In computer science, a number, usually a reasonably small positive integer, sometimes zero, used to indicate the position within an array that an instruction is intended to access.

An example, in BASIC:

```
DIM A(10) ' Dimension an array A to contain 10 elements
FOR I = 1 TO 10
A(I) = I ' Set each element of A equal to its location index
NEXT I
```

Using A(I) to access, say, the 2nd element of array A when I = 2, is called indexing the array.

The index of a subgroup H of a finite group G is the number of left (or right) cosets of G modulo H. The index is often written as [G:H] but there is no reason to introduce this extra notation because [G:H] = o(G)/o(H), where o(X) is the order of (number of elements in) the group X. The fact the [G:H] = o(G)/o(H) is called Lagrange's Theorem, but I see no reason that such a trivial fact should be named after someone.

If x is an element of G, the left coset of x modulo H, xH, is the set of all elements xh where h is in H. The right coset, Hx, is the set of all elements hx where h is in H. The left coset space, G/H, is the set of all distinct left cosets modulo H.

Proof that [G:H] = o(G)/o(H):

For any element g in G, gH contains exactly o(H) elements. This follows from the fact that gx = gy implies that x = y (see my writeup in group theory).

If f in G is not in gH, then no f' in fH can be in gH. This is easy to see. Assume some f' is in gH but f is not. Let f' = fh1 and an element in gH be gh2. If fh1 = gh2, then f = gh2h1-1 so f is in gH, a contradiction.

Therefore o(G) = [G:H]o(H) and Lagrange's Theorem is verified.

Stocks are among the most popular instruments to invest money in. There are many different stocks; even in small countries like the Netherlands, there may be more than 100. Now, for many reasons, people want to have an idea how well the general stock market is doing. For this reason, people have invented the stock market index, or index for short.

An index is the value of a collection of stocks. The publisher of the index, often the stock exchange, selects a certain number of companies. This number is often fixed, such as 225 in the Nikkei index, 25 in the AEX index, or 100 in the FTSE index. Of these companies, a certain number of shares is bought. We then add the value of all these shares together to come up with a value of the index.

Let’s build or own index to illustrate this. To keep things simple, we want to build an index out of only three companies, but these three should be representative of three major sectors of the economy, namely energy, banking and IT. Let’s pick Exxon, Citigroup and Google as an example. Now, we could make an index by just taking one share of Exxon, one of Citigroup and one of Google. Exxon is worth \$70.12 a share, Citigroup is worth \$ 5.23 a share, Google is worth \$ 464.75 a share, making our index worth \$540.10. Now, imagine that Citigroup were to double overnight. This is of course great news for many people owning the stock, and should have a large impact on our index. Unfortunately, our index will rise by less than 1%: it will be worth \$ 70.12 + \$ 10.64 + \$ 464.75 = \$545.33.

The solution to this is by taking multiples of each stock. Let’s say we take one Google share, add 10 shares of Exxon to that (because Exxon is a bigger company) and add 50 shares of Citigroup (because Citigroup is a lot smaller – around 20 times smaller than Exxon at the moment I’m typing this). This would make our index worth 1427.45.

One finishing touch we could add is to scale our index a bit. This is often done in practice. If one, for instance, has 50 shares, adding even a small number of each share produces an index value that is very large. Therefore, many indices are determined as a set of stocks that make up a certain number of indices, for instance 100. In our case, we could say that our stock basket – that’s the technical term for it – of 28 shares of Google, 602 shares of Exxon and 5895 shares of Citi. This package of shares is almost \$ 100.000, and we could say we have 100 indices of 1000.

With this, we can compute both how much one cent and one percent move in one of the stocks changes the index. For instance, a 1 cent move in Google moves our index by 28/100, or 0.28 points, while a 1 cent move in Citi moves our index by 5895/100, or 5 points. A 1 percent change in Google, on the other hand, changes our index by 28 * 0.01 * 464.75/100 = 2.7 points, while a 1 cent change in Citi changes our index by 2895 * 0.01 * 5.23 /100 = 3.8 points. We hence see that the weight of Google and Citi in the index is nearly the same, even though the share price of both companies is very different.

Having seen how the value of an index is computed, we essentially know what it is. The next step is trading one. It’s all nice to hear the Eurostoxx 50 index rose by 13 points, but how can we profit from this? There are three common ways of going long an index.

The first and most obvious way is just buying all the shares in an index, in the right proportion. This is a bit of a hassle for three reasons. Firstly, we might need to trade shares in a lot of different companies. That is a lot of work, and also costly, as exchanges often charge a fee per trade. Buying shares in 225 companies, like the Nikkei has, means paying this fee 225 times. Other transaction costs could also accrue. Secondly, as mentioned, one basket is often multiple times the index. For instance, the AEX index is currently at 300. However, the basket is 100 times the index – and that basket contains a fractional number of shares, so to be precise, we need 4 basket, or 120.000 euros to buy all shares. That’s a considerable amount of money, and often too much to invest in a single index. Thirdly, the composition an index can change as companies shrink, grow, get taken over or go bankrupt. This means that keeping the basket up-to-date requires work. As a final point, not all indices even have futures.

The second, and far more common way, is to buy a futures contract on the index. Such a futures contract gives us the right and obligation to buy the index at a point in time in the future, for the price we currently pay. In practice, what happens is that we receive or pay money every day for changes in the index, in order to build a buffer. There are other fine points to trading futures, such as margining, the interest and dividend exposure of futures, and the fact that futures will only give the exposure until the time when we are in effect “buying” the index. These points are deserving of a write-up on their own, and those who don’t understand these finer points are well advised to stay away from futures, as it is really easy to lose an enormous amount of money. A second, more pragmatic point, is that futures are often a few times the index. For instance, one DAX future is 25 DAX indices, so it is an exposure of around 25 * 5500 = 138.000 euros. Professionals love futures, but for private investors, they are not the instrument of choice.

