Indices

This page is all about indices (Note: the plural of "index" is "indices").

An index simply measures the performance of a group of securities.  The membership of securities in the group may change from time to time, according to the circumstances of each different security.

Indices are usually used for the purpose of benchmarking (see the Benchmarks page for further details).  For an equities fund, one might simply use an equities index as the benchmark.  For bonds containing multiple asset classes, one might use a weighted sum of different indices to create the benchmark.

Even though indices and benchmarks are separate topics, it is usually true that a good index is one that is suitable for use as a benchmark.

Basic Principles of Index Calculations

From the perspective of performance measurement and attribution, the salient information about any index falls mostly into the following three categories:

1. The list of securities that are contained in the index.  This list may change from time to time, as securities are added to or deleted from the index.  A security might enter the index because it is listed on the exchange (or, in the case or a fixed income security, because it is issued).  A security may leave the index because it ceases to trade on the exchange (or, in the case of a fixed income security, because it matures).  These reasons for entering of leaving an index are relatively objective.  However, there is another whole category of subjective reasons why a security might enter or leave an index.  These subjective reasons often apply when committees make decisions about what securities should be in an index from time to time.
2. The weight for each security in the index on each day.
3. The security-level daily returns.

A fourth characteristic that is relevant mostly for international indices is currency.  Each security is valued in its local currency (e.g. US Dollars for Google, and British Pounds for Vodafone).  However, when the index includes securities with more than one different local currency, there is a need to express the valuations and returns for the index in some common form.  This can be done by converting all the security-level data into one particular currency.  This can be done at the spot rate, or (to add a bit more complexity) it can be done by a hedging calculation (look here for an explanation of currency hedging).  Normally, investors in each different currency would be interested in using indices that have been converted (either using the spot rate or a forward rate) into their home currency.

The notes below include a more detailed discussion of the four characteristics we have just described.

Before considering the details of these characteristics, here is a simple example of how index calculations work.

Simple Single Currency Example

This example shows how to calculate one day's data for an index containing only three stocks.  While "real world" indices are actually bigger and more complicated than this, these real-world index calculations are merely a set of refinements gradually added to the basic calculation method shown here.

The following table shows the input data for our index calculations:

This is a single-currency index.  We can identify the three relevant characteristics  that are needed for index calculations:

1. The first column of the table contains the list of securities in the index.

2. For this index, we will calculate the weight of each security using the market capitalization of each security.  The market capitalization is the number of shares on issue, multiplied by the price.  So, for example, we might calculate the market capitalization for security DEF as 100 x $15.75 = $1,575.  The sum of the market caps for all three stocks is $8,755.  Hence the weight for security DEF is 1575 / 8755 = 17.99%

3. The daily returns are simply today's closing price divided by yesterday's closing price, less one.  For example, the daily return for ABC is ( 21.80 / 21.20 ) - 1 = 2.83%.

The calculations for this index are as shown in the following table:

This table shows all of the information that is necessary for doing performance measurement and attribution with this index.  The index return is simply a weighted sum of the security-level returns.  In this particular case (as is most common), the security weights are in proportion to the market capitalization of each security.

Putting it mathematically, the total index return is given by the index return formula:

where:

In the media, one often sees indices presented as a particular number (e.g. "The Dow Jones Industrial Average today went up 0.74% to 11,517".  Index values are simply a way of recording a cumulative return.  If the index went up 0.74% to 11,517, one could infer that the previous value had been 11,432 (since 11,432 x (1+0.74%) = 11,517).  The advantage of an index value is that it makes it simple to calculate (in one step) a return over any period.  For example, if the Dow Jones Industrial Average went up to 14,396 over the next three years, one could infer that the return over that period was (14,396 / 11,517 ) - 1 = 25%.

Here is a spreadsheet showing the data and calculations for this example.

What Didn't This Example Cover?

It is perhaps obvious to state that a "real world" index would include a lot more securities, and it would need to be calculated every day, every month, year after year.  That presents a big practical data management problem, and it is the reason why nobody in their right mind would attempt to calculate a "real world" index in a spreadsheet.

