# Index Definitions & Calculations

Calculations and definitions that are used in or are useful for understanding the CRSP Research Indexes are included in this section. The items are listed alphabetically: Income Return, Index Count, Index Level, Index Return, Index Weight, Rebasing Index Levels, Scholes-Williams Beta, Standard Deviation, Total and Used Counts, and Total and Used Values.

### Income Return

Income Return is the return on the ordinary dividends paid to shareholders of a security. It is the ratio of the amount of ordinary dividends since the end of the previous period up to and including the end of the period of interest to the price at the end of the previous period. It is similar to a dividend yield.

Income Return is calculated by CRSP as the difference of the Total Return and Capital Appreciation Return, as follows. irett=tretarett where:

• irett is the income return for time t
• tret is the total return for time t,
• arett is the capital appreciation return for time t.

### Index Count

Index Count is the count in an index for a time period is the number of securities in the portfolio during the time period. Rules are based on the specific index or portfolio methodology. See Total and Used Counts for more details.

### Index Level

Index Level is the value of an investment relative to its value at one fixed point in time. Index Levels allow convenient comparison of the relative performance of the different portfolio or asset classes. Differences arise when indexes are based on different underlying databases such as daily and monthly CRSP stock products.

The initial date and value are set arbitrarily, but must be consistent if comparing multiple indexes. The Index Level for any series at any time after the initial point indicates the value at that time of the initial value invested at the initial point. The Index Level for any series at any time before the initial point, indicates the value invested at that time that will result in the initial value at the initial point. The Index Level of a series missing prior to its first available return. Let:

• It = Index Level for any series at time t
• Rt = return for the period t-1 to t
• F = First Return. The time of the first non-missing return of the series
• D = Initial Date. An arbitrary date where the level is set to the initial value
• L = Initial Level. An arbitrary value the level is set to on the initialization date

then

• if t = D, then It = L
• if t > D, then It = It-1*(1+ Rt)
• if t < D, then It • if t-1 < F then It is set to missing- Note: Missing values are file format specific.

Defined CRSP indexes use the following initial dates and levels:

 CRSP Stock File Indexes initial level 100.00 initial date December 29, 1972
 CRSP Cap-Based Portfolios initial level 1.00 initial date December 31, 1925
 CRSP US Government Treasury and Inflation Indexes initial level 100.00 initial date December 29, 1972

Publicly available indexes such as for the S&P 500 Composite and NASDAQ Composite have initial values set by their creators and differ from the CRSP initializations.

### Index Return

An Index Return is the change in value of a portfolio over some holding period. The return on an index (Rt) is calculated as the weighted average of the returns for the individual securities in the index: where:

• Rt is the index return
• wi,t is the weight of security i at time t.
• ri,t is the return of security i at time t. (see section xxxx of Stock guide for Security Return Calculation)

In a value-weighted index, the weight (wi,t) assigned is its total market value; see Index Weight below. In an equally-weighted index, the weight is equal and by convention wi,t is set to one for every stock. Such an index would consist of n stocks, with the same dollar amount invested in each stock.

The security returns can be total returns or capital appreciation (returns without dividends). This determines whether the index is a total return index or a capital appreciation index.

In an index where the individual components are not known, but an index level is available from an external source, such as the Standard & Poor’s 500 Composite Index, the return Rt is calculated as follows: Rt is the index return for time t

It is the index level at time t

It-1 is the index level at end of the previous period (time t-1)

### Index Weight

The weight of an index for a time period is the total market value of the securities in the index at the end of the previous trading period. Vt = ∑(wi,t)= ∑(vi,t) where: vi,t = pi,t-1* si,t-1 in which:

• vi,t is value of security i at time t
• pi,t-1 is the price of security i at the end of the previous trading period (time t-1).
• si,t-1 is the number of shares outstanding of security i at the end of the previous trading period (time t-1).

