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Asset-Weighted Standard Deviation

You are interested in measuring the variability of the individual fund returns for a particular period around the composite average return for that period. The standard deviation of the portfolio returns around the weighted average composite return is an indication of the variability of fund returns within the composite.

With the standard formula for calculating standard deviation, you divide the deviations from the mean by the number of returns for the period. This has the effect of equal weighting the deviations from the mean. You can instead calculate an asset-weighted standard deviation that is consistent with the asset-weighted composite return. You do this by assetweighting the deviations from the mean return.
The following figure shows the calculation of the asset-weighted standard deviation for the sample composite for November.

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The asset-weighted standard deviation for November is 7.00%. Because this standard deviation is asset-weighted, larger funds contribute more to this variability measure than smaller funds with a similar return. For example, Fund 10 has the worst return, 0.58%, and the return furthest from the average, but it is a small fund that comprises only 2% of the composite. The effect of Fund 10's return is lessened because the squared return deviation is weighted by the asset size. The higher the asset weighted standard deviation, the more variability there was in the return to the average dollar invested in the strategy. With an average return of 13.60% and a standard deviation of 7.00%, you would expect the returns to approximately 2/3 of the dollars invested in the strategy to fall in between 6.60% and 21.60%.

While this example shows how to calculate the asset weighted standard deviation for a single period, there is a problem introduced when you calculate the statistic over multiple periods. The formula uses a weight to establish the individual fund's contribution to the composite standard deviation. You can use the beginning of period weight, but there may be portfolios that start or leave the composite after the first period. Because of this, when you calculate a multiperiod asset-weighted standard deviation, you typically include only the portfolios that are in the composite for the complete time period. The example in the following figure illustrates the calculation of a multiperiod asset weighted standard deviation over the two-month period.

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The two-month composite asset-weighted standard deviation is equal to 13.41%. Only the 8 funds present in the composite for both months are included in the calculation, that is, the summation in the standard deviation calculation is over all the constituents that have been in the composite for the whole period. The individual fund returns are the subperiod linked returns for the two-month period. Notice that the average composite return used to determine the deviations from the average is 14.73%. This is different from the actual reported composite return equal to 15.49%. This is because the mean return used for calculating the multiperiod asset weighted standard deviation includes only the funds in the composite for the complete period.

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Quartile Dollar Dispersion

In addition to the asset-weighted mean and standard deviation, you can further describe the asset-weighted distribution of returns by calculating asset-weighted first and third quartile returns. Quartile dollar dispersion is the rate of return for the best and worst performing 25% of dollars invested in the composite during the period. The return on the best performing 25% of dollars is sometimes called Quartile Dollar Dispersion 1, or QDD1. The return on the worst performing 25% of dollars is Quartile Dollar Dispersion 4, or QDD4.

To calculate the Best QDD:

1. Select the funds that were in the composite for the whole period.
2. Order the fund returns from high to low.
3. Calculate a quarter of the composite market value.
4. Starting with the best performing fund, accumulate the portfolios up through the quarterly value from step 3.
5. Weight the portfolio returns by their percentage of the quarterly value. For the fund that puts the accumulated market value over the quarterly market value, use only the portion of the market value required to foot to the quarterly market value.
6. Calculate the QDD by summing the weighted returns.

You calculate the Worst QDD in the same way, except that you use the mirror of the process to accumulate the worst performing portfolio returns. The following figure shows the calculation of the best and worst quarter dollars under management for the sample composite over the two-month period.

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Notice that in the example, the Best QDD equals the contribution of the best performing fund plus a portion of the next best performing fund.

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Equal-Weighted Composite Statistics

One of the drawbacks to using a weighted average return statistic to represent the historical performance of an investment strategy is that not only is it possible for a particular fund to achieve a return different than the reported weighted average return, but it is also possible that no client actually got a return close to the reported return. Because managers usually do not disclose the individual constituent fund returns and weights in their presentations to prospective customers, you need additional statistics to understand dispersion of return around the average experienced by individual clients during the period (Assuming that each investor has a separate portfolio).