However, there also a few concepts that this index didn't cover:

• Corporate actions, such as dividends and share placements, can alter the number of shares on issue (hence also the market cap) and/or the return of a particular security.
• Once currencies are involved, one needs to introduce currency conversions (at the spot rate), or alternatively, currency hedge calculations (which will in turn require risk-free cash rates for each currency).
• Shares are perhaps the easiest asset class for which to create an index.  For bonds, there are additional difficulties to consider.  Bonds tend to be illiquid, and a particular bond issue with a big capitalization may not even trade on any given business day.  This can make it difficult to select a sensible closing price for a bond.  In the absence of a recent trade, one might choose to use a bond valuation model in order to work out a reasonable valuation for the bond.

Despite all these possible added complexities, the essential part of index calculations is as simple as the example shown above.  Providing that one is clear on these basic principles of index calculations, it is not too difficult to fold-in some of the elaborations required in the real world.  We will give examples of this below.

Determining Which Securities Go In the Index

The securities contained within an index at any given time are known collectively as the index constituents.

In principle, it should not be very difficult to determine which securities go into an index.  Moreover, deciding which securities will go into an index is the job of the person who creates the index, rather than a person who uses the index.

For the standpoint of an index user, there are four issues that are relevant:

• Firstly, knowing the index constituents is very useful when you are trying to decide whether a particular index will be appropriate for benchmarking a particular portfolio.
• Secondly, if you are designing or using systems for doing attribution or other security-level portfolio analysis, it can be helpful to have a broad understanding of how frequently (and by how much) the index constituents change.
• Thirdly, for anyone contemplating the use of a fixed income index, it is useful to know from the outset that some fixed income index firms will not even publicly reveal the constituents of their indices.  This makes for an extremely murky situation when anybody wishes to do detailed comparisons between their portfolios and the index.  It also makes fixed income performance analysis much more difficult, expensive, and opaque than it needs to be.  This state of affairs seems to be a legacy of past days, and it will not be surprising if the leading fixed income index vendors become less relevant to the world around them, either because investors switch to vendors who are more transparent, or alternatively because investors simply find a way to bypass the lack of cooperation from some leading vendors (See the paper by Dr Andrew Colin that appears below).
• Fourthly (and finally), the popular press frequently report that securities which are admitted to large well-known equity indices such as the Dow Jones Industrial Average (DJIA) or the Standard & Poors 500 (S&P 500) increase in price due to the announcement of their admission (for example, this was widely reported when Google was admitted to the S&P 500).  Whether or not this is true, investment managers should be cognizant of the fact that there is a widespread expectation that it will be significant for a security's returns if it is added to (or deleted from) a prominent index.

In terms of the practical process that is used for determining whether a security should be in an index, this is usually a combination of objective fact (e.g. the security is only likely to be considered if it is listed and actively trading on an exchange), and subjective opinion.  Another way of looking at this is that the design of an index can be to some extent descriptive (i.e. simply trying to reflect what the market is doing), or to some extent normative (i.e. trying to capture a list of "good" stocks).

Many professional investors want indices that are purely descriptive: they want to use indices as a measure of what the overall market is doing.  Indeed, finance theory mostly defines an index as a broadly-based measure of the market's performance.

However, there is a strong tendency in index design to create indices so that membership of the index confers glory upon the exclusive club of companies who are added to the constituent list.  Some examples of this are:

• Dow Jones Industrial Average: "Today's 30 constituents are chosen by Dow Jones and the Wall Street Journal to represent a balanced selection of blue-chips. In recent years, the constituents have been gradually altered, to reflect the shift in the US economy away from traditional manufacturing towards computers and service industries." (http://www.moneyextra.com/dictionary/dow-jones-industrial-003574.html)
• Nasdaq Dividend Achievers: "The NASDAQ Dividend Achievers Total Return Index is a market capitalization weighted index designed to track the performance of companies that are listed on NASDAQ and meet the Dividend Achievers requirements of increased annual regular dividend payments for the last 10 or more consecutive years. This index is calculated on a total return basis." (http://www.nasdaq.com/reference/IndexDescriptions.stm)
• S&P 500 Dividend Aristocrats: "... is designed to measure the performance of S&P 500 constituents that have followed a managed-dividends policy of consistently increasing dividends every year for at least 25 years". (S&P 500 Dividend Aristocrats Index Methodology, p. 2, May 2nd 2005, available at www.standardandpoors.com). 

