Paul Merriman: How does Vanguard justify, with this great family of index funds, also having … actively managed funds?

John C. Bogle: [A] lot of investors, no matter how persuasive the case for indexing is (and it's overpoweringly persuasive), just don't quite get it. They want a little more activity. So what we tried to do was pick good managers … [and deliver] the other big advantages of indexing: no sales commissions, very low expense ratios and low portfolio turnover. An article done by some professors at Duke University about a year ago [Reinker and Tower (2005)] showed that our active managers … actually did a hair better than [similar] index funds. On the other hand, if we had started the comparison a little bit later, the active managers would have done a little bit worse. But I think it's a valid strategy. What can I do and tell you? I'm still 80% indexed. [Bogle, 2006b, approximate transcription]

Dan Wiener: Vanguard wants you to ‘believe' in indexing. Your faith in indexing is the cornerstone of their business. But it's a lie… The big famous index funds at Vanguard have chronically underperformed over the last few years, exposing conservative investors to the worst risks of bear markets… Indexing is a great business—but it's a lousy investment! [Wiener (2007a)]

Introduction

Is John Bogle right that Vanguard's index and managed funds are comparable? Or is Dan Wiener right that Vanguard's index funds, especially the big index funds, underperform? And how should Vanguard investors choose between Vanguard's index funds and its managed funds?

To answer that last question, we examined the returns of 20 Vanguard active funds against comparable index benchmarks over the four-year period stretching from January 2003 to December 2006. Each Vanguard managed fund produces the same return as a tracking basket of comparable index funds, plus a differential. If the average differential, which we refer to as α, is positive, the managed fund is superior to the tracking index fund basket, which from now on we refer to as the tracking index.

In this paper, we calculate two types of α for Vanguard's managed funds. Our preferred measure is the geometric α. If the α for a managed fund is positive, that fund is superior to its tracking index.
What do we find? Managed funds that are closely tracked by their tracking basket were beaten by those baskets. Meanwhile, those that were more loosely tracked by their basket beat those baskets.
In addition, to determine if investors can use this information to make smart decisions, we asked whether α's in one period predict α's in the next period. They did not. We also asked whether managers made prescient style choices. They did not.

Daniel Wiener in his The 2006 FFSA Independent Guide to the Vanguard Funds (p.186) provides correlations of returns between different Vanguard equity funds, as a way of helping investors put together a portfolio of funds with reduced total risk. A complementary tool is the one we provide, the tracking index fund basket, defined as that collection of index funds that best tracks the returns of the managed funds. This tracking index basket enables one to describe the return of a managed fund as the return of a basket of index funds plus a constant term, which may be positive or negative. Investors who find that one of their managed funds substantially duplicates one of their index funds may wish to lighten their holdings of one or the other in order to maintain portfolio diversification. 1, 2

Method

We wish to describe each of Vanguard's managed funds in terms of its comparable index funds. The index funds we use are the entire collection of 12 diversified equity funds available over the four-year period from January 2003–December 2006. They are:

• Vanguard 500 Index
• Vanguard Emerging Markets Stock Index
• Vanguard European Stock Index
• Vanguard Extended Market Index
• Vanguard Growth Index
• Vanguard Mid-Cap Index
• Vanguard Pacific Stock Index
• Vanguard Small-Cap Index
• Vanguard Small-Cap Growth Index
• Vanguard Small-Cap Value Index
• Vanguard Total Stock Market Index
• Vanguard Value Index

Each index fund represents an investment in a segment of the stock market, i.e., a particular style of investment.

The 20 Vanguard managed funds we consider are:

• Vanguard Capital Opportunity
• Vanguard Capital Value
• Vanguard Dividend Growth
• Vanguard Equity Income
• Vanguard Explorer
• Vanguard Global Equity
• Vanguard Growth and Income
• Vanguard Growth Equity
• Vanguard International Explorer
• Vanguard International Growth
• Vanguard International Value
• Vanguard Mid-Cap Growth
• Vanguard Morgan Growth
• Vanguard PRIMECAP
• Vanguard Selected Value
• Vanguard Strategic Equity
• U.S. Growth
• U.S. Value
• Windsor
• Windsor II

Vanguard funds come in a variety of share classes: Investor, Admiral, Institutional, etc. The Investor share class carries the highest expense ratio, and is available with the lowest initial investment. Because certain funds do not have Admiral or Institutional shares, we use the Investor shares in our analysis in the interest of uniformity.

