The material in this section was initially prepared by the author based on Morningstar's descriptions and tested for accuracy with experiments using Morningstar's CD-ROM data to replicate selected results. An early draft of the material was reviewed by Morningstar and revised to correct errors. Subsequently, new material was added. I am grateful for the assistance provided by Paul Gozali, Don Phillips and Paul Torregrosa of Morningstar. Any errors, of course, remain mine.
Morningstar provides a great deal of useful information in its print publications and in its Principia Plus CD-ROM data service. We focus here on six measures of performance that take into account both a measure of historic average or compound return and a measure of historic risk and/or benchmark performance. In order of increasing complexity, they are:
The first three measures are absolute and cardinal -- each fund receives a numeric rating that is not dependent on the performance of other funds. The other measures are relative and ultimately, ordinal. Each is built in three stages. In the first stage, a cardinal measure is determined for each member of a group. In the second stage these measures are converted to ordinal ranks, indicating relative positions within the group. In the final stage, the ranks are used to assign each fund a rating from 1 to 5, with 5 assigned to the best funds in the group and 1 to the worst.
The first four measures summarize aspects of fund performance over the most recent three years, based on total returns without considering any load charges. The last two measures take load charges into account.
The succeeding sections describe a number of subsidiary calculations that serve as components of the six measures of interest. To avoid ambiguity and to follow in the tradition of Morningstar's nomenclature, we prefix some of the measures with "ms". We also highlight the sections describing such measures in red.
A primary ingredient in all the Morningstar calculations is the set of historic monthly rates of return for the funds. The rate of return for fund i month t is calculated as follows:
Rit = (EVit + EVDit - EVi,t-1) / EVi,t-1
where:
EVit = closing net asset value on the last trading day of month t
EVDit = closing net asset value on the last trading day of month t of distributions taken as shares in month t
EVi,t-1= closing net asset value on the last trading day of month t-1
In general, distributions are reinvested in the fund's shares at the closing net asset value on the "ex-distribution day".
For many calculations, excess returns are utilized. The excess return for fund i in month t is the difference between the fund's return for the month and the return on a 90-day U.S. Treasury bill for that month:
ERit = Rit - Bt
where:
ERit = the excess return on fund i in month t
Bt = the return on a Treasury bill in month t
While Morningstar does not provide the source of its Treasury bill returns, they are very similar to those of Salomon Brothers' monthly returns on 90-day Treasury bills, averaging slightly under 25 basis points (.25%) less per year over the period from 1987 through 1996. During this ten-year period the two series were also highly correlated, with a correlation coefficient of 0.981.
One measure of performance over a period is a simple arithmetic average of a fund's monthly returns, calculated by summing the fund's returns for all the months, then dividing by the number of months:
AMRi = ( 1 / T ) * sumt=1..T { Rit }
where:
AMRi = the arithmetic monthly average return on fund i during period T
sumt=1..T { } denotes the sum of the expressions in the brackets with t = 1, 2,..,T
T = the number of months in the period
Both here and in subsequent equations we omit the time period covered (T) in order to simplify the notation. Most of the calculations cover the last three years, with T = 36.