To do this, you can calculate composite average and dispersion statistics on an equal-weighted basis. While the asset-weighted average statistics provide a measure of the average dollar invested during the period, the equal-weighted statistics measure the dispersion of performance to the average fund. You measure equal-weighted dispersion using standard descriptive statistics including the following:

• Mean and standard deviation of returns (equal weighted)
• High, low and range of returns
• First quartile, median, and third quartile returns

The example in the following figure illustrates the calculation of equal weighted composite descriptive statistics over the two-month period.

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Notice that you continue using the convention of including in the calculations only those funds that were in the composite for the entire (in the example) two-month period. The mean return of 9.77% is the equal-weighted average return using the eight funds that were constituents of the composite over both months. The dollar weighted average return for the composite was significantly higher at 15.49% because several large portfolios did better than the rest during the two-month period, and therefore heavily influenced the weighted average composite return.

There are several alternative algorithms available for calculating equal-weighted composite descriptive statistics like quartile, decile, and percentile return. The formulas used by Eagle Performance are documented here.

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Percentiles

The algorithm for calculating Percentiles (whether Quartile, Decile or Custom quintiles) is to rank the observations and then determine which observation to use as the n-tile break using:
n + 1) p = i + f

Where:

• n = the number of observations (constituent portfolios)
• p = the percentile value divided by 100
• i = the integer part of (n + 1)p
• f = the fractional part (n + 1)p

For Best Performing percentiles, you order the observations in descending order, for Worst Performing percentiles, you order observations in ascending order. Then you select the observation x_i to use by looking at the value for f:

If f = 0.0 then Percentile Value = x_i
If f = 0.5 then Percentile Value = (x_{i +} x_{i +1})/ 2
If f < 0.5 then Percentile Value = x_i
If f > 0.5 then Percentile Value = x_{i +1}

The following table lists some examples for a composite made up of 9 constituent portfolios for the 1-year period ended 12/31/2000.

The following table shows this example ordered descending.

The following table defines the example composite calculations.

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Cumulative Equal-Weighted Return

The statistics calculated in the previous example use only the funds that are in the composite for the complete period, thus you excluded funds 6 and 7. You can also calculate an equal weighted return that includes all of the portfolios.

To calculate a cumulative equal-weighted return including all of the portfolios:

1. Calculate an equal weighted composite return for each period.
2. Link the equal weighted composite returns across periods. The example in the following figure illustrates the calculation of the cumulative equal weighted return using all of the constituent portfolios.
   
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Composite Membership Statistics

For some of the dispersion statistics you calculated, you only used the funds that were constituents of the composite over the entire linked period. To assist in the analysis of the performance presentation, you can include other statistics about the number of portfolios that were included in the calculation of the composite statistics, including the number of portfolios that were added to, removed from, and remaining in the composite for the complete period. These composite membership statistics can provide some insight into the rate of client turnover or strategy changes.

It is possible for a composite to have the same number of portfolios at the beginning and end of a yearly period, but given client turnover, the actual funds used in the computation of the composite returns and descriptive statistics could have changed drastically during the period.

Finally, GIPS requires disclosure of the percentage of the firm's total assets represented by the composite at the end of the period. Care must be taken to avoid double counting when the total net assets are calculated. Example statistics are shown in the following figure.

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Composites of Composites

The Composite Analysis report allows enumeration of composites of composites as an option in the report profile to enable drill through to the lowest level in a composite of composites entity structure when determining the entities to use for selecting the performance data to support the dispersion calculations.

When a Composite Analysis report is submitted, it queries the ENTITY_RANGE_DETAILS table to determine the underlying member entities (constituents) for each composite that was chosen in the profile. The performance data required is then retrieved for these entities and used in the dispersion statistic calculations.

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Enumeration of Composites

The report includes an option to Enumerate Composites in the report profile window. When this option is selected, the process traverses the multiple levels of a composite entity structure down to the lowest account level that exists. In PACE, the lowest account level can be any of the following entity types: Portfolio, Benchmark, or Sub-Portfolio. At this level, the process gathers the performance data necessary to calculate the composite's dispersion statistics. This is the same as the Enumerate Composites option found in the Performance Return and Composite Weighted return calculation reports.

