Mehmet E. Akgul
Author
Covers investment, financial analysis and related financial market issues for BrightHedge. He has extensive experience in portfolio management, business consulting, risk management, and accounting areas.
How are Market Indexes Calculated? – Definition & Examples
How are market indexes calculated?
How are price return and total return measured in market indexes?
What are index weighting methods?
How is each index weighting scheme calculated?
What are the terms of rebalancing and reconstitution?
If you’re looking for answers to these questions, this article will help you understand market indices.
- by Mehmet E. Akgul
- July 18, 2019
Investors need to analyze security markets and compare return metrics continuously to assess their portfolios, portfolio managers, or find opportunities in the markets. Market indexes, such as Dow Jones Industrial Average (DJIA), Standard & Poor’s 500 (S&P 500), Nikkei 225, Hang Seng, FTSE 100, Dax, and CAC 40 provide basic securities performance metrics in financial markets.
Recent above-average returns of low-cost index funds boost the popularity of market indexes.
The table below indicates some well-known indexes showing different asset classes and weighting methods.
| Index | Weighting Method |
|---|---|
| S&P 500 | Capitalization-weighted |
| DJIA (30 stocks) | Price-weighted |
| FTSE 100 | Capitalization-weighted |
| Russell 2000 | Capitalization-weighted |
| CAC 40 | Capitalization-weighted |
| Topix (varies) | Capitalization-weighted |
| Nikkei (225 stocks) | Price-weighted |
| Hang Seng (50 stocks) | Capitalization-weighted |
| Dax (30 stocks) | Capitalization-weighted |
| Ibex 35 | Capitalization-weighted |
| Shanghai Composite (varies) | Capitalization-weighted |
| MSCI All Country World Index | Capitalization-weighted |
| Bloomberg Barclays Global Aggregate Bond Index | Capitalization-weighted |
| iShares Long-Term Corporate Bond ETF | Capitalization-weighted |
| Invesco S&P 500® Equal Weight ETF | Equal-weighted |
| SPDR S&P Biotech ETF | Equal-weighted |
| iShares Edge MSCI USA Quality Factor ETF | Fundamental-weighted |
Price Return Index and Total Return Index
There are two versions of return calculation for the same index.
Price return index indicates the prices of securities within the index.
Total return index exhibits price return index + dividend return + interests + other distributions.
At the beginning of index creation, the values of price and total return indexes are equal. In the following periods, the total return index value is higher than the price return index with the effect of non-price variables (dividend, interest, and other distributions).
The Value of a Price Return Index
The value of a price return index formula is as follows:
\(V_{PI}=\frac{\sum\limits_{i=1}^N P_i n_i}{D}\)
where:
\(V_{PI}\) = value of the price return index
\(N\) = the number of securities in the index
\(P_i\) = the unit price of security \(i\)
\(n_i\) = the number of units security \(i\) retained in the index
\(D\) = the value of the divisor
The Value of a Total Return Index
The value of a total return index formula is as follows:
\(V_{TI}=\frac{\sum\limits_{i=1}^N (P_i + Div_i + Int_i + Dist_i) n_i}{D}\)
where:
\(V_{TI}\) = value of the total return index
\(N\) = the number of securities in the index
\(P_i\) = the unit price of security \(i\)
\(Div_i\) = the dividend of security \(i\)
\(Int_i\) = the received interest of security \(i\)
\(Dist_i\) = the received distributions of security \(i\)
\(n_i\) = the number of units security \(i\) retained in the index
\(D\) = the value of the divisor
Comparison of Price Return Index and Total Return Index
| Period | Price Return (%) | Income Return (%) (dividend, interest, etc.) | Ending Value (Price Return Index) | Ending Value (Total Return Index) |
|---|---|---|---|---|
| 0 | 10,000.00 | 10,000.00 | ||
| 1 | 4,00 | 1,50 | 10,400.00 | 10,550.00 |
| 2 | 3,00 | 1,00 | 10,712.00 | 10,972.00 |
| 3 | -2,00 | 0,50 | 10,497.76 | 10,807.42 |
Index Weighting Schemes
Price-Weighted Indexes