For some purposes, statistics for a fund's excess returns are used. The average monthly excess return is computed in the same manner as the average monthly return, but using excess returns rather than total returns:
AMERi = ( 1 / T ) * sumt=1..T { ERit }
where:
AMERi = the arithmetic monthly average excess return on fund i during period T
A traditional measure of performance in the mutual fund industry is the cumulative value of $1 compounded over a specified number of periods. Morningstar uses such measures on both before-load and after-load fee bases. The value relative at the end of T periods not taking any load charges into account is:
VRi = prodt=1,..T { 1 + Rit }
where
VRi = the T-period value relative for fund i
prodt=1,..T { } denotes the product of the expressions in the brackets with t = 1, 2,..,T
Morningstar uses the term Mean for the annual rate of return that would provide the same T-period value relative as did the actual monthly returns for a fund over the T periods. If there are T months in the period, the number of years (y) will equal T/12. The desired value is then found by solving the equation:
( 1 + msMRi ) y = VRi
where:
msMRi = the Morningstar mean return for fund i
y = T/12 = the number of years in the period
A statistic summarizing the historic variation of monthly returns during a period is the standard deviation. This is calculated by taking the square root of the average of the squares of the deviations of the returns from their average value:
VMRi = ( 1 / T ) * sumt=1..T { ( Rit - AMRi ) 2 }
SDMRi = sqrt { VMRi }
where:
VMRi = the variance of monthly returns for fund i during period T
SDMRi = the standard deviation of monthly excess returns for fund i during period T
sqrt { } denotes the square root of the expression in the brackets
To some purposes it is useful to measure the standard deviation of a fund's monthly excess returns, computed as:
VMERi = ( 1 / T ) * sumt=1..T { ( ERit - AMERi ) 2 }
SDMERi = sqrt { VMERi }
where:
VMERi = the variance of monthly excess returns for fund i during period T
SDMERi = the standard deviation of monthly excess returns for fund i during period T
Morningstar's reported standard deviation of fund returns uses both the monthly standard deviation and monthly mean return to compute an annualized value, assuming compounding of the monthly returns and zero serial correlation. The formula is:
msSDi = sqrt { [ SDMRi2 + ( 1 + AMRi ) 2 ] 12 - [ ( 1 + AMRi ) 2 ] 12 }
where:
msSDi = Morningstar standard deviation of fund i
For some calculations, Morningstar compares the cumulative value of $1 invested in a fund with the cumulative value of $1 invested in Treasury bills. The latter is:
VRB = prodt=1,..T { 1 + Bt }
where:
VRB = the T-period value relative for Treasury bills
The excess T-period Value Relative for fund i is then:
EVRi = VRi - VRB
where:
EVRi = the excess value relative for fund i after T periods
To compute a mean excess return, Morningstar finds the annual rate of return that, if compounded, would have produced a value equal to one plus the excess T-period value relative:
( 1 + msMERi ) y = 1 + EVRi
where:
msMERi = the Morningstar mean excess return for fund i
y = T / 12 = the number of years in the period
Morningstar computes an annualized standard deviation based on a fund's excess returns using a formula similar to that used for total returns:
msSDERi = sqrt { [ SDMERi2 + ( 1 + AMERi ) 2 ] 12 - [ ( 1 + AMERi ) 2 ] 12 }
where:
msSDERi = Morningstar standard deviation of fund i's excess returns
To compute its version of the Sharpe Ratio for a fund, Morningstar divides its mean excess return measure by its standard deviation measure:
msSRi = msMERi / msSDERi
where
msSRi = the Morningstar Sharpe ratio for fund i
In early 1997, Morningstar assigned each fund in its database to a category. Each category was, in turn, assigned to one of four asset classes. The tables below show the categories included in each of the four asset classes at the time.
Domestic Equity
Large Value Large Blend Large Growth Medium Value Medium Blend Medium Growth Small Value Small Blend Small Growth Specialty - Precious Metals Specialty - Natural Resources Specialty - Technology Specialty - Utilities Specialty - Health Specialty - Financial Specialty - Real Estate Specialty - Communication Specialty - Unaligned Domestic Hybrid Convertibles
International Equity
Europe Latin America Diversified Emerging Markets Pacific Pacific ex-Japan Japan Foreign Stock World Stock International Hybrid
Taxable Bond
Long-Term Government Intermediate-Term Government Short-Term Government Long-Term Bond Intermediate-Term Bond Short-Term Bond Ultrashort Bond High-Yield Bond Multisector Bond International Bond
Municipal Bond
Muni National - Long-Term Muni National - Intermediate Muni Single State - Long-Term Muni Single State - Intermediate Muni Bond - Short-Term
Of particular interest are the first nine categories in the Domestic Equity asset class, termed diversified equity funds by Morningstar. Beginning in later 1996, each equity fund that was considered sufficiently diversified was assigned to one of these nine classes, with the choice of category "... based on an average of the portfolio's characteristics over the past three years".1 Generally, such assignments are likely to be related to the average historical style box classifications of the funds over the previous three years.(the style box classification of a fund at any given time is based on the average price-to-book value, average price-to-earnings ratio, and market-capitalization of its most recently reported equity holdings). However, Morningstar "... may occasionally move a fund to a different category than its historical statistics would indicate if the fund has made a dramatic shift in style."2 but in such a case will not calculate relative-to-category measures until the fund has been in the new category for 18 months. In general, the category to which a fund is assigned is likely to reflect its average position over the previous three years on both the value/growth (low to high average stock price-to-book and price-to-earnings) and large/small (large to small average stock market capitalization) spectrums.
