Market Indexes

a wee note about Market Indexes

To create a Market Index we (normally) take the weighted average of N stock prices: P₁, P₂, P₃, ... P_N.
For the TSE 300, N =300 (the number of stocks in the average).

If the weights are called W₁, W₂, W₃ ... W_N, then the weighted average is just:

{ W₁P₁ + W₂P₂ + W₃P₃ + ... + W_NP_N }/{ W₁ + W₂ + W₃ + ... + W_N}

For example, if N = 12 and all the Weights = 1, then the Index is the simple, garden-variety, equally-weighted average that we all know and love, like so: ( P₁ + P₂ + P₃ + ... +P₁₂ )/12
... and (surprise!) that's the original Dow Jones Industrial Average (named after a fella called ... uh ... Dow Jones or ... well, take a look at Dow Stuff).

Okay, but the DOW is unusual. Often, the weights are market capitalizations for each company represented in the Index. For example, two communication companies represented in the TSE 300 are Aliant and BCE. (I picked these 'cause they look like they're the first two in the TSE 300, alphabetically speaking. They ain't, but it looks good, eh?)

Aliant:: the number of outstanding shares is 127.56M (M = million) and the stock price (as I write this, on April 1/00) is $39 so
the market cap is: 127.56M x 39 = $4,975M
BCE:: the number of outstanding shares is 643.8M (M = million) and the stock price is $181 so
the market cap is: 643.8 x 181 = $116,530M

Their presence in the TSE 300 might look like so:

TSE 300 Index = { (4975)(39) + (116530)(181)+ ... } / (4975+116530+ ...)

assuming that the TSE 300 is a standard market-cap-weighted index ... which it ain't. (See below.)

Now comes the heavy stuff for a standard market-cap-weighted index, as described above. (Time to turn back?)
I'll use Σ to represent sums, so
Σ(W P) means the sum W₁P₁ + W₂P₂ + ... + W_NP_N and
Σ(W) means W₁ + W₂ + ... + W_N and we get the simple prescription:
Market Index = Σ(W P) / Σ(W)

Of course, if the number of outstanding shares for each company is denoted by n₁, n₂, ... n_N, then the market capitalizations are n₁P₁, n₂P₂, ... n_NP_N. For example, if Aliant and BCE are the first two of the 300 companies in the TSE 300, then :

Aliant:: n₁ = 127.56M, P₁ = $39 and W₁ = n₁ x P₁ = 127.56M x 39 = $4,975M
BCE:: n₂ = 643.8M, P₂ = $181 and W₂ = n₂ x P₂ = 643.8 x 181 = $116,530M

Okay, since W = nP for each company [ mkt cap = (# shares) x (stock price) ] that means that

Market Index	= Σ(nP x P) / Σ(nP)
	= Σ(nP²) / Σ(nP) 'cause P x P = P²
	={Σ(nP²) / Σ(n) } x {Σ(n) / Σ(nP) } where we divide then multiply by Σ(n)
	={Σ(nP²) / Σ(n)} /{Σ(nP) / Σ(n) } instead of multiplying by the second factor, we divide by its reciprocal

You'll recognize this as the ratio of two weighted averages:

Market Index = Average (Stock_Price²) / Average (Stock_Price)

where the weights in each average are the number of outstanding shares, n. (I dunno 'bout you, but this surprised me!)

In fact, a Market Cap Weighted Index is normally non-standard; it doesn't use the standard formula for "weighted average of stock prices" as described above. In fact they are calculated by an even simpler prescription than
Market Index = Σ(W P) / Σ(W)

For example, I said that the TSE 300 Index used this prescription (normal market-cap-weighted). I lied. The TSE index actually uses an even simpler formula, namely:

TSE300_Index = 1000 Σ(n P) / Divisor

where the n is the number of outstanding shares (the "float"), P is the stock price and Divisor changes when a stock is dropped/added/etc. to the set of 300 stocks. (You'd think this'd be called float-weighted, but ...)

If we set 1000/Divisor = C (for sanitary reasons), we can write:

TSE300_Index = C { n₁P₁ + n₂P₂ + n₃P₃ + ... + n₃₀₀P₃₀₀ }

Now it's reasonable to ask (for example):

"What fraction of the Index does stock P₁ represent?"