### Rebasing Index Levels

It is possible to rebase an index to make index levels of two index level series comparable. To rebase an index, choose a new initial date and value, find the current index level on the new initial date, and multiply the levels on all dates by the new initial value divided by the old initial date index level: where:

• It = Original Index Level for the series at time t
• Nt = New Index Level for the series at time t
• D = New Initial Date.
• ID = Original Index Level for the series on the new initial date
• L = New Initial Level.

### Scholes-Williams Beta

Beta is a statistical measurement of the relationship between two time series, and has been used to compare security data with benchmark data to measure risk in financial data analysis. CRSP provides annual betas computed using the methods developed by Scholes and Williams (Myron Scholes and Joseph Williams, “Estimating Betas from Nonsynchronous Data,” Journal of Financial Economics, vol 5, 1977, 309-327).

Beta is calculated each year as follows:

where: • βi is the Beta for security i for the year being calculated
• ri,t is the return of security i at day t
• lri,t = ln(1+ri,t ) is the natural log of the return of security i at time t+1 or the continuously compounded return.
• Mt is the value-weighted market return at time t
• lMt=ln(1+Mt ) is the natural log of the value-weighted market return at time t+1 or the continuously compounded return.
• M3t = lMt-1+ lMt + lMt+1 is the three-day moving window of the above market return
• ni is the number of non-missing returns for security i during the year

where the summations are over t and include all days on which security i traded, beginning with the first trading day of the year and ending with the last trading day of the year. There are two index families based on Scholes- Williams Beta calculations: NYSE/NYSE American and NASDAQ-only.

In the NYSE/NYSE American family, only trading prices are considered in the beta calculation, and a security must have traded half the days in a year to be given a non-missing beta for that year. The index used in the calculation is the total returns on the Trade-only NYSE/NYSE American Value-Weighted Market Index.

Betas for the NASDAQ family do not use the standard Scholes-Williams trade-only data restriction, since most NASDAQ securities were not required to repoRt transactions until 1992. Removing bid/ask averages would restrict NASDAQ data to only NASDAQ National Market securities after 1982. NASDAQ returns based on bid/ask averages have different characteristics from trade-based returns, and betas are provided for comparison. NASDAQ betas are based on the total returns on the NASDAQ Value-Weighted Market Index.

### STANDARD DEVIATION

Standard Deviation is a statistical measurement of the volatility of a series. CRSP provides annual standard deviations of daily returns using the following calculations: where:

• σi is the standard deviation for security i for the year being calculated
• ri,t is the return of security i at time t
• ni is the number of non-missing returns for security i during the year

where the summations are over t and include all days on which security i had a non-missing return, beginning with the first trading day of the year and ending with the last trading day of the year. A security must have valid returns for eighty percent of the trading days in a year to have a Standard Deviation calculated. There are two families of indexes provided by CRSP with annual standard deviations as the statistic, the NYSE/NYSE American Standard Deviation Portfolios and the NASDAQ Standard Deviation Portfolios.

### Total Counts (totcnt) and Used Counts (usdcnt)

Total Counts and Used Counts are provided for all indexes and portfolios. The following table identifies differences.

Total Count Used Count
Current Day closing price required for inclusion Previous day & current day closing prices required for inclusion
On same date the Total Count will always be greater than or equal to the Used Count. The difference will be the number of securities with missing prices on the previous day (usually adds). The Total Count on Day t will be greater than or equal to the Used Count on Day t+1. The difference will be the number of securities with missing prices on t+1 (usually the drops)
Total Count will fluctuate throughout the year. Used Count will fluctuate throughout the year.

### Total Value (totval) and Used Value (usdval)

Total Value and Used Value are provided for all CRSP stock indexes. The following table identifies differences.

Total Value Used Value
Current Day market value of eligible securities - price and shares for the current day are required for inclusion For value-weighted indexes, this is the Index weight - market value of eligible securities with - price for the current day and price and shares for the previous day are required for inclusion
On same date the Total Value will always be greater than or equal to the Used Value.