To compute a second measure of performance, Morningstar performs a regression analysis comparing the monthly excess returns on a fund over the last 36 months with the excess returns on a standard index. For funds in the Domestic Equity and International Equity classes, the index utilized is Standard and Poor's 500 stock index. For funds in the Taxable Bond and Municipal Bond asset classes, the index used is Lehman Brother's Aggregate Bond Index.
The regression equation may be written as:
ERit = ai + msBetai * ERindex,t + eit
where:
ERit = the excess return on fund i in month t
ERindex,t = the excess return on the index in month t
ai = the regression intercept
msBetai = the regression slope coefficient
eit = fund i's residual return in month i
As in any such regression analysis, the slope coefficient can be computed by dividing the covariance of the variables by the variance of the independent variable. Here:
msBetai = cov ( ERi , ERindex ) / Var ( ERindex )
where:
msBetai = Morningstar's beta for fund i
As the notation indicates, this gives Morningstar's Beta coefficient for fund i.
The intercept from the regression used to compute the Morningstar beta for a fund provides a measure of fund performance, since it represents the mean difference between the fund's excess return and that of a strategy using an index, levered up or down to have the same beta value relative to the underlying index. To produce its measure of alpha, Morningstar annualizes the regression intercept using compounding, so that:
1 + msAlphai = ( 1 + ai ) 12
where:
msAlphai = Morningstar's alpha for fund i
Morningstar R-squared
The explanatory power of the regression equation fit by Morningstar to compute alpha and beta values is measured by the R-squared value. This is equal to one minus the ratio of (1) the variance of the residuals from the equation to (2) the variance of the fund's excess returns:
1 - msR2i = var ( ERi - Betai * ERindex ) / var ( ERi )
where:
msR2i = Morningstar's R-squared for fund i
Morningstar Best-Fit Alpha
In addition to the regression analysis used to produce its alpha measure, Morningstar regresses each fund's excess returns for the last 36 months on a number of other indices, one at a time. It then selects the index that produces the best fit (highest R-squared value) for each fund. The alpha value from this regression is reported as the fund's best-fit alpha. The procedure is identical to that used to compute the alpha value for the fund, with the best index used instead of the standard index for the fund's asset class. In some cases the two indices are the same and the best-fit alpha will equal the (regular) alpha. In other cases the indices differ, producing potentially different values of alpha.
In early 1997, twenty-one indices (termed secondary, specialized benchmarks by Morningstar) were used in this analysis. Of these, 14 were used for stock funds and 13 for bond funds. The following table lists the indices and indicates which ones were tested for each of the two types of funds.
