the two indices:

>Okay, that's for something volatile. How about the S&P 500?
I just want to provide some feeling for the difference. I don't want to consider all the stocks on the Nasdaq or the S&P 500.

>Try some DOW stocks.
Good idea. Here's a picture of a DOW 10 Index.

>Fictitious, eh? You're assuming no change in number of shares.
Right.
>So? What's your conclusion?

Note the sensitivity of an Index with respect to changes in a stock price. If the number of shares of the N stocks are S1, S1, etc., then the Index is:

V = S1P1 + S2P2 + ... + SNPN     and we note that stock#1 represents a fraction S1P1/V of the Index
If the price of stock#1 changes by ΔP1 it generates a change in the Index according to the number of shares of that stock, namely:
ΔV = S1 ΔP1     so that     ΔV/V = { S1P1/V } ΔP1/P1
Note (again!) that S1P1/V is the fraction of the Index represented by stock#1 and a fractional change ΔP1/P1 in the price of stock#1 produces this fractional change in the Index: ΔV/V, as given above.
1. For an EQUAL WEIGHT Index each stock is equally represented in the Index. A 1% change in the price of stock #1 has exactly the same effect on the Index as a 1% change in stock#2 or stock#3 etc..
The little companies have just as much weight as the big guys.

2. For a MARKET-CAP WEIGHTED Index the stocks are represented in proportion to their market capitalization. The largest companies produce the most dramatic changes in the Index (like the S&P or TSE).
If the changes in stock prices of the largest companies are gradual (lethargic?), the Index will tend to change gradually.
 Because some of the smaller companies on our DOW 10 had big gains, the EQUAL WEIGHT Index done good. Look at Microsoft (MSFT) and Intel (INTC) and Citigroup (C) Back in 1990, these were the little guys with a total market capitalization of maybe 10% of the ten-stock total ... but look at their influence on an EQUAL WEIGHT Index !!
I should point out that the DOW Jones Industrial Index is different from either of the two Indexes we're considering. A stock's representation on the DOW is proportional to its price. Stocks with big prices have big influence. Intel, for example, has a small stock price so has little influence on the DOW ... in spite of the fact that it's one of the largest companies on the DOW.

>So? What's your conclusion?
Conclusion? I have no conclusion.
>Then why did you ...?
It's fun, interesting, edifying. Isn't it?
>No. Besides, suppose one of the stocks splits into three pieces? With equal-weighted you'd then have to buy equal dollar amounts of THREE stocks instead of one and that outfit would have TRIPLE the weight in your Index.
My Index? It isn't my Index. Besides, we could limit ourselves to just N stocks and pick just one of the three pieces. Then ...
>But wouldn't that cause an abrupt change in your Index?
Stop calling it my Index. Anyway, the multiplier "A", originally set to 1000 (for example), would be modified to eliminate any discontinuous change in the Index. That's what happens to other indices ... the Nasdaq 100 or DOW 30 or TSE 60. Stocks are added or deleted all the time. Originally the DOW just added the prices of a dozen stocks and divided by 12. How's that for a novel idea for an Index? Just the average price. Now there are thirty stocks on the DOW and this divisor keeps changing and it's now something like 0.2 and ...

>So? What's your conclusion?
Conclusion? I have no ...
>Right, I know! It's fun ... edifying. But which do you like best?
Market Cap Weighted. If you calculated an equal weight "supermarket index" according to the prices of bread and caviar, would changes in the prices of these items represent the change in the price of food? Or would you rather have a "sales-weighted" index which would reflect how much money is spent on these items. It's like the Consumer Price Index. It weights houses more than milk. It's like ...

>zzzZZZ