# Level 1 CFA® Exam:

Market Indices

A security market index is a measure that shows trends in a market or a segment of a market. An index can also represent any group of instruments, such as a class of assets. The value of an index is calculated regularly based on the prices of its constituent securities.

We can distinguish between two types of indices:

- price return indices, which only reflect changes in security prices, and
- total return indices, which also reflect dividends, interest, and other cash flows.

### Value of Price Return Index

\(V = \frac{\Sigma^{N}_{i=1}w_{i}\times P_{i}}{D}\)

- \(V\) - value of the price return index
- \(w_{i}\) - number of units of constituent security "i"
- \(N\) - number of constituent securities in the index
- \(P_{i}\) - unit price of constituent security "i"
- \(D\) - divisor

The role of a divisor is to scale the value of an index to a convenient value. For example, if the value of the numerator in the formula was USD 15.9 billion, we could use a divisor of USD 100 million to scale the index value to 159.

Another role of a divisor is to assure that an index is a continuous measure, so for example when one company in the index is replaced by another company with different capitalization, the value of the divisor will be changed to make the values of the index comparable over time.

Note: value of the total return index cannot be computed if we don’t know the values of dividends or interests from the previous periods.

### Price Return vs Total Return

\(PR=\frac{V_{1}-V_{0}}{V_{0}}\)

- \(PR\) - price return of an index
- \(V_{1}\) - value of the price return index at the end of the period
- \(V_{0}\) - value of the price return index at the beginning of the period

\(PR_{i} = \frac{P_{i1} - P_{i0}}{P_{i0}}\)

- \(PR_{i}\) - price return of a constituent security "i"
- \(p_{i1}\) - price of constituent security "i" at the end of the period
- \(p_{i0}\) - price of constituent security "i" at the beginning of the period

\(PR=\Sigma^{N}_{i=1}w_{i}\times PR_{i} = \Sigma^{N}_{i=1}w_{i}\times (\frac{P_{i1}-P_{i0}}{P_{i0}})\)

- \(PR\) - price return of an index
- \(PR_{i}\) - price return of constituent security "i"
- \(N\) - number of individual securities in the index
- \(w_{i}\) - the weight of security "i"
- \(P_{i1}\) - price of constituent security "i" at the end of the period
- \(P_{i0}\) - price of constituent security "i" at the beginning of the period

\(V_{T} = V_{0}\times (1+PR_{0;1})\times (1+PR_{1;2})\times \ldots\times (1+PR_{T-1;T})\)

- \(V_{T}\) - value of the price return index at time "T"
- \(V_{0}\) - value of the price return index at inception
- \(PR_{T-1;T}\) - price return on the index over period from "T-1" to "T"

\(TR= \frac{V_{1} - V_{0} + I_1}{V_{0}}\)

- \(TR\) - total return of an index
- \(V_{1}\) - value of the price return index at the end of the period
- \(V_{0}\) - value of the price return index at the beginning of the period
- \(I_1\) - total income from constituent securities over the period

\(TR= \frac{P_{i1} - P_{i0} + I_{i}}{P_{i0}}\)

- \(TR\) - total return of a constituent security "i"
- \(P_{i1}\) - price of constituent security "i" at the end of the period
- \(P_{i0}\) - price of constituent security "i" at the beginning of the period
- \(I_{i}\) - total income from security "i" over the period

\(TR= \Sigma^{N}_{i=1}w_{i}\times TR_{i} = \Sigma^{N}_{i=1}w_{i}\times (\frac{P_{i1} - P_{i0} + I_{i}}{P_{i0}})\)

- \(TR\) - total return of an index
- \(TR_{i}\) - total return of a constituent security "i"
- \(w_{i}\) - the weight of security "i"
- \(P_{i1}\) - price of constituent security "i" at the end of the period
- \(P_{i0}\) - price of constituent security "i" at the beginning of the period
- \(I_{i}\) - total income from security "i" over the period
- \(N\) - number of securities in the index

\(V_{T} = V_{0}\times (1+TR_{0;1})\times (1+TR_{1;2})\times \ldots\times (1+TR_{T-1;T})\)

- \(V_{T}\) - value of the total return index at time "T"
- \(V_{0}\) - value of the total return index at inception
- \(TR_{T-1;T}\) - total return on the index over period from "T-1" to "T"

Remember that the changes of the price return index depend only on the changes in the prices of the constituent securities. However, in the case of the total return index, the total return of an index is taken under consideration.

The first step in creating a market index should be to specify the market it is supposed to reflect. Once we've done that, we should carefully examine the market and specify the number of constituent securities as well as their qualities.

The next step is the selection of securities for the index. We have two options here. One is to simply include all the securities in the market. The other is to select a representative sample of the market, that is a part of the companies that make it up and best represent its characteristics.

Finally, we have to choose the weighting method and the initial value of the index.