ID Stk Bds Index LB Agg Y Lehman Brothers Aggregate Bond Index LB Int Y Lehman Brothers Intermediate-Term Treasury Index LB L-T Y Y Lehman Brothers Long-term Treasury Index LB Govt Y Lehman Brothers Government Bond Index LB Corp Y Lehman Brothers Corporate Bond Index LB Mtg Y Lehman Brothers Mortgage-Backed Securities Index LB Muni Y Lehman Brothers Municipal Bond Index FB HY Y Y First Boston High-Yield Bond Index S&P500 Y Y Standard and Poor's 500 Index SP Mid400 Y Standard and Poor's MidCap 400 Index Russ 2000 Y Y Russell 2000 Index Wil 4500 Y Y Wilshire 4500 Index SB World Y Salomon Brothers Non-U.S. Dollar World Government Bond Index MSCI AllCtry Y Y Morgan Stanley Capital International All Country Index MSCI EASEA Y MSCI Europe, Australia and South East Asia Index MSCI Eur Y Morgan Stanley Capital International Europe Index MSCI Pac Y Morgan Stanley Capital International Pacific Index MSCI PacxJp Y Morgan Stanley Capital International Pacific ex Japan Index MSCI WdxUS Y Morgan Stanley Capital International World ex U.S. Index JSE Gold Y JSE Gold Index Wil REIT Y Wilshire Real Estate Investment Trust Index
Morningstar also reports beta and R-squared results from the best-fit regression, giving three measures:
msBFAlphai = Morningstar's best-fit alpha for fund i
msBFBetai = Morningstar's best-fit beta for fund i
msBFR2i = Morningstar's best-fit R-squared for fund i
To provide a normalized measure of returns of funds within each category, Morningstar divides each fund's excess 36-month value relative by a base figure. For each category, the base is the larger of (1) the average excess 36-month value relative for the funds in the category or (2) the 36-month value relative for Treasury bills - 1:
msCReti = EVRi / RetBasec(i)
where:
msCReti = Morningstar's category return for fund i
RetBasec(i) = the return base for fund i's category
and:
RetBasec(i) = max { avgj in c { EVRj } , VRB - 1}
where:
avgj in c { } indicates the average of the term in the brackets for funds (j) in category c
Since the excess value relative equals the value relative minus the value relative for bills, the return base will equal the average excess value relative only if the average increase in value for the category exceeds two times the increase in value for Treasury bills:
if avgj in c { VRj - 1 } > 2 * ( VRB - 1)
RetBasec(i) = avgj in c { EVRj }
else
RetBasec(i) = VRB - 1
If the category returns are high enough for the first of the two formulas above to be utilized, the average of the category returns for the funds within a category will equal 1.0. Otherwise, it will be less than 1.0.
Average Monthly Loss
Morningstar's measure of risk is based on the extent to which a fund's returns fell below those of Treasury bills. The first step in such calculations involves the calculation of a fund's loss in each month, defined as the smaller of its excess return and zero:
Lit = min { ERit ,0 }
where:
Lit = fund i's loss in month t
The Average Monthly Loss for fund i is its average loss over T periods:
AMLi = sum ( Lit ) / T
where:
AMLi = fund i's average monthly loss over T periods
To compute a measure of relative risk for each fund in a category, Morningstar divides each fund's average monthly loss by the average of such values for all funds in the category:
msCRiski = AMLi / RiskBasec(i)
RiskBasec(i) = avgj in c { AMLj }
where:
msCRiski = Morningstar's category risk for fund i
RiskBasec(i) = the risk base for fund i's category
While the average monthly loss for a fund will typically be negative, so will the risk base for its category. Thus the Morningstar category risk values will all be positive and the values for the funds within a category will average to 1.0.