In contrast, the FTSE indices aim to be mainly descriptive.  The FTSE Global Guide to Calculation Methods states: "The primary purpose of the indices is to reflect movements in the underlying market accurately". (p.5, Version 1.5, available at www.ftse.com).

Somewhat polemically, Russell also eschew arbitrary decisions on index constituents: "Selecting stocks is a money manager's job, not an index's job. We think indexes should be as neutral and as representative of the market as they can be and we think investors want that too." (See below for a reference to this page).

There are many other practical issues concerned with how index firms decide upon index constituents.  For example, if a listed company ceases to trade on an exchange (maybe because it has been acquired, or has gone insolvent), there is little option but to immediately exclude the company from the index.  On the other hand, when new companies list on the exchange (often called an IPO), their shares may not be immediately included in indices.  For example, Russell state on their web site that they include new companies every quarter.  Standard and Poors seem to decide more idiosyncratically on index constituents — although they might prefer to describe this as applying "quality" criteria to determining index constituents.

To highlight some of these differences in determining index constituents, we can consider Russell's summary of differences between S&P 500 constituents and Russell 1000 constituents (http://www.russell.com/us/indexes/us/followtheleader.asp).  According to Russell, "110 of the largest 500 companies in the U.S. [are] not included in the S&P 500".  Their list (dated 30th June 2005) includes companies such as Kraft Foods Inc., Legg Mason Inc., Neiman Marcus Group Inc., and Pixar.  Russell summarize the difference thus: "Many think the S&P represents the 500 largest companies in America. Not so. It's a selection from the market, chosen by a closed-door committee. ... We don't pick the stocks in the Russell indexes — the market does".

Some of the differences between different indices can involve reasonably complicated issues concerning calculation methods.  However, even a simple issue such as deciding on the constituents can very greatly from one index company to the next.

Determining the Weight of Each Security On Each Day

After deciding on the index constituents, one needs to decide on the weight that each of them will have on the index each day.  There would be many ways of tackling this problem.  Three popular approaches are:

1. Weighting by market capitalization: this is the most popular approach, and it makes a lot of sense.  Each security is weighted by its market capitalization.  For example, if the index includes two stocks whose price is $20, and the Stock1 has nine times more shares on issue than the Stock2 does, the weights will be 90% for Stock1, and 10% for Stock2.  Market cap weighting reflects the sensible idea that the "average investor" (if there is such a thing) holds each share in proportion to its market cap.  Looking at it another way, if every investor chose to fully replicate the index, they could do this with a cap-weighted index.  However, if the index was (say) equal-weighted, it would not be possible for everybody to replicate it.  In the simple example we just discussed, if everybody tried to buy Stock1 and Stock2 in equal amounts, they would quickly discover that Stock2 was not available in the same quantity as Stock1.  Most market indices around the world follow some kind of cap-weighting scheme.  Some examples would be the MSCI World Index, the FTSE All World Index, the Russell 1000, and the S&P 500.

   We will later consider the "free float" idea, which deals with cases where substantial number of shares on issue are not available for trading because they are the long-term holdings of governments, founders, etc.