Since P₁ provides a term C n₁P₁ to the sum, we can divide to get the fraction:

C n₁P₁/TSE300_Index

At this very moment (Sep 12/00) TSE300_Index = 10,557 and one component, namely Nortel has a stock price of
P₁ = $94.50 and the number of outstanding shares is about n₁ = 3,000,000,000 = 3 x 10⁹ (that's 3 billion!) so we get the fraction:

C (3 x 10⁹)(94.50)/10,557 = 27 x 10⁶ C

Now, if I only knew C = 1000/Divisor ... but since

27 x 10⁶ C = 27 x 10⁶ (1000/Divisor) = 27 x 10⁹/Divisor
is a fraction (less then "1", right?) then Divisor must be ... uh ...HUGE!

So, I got in touch with Standard & Poor's (they're in charge of computing the TSE 300 Index) and they kindly faxed me the current value of the Divisor (as of Sep 12/00). Are y'all ready for this?

TSE 300 Divisor = 92,251,469,343

Whooeee!

Where were we? Oh yes, the Nortel fraction (as of Sep 12/00) was:

(27 x 10⁹)/(92.251469343 x 10⁹) = .29 or 29%

P.S. This means - and I'll leave out the math - that a 1% change in P₁ results in a .29% change in the Index.

A convenient formula for computational purposes can be gleaned from the above result, namely:

C (3 x 10⁹)(94.50)/10,557 = (1000/Divisor)(3 x 10⁹)(94.50/10,557 )

so, we stick in the value of the Divisor and get:

Nortel_Fraction = 32.5(94.50/10,557) = 32.5 (Nortel_Price/TSE300_Index)

so, on Sep 13/00, when Nortel closed at 99.25 and the TSE300 closed at 10,750 the fraction was

32.5 (99.25/10,750) = .30 or 30%

Of course, that 32.5 is just for Nortel. However, if that number hasn't changed since Jan, 2000 (meaning the number of outstanding shares and the magic divisor haven't changed and no changes in the components of the TSE), then without Nortel, the TSE300 gains (to Oct 25/00) would look like so, where we've used the gains in TSE(1-Nortel_Fraction) as the Nortel_Fraction changed, day to day, because of the change in Nortel_Price:

Although the TSE-sans-Nortel would NOT change to TSE (1-Nortel_Fraction), because the divisor would change as well, nevertheless, the percentage changes in TSE (1-Nortel_Fraction) should be the same as the percentage changes in TSE-sans-Nortel because the divisor cancels out when computing percentage changes. In case the chart looks strange, on Oct 25, Nortel dropped about 26% on that day ... and the TSE dropped about 8% (like, about, 30% of 26%), but almost all of the 840 point drop in the TSE was due to Nortel. Without Nortel, the drop in the TSE299 would have been, maybe 60 points!

Okay, for any stock of the TSE 300 we got us a formula:

Stock_Fraction = (1000/Divisor) n₁P₁/TSE300_Index

so if we substitute Divisor = 92.251 (in Billions) and we stick in the Market_Cap = n₁P₁ (also in $Billions) we get the magic formula:

TSE300 Stock_Fraction = 10.8 Market_Cap/TSE300_Index

Example:

If
Royal Bank of Canada has a Market Cap of 27 Billion.
The TSE300 = 10,750.
Then RY accounts for 10.8 (27)/(10750) = .027 or 2.7% of the TSE 300 Index.

Neat, eh?

To summarize:

If n represents the number of outstanding shares traded (the "float") and P represents the stock price
and M = nP denotes the market capitalization, then:

DOW_Industrial_Index = A { P₁ + P₂ + P₃ + ... + P₃₀ }
S&P500_Index = B { n₁P₁ + n₂P₂ + n₃P₃ + ... + n₅₀₀P₅₀₀ }

TSE300_Index = C { n₁P₁ + n₂P₂ + n₃P₃ + ... + n₃₀₀P₃₀₀ }

where A, B and C are constants that change from time to time ... if a stock is replaced on the index or maybe a stock split.

Notes:

For the DOW, the value of A is about "5" so a $1.00 increase in any stock will increase the DOW by 5 points.
For the TSE300, n will double after a 2-for-1 stock split, but P will halve, so nP don't hardly change ... so there's no need to adjust the value of C.