Investors are concerned with real returns, so we adjust nominal returns by the Consumer Price Index (provided in the Morningstar Principia Pro disks) to get real returns. Henceforth, when we refer to returns, we mean real returns. The formula used for the conversion is:

1 + R = [1+ N]/[1+I]

where R is the real rate of return, N is the nominal rate of return and I is the rate of inflation in the consumer price index, with all numbers expressed as a fraction per month.
To describe the return of a managed fund in terms of the index fund returns, we regress the monthly return of the fund on the monthly returns of all of the index funds, while constraining all of the coefficients of the index funds to be nonnegative and to sum to one. Using the Vanguard PRIMECAP fund as an example, we get:

where R denotes monthly (real) return in percentage points. This regression says that PRIMECAP is an asset whose return is best described as the return of a basket of index funds consisting of 37 percent Vanguard 500 Index fund, 1 percent Vanguard Emerging Market Index fund, 29 percent Vanguard Growth Index fund, 9 percent Vanguard Mid-Cap Index fund and 24 percent Vanguard Small-Cap Growth Index fund, with an additional return of 2.83 percent per year (0.236 percent per month), and a random term, where the index basket is rebalanced at the beginning of each month.

In this case, we say the managed fund outperforms the tracking index basket by 2.83 percent per year, the arithmetic α. These results are recorded in Figure 1.3
We use the statistical program R and its package mgcv, which are freely available online,4 to perform all of the calculations. In addition to the regression coefficients, we compute R2 values (interpreted as the proportion of variability in the data explained by the model) as one minus the ratio of the sum of squared errors to the sum of squares of the returns, as well as t-statistics for the significance of the intercept term using the standard formulas for linear models.5 An easier approach for those unfamiliar with R is to follow Tower and Yang (2007) and program the calculation using Microsoft Excel's solver.

Next we use the regression coefficients to calculate the return of the tracking index basket over the period. We calculate geometric α as the geometric average monthly return of the managed fund minus that of the index basket.6 In the figures, we multiply it by 12 to put it on the more familiar annual basis. The geometric α differs from the intercept of the regression (the arithmetic α) because geometric α is the difference in geometric returns, while the intercept of the regression is the difference in arithmetic returns. The arithmetic α is the amount by which the expected one-period return of the managed fund exceeds the expected one-period return of the tracking index.7 Figures 1 and 2 show the two α's to be similar.

Our use of the tracking basket is our attempt to deal with Kizer's [2005] point that in assessing managed funds versus index funds, one must compare managed performance with index performance of comparable style.
We did not bother to risk-adjust. Instead, we simply report the standard deviations of monthly returns of the fund and the tracking index. Given α, the risk-averse investor will prefer the fund with the lower standard deviation. 8, 9

Results

Figures 1 and 2 present the calculations from which most of our conclusions flow. Figure 1 provides calculations for the period of our study taken as a whole: January 2003 through December 2006. Figure 2 presents the calculations from the first half of our study, 2003-2004, and the last half, 2005-2006. Both list the managed funds vertically and the index funds horizontally. The figures indicate the composition of the tracking indexes, geometric and arithmetic α's, t-stats for the arithmetic α's, R2s for the regression of the managed real returns on the index real returns, and standard deviations of the returns.10

Our data covers years in which the S&P 500 Index was rising. During such years, index funds tend to beat managed funds by more than when the index is falling, because index funds tend to be more fully invested in equities than managed funds.11

Adequate Comparisons Require Multiple Indexes

Adequate comparisons of managed funds and index funds require multiple indexes. Using only the S&P 500 Index as a benchmark for managed funds is a misguided strategy.

Figures 1 and 2 show that all tracking baskets require at least two different index funds, and no single index (neither the S&P 500 nor any other) is consistently used across all managed funds. However, we do observe indexes like Small Cap Value and Total Stock that add little explanatory power.

Insights From The Tracking Basket

The tracking basket provides insights into the managing styles of funds. Most of the results from the linear regression are not surprising, i.e., that Growth funds invest heavily in securities that are highly correlated with the Vanguard Growth Index fund. However, some striking patterns arise, shedding light into the managing style of funds. For example, drawing on Figure 1, the Vanguard Windsor II fund invests 92 percent of its assets in securities linked to the Vanguard Value Index fund, while the original Vanguard Windsor fund diversifies its investment, investing only 46 percent in Value index linked securities, with 28 percent in Growth and 9 percent in Small-Cap Growth. The Morningstar Principia Pro disk lists both of these funds as large value funds, showcasing the shortcomings of a simple assessment.

Similarly, the disk lists the Vanguard Capital Opportunity fund as a Mid-Cap Growth fund, while we find that its major holdings are small-cap-growth-linked securities. This is also true of the Vanguard Mid-Cap Growth fund. We suspect both are explained by the absence of a Vanguard Mid-Cap Growth Index fund until recently; the launch of that fund came too late to include in the study.

Also, the Vanguard International Value fund and Vanguard International Growth fund follow similar strategies, with European and Pacific stocks making up around 55 percent and 24 percent of their portfolios, respectively. The main difference is the higher proportion of emerging markets securities held by the Vanguard International Value fund. If international growth and value index funds existed, then the differences between these two funds would be more marked.

The only fund that tracks the S&P 500 closely is the Vanguard Growth & Income fund, with 84 percent of its capital invested in securities highly correlated with the index.

Figure 1

Social ''Index'' Fund

We find that the Vanguard FTSE Social Index investor class has an average short-period α of -0.52 percent per year, and it is riskier than the tracking index basket in both