    

2. Equal Weighting: A very simple approach is to start the index with equal weights.  Due to the different returns of different stocks, the weights will diverge.  For example, if the index starts with equal weights (50% each) in Stock1 and Stock2, and the returns of these stocks over the first day are 10% and 0% respectively, then the weights at the end of the Day 1 will be 52.38% and 47.62%.  Taking the weights back to 50% apiece at the start of Day 2 would imply a rebalancing cashflow out of Stock1 into Stock2.  To keep these rebalancing cashflows within reasonable bounds, one might choose to make them less frequently (e.g. quarterly or annually) rather than every day.  The calculations involved in rebalancing an index are essentially the same as those involved in rebalancing a weighted-sum benchmark.  See the section on Benchmark Rebalancing for details of this.  The S&P 500 Dividend Aristocrats is an equal-weighted index that is rebalanced quarterly.

3. Price Weighting: Yet another approach to weighting securities would simply be to weight them by price.  This approach has become infamous  due to the Dow Jones Industrial Average (DJIA).   Consider two stocks that are currently constituents of the DJIA.  3M Price has a share price of $80.39, while its market cap is $60.7 Billion. General Electric  has a current price of $33.47 and its market cap is $350.8 Billion.  It is clear that General Electric (GE) has a market capitalization about six times that of 3M.  However, the 3M share price is about 2.4 times greater than GE's.  That's what counts for purposes of the DJIA.  Consider how the calculation would work if these were the only two securities in the index.  Today's value for the index is $80.39 + $33.47 = $113.86.  Suppose that, on the next day, GE goes up by 10%, while 3M has a zero return.  In a cap-weighted index, the index would go up by about 8.52%, due to GE's dominant market cap.  However, using the price-weighted calculation, market cap is irrelevant: the weight is determined by the starting price.  Tomorrow's index value would be $80.39 + $36.82 = 117.21.  Dividing this by today's index value (and subtracting one), leaves us with an index return of 2.94%.  This shows how, even though GE is by far the larger company, its impact on the index is very slight because of its low dollar share price.

   The price weighted calculation that the DJIA uses is exceptionally anachronistic.  It doesn't reflect any modern knowledge about how capital markets work.  In many ways, the DJIA is a "living fossil", because it reflects very crude principles of index design, yet for some reason it is very widely-quoted by the press.  It is hard to think of any reasonable argument in favor of the price weighting approach.

   For his column in Slate, Daniel Gross was able to find a delightful quote that clarifies the strange properties of price-weighted indices such as the DJIA:

   "In fact, 3M counts for more in the index than General Electric, Citigroup, Exxon Mobil and Microsoft will after the latter's share split," Barron's perpetually sharp Michael Santoli noted last week. "This is a stark distortion considering that those four heavies have a combined market value of more than $900 billion, compared with $50 billion for 3M."

Before moving on from the topic of weights, let's just clear up a point of confusion that might sometimes arise.  Can weights depend in any way on the currency perspective of your calculations?  After all, returns and market values are different when expressed in different currencies.  So what about weights?  Fortunately, the answer is very simple.  To properly calculate a weight, you must ensure that the numerator and denominator are the same currency.  Then you divide to obtain the weight.  Calculating the weight in any other currency would make absolutely no difference (providing that your currency conversions are done using a consistent matrix of FX rates), since the numerator and the denominator in the weight calculation would both change by exactly the same factor during that currency conversion.  To use an analogy, a mixture of 50% water and 50% ethanol is still 50/50 whether you measure it in litres, US gallons, imperial gallons, or teaspoonfuls.  This is a very minor point, but navigating the sea of multicurrency indices is easier if you understand that there is no such thing as converting a weight from one currency to another.

Refinements on Cap Weighting

Since capitalization weighting is the most popular approach to deciding security weights, it is perhaps no surprise that various refinements have been proposed.  The main refinement is to use a company's "free float market capitalization", rather than its "full market capitalization".  The idea here is simple.  To give an example, the traditional telecommunications provider in Australia is a company called Telstra.  Telstra is listed on the Australian Stock Exchange, but it is still approximately 50% owned by the Australian federal government.  The free float idea is simply to exclude from Telstra's "free float market cap" the shares that are owned by the federal government.

This idea of free float adjustment is an intuitive and consistent extension of the basic concept of cap-weighting an index.  It just brings the (free float) cap-weights into alignment with what an investor actually could buy on the market if they chose.  Once again, it makes it possible for every investor (in a hypothetical world) to satisfy their choice to hold every index security at index weight.  This would not be strictly possible without the free float enhancement to cap-weighting.