Each member security is weighted in the percentage of its price per share in the index. High-priced securities have higher weights in the index return calculation. The price-weighted index formula is as follows:
\(w_i^P=\frac{P_i}{\sum\limits_{j=1}^N P_j}\)
where:
\(w_i^P\) = weight of security \(i\)
\(P_i\) = price of security \(i\)
Example of Price-Weighted Index
| Security | Beginning Price | Beginning Weight (%) | Ending Price | Ending Weight (%) | Dividends Per Share | Price Ret. x Beg. Weight % | Total Ret. x Beg. Weight % |
|---|---|---|---|---|---|---|---|
| ABC | 20.00 | 26.66 | 24.00 | 30.00 | 2.50 | 5.33 | 8.66 |
| DEF | 10.00 | 13.33 | 8.00 | 10.00 | 1.00 | -2.66 | -1.33 |
| GHI | 40.00 | 53.33 | 42.00 | 52.50 | 0.00 | 2.66 | 2.67 |
| XYZ | 5.00 | 6.67 | 6.00 | 7.50 | 0.50 | 1.33 | 2.00 |
| Total | 75.00 | 100 | 80.00 | 100 | 6.66 | 12.00 | |
| Index | 18.75 | 20.00 |
How to Calculate Price Return and Total Return of Price-Weighted Security
Price return of ABC security;
\(\frac{Ending\ Price}{Beginning\ Price}\ Beginning\ Weight = Price\ Return\)
\(\frac{24.00}{20.00}\ 26.66\% = 5.33\%\)
Total return of ABC security;
\(\frac{Ending\ Price + Dividends\ Per\ Share}{Beginning\ Price}\ Beginning\ Weight = Total\ Return\)
\(\frac{24.00+2.50}{20.00}\ 26.66\% = 8.66\%\)
Advantages
- Simplicity
- Long historical records of price-weighted indexes
Disadvantages
- May not consider high market value companies
- High-priced securities have higher weights
- Stock splits change security prices
Capitalization-Weighted Indexes
Each member security is weighted according to the total market capitalization of the securities in the index. The market capitalization of each security is determined by multiplying the market price of a security and outstanding shares of a security. The capitalization-weighted index formula is as follows:
\(w_i^M=\frac{P_iQ_i}{\sum\limits_{j=1}^N P_jQ_j}\)
where:
\(w_i^M\) = weight of security \(i\)
\(P_i\) = price of security \(i\)
\(Q_i\) = number of shares of security \(i\)
Example of Capitalization-Weighted Index
| Security | Beg. Price | Shares O/S | Beg. Market Cap | Beg. Weight (%) | Ending Price | Div. Per Share | Ending Market Cap | Price Ret. x Beg. Weight % | Total Ret. x Beg. Weight % |
|---|---|---|---|---|---|---|---|---|---|
| ABC | 20.00 | 4,500 | 90,000 | 36.00 | 24.00 | 2.50 | 108,000 | 7.20 | 11.70 |
| DEF | 10.00 | 7,000 | 70,000 | 28.00 | 8.00 | 1.00 | 56,000 | -5.60 | -2.80 |
| GHI | 40.00 | 1,000 | 40,000 | 16.00 | 42.00 | 0.00 | 42,000 | 0.80 | 0.80 |
| XYZ | 5.00 | 10,000 | 50,000 | 20.00 | 6.00 | 0.50 | 60,000 | 4.00 | 6.00 |
| Total | 250,000 | 100 | 266,000 | 6.40 | 15.70 | ||||
| Index | 10,000 | 10,640 |
How to Calculate Price Return and Total Return of Capital-Weighted Security
Price return of ABC security;
\(\frac{Ending\ Price}{Beginning\ Price}\ Beginning\ Weight = Price\ Return\)
\(\frac{24.00}{20.00}\ 36.00\% = 7.20\%\)
Total return of ABC security;
\(\frac{Ending\ Price + Dividends\ Per\ Share}{Beginning\ Price}\ Beginning\ Weight = Total\ Return\)
\(\frac{24.00+2.50}{20.00}\ 36.00\% = 11.70\%\)
Advantages
- More objective and explicit measure
- Limited rebalancing
- Investors can hold securities easier than other weighting schemes
Disadvantages
- Overpriced securities dominate the index
- Large-cap securities are represented more
Equal-Weighted Indexes
Each member security is weighted equally. The equal-weighted index formula is as follows:
\(w_i^E=\frac{1}{N}\)
where:
\(w_i^E\) = weight of security \(i\)
\(N\) = the number of securities in the index
Example of Equal-Weighted Index
| Security | Beg. Price | Shares O/S | Beg. Market Cap | Beg. Weight (%) | Ending Price | Div. Per Share | Ending Market Cap | Price Ret. x Beg. Weight % | Total Ret. x Beg. Weight % |
|---|---|---|---|---|---|---|---|---|---|
| ABC | 20.00 | 125.00 | 2,500 | 25 | 24.00 | 2.50 | 3,000 | 5.00 | 8.13 |
| DEF | 10.00 | 250.00 | 2,500 | 25 | 8.00 | 1.00 | 2,000 | -5.00 | -2.50 |
| GHI | 40.00 | 62.50 | 2,500 | 25 | 42.00 | 0.00 | 2,625 | 1.25 | 1.25 |
| XYZ | 5.00 | 500.00 | 2,500 | 25 | 6.00 | 0.50 | 3,000 | 5.00 | 7.50 |
| Total | 10,000 | 100 | 10,625 | 6.25 | 14.38 | ||||
