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Construction of indices: An investment index represents the

relative changes in the share/stock prices of the constituent companies or stocks which make up the index. The various methods of averaging those relative price changes are dealt with below.


here are many investment indices in use throughout the world,

T with each index having its own particular purpose and use. Each index will be constructed using certain eligibility criteria to determine the constituent companies from time to time.


Weighted arithmetic indices

Most investment indices are calculated on a weighted arithmetic average basis,


For investment indices, the weights used are

the market capitalisations of the constituents, usually the market capitalisations of the constituents at time 0. In the same way that a typical portfolio will be weighted towards the bigger companies, the index will be weighted towards the bigger companies.
The constant K is fixed when the index is first set up. Often it is fixed so that the index starts at a round number such as 100 or 1,000. For example, the FTSE 100 had a base value of 1,000 when it was started on 31 December 1983.


A new index is to be constructed in a share market that contains just two shares, in Companies A and B. At the starting date of the index, time 0, there are 100 shares in each company. Company A shares are priced at 3, whilst Company B shares are priced at 1.
. (i)  What is the market capitalisation at time 0? 

. (ii)  What are the weights to be used in the above formula? 

. (iii)  What value of the constant K is required if the index is to have a starting value of I (0) = 100 ? 

. (iv)  What is the value of the index at time 1 if the share prices have by then increased to 3.2 and 1.1 respectively?



The weights are updated

each time the number of shares issued by a constituent company changes and continuity is maintained by “chain-linking” the index on the new capital to that of the previous index.
It is now becoming common practice to restrict the weights to reflect the level of “free float” of shares available for purchase, thereby eliminating strategic holdings.
The free float of a share is the proportion of the shares that are freely available for purchase on the open market. It therefore excludes shares that are held for strategic purposes – eg by holding companies in subsidiaries – and are thus highly unlikely to be sold.


Chain-linking :The chain-linking process is set up so that:

  the index reflects the new market capitalisation of the constituents 

  the index value is not disturbed by the change in capital structure of its constituents. 
This is appropriate because if the aim of the index is to reflect the investment performance of its constituents, then the index value should change only in response to that investment performance and not in response to injections or withdrawals of funds into or out of the market itself.


Rights issues, for example, will

increase both the number of shares and the total market capitalisation. However, this does not mean that the index should be increased. After all, the value of a portfolio of shares would not jump upwards. For a portfolio to remain fully representative of the whole market, some of the holdings would have to be sold in order to take up an appropriate proportion of the rights issues.


List three circumstances in which chain-linking would be required.



Total return indices

In many cases, a measure of the total return, including income received, is required on an investment class. To provide this measure, a total return index is calculated from the capital index described above. Total returns can be calculated using ex-dividend adjustments or yield figures.


XD adjustment

The ex-dividend or XD adjustment represents the amount of income that has been received since the start of the year by the capital value index. It is assessed in the same terms as the main index. For example, on 12 December 2001, the index value for the FTSE 100 Index was 5,160.8. On the same day, the XD adjustment was 134.24 (ie 2.6% of the index value). In other words, by mid-December the dividend payments that had been paid by the FTSE 100 companies since 1 January 2001 were equal to 2.6% of their total market capitalisation.
To allow for the effect of investment income it is necessary to make assumptions about the time that the income is reinvested and whether it is reinvested net or gross of tax.
An allowance may also be required for the expenses of reinvestment.
Assuming that the dividend or interest payment is reinvested back in the index on the ex-dividend date, and that is it is added to the current market capitalisation, the corresponding increase in the index value would be the investment income divided by the base value,


The XD adjustment is t

he accumulated total of each constituent over the calendar year as each constituent company declares a dividend. It is returned to the value zero at the beginning of the year and a new accumulation is started.
This implicitly assumes that:
  dividends are subject to the rate of tax (if any) assumed in the calculation of the index 

  there are no expenses or losses incurred in reinvesting the dividends.
The figure calculated using the above expression is sometimes referred to as the holding 
period return, which is more generally defined as:


Explain how we can use the capital value index and the XD adjustment to estimate the total return yielded by an investment market over the period from time t to time t+1.



The holding period return is sometimes used

an approximation to the internal rate of return, but is inaccurate because it fails to allow for the fact that in practice part of the total return comes from the reinvestment of the income (d). However, by linking successive holding period returns together we can generate a total return index series. Thus, the total return index value at time t is related to the total return index value at time t–1 as follows:


Yield adjustment

For many published equity indices ex-dividend adjustments are not available but figures for the net dividend yield on the index are. The yield figures can then be used to estimate the dividend income over the period in order to calculate total return.

The income received over shorter periods can be estimated by taking the relevant proportion of the annual amount but this will only give an approximation to the true value as income is not generally received uniformly over the year.
Having estimated the income in this way, a total return index series can then be calculated as described above.


Unweighted arithmetic indices

An unweighted, or price-weighted, arithmetic index is the arithmetic average of the relative price changes of the constituents. It is unsuitable as a benchmark for dynamic institutional portfolios. 
Although such indices are rather crude and generally inappropriate for performance measurement work, several of the world’s most famous indices are constructed using this method, eg the Dow Jones Industrial Average, the Nikkei 225.


Geometric indices

A geometric index is based on the geometric mean of the relative price changes of the constituents


An unweighted geometric index is

easy to calculate as only price data is required. It gives an indication of short-term price movements, but is totally unsuitable as a benchmark for investment strategy or portfolio investment measurement.
For example, if the price of one constituent falls to zero, then so does the index. Hence, constituents need to be changed, when necessary, to avoid this happening.


Would an (unweighted) arithmetic index rise faster or slower than a geometric index in a rising market?



uses to which indices can be put include:

  a measure of short-term market movements 

  providing a history of market movements and levels. 
All of the types of index described above can be used to give at least a broad indication of both whether the level of a particular investment market has increased or decreased, and the extent to which it has done so. 

  as a tool for estimating future movements in the market, based on past trends. 
Analysing how the investment market, as represented by the index, has moved in the past is likely to help us to predict how it may move in the future. Such technical analysis is discussed in detail in Chapter 21. 

  as a benchmark against which to assess the investment performance of portfolios. 
We discuss how this may be done in Chapter 15 of this course. 

  valuing a notional portfolio. 
This was discussed in detail in Subject A301. Recall that the fund split used could reflect either the actual split between equities and bonds (broad discounted cashflow approach) or an entirely notional split (notional portfolio approach). 

  analysing sub-sectors of the market. 
Many market-wide indices are built up from sub-indices, each of which represents a particular sub-sector of the total market. As an example, recall that in Chapter 13 we described how the FTSE industry classification system breaks down the UK equity market into ten industry groups that are further divided into sectors.

  as a basis for index funds that track the particular market. 
As the name suggests, index tracker funds aim to track or replicate the performance of an investment index. Thus, in order to construct an index tracker fund, we need an appropriate index to track. We discuss index tracking and index tracker funds in Chapter 20 of this course. 

  to provide the basis for the creation of derivative instruments relating to the market or a sub-section of the market. 
Many standardised derivative contracts are based on widely used indices, as opposed to individual securities. Such contracts are particularly useful if we wish to use derivatives to manage the risks arising from movements in an entire investment market. 
In particular for government bond indices:
  a standard against which yields on other fixed interest investments can be assessed 

  approximate valuation of a fixed interest portfolio.
The price indices can be used to obtain a quick, albeit approximate, valuation of 
a fixed interest portfolio without reference to all of the individual bond prices. 

  providing a picture of general yield structures of fixed interest investments. 
Yield indices provide a quick summary of the yield curve. 

  Yield indices allow comparison to be made with yields on ordinary shares as a measure of the yield gap between bonds and equities. 
Again, this is because yield indices summarise the yield curve.


Relevance of indices

  The investor’s investment objectives may be specified with reference to one or 
more indices – eg track or outperform a particular index. 

  Indices can be used in the development of the appropriate portfolio to best achieve those investment objectives – by helping us to predict the future possible returns on a particular investment market or sector. 

  Indices can be used to value the portfolio and can provide a benchmark against which to monitor its performance – as part of monitoring the investment experience. 
Fit for purpose 
When using any index it is vital to ensure that the index chosen is fit for purpose. In particular where an index is used to model a real world situation, it is important to ensure that the constituents of the index are a good match for the real world situation being modelled. 
This is particularly true when using indices compiled by commercial institutions. Such indices may have been established primarily for the sponsoring institution’s internal purposes, and later put into the public domain as a marketing tool. 
One example is that corporate bond indices produced by UK merchant banks often have a greater exposure to debt from financial institutions than does the bond market as a whole. Using such an index without care during the bond market turmoil in 2008-2009 could have generated significant model risk.


FTSE UK Index Series

This is a series of indices covering the whole quoted UK equity market. All the indices used to be calculated on a weighted arithmetic average basis with the market capitalisations as the weights. As such they may be suitable for performance measurement purposes.
From June 2001 the weightings of all FTSE constituents were altered to reflect the availability of stock in the market. Where the actual “free float” is 5% to 15%, this percentage (rounded up to the next whole number) is used. Otherwise weightings are in bands according to the next higher of 20%, 30%, 40%, 50%, 75% and 100%.
The free float weighting of a share is the proportion of the company’s equity that is available for trading by the public. It therefore excludes any strategic holdings by holding companies in subsidiary companies.
In addition to the capital and total return index numbers:
  average net dividend cover 

  actual dividend yield 

  price earnings ratio 

  ex-dividend adjustment 
are given for each of the indices in the series. 
The dividend cover and dividend yield are based on the most recent year’s profits and declared dividend figures respectively, updated for interim changes and for any statements by companies forecasting future earnings and dividends. A euro index value is also computed. 
Data for the FTSE UK Index Series is published each day in the Financial Times. The indices are calculated by FTSE International Limited. Note that since July 1997, the total return index has been calculated on a net basis – ie assuming reinvestment of net dividends. 


FTSE 100 Index

This consists of the 100 largest quoted companies by market capitalisation, accounting for about 80 per cent of the total UK equity market capitalisation. It is the main indicator of short-term movements in the UK equity market and is used extensively as a basis for investment products, such as derivatives and exchange- traded funds. 
The capital value is the one that is widely quoted. It was started on 31 December 1983, with a base value of 1,000. At the close of play on 31 December 2014, the value was 6,566.10. 
The index is calculated on a real-time basis, ie continuously during the day, effectively every minute, and is based on the last trade prices. For continuity and administrative reasons, the constituents are changed once a quarter in accordance with predetermined rules. The index is also used as a basis for stock index derivatives, namely the FTSE 100 Index Futures and the FTSE 100 Index Options. 


 FTSE 250 Index

This index covers the 250 companies ranking below the top 100 companies by market capitalisation. 
Its constituents are also changed every quarter. It is calculated on a real-time basis and is another base for stock index derivatives.


 FTSE 350 Supersectors Index

This index combines the 100 and 250 indices. 
The 350 Index constituents account for over 90 per cent of the total UK equity market. Sub-indices are also calculated for high yielding and low yielding stocks.


FTSE SmallCap Index

This index covers all companies below the top 350 companies with a market capitalisation greater than a certain limit and whose shares are actively traded. It currently represents around 2% of the UK market capitalisation. 
The number of constituents is not fixed as with the previous indices – the chain-linking process enables the number of constituent companies to be varied. The index is calculated at the close of each day. 


 FTSE All-Share Index

This comprises the 350 and the SmallCap indices. It accounts for around 98 – 99 per cent of the total overall market capitalisation, and is calculated and updated every minute. 
Sub-Indices are calculated for industrial sub-sectors in accordance with the FTSE industry classification system described in the previous chapter.


FTSE Fledgling Index

This index consists of the remaining, sufficiently marketable, quoted companies which are too small to be included in the SmallCap index. 
As with the SmallCap index, each of the constituent companies must fulfil eligibility requirements concerning the level of active trading in their shares.


 FTSE Aim Index Series

This series of indices covers companies traded in the UK Alternative Investment Market. These are companies which are too small or too new to apply for a listing. The indices are calculated on a free-float basis.


Dow Jones

It is a very crude index because:
  it is based on only 30 shares 

  there is no weighting by market value. 
The basic principle of calculation is that you add up the share prices of all the constituents and divide by 30. Therefore, if the price per share of the company with the most costly share rises by 10%, it will have a disproportionate impact on the index – even if the company happens to be the smallest of the thirty companies. 
The divisor does get adjusted to reflect capital changes such as scrip issues. Consequently, the divisor now bears no resemblance to 30. However, the basic principle of summing the prices of the thirty shares and dividing by a constant remains. This is the best known and most widely quoted of the New York indices.
The Dow Jones Industrial Average, commonly known as the Dow Jones index, is made up of 30 shares. It is an unweighted arithmetic index. It is therefore unsuitable for performance measurement calculations.
It provides a quick guide to shares in the industrial sector, but it is not representative of the American equity market as a whole. It is, however, very widely reported.