Index weighting is about determining the proportion of individual securities in an index. We distinguish among:

- price weighting,
- equal weighting,
- market-capitalization weighting,
- float-adjusted market-capitalization weighting, and
- fundamental weighting.

In price weighting, the weight of a security in an index is computed using this CFA exam formula:

\(w_{i} = \frac{P_{i}}{\Sigma^{N}_{i=1}P_{i}}\)

- \(w_{i}\) - weight of security "i" assuming price weighting
- \(P_{i}\) - price of a constituent security "i"

As you can see from the formula, the weight of a security is the quotient of the price and the sum of the prices of all securities in the index. Let's go through an example of how price weighting works.

A price-weighted index includes 4 constituent securities.

Green Company | IT Company | RGD Company | Pear Company | |
---|---|---|---|---|

Beginning-of-period price | 130.4 | 34.4 | 58.6 | 76.3 |

End-of-period price | 150.2 | 36.3 | 36.7 | 92.3 |

What is the value of the index at the beginning of the period, at the end of the period, and what is the value of the price return? Assume that the divisor is equal to 10.

(...)

A price-weighted index includes 4 constituent securities.

Green Company | IT Company | RGD Company | Pear Company | |
---|---|---|---|---|

Beginning-of-period price | 130.4 | 34.4 | 58.6 | 76.3 |

End-of-period price | 150.2 | 36.3 | 36.7 | 92.3 |

Total dividends | 13.4 | 0.0 | 2.3 | 0.0 |

What is the value of the index at the beginning of the period, at the end of the period, and what is the value of the total return? Assume that the divisor is equal to 10.

(...)

The other weighting method is equal weighting, in which all the constituent securities of an index have exactly the same weights:

\(w_{i} = \frac{1}{N}\)

- \(w_{i}\) - weight of security "i" assuming equal weighting
- \(N\) - number of securities in the index

In other words, we assume that the same amount of money is invested in each of constituent securities.

Total dividends | Beginning-of-period price | End-of-period price | |
---|---|---|---|

IT Company | 1.4 | 46.36 | 46.89 |

Green Company | 0.16 | 26.34 | 23.01 |

RGD Company | 2.35 | 23.86 | 21.46 |

What is the value of the total return on an index consisting of these 3 securities?

(...)

The market-capitalization weighting is given by the following formula to be used in your level 1 CFA exam:

\(w_{i} = \frac{Q_{i}\times P_{i}}{\Sigma^{N}_{j=1}Q_{j}\times P_{j}}\)

- \(w_{i}\) - weight of security "i" assuming market-capitalization weighting
- \(Q_{i}\) - number of shares outstanding of security "i"
- \(P_{i}\) - share price of security "i"
- \(N\) - number of securities in the index

As you can see we divide the market capitalization of a given company by the sum of market capitalizations of all companies that constitute the index.

Let's analyze a capitalization-market index consisting of three securities.

Shares outstanding | Beginning-of-period price | End-of-period price | |
---|---|---|---|

IT Company | 1,000 | 46.36 | 46.89 |

Green Company | 500 | 26.34 | 23.01 |

RGD Company | 700 | 23.86 | 21.46 |

What is the value of the price return?

(...)

Another weighting method is float-adjusted market-capitalization weighting, which can be expressed using this formula that you can use in your level 1 CFA exam:

\(w_{i} = \frac{f_{i}\times Q_{i}\times P_{i}}{\Sigma^{N}_{j=1}f_{j}\times Q_{j}\times P_{j}}\)

- \(w_{i}\) - weight of security "i" assuming float-adjusted market-capitalization weighting
- \(f_{i}\) - fraction of shares outstanding in the market float
- \(Q_{i}\) - number of shares outstanding of security "i"
- \(P_{i}\) - share price of security "i"
- \(N\) - number of securities in the index

As you can see, it is very similar to the formula for the market-capitalization weighting but for adjusting the capitalizations of companies for their shares outstanding in the market float.

So, if a part of the shares is not actively traded because it is for example held by a major investor that is not willing to trade it, it will not be included in the market float.

As the name 'fundamental weighting' suggests, fundamental weighting is based on fundamental values and ignores security prices:

\(w_{i} = \frac{F_{i}}{\Sigma^{N}_{j=1}F_{j}}\)

- \(w_{i}\) - weight of security "i" assuming fundamental weighting
- \(F_{i}\) - given fundamental size measure of company "i"

Both the price weighting and equal weighting are simple to use and understand, so it is definitely their advantage. However, they have their drawbacks too.

The price weighting leads to overweighting the shares with higher prices and underweighting the shares with lower prices in the index. Consider such an example: we have an index that consists of two companies A and B and the price of one share of company A is USD 1 and the price of one share of company B is USD 99. The weight of company A will be 1% and the weight of company B will amount to 99%. It is an extreme example, however, it shows well that company A has an impact on the value of the index only in theory, whereas in practice company B dominates the index.