| Index | 10,000 | 10,625 |
How to Calculate Price Return and Total Return of Equal-Weighted Security
Price return of ABC security;
\(\frac{Ending\ Price}{Beginning\ Price}\ Beginning\ Weight = Price\ Return\)
\(\frac{24.00}{20.00}\ 25.00\% = 5.00\%\)
Total return of ABC security;
\(\frac{Ending\ Price + Dividends\ Per\ Share}{Beginning\ Price}\ Beginning\ Weight = Total\ Return\)
\(\frac{24.00+2.50}{20.00}\ 25.00\% = 8.13\%\)
Advantages
- Large-cap securities are underweighted and more diversified
Disadvantages
- High concentration of small-cap securities may result in fluctuating returns
- Less liquid securities due to small-cap bias
- Frequent rebalancing
Fundamental-Weighted Indexes
Each member security is weighted according to metrics such as cash flow, revenues, earnings, dividends, book value to stabilize the disadvantages of capitalization-weighted indexes. The fundamental-weighted index formula is as follows:
\(w_i^F=\frac{F_i}{\sum\limits_{j=1}^N F_j}\)
where:
\(w_i^F\) = weight of security \(i\)
\(F_i\) = fundamental measure of security \(i\)
Example of Fundamental-Weighted Index
| Security | Beg. Price | Total Sales | Beg. Weight (%) | Ending Price | Div. Per Share | Ending Weight (%) | Price Ret. x Beg. Weight % | Total Ret. x Beg. Weight % |
|---|---|---|---|---|---|---|---|---|
| ABC | 20.00 | 2 Bil | 73.39 | 24.00 | 2.50 | 77.17 | 14.68 | 23.85 |
| DEF | 10.00 | 500 Mil | 9.17 | 8.00 | 1.00 | 6.43 | -1.83 | -0.92 |
| GHI | 40.00 | 200 Mil | 14.68 | 42.00 | 0.00 | 13.50 | 0.73 | 0.73 |
| XYZ | 5.00 | 300 Mil | 2.75 | 6.00 | 0.50 | 2.89 | 0.55 | 0.83 |
| Total | 54,500 Billion | 62,200 Billion | 14.13 | 24.50 | ||||
| Index | 10,000 | 11,412.84 |
How to Calculate Price Return and Total Return of Fundamental-Weighted Security
Price return of ABC security;
\(\frac{Ending\ Price}{Beginning\ Price}\ Beginning\ Weight = Price\ Return\)
\(\frac{24.00}{20.00}\ 73.39\% = 14.68\%\)
Total return of ABC security;
\(\frac{Ending\ Price + Dividends\ Per\ Share}{Beginning\ Price}\ Beginning\ Weight = Total\ Return\)
\(\frac{24.00+2.50}{20.00}\ 73.39\% = 23.85\%\)
Advantages
- Less overpriced securities than capitalization-weighted indexes
- More small-cap representation
Disadvantages
- Subjective weighting scheme
- Some metrics may not be suitable for some securities
- Investors may not be aware of index construction methodology
Rebalancing and Reconstitution
Rebalancing
Rebalancing is the process of adjusting the weightings of the member securities in the index. The index’s original or desired level of weighting or risk is maintained by periodically (usually quarterly) buying and selling securities.
- Price-weighted indexes need not be rebalanced because these indexes are price-adjusted.
- Capitalization-weighted indexes rebalance according to market capitalization. However, mergers, acquisitions, liquidations, and other security-related events are reasons for rebalancing adjustments.
- Equal-weighted indexes rebalance after market-cap changes of each security. The equal-weighted index is not equally weighted after the inception of the index.
- Fundamental-weighted indexes rebalance according to fundamental metrics. Some metrics of fundamental-weighted indexes may not be publicly available and not easy to follow.
Reconstitution
Reconstitution is the process of buying and selling the member securities in the index to reflect the current style and market cap. The addition and removal of securities during the reconstitution process of the index may have a significant impact on security values.
Knowing the basic understanding of security market indexes’ structure and management helps investors and users make decisions.
Disclosure: I do not have any of the securities mentioned above. This article expresses my own views, and I wrote the article by myself. I am not receiving compensation for it. I have no business relationship with any company whose security is mentioned in this article.
Source: Improved Beta? A Comparison of Index-Weighting Schemes – EDHEC-Risk Institute
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Mehmet E. Akgul