We call the group of funds used for performance comparison a perfor- mance universe. A universe is a list of portfolios similar to ours and includes their returns and other statistics over different time periods. We can create our own universe, or we can obtain precompiled performance data from universe publishers. There are separate universes available for funds marketed to institutional and retail investors. Publishers of retail fund universe data include Morningstar, Lipper, and Micropal. We can use their data to compare mutual funds. Pension consultants, custodi- ans, and publishers maintain institutional fund universes. Some publish- ers of institutional fund universe databases include Russell-Mellon, Wilshire (TUCS), and the WM Company. In addition to the published universes, pension consultants, fund companies, and other organiza- tions maintain inhouse universes for research and competitor analysis.
While each performance universe has its own distinguishing charac- teristics, the methodology used to create and rank funds within them are largely the same. The main task of a universe publisher is to periodically
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gather data on the funds in which it is interested. Some of the universe publishers independently calculate the returns; others obtain returns precalculated by managers, plan sponsors, or their custodians. The retail universes calculate their own returns using the published NAVs and distribution data. Quarterly periodic returns are the norm for insti- tutional universe products, but some are updated more frequently.
The first step in comparing our fund to a peer group is to select the appropriate universe. Within each universe funds are grouped by cate- gory into subuniverses. Subuniverses are created by filtering the universe of funds to derive a list of comparable portfolios by:
■ Asset class and strategy. For example, domestic equity, international equity, and real estate.
■ Type of investor. For example, funds managed on behalf of corporate or public defined benefit plans, endowments, and foundations.
■ Type of manager. For example, funds managed by asset management firms, banks, or insurance companies.
Within the appropriate subuniverse, funds can be further filtered to create a customized list of appropriate funds for comparison. For exam- ple, we could take one of the institutional fund universe data sets for funds that are being managed on behalf of corporate pension plans and then filter it to exclude funds under a certain size, and then exclude funds that do not yet have a three-year track record. We can then com- pare and contrast the funds. For example, we could compare the perfor- mance of our small-capitalization company separate account manager to all of the small-capitalization funds that are included in the universe.
We can rank and compare the performance of the funds in the nar- rowed resulting list. We can compare performance by any return, risk, or risk-adjusted return statistic. Data are available over several time periods; so we could, for example, compare funds by year-to-date cumu- lative return, or three-year annualized returns. In addition to perfor- mance statistics we can compare other characteristics of the fund, such as the fund’s asset allocation weightings, average credit quality, or dura- tion.
Rank and Order Statistics
We are interested in our relative performance within the universe. Was our fund a top performer? Or was it a middling or poor performer? We use rank and order statistics to evaluate the relative performance of a fund within a universe. Rank and order statisticslike medians, quartiles, and percentiles provide a way of describing the relative position of a
Measuring Relative Return 101
particular observation within a data set. Suppose we are interested in a peer group comparison of 1-year returns for a group of funds and that our fund had a 6.21% return in the period. Exhibit 7.1 shows the set of 1-year returns for a universe of 20 funds, sorted from high to low. We chose 20 for clarity of the examples, and while it depends on the market and strategy, the typical universe has more funds.
Once the returns are sorted, we can evaluate our relative position.
The first step is to determine the median return. The median is the return for the middle fund in the universe. Half of the funds will have performance above and half below the median. By looking at the perfor- mance of our fund versus the median we can see whether our fund per- formed better or worse than the average fund. Exhibit 7.2 shows the median universe return was 3.56%.
Note that by constructing the universe in this way we are equal weighting each portfolio return in the comparison. That is, we are not weighting the funds by their asset size or other criterion. We determined the median by looking for the middle observation in the sorted array of returns. We find the rank position of the middle value by taking (N+ 1)/
2 = 10.5 where N is the total number of return observations. If the total number of funds is an even number, then we can take the median as the average of the two middle observations. If Nis odd, the middle observa- tion serves as the median. In our case the median lies halfway between the tenth and eleventh observations.
EXHIBIT 7.1 Peer Group Universe Returns
EXHIBIT 7.2 Universe Median
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We can further refine the comparison by splitting the data set into four groups, or quartiles. The first quartile return is the middle return earned by the funds that performed better than the median return. In investments industry practice, first quartile performance denotes the performance of the best quartile. Performance within the first quartile indicates that the fund was a top 25% performer for the period. 25% of the return observations will be higher than the first quartile return and 75% will be below. In other contexts, first quartile is used to reference the worst performing quartile. The Excel quartile function uses the con- vention 1 = worst and 3 = best quartile.
The third quartile return is the performance of the fund halfway between the median return and the fund with the poorest performance for the period. Performance below the third quartile return indicates that the fund performance ranked in the bottom quartile. A measure of the dispersion, or variability, around the median return is the semi-inter- quartile range, which is the difference between the first and third quar- tile returns. The semi-interquartile range is an appropriate measure of variability when we are using the median as the indicator of the middle return. We discuss measures of dispersion where the average return is instead measured by the mean in Chapter 20.
Exhibit 7.3 illustrates the calculation of universe quartile returns.
Notice that the position of the best quartile return is 5.25 values from the top return ((20+1)/4). We used linear interpolation to calculate the return that corresponds with this position by taking the return at the sixth position (6.21%) and adding 0.75 × the difference between the fifth and sixth returns, or 0.05 (0.75 × (6.28 – 6.21)). This method is consistent with the way we calculated a median return by averaging the two middle observations. There are other ways of interpolating a return that falls at a noninteger rank. For example, the Excel quartile function returns 6.23%, which strikes us as strange because it is closer to the sixth observation than the fifth.
We can also determine the quartile ranking of a fund without calcu- lating a quartile return. Equation (7.1) shows how we can take the rank position to calculate the quartile ranking of a portfolio.
EXHIBIT 7.3 Universe Quartile
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EXHIBIT 7.4 Determining Quartile from Rank
(7.1)
Where ceiling is a function taking the result and rounding it up to the next highest integer value. Exhibit 7.4 uses this formula to deter- mine that our fund falls within the second quartile.
We did this by taking the rank order of the portfolio, multiplying it by the number of groups, and dividing the result by the total number of observations + 1. We then round the result up to the next highest integer value, which equals 2, or second quartile.
We can collect the funds into any number of groupings. Quantileis the generic term for these groupings, where quartiles = 4, quintiles = 5, decile = 10, percentiles = 100, and so on. Analysts commonly group portfolios into quintiles, or five groups, and deciles, or 10 groups, in addition to quartiles. We can generalize Equation (7.1) to find the Quantile rank of a portfolio:
(7.2)
A generic algorithm for calculating percentile returns is to:
1. Sort the array of returns from highest to lowest.
2. Compute the location of the N-tile within the array by:
(7.3)
3. If the quantile location is an integer value, then the percentile equals the value at that location. If it is not then we use linear interpolation to obtain the value. We do this by taking the two returns that bound the N-tile location, take the difference between them, and multiply by per- centile value. Then we add this result to the lower value.
Quartile ranking Ceiling Rank position×4 Number of observations+1 ---
=
Quantile ranking Ceiling Rank position×Number of groups Number of observations+1 ---
=
Quantile location Percentile ---100
Number of observations+1
( )
×
=
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EXHIBIT 7.5 Quantile Returns
EXHIBIT 7.6 Universe Comparison
Exhibit 7.5 shows the calculation of various quintiles for our sam- ple universe.
To calculate the n-tiles, we first converted them into percentiles. A percentile return P% for a data set is the value that is greater than or equal to (1 − P%) of the returns, but is less than P% of the returns (holding to our convention that the best quantile is the highest return).