Many important indices these days use some form of free float cap weighting.

In their FTSE All-World Index Series Guide To Calculation, FTSE define "Free Float" in the following way:

"Free float is the proportion of shares tradable within the market place for a given stock. The free float adjustment which FTSE makes within its indices is to reflect situations where a party owns a proportion of a line of stock and that proportion is unlikely to be for sale." (7.1.1)

Another explanation of free float weighting is provided by the Bombay Stock Exchange.

Security Level Returns

In the Index Return Formula shown above on this page, index return calculation is all about the weight of each stock and the return of each security.  This security-level perspective is a very natural way for investment performance analysts to view things.  Security-level returns are an essential ingredient for:

• Stock Level Attribution;
• Sector Attribution with a drill-down to the security-level;
• Understanding in detail how the index is calculated;
• Calculating custom indices; and
• Many different sorts of research and market analysis.

It therefore seems completely obvious that commercial index data services would (at least optionally) include security-level returns.

However, this is not the perspective that has generally been adopted by index companies.

Indeed, many index companies say that they are unable to provide security-level returns, since they do not calculate that information.  There are a few noble exceptions to this generalization.  From the perspective of a performance analyst, this seems utterly beyond belief for three reasons:

• Firstly, performance analysts naturally think about indices as being a weighted sum of security-level returns.  They do this with good reason, since it is a central axiom of performance attribution that the index return is equal to the weighted sum of security-level returns.
• Secondly, it may not be obvious how one could calculate the correct value of an index without working out the security level returns.  After all, it is universally agreed that the index will only be correct if it agrees with the security level weights and returns.
• Thirdly, it is a mystery (to us at least) why an index company should charge tens of thousands of dollars per year for index data, when one of the most vital components (the security-level returns) is completely missing.  This seems as dysfunctional as an automotive company selling cars with no engines in them.  Having said that, plenty of people seem happy to pay for these cars with no engines — so maybe we performance analysts have unreasonably high expectations in hoping that index data would include security-level returns.

To unravel the mystery of why index companies don't seem to put a very high value on security-level returns, it seems vital to understand something about how they can calculate the index returns without even knowing (so they say) the security level returns.

How actually do index companies calculate indices?  Each different index company provides some information about this on their web sites.  However, in some cases, understanding exactly how the calculations work seems more difficult than discovering the secret formula for Coca Cola essence.

Fortunately, some companies do provide clear documentation on their calculations.  The Guide to Calculation Methods for FTSE All World Indices (Version 1.5, January 2005) provides an explicit description of the calculations.  We will summarize a couple of key points here (this is a complex topic: for a definitive reference, please consult the publications provided by each different index company).

 The key point is that an index company thinks of index calculations rather like one can think of portfolio calculations.  Of course one can calculate a portfolio return without knowing the returns for all the securities in the portfolio — one simply needs to know the total-level market values and cashflows.  In the same way, one can calculate the returns for an index by summing the market caps for all of the securities in the index.  This is in fact a very simple exercise, except for corporate actions such as dividends, issuing new shares, returns of capital to stockholders, etc.  Collectively these exceptional cases are known as "capital changes".

In a float-adjusted cap-weighted index such as FTSE All World, the effective capitalization of a stock is the product of three variables:

1. the number of stock on issue;
2. the free float factor for the stock (this will be a number between 0.0 and 1.0); and
3. the price of the stock.

So, for example, if Stock XYZ had a million shares on issue, and its free float factor was 75%, and the stock price was $4.00, this would multiply out to a float-adjusted capitalization of $3 million.  This calculation can be done for all the stocks in the index.  The total float-adjusted capitalization of the index may sum to $100 billion.  If, on the following day, this value went up to $101.030 billion (in the absence of any capital changes), the index return for that day would be 1.030%.  Apart from the complexities of capital changes, (and of course also leaving aside issues such as currency conversions) this is all there is to it when you want to calculate an index.