Morningstar combines a fund's category return and category risk to compute a category risk-adjusted rating:
msCRARi = msCReti - msCRiski
where:
msCRARi = Morningstar's category risk-adjusted rating
These values are then ranked and converted to category rating percentiles, with the largest value assigned to the 100'th percentile and the lowest to the first percentile. Finally, category ratings are assigned, based on the percentile rankings of the funds within the category, as follows:
From Percentile To Percentile Category Rating 0 10 1 10 32.5 2 32.5 67.5 3 67.5 90 4 90 100 5
For purposes of its"star" ratings, Morningstar uses returns that take into account any "maximum front-end load fees, applicable deferred loads, and applicable redemption fees". The goal is to compute the "cash-out" value at the end of T periods, assuming an initial outlay of $1. For example, if a fund requires a maximum front-end load fee equal to Li of the initial outlay, the load-adjusted T-period value relative will be:
LAVRi = ( 1 - Li ) * VRi
where:
LAVRi = fund i's load-adjusted value relative for T periods
For a no-load fund, Li = 0 and LAVRi = VRi
Morningstar's load-adjusted return is the load-adjusted counterpart to the Morningstar mean return. It is the annual rate of return that would provide the same T-period load-adjusted value relative as did the fund. If there are T months in the period, the number of years (y) will equal T/12. The desired value is then found by solving the equation:
( 1 + msLARi ) y = LAVRi
where:
msLARi = the Morningstar load-adjusted return for fund i over T periods
y = T/12 = the number of years in the period
A fund's Load-adjusted excess T-period value relative is calculated by subtracting the ending value of $1 invested in Treasury bills over T periods from that obtained from the fund, after adjusting the latter for any load charges:
LAEVRi = LAVRi - VRB
where
LAEVRi = the Load-adjusted excess value relative for fund i after T periods
To normalize load-adjusted excess returns for a period, Morningstar divides each fund's Load-adjusted excess value relative by a base figure. This differs from the calculations used for category returns, however, since the base is determined by averaging the comparable values for all funds within the asset class in which the fund is placed:
msReti = LAEVRi / ClassRetBaseAC(i)
where:
msReti = the Morningstar return for fund i
ClassRetBaseAC(i) = the return base for fund i's asset class
and:
ClassRetBaseAC(i) = max { avgj in AC { LAEVRj } , VRB - 1}
Here, the return base will equal the average load-adjusted excess value relative only if the average increase in value for the asset class exceeds two times the increase in value for Treasury bills:
if avgj in AC { LAVRj - 1 } > 2 * ( VRB - 1)
ClassRetBaseAC(i) = avgj in AC { LAEVRj }
else
ClassRetBaseAC(i) = VRB - 1
The Morningstar Risk of a fund is computed by taking the ratio of its average monthly loss during a period to the average of such values for all funds in its asset class:
msRiski = AMLi / ClassRiskBaseAC(i)
ClassRiskBaseAC(i) = avgj in AC { AMLj }
where:
msRiski = The Morningstar risk for fund i
ClassRiskBasec(i) = the risk base for fund i's asset class
To compute a risk-adjusted rating for a fund, Morningstar subtracts the Morningstar risk value from the Morningstar return value:
msRARi = msReti - msRiski
where:
msRARi = Morningstar's y Risk-adjusted rating
Where data are available, Morningstar computes risk-adjusted ratings for a fund using 3, 5 and 10-year periods (1 year values are computed as well, but of less interest here) . For any given period (say, 3 years), the risk-adjusted ratings for all funds within an asset class are ranked and converted to percentiles, then stars are assigned using the same cutoff points employed for category ratings:
From Percentile To Percentile Stars 0 10 1 10 32.5 2 32.5 67.5 3 67.5 90 4 90 100 5
Of particular interest in this connection are the 3-year risk-adjusted ratings and the associated 3-year star ratings, since they are based on data from the same period used for the other performance measures and are available for the most funds.
In addition to the star ratings for specific periods, Morningstar calculates an overall star rating for each fund for which there are at least 36 months of data. The first step involves the determination of a star rating for each of the time periods (3, 5 and 10 years) available for the fund in question. For each fund, a weighted average of the number of stars in each time period is then taken, with the weights depending on the months for which data are available, as follows:
Months 3 year RAR 5 year RAR 10 year RAR 36 to 59 1.00 0 0 60 to 119 0.40 0.60 0 120 or more 0.20 0.30 0.50
The rounded value of the result is then reported as the fund's overall star rating.
This section has summarized six substantially different performance measures computed routinely by Morningstar:
The first five measures are based on fund returns over the most recent 36 months and thus provide different ways of analyzing a given body of data. The last covers periods of 3 years for some funds, 5 for others, and 10 for yet others using a procedure similar to that employed for the 3-year ratings, but on a different body of data. In subsequent sections, the emphasis will be on the first five measures.
1. Amy C. Arnott, Morningstar Mutual Funds. December 6, 1996.
2. ibid.
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