Stock Indexes another kind of DOW

If we wanted to calculate some kind of weighted average of the numbers x, y and z we'd usually use:

[1]       A₁ = αx + βy + γz   where α + β + γ = 1.

>Sounds good to me.
In fact, it has the property that if any of the three variables change by some constant "c", the change in the A₁ is proportional to the weight assigned to that variable.

>So the weights assign some importance to the variable, right?
Yes. If we want x to be twice as important as y in generating the Average, we could make α = 2 β.
Indeed, if x, y or z increased by c, then A₁ would increase by cα , cβ and cγ respectively.

>That's good, right?
Sometimes ... for many situations, but maybe not if we're calculating an Index which measures the performance of three stocks: x, y and z.

Suppose you wanted our Index to change by an amount proportional to the weights when the variables changed by a constant percentage.
Then, how would you define the Index?

>I have no idea.

One good thing about the Average A₁ = αx + βy + γz?
If every component changed by 1% then this Average would change by 1%.

Here are the conditions we'd like our Index to satisfy:

1. If all components changed by r%, our Index will change by r%.
2. An r% percentage change in any one component will change our Index I by r%, multiplied by the weight attached to that component.

>I think it'd depend upon the weight attached to that component.
Okay, suppose that one component was weighted as 5.6% of the Index.

>Then I think the Index should change by just 5.6% of that 1%. Doesn't the garden variety A₁ do that?
No. It satisfies condition 1, but not condition 2. In fact, a change of r% in x would change A₁ by: α r x.
We're asking that our modified Index, I, change by: α r I.
That is, if all components changed by r%, then the TOTAL change in our Index would be r% ... according to condition 1.
However, if only one component changed, our Index should change by a fraction of the change in that component; the fraction being equal to the weight assigned to that component.

Okay, we'll do it this way: Our Index will be a function of x, y and z and the three weights α, β and γ.     I = f(x,y,z,α,β,γ)

1. To satisfy condition 2 we'll need:
   • f((1+r)x,y,z,α,β,γ) - f(x,y,z,α,β,γ) = α r f(x,y,z,α,β,γ) ... for small values of r.
   • Let Δx = rx so we can rewrite the equation as:   f(x+Δx,y,z,α,β,γ) - f(x,y,z,α,β,γ) = α r f(x,y,z,α,β,γ)
   • Write this equation like so:   Δf = α r f.
   • Divide this equation by Δx = rx to get:     Δf / Δx = α f / x
   • Let r → 0 so Δx → 0 and get: ∂f/∂x = α f / x     (with similar equations for y and z).
   • That'd give:     ∂ log(f) /∂x = α / x = α ∂log[x] /∂x ... so f = α log(x) + stuff independent of x
   • This implies that log[ f(x,y,z,α,β,γ) ] = α log[x] + β log[y] + γ log[z] + constant = log[ x^αy^βz^γ]+ constant
   • Now we have:     I(x,y,z,α,β,γ) = K x^αy^βz^γ
2. Condition 1 is automatically satisfied for this Index.
   • Replace x, y and z by (1+r)x, (1+r)y and (1+r)z and get: I_new = K (1+r)^α+β+γx^αy^βz^γ = (1+r)I_old     since α+β+γ = 1.

>Interesting, but what good is this Index?
Uh ... good question. Remember that the DOW is the equally weighted, garden variety average of 30 stock prices.
That is, DOW is the sum of the thirty prices multiplied by some DOW multiplier (which is modified each time there's a stock split so the DOW doesn't change abruptly).
Just think of what the DOW would be like if, instead of

      DOW = K [ P₁ + P₂ + ... + P₃₀ ] ... with some appropriate value for K
we'd use:
      gDOW = K [ P₁P₂P₃₀ ]^1/30 ... since all weights are equal to 1/30.

Note:
K is some magic DOW multiplier which changes the sum of prices into the DOW Index.
As I write this, the sum of 30 stock prices is $1293 and the DOW is 13,900 so the mutiplier is: K = 13900/1293 = 10.75

>gDOW? Why gDOW?
The g stands for geometric mean.
>Or gummy?

In the old current DOW Index, a 1% change in any stock price would change the Index by an amount proportional to the stock price
... so a 1% change in BA, with a current price of $77.80, would be over 4 times more important than a 1% change in INTC with a current price of just $17.50.

>So if you want your stock to be a VERY important component of the DOW, don't split 2-for-1, eh?
Exactly. I remember, in 1999, when American Express was the the Big Cheese

then AmEx split

Where-O-where is American Express now? See the DOW.

Let's see the DOW and the (modified) gDOW over the last twenty years (where we've chosen K, in the gDOW, so that they're equal ... 20 years ago):

>Hey! I like the new DOW!
Nice, eh? And the volatility is about the same, too.

In general:

The geometric Index for n stocks is: gI = K [ P1α P2β P3γ ... Pnω ] where the weights add to 1:   α + β + γ + ... + ω = 1.

>So you're talking the geometric mean of the prices, right?
Only if α = β = ... = ω = 1/N, as in the equally-weighted DOW. Interesting, eh?
>Not particularly.
But check it out.
Suppose that the N components changed by r₁, r₂ ... etc. and our Index changed from gI_old to gI_new.
Then:
    gI_old = K P₁^α P₂^β... P_n^ω
and (replacing each component by their changed values):
    gI_new = K (1+r₁)^αP₁^α (1+r₂)^βP₂^β ... (1+r_n)^ωP_n^ω
and dividing gives
    gI_new / gI_old = (1+r₁)^α(1+r₂)^β...(1+r_n)^ω
and that's the product of the "weighted" gain factors for each component.

>But if all the r's are 0 except the first, does your gIndex change by ...
Uh ... hold on. I did mention that we're talking small changes in the stock prices.
If that's the case, then (1+r₁)^α ≈ 1+αr₁.
That means that our gI changed by a percentage equal to αr₁, just like we wanted.

>We? What's with this "we"? It's just like you wanted. Besides ... where's the spreadsheet?
Here:

You pick 30 stocks as well as some Benchmark Index (like the DOW = ^DJI) and click a button to download Prices from Yahoo.
The spreadsheet calculates some neato stuff like Beta and Correlation with the benchmark and ...

>And all the weights are equal?
Uh ... not exactly. When you download the spreadsheet (by clicking on the picture, above) they're all eqal to 1/30 ... but you can change 'em in cells X1, Y1, etc.

Note:
It's "usual" to generate an Index (like the S&P500 ot the TSX) as the sum of the market capitalizations ... multiplied by some magic constant. (See: market Indexes).
These are often referred to as a Market Cap Weighted Index or (worser) a Market Cap Weighted Average of Stocks.
What on Earth does "Market Cap Weighted Average of Stocks" mean? Can you average "stocks"? What would be the "weighted average of people" or the "weighted average of bananas"?

If the mkt caps of three stocks were M₁, M₂ and M₃, then an Index proportional to their sum, namely: Index = C (M₁+M₂+M₃) would be a mkt cap Index.
It involves a simple sum of mkt caps ... yet it's called a "weighted average" (for reasons that escape me).
Nevertheless, if the first stock increased by r%, then this Index would increase by: C r M₁ and the fractional change in the Index would be: r M₁ / (M₁+M₂+M₃).
That's a change proportional to the "weight" attached to the first stock IF the mkt caps were regarded as the "weights".

Moral?
These Indexes should be called Market Cap Indexes ... with the words "weighted" and "average" annihilated.