(...)

Finally, let’s also say a few words about the disadvantages of fundamental weighting. For example, if the weights depend on earnings yields, the companies with higher earnings yields will have greater weights in the index. However, we know that earnings depend on accounting assumptions and actions taken by companies. So, fundamental weighting may lead to a situation where some companies in the index will get overweight and some other companies will get underweight.

### Index Rebalancing

Since the market prices of the securities in an index change, their weights need to be adjusted. The process of doing so is called index rebalancing and is conducted regularly by index providers.

The purpose of rebalancing is to keep the weights of the securities in an index consistent with the adopted weighting method. Note that from the types of weighting methods we’ve discussed in this lesson the price weighting index doesn’t need rebalancing and the equal weighting index needs to be rebalanced the most often.

### Index Reconstitution

Index reconstitution is about removing constituent securities to substitute them with ones that are closer to the characteristics of the index. An index is reconstituted on a specified day (for example in each quarter). A decision is made if a security should remain in the index or be removed.

As financial markets and financial engineering are developing, market indices serve more functions than before. Market indices:

- measure changes in the values of securities,

are gauges of market sentiment, - provide information on the rates of return and risk that can be used in financial models, and
- serve as benchmarks for financial instruments and portfolios, as well as for ETF funds and index funds that follow these benchmark indices.

Various financial models use data on historical returns on assets and risks associated with the assets. Such data are often based on market indices. One example is systematic risk, used in the CAPM model.

To assess the rate of return on investment, we need to compare it against other values. For example, our point of reference can be the results of other instruments or portfolios. Alternatively, we can compare the results of an investment against a benchmark market index. For example, an American investor who buys and sells shares in the American market can compare his results to the S&P 500 index.

Most actively managed funds yield worse results than their benchmark indices. For this reason, a trend has emerged to invest in market indices rather than in actively managed funds. To make things easier, instruments have been developed that follow changes in index prices. They include ETFs and index funds. The managers of such funds try to follow index results as closely as possible. We can therefore say that some market indices are benchmark indices for those funds.

Let’s categorize indices according to the types of assets and securities they are consisted of. We distinguish among:

- equity indices,
- fixed-income indices,
- commodity indices,
- real estate investment trust indices, and
- hedge fund indices.

### Equity Indices

Since the equity market is very diverse, there are plenty of options for creating equity indices. Equity indices include broad market, multimarket, sector, or style indices.

(...)

### Fixed-Income Indices

Indices representing fixed-income securities are created based on a variety of criteria. They include the issuer's sector, country of origin, type of financing, the currency of payments, maturity, or the issuer's credit quality. One problem for those constructing fixed-income indices is bond rollover and expiry. Additionally, a fixed-income market is a dealer market, so it may be necessary to obtain price information from dealers.

### Commodity Indices

Commodity indices can represent all types of commodities, e.g. precious metals, agricultural products, energy products, and industrial metals.

### Real Estate Investment Trust Indices

We distinguish between two categories of indices associated with the real estate market. The first category is indices associated with markets for real estate securities. These indices consist of stocks issued by property developers. The other represents markets for real estate. Real estate security markets are characterized by high liquidity, whereas the liquidity of real estate markets is very low.

Real estate indices include appraisal indices, repeat sales indices, and real estate investment trust (REIT) indices.

### Hedge Fund Indices

Hedge funds are aggressive investment funds that use leverage and a wide range of investment strategies. Given the fast growth of this market, many indices have emerged in recent years that reflect hedge fund return changes. One such index is the Van Global Hedge Fund Index. Indices of this type are usually created based on databases kept by research organizations. They reflect the results of hedge funds either globally or for specific strategies.

- A market index is a tool for gauging market trends.
- We can distinguish between two types of indices: price return indices and total return indices.
- The role of a divisor is to scale the value of an index to a convenient value and to assure that an index is a continuous measure.
- Index weighting is about determining the proportion of individual securities in an index. We distinguish among price weighting, equal weighting, market-capitalization weighting, float-adjusted market-capitalization weighting, and fundamental weighting.
- The price weighting leads to overweighting the shares with higher prices and underweighting the shares with lower prices in the index.
- In the case of equal weighting, the smaller companies have a relatively greater impact on the index value, than the bigger ones.
- In market-capitalization indices overvalued companies have a bigger impact on the index than they should have.
- The purpose of rebalancing is to keep the weights of the securities in an index consistent with the adopted weighting method.
- Index reconstitution is about removing constituent securities to substitute them with ones that are closer to the characteristics of the index.
- Broad market indices represent the situation in the market as a whole.
- Instruments in multimarket indices are those traded in more than one market.
- Sector indices include instruments representing a particular sector of the economy.
- Style indices include securities with similar characteristics.