A third way of buying an index is buying a tracker. In essence, when buying trackers, someone will use the money to buy index baskets. Because trackers are bought and sold in large amount, economy of scale applies, and the person buying and selling the baskets can do so in such large volumes that his costs per basket are modest. Buying a tracker does have a disadvantage: an annual fee is charged. This fee, however, is typically small, around 0.2 to 0.5% of the value of the tracker. For a private investor, this is typically a lot less than the trading costs that would be incurred by buying the basket. For additional convenience, trackers may represent only a fraction of one index. For instance, it may require 100 27-euro Eurostoxx trackers to "build" one Eurostoxx index. Because trackers pay dividends at different times than the stocks in the basket, there might be a slight difference in value between the set of trackers and the index. We'll see below that this is not a real issue in practice. It is noted that note all indices have trackers; on the other hand, some indices have multiple trackers.

One might ask whether all these three products have the same value. The answer is a rather firm yes. There are many market participants who can compute the value of a future rather precisely from the index basket and vice-versa. Something similar goes for trackers. If a profit can be made by for instance selling the future and buying the basket, they will do this. Because there are so many people watching this, even a tiny profit is enough to trigger a trade, as otherwise, someone else will do it. People doing this are called arbitrageurs. They typically trade huge volumes for tiny profits, guaranteeing an efficient market and receiving a modest profit from it.

We now know what an index is, and even how to trade one. This is quite useful to know, because there are many different indices. In general, indices are sorted by four criteria:

• Location: This can be one or more countries, a continent or the entire world. For instance, the DAX-index contains German stocks, whereas the Eurostoxx –index contains Euro-denominated stocks
• Size: of the companies: There are many indices for big companies, but also indices for mid cap and small cap stocks. For instance, the MDAX index consists of German stocks that are one size too small to be in the DAX-index
• Sector: There are many sector indices, such as the Nasdaq-index (technology), but also gold mine indices.
• Product: In the discussion above, I’ve focused on stock indices. Stock indices are not the only type of index around; we can build an index around a pool of bonds to create a bond index, for instance.

In summary, indices are tools to indicate how (a part of) the (stock) market is behaving. Investing in indices is not straightforward, but handy proxies for an index may exists. Investors should not only decide what index they want to invest in and for what size, but also what proxy to use.

One small thing: this isn’t investment advice. So don’t come complaining when you make an investment decision based on this write up and it goes wrong. By the way, exchange websites provide excellent

Sources:

• I looked up the present stock values at Yahoo Finance.

In"dex (?), n.; pl. E. Indexes (#), L. Indices (#)(&?;). [L.: cf. F. index. See Indicate, Diction.]

1.

That which points out; that which shows, indicates, manifests, or discloses.

Tastes are the indexes of the different qualities of plants.
Arbuthnot.

2.

That which guides, points out, informs, or directs; a pointer or a hand that directs to anything, as the hand of a watch, a movable finger on a gauge, scale, or other graduated instrument. In printing, a sign [⇒] used to direct particular attention to a note or paragraph; -- called also fist.

3.

A table for facilitating reference to topics, names, and the like, in a book; -- usually alphabetical in arrangement, and printed at the end of the volume.

4.

A prologue indicating what follows. [Obs.] Shak.

5. (Anat.)

The second digit, that next to the pollex, in the manus, or hand; the forefinger; index finger.

6. (Math.)

The figure or letter which shows the power or root of a quantity; the exponent. [In this sense the plural is always indices.]

Index error, the error in the reading of a mathematical instrument arising from the zero of the index not being in complete adjustment with that of the limb, or with its theoretically perfect position in the instrument; a correction to be applied to the instrument readings equal to the error of the zero adjustment. --
Index expurgatorius. [L.] See Index prohibitorius (below). --
Index finger. See Index, 5. --
Index glass, the mirror on the index of a quadrant, sextant, etc. --
Index hand, the pointer or hand of a clock, watch, or other registering machine; a hand that points to something. --
Index of a logarithm (Math.), the integral part of the logarithm, and always one less than the number of integral figures in the given number. It is also called the characteristic. --
Index of refraction, or Refractive index (Opt.), the number which expresses the ratio of the sine of the angle of incidence to the sine of the angle of refraction. Thus the index of refraction for sulphur is 2, because, when light passes out of air into sulphur, the sine of the angle of incidence is double the sine of the angle of refraction. --
Index plate, a graduated circular plate, or one with circular rows of holes differently spaced; used in machines for graduating circles, cutting gear teeth, etc. --
Index prohibitorius [L.], or Prohibitory index (R. C. Ch.), a catalogue of books which are forbidden by the church to be read; the index expurgatorius [L.], or expurgatory index, is a catalogue of books from which passages marked as against faith or morals must be removed before Catholics can read them. These catalogues are published with additions, from time to time, by the Congregation of the Index, composed of cardinals, theologians, etc., under the sanction of the pope. Hook. --
Index rerum [L.], a tabulated and alphabetized notebook, for systematic preservation of items, quotations, etc.

In"dex (?), v. t. [imp. & p. p. Indexed (?); p. pr. & vb. n. Indexing.]

To provide with an index or table of references; to put into an index; as, to index a book, or its contents.

In"dex, n.

The ratio, or formula expressing the ratio, of one dimension of a thing to another dimension; as, the vertical index of the cranium.

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