This functionality supports reports submitted for a single composite, multiple composites, or Lists of composites. The Multicurrency Composites and Do Not Calc if Missing Data options also support this feature. In both cases, the enumeration of the composites constituents happens independently from the calculations specific to these options. This option defaults to No.

In PACE, composites created for GIPS performance presentation have an entity type of ACOM, and are called Performance Composites. The following Entity Type codes are referred to throughout the rest of this section. See the following table for translations.

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Enumeration Rules

Assumptions:

• Performance Composites only contain member entities of the following types: ACOM, PORT, SUB, and INDX.
• Entity IDs must not be duplicated within a Performance Composite. If ACOM1 contains ACOM2 and ACOM2 contains PORT1, then ACOM1 should not also contain PORT1.
• If duplicate entities do exist and at least one is eligible for inclusion based on its start and stop dates in its parent composite, it will be included as a constituent of the composite.
• If duplicate entities exist and both are eligible for inclusion, only data for one will be utilized in the composite calculation.
• There could be multiple levels of entities to traverse through to find the final list of lowest level entities to query the database for (that is Composite made up of Composites that are also made up of composites and so on). There should be no limit to the levels within the Composite Entity structure.
• Enumeration is used to drive how the supporting data for the calculations is gathered. Everything else within the reports remains as it is today.

The following table shows an example of Performance Composite enumeration for Annual Calculation Period 2001 (01/01/2001 to 12/31/2001).

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Determination of Composite Constituents

With the Enumerate Composites option selected in the report profile:

• The final constituent entity list for the profile composite ACOM1 would include the following five entities: PORT1, PORT5, PORT3, PORT4, and SUBPORT1.
• Port2 is not included due to the stop date of 1/1/00.

Without the Enumerate Composites option selected in the report profile:

• The final constituent entity list for the profile composite ACOM1 would include the following three entities: PORT1, PORT5, and ACOM2.
• Port2 is not included due to the stop date of 1/1/00.

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Enumeration Process

For each Date Range period in the Composite Analysis report:

1. Evaluate member inclusion for all L1 member entities.
2. Of the eligible L1 member Entities, determine if any are of entity type ACOM.
3. For eligible ACOM entity types, evaluate their member entities' eligibility for inclusion based on their start and stop dates in the L1 ACOM entity. The same inclusion rules used at L1 should be followed here (that is L2 entity must be a member of L1 entity for the entire calculation period to be included in the calculation).
4. Only include L2 members that were part of eligible L1 ACOM Entity for the entire calculation period.
5. Determine list of entities to process as constituents of the profile composite for the period.
6. Check for duplicate entity Ids in the list.
7. If any duplicate entities exist, only include one of the entities in the final constituent entity list
8. Use the final constituent entity list to gather performance data for the profile composite.
9. Once data is gathered as noted above, all composite analysis calculations and processes should operate in the same manner as they normally do today.
10. The drill-through capability for each date range should show the final constituent entity list and their returns.

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Dispersion Statistics Calculation

The Composite Analysis report provides on-the-fly calculation of common composite dispersion statistics. It also will report multiperiod composite returns, market values and other Composite performance data that is stored for the Composite entity in the PERFORM database. The Composite Analysis report:

• Accurately identifies the constituent portfolios of the composite for the various periods of the report.
• Can be configured to calculate composite dispersion statistics for several months, quarters, years, since inception or any customized period.
• Calculates these statistics based on the constituent portfolio data stored in the PERFORM database.
• Can also pull out stored Composite Data and link Composite sub-period returns to report multiperiod returns.
• Can report Benchmark data with the Composite.

The Composite Analysis report is configurable. When configuring the statistic fields, the data mapped from the PERFORM database as the inputs to the calculation can be completely customized. For example, you can calculate asset weighted dispersion statistics using various weighting factors such as Beginning Market Value, or Average Invested Balance (ABAL).

With the enumerate composite feature, the data mapped from the PERFORM database as the inputs to the calculation would be selected for the final constituent entity list as determined by enumerating through the profile composite down to the lowest level entity. See the following figure.

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