Putting these concepts into algebraic form, we can think first about what happens to an index as it develops from the start date, during the happy period before any capital changes have taken place. 

The initial float-adjusted capitalization for the index will be:

In other words, it is simply the sum over all the stocks of:

• the number of shares on issue for that stock on day 0; multiplied by
• the price of that stock on day 0; multiplied by
• The free-float adjustment factor for that stock on day 0.

This is just an algebraic restatement of what we said a few paragraphs ago.

As we move forward in time, things are fairly simple when there are no capital changes.  During this happy period, we can work out the cumulative index thus:

The index for a country c on day t is:

NOTE that in the numerator of this formula, P has the subscript t rather than 0.  This is because we are getting the index value on day t.

It is a fairly trivial piece of algebra to show that the index for a country c on day t can also be expressed as:

This form of the equation make it clearer that this calculation is, in fact, implicitly capturing stock-level returns (the stock-level return appears at the right hand side of the numerator).

When capital changes take place, the calculations for day t's index value introduce a new variable that reflects these capital changes.  The denominator in the index value calculation formula changes substantially:

• Sum these N*P*F terms across all
• Do a

1. the processing power required by the index firm to do security-level calculations is now relatively inexpensive; and
2. people who use index data now have the processing power available to use security-level data in portfolio performance analysis.

Security-level returns are available from some index companies, but not from others.  To us, it seems obvious that security-level returns are a very useful thing to have.  There seems to be a gradual trend for more index companies to make security-level returns available to their customers.  If you are buying index data, it's worth checking carefully what is included in the deal.

Index Calculations Example

To provide a simple illustration of how index calculations work, consider an index that only contains one security.  The purpose of this example is to demonstrate how it is possible to deal with capital changes, and with changes in the stock price.  The solution for multiple securities is the same, except that one sums the data for all securities before calculating the index, as described above.

As another simplification, we exclude free float factors from this example.  It is fairly clear how one could add them to the calculation.

The numbers for this example are shown in the following table:

The scenario is deliberately simple.  The only security in the index has 100 shares on issue.  The index starts from day 0.  There are two notable developments during the six days shown in this table:

1. The company issues 10 new shares to some of the existing shareholders after the close of business on day 2.  The new shares are issued at the current market price, so the effect on the index performance should be zero.  This example shows how the capital change variable adjusts the index calculation so that the capital change doesn't spuriously affect the index returns.  Without the capital change variable, a naive calculation might show that the index had gone up by 10% simply because 10% more shares had been issued.  (Note that there are other scenarios where the number of shares goes up by 10%, but the calculation is different.  For example, if each existing stockholder received one extra share for each share they already held — with no payment involved — it would be likely that the stock price would drop by about 9%.  In this case, the capital change variable would be zero.  If the stock price dropped by exactly one-eleventh, the market cap would neither fall nor rise, and the index would still be exactly 1.00).  The point here is that the capital change variable has to reflect knowledge about all the circumstances of the company's change in capital.
2. On day 5, the stock price finally increases by 5%.  The calculations show that this flows through normally into the index values.  The index goes up by 5%.

Normally, an index company might choose to use a base of (say) 100 for an index.  This would mean that all the index values would simply be multiplied by 100.

This numbers and calculations for this example appear in the spreadsheet IndexCalculations.xls.

One could extend this example to include multiple stocks, the free float factor, and multiple currencies.  But the basic principles would remain the same.

Currency Issues with Indices

It is unavoidable to deal with multicurrency issues when calculating or using international indices.  Some useful pieces of terminology are:

• Local currency. This refers to the currency in which a security is usually priced.  For example, for General Motors, this would be USD, while for Sony, this would be JPY.  Evidently, when an index is constructed out of securities with differing local currencies, the need arises to convert all the data into one single currency in which all the data can be combined.
• Base Currency.  This is the single currency in which all the data are combined.  Because the US has long been a financial superpower, the US Dollar is often chosen as the base currency for doing index calculations.
• Third currency.  Once market capitalizations and returns for an index have been calculated in a single currency (for example USD), it is then a fairly trivial calculation to convert them into any other currency.  Sometimes this is (imprecisely) described as a "third currency", to distinguish it from local currency and base currency.
• Converting from one currency to another.  Currency conversions are a basic calculation that is used a lot in any kind of multicurrency analysis.  An ordinary currency conversion simply applies the exchange rate that was applicable at the time of the conversion.
• Hedging from one currency to another.  Sometimes, to reflect a technique which is commonly used by portfolio managers, an index will be calculated on the basis that the local currencies are all hedged back into the base currency.  For an explanation of currency hedging in portfolio management, see the explanation of currency hedling on the Multicurrency Attribution page.

All of these concepts (except for hedging) are demonstrated in the sample spreadsheet IndexCurrencies.xls.

Let us work through that example, to examine some of the nuts and bolts of multicurrency index analysis.

Qantas Airways Ltd is a company listed on the Australian Stock Exchange (ASX).  The number of shares on issue is 1,955,035,444.  Some daily closing prices for Qantas appear in the second column of the following table:

On each day, the market capitalization for Qantas (in the local currency — Australian dollars (AUD)) is simply the shares on issue multiplied by the local price.  The third column shows the local currency market caps.

The fourth column shows the market cap for each day in US Dollars (USD).  This is simply calculated using the exchange rate (rightmost column).

The fifth column shows the local currency return on each day.  Since there were no dividends or other corporate actions during this period, this is simply today's price divided by yesterday's price (less one).

Before moving on to the next column, let's try to get clear on exchange rate returns.  The AUD/USD rate of 1.3559 on 20th April shows that it took AUD $1.3559 to buy a US dollar on that date.  Needless to say, the rate for converting the other way was the inverse of this (i.e. 1/1.3559 = 0.7375).  Taking the inverse of an exchange rate seems absolutely elementary (because it is), but sometimes this can lead to confusion.  For example, the AUD/USD return on 21st April was (1.3435/1.3559) - 1 = -0.91%  The USD/AUD return was (1.3559/1.3435) - 1 = 0.92%.  These FX returns can be useful in converting one way or the other between USD and AUD returns, so long as one can remember which way the conversion is going!

The sixth column shows the USD return on each day.  This is simply the AUD return multiplied by the USD/AUD return.  For example, on 21st April 2006, the AUD return for Qantas was 1.14%.    The USD/AUD return (as we saw in the preceding paragraph) was 0.92%.  When you compound these together, you obtain the USD return for Qantas: 2.07%.

We have now assembled all the building blocks required for constructing a multicurrency index.

The task now is to construct the index.  For this, we selected three different stocks with three different local currencies:

1. Hopewell Holdings (HKD);
2. Rolls Royce Group Plc (GBP); and
3. Qantas Airways Ltd (AUD).

We will construct a cap-weighted index (no free float factors) for these three stocks.

To support the index calculations, we prepared the data for each stock in just the same way that we have described preparing the Qantas data.

The common thread for bringing together the data about all these different stocks is the USD market caps (of course, one could choose any single country — it doesn't have to be USD).  Each company's weight in the cap-weighted index is given by the company's USD market cap, divided by the sum of all the USD market caps on that particular day.  Thus we have a consistent set of weights for the close of business on each day.

Note: When capital changes take place, the market cap for a stock before the start of business on day t may differ from the market cap at the close of business on day t-1.  The former is in fact the correct value to use.  We omitted this detail from the example, since it doesn't affect the key principles.

The local currency returns for the index are a weighted sum of the local currency returns for each stock.  The USD returns for the index are a weighted sum of the USD returns for each stock.  It is useful to keep track of both these sets of returns, since (as we discuss above) one cannot be readily obtained from the other.  However, it is a trivial matter, once one has the USD returns, to convert them into AUD, GBP, HKD, or any other currency of one's choosing.

In Summary, the weights, local currency returns, and base currency returns are shown in the following two tables: