In this workout. We are asked to construct an equally weighted a price weighted and a market capitalization weighted index for the following four stocks, assuming the index value at the start of the period is 1,000. So we've got a four stocks Typhon and kidna. Sila and stymphalian and we've got the share price at the start of the period and then we've got the number of shares they have and we've got a share price then the period and the same number of shares for each company. Before we go into analyze the performance of an index. Under these three different index construction methods. Let's just have a look at the performance of each stock on its own. So we take the type 1 ending share price and divided by the beginning share price and take one away and put a percentage of cell style around that then we can see the Titans gone up by six percent. a kidnap 10.3 Sila by 15.4 and stemphalion by 33.3 so these different companies have performed quite differently in terms of their overall return and the index is going to average those returns out some way.

The first construction method that we're going to look at is an equally weighted portfolio. Now what we have here to begin with is a index that starts offered a thousand points, and we need to identify how many shares of each constituent company. We need to hold to give us an equal weighting. So the way to do that is to just take the overall index points and divide it by the four being the number of stocks that we have. To show that each of our stocks needs to represent 250 index points.

One way we can think about this is that each of these points represents one dollar and as a result we could then identify how many shares we need to hold to give us that 250 points or 250 dollars worth of value for type 1. So 250 index points 250 dollars worth of points. If you like are going to be represented across these shares that have 25 dollars as their share price. So as a result, we need to hold 10 shares in Typhon at 25 each to give us 250 points. if we then do the same thing for Echidna and seller and stymphalian you can see most dramatically for stymphalian where they have a very low share price only 0.3. They need to hold a far greater number of shares 833.3 shares to give us this 250 of value. So we have a different number of shares for each company in our index. So that all together they have the same weight in the index.

But with any to do to identify the value of the index at the end of the period. Is take the number of shares that we have. And multiply it by the share price at the end of the period. So type ones share price has gone up. At the beginning of the period we had 10 shares in our portfolio that represents the index.

10 shares each of them were 25 again with the 250 event. Now we still got 10 shares in the index, but they now given our 26.5 each give us 265 of index points. And if we copy that down across the entire portfolio, we can see that each of these has grown in value. and if we just add them all up we will then get the overall value of the index at the end of this period now we can see that the value index has increased but other things to notice this equally weighted index at the end of the period is now no longer equally weighted. This is a drawback of those equally weighted indexes in that. They need to be rebalanced back to the equally weighted over time. If we look at the return the indexes generated. Value at the end divided by value the beginning minus one.

This will give us a return of 16.2% and if we look at the returns of the individual companies, we can see it broadly sits somewhere in the middle with biased up with a little bit towards Tim failing with their really high 33.3% return. But also brought down by the 6% of python.

Okay, so that's 16.2% and average a straightforward average of the returns of the four constituents in the index. If we then gone to look at the next index construction method, which is a price pointed index. The way that this index works is that we effectively have one share of each of the constituent companies within our index. So the share price we can pick up of each individual company and then we can just copy that down. What we now need to identify is the weight that each company has in the index. And since we are with a price weighted portfolio effectively constructing a portfolio where we have one share of each company within our portfolio all we need to do to get the weights. Is to add up the component parts to get if you like the overall value of our portfolio that contains one share of each of these companies.

So the overall value of that portfolio where we have one share of each of the constituent companies in our portfolio would be the 42.5. Well we now need to do is to convert these prices of our shares prices of the constituents in our portfolio into a number of index points for those constituents. So we're effectively saying what waiting to these have in the index and we can say that. Type 1 has 25 dollars out of the overall 42.5 dollars value of the portfolio and if we convert that into a number of index points by multiplying it by the overall value of the index at the beginning of a thousands. That will tell us that type 1 is going to end up with 588.2 index points. Over half of the index therefore being represented by Typhon company because they have the highest price per share. And if we copy this down we can then say that there's much smaller weights from the other companies stim failure. For example only has 7.1 index points less than 1% of the weight in the index comes from this one company. if we then identify the share price at the end of the period for each company, that's just Brought down from above the way that we can now calculate our index points is just to look at the growth in the share price and apply it to our index points. So if we take the ending share price divided by the beginning share price and multiply that by the number of index points, this is giving a growth rate to those index points.

And if we add those all up we can then get to a closing index value. Now you can see properly straight away that the growth in the value of this price weighted index is much lower than the growth of the equally weighted Index. This index is only shown a return of 8.7% But this is because we are biased towards companies that have the higher share price. most notably Typhon And the type on individual stock price performance is only 6% the lowest of all the four companies. So we're heavily biased towards the company that has performed most poorly and as a result, we get a low oval return for our price weighted index. this indicates that price waited indices are biased towards companies which have a high stock price.

Our final kind of index is a market capitalization weighted index and we need to follow broadly the same approach as the price pointed index but our portfolio that represents the index is not made up of one share of each of the constituent companies, but rather it's effectively made up of all of the shares. each of those constituent companies So to derive our weightings, we need not the share price, but the market capitalization. So we need to take for Typhon the opening share price and multiply that by the opening. number of shares To give us the market capitalization for a Typhon. We can then copy that down for all of the four companies.

And then to get the overall value of this portfolio that represents the index that has all of the shares of the four constituent companies. This portfolio would then be worth $413,500? We can convert this into index points. by taking the market capitalization of each company and dividing it by the overall market capitalization of the whole portfolio of the whole markets. and multiplying that by our opening number of index points and here we can see that's Typhon does have the biggest market share still but it's not as big as for the price waited index fill up has a substantial degree of influence. We just go back to look at why that is the case. They do have a lower share price. And Typhon, but a much greater number of shares in issue. Which helps explain why they have that lower share price.

So if we now go on to calculate the performance of the market capitalization weighted index we need the market cap at the end of the period. Which we can get by taking the share price and multiplying it by the number of shares at the end of the period for each of our four companies.

And then we can use that to calculate a growth rates. So we take the ending value divided by the beginning value. That's one plus our rate of return if we multiply it by the opening number of index points. And then copy this down we can apply it to all of the companies and add it all up to get our closing index.

The return we get for the index in this instance with our market capitalization weighted index. Is now 11.7% Much higher than we got for the price weighted index because we now have a much greater weight coming through from the Silla Corporation.

Or from Sila gmbh. And silat gmbh generated the second highest rate of return at 15.4% So in the market capitalization weighted index, we're getting a much greater influence coming through from Cilla with their relatively High rate of return to give us this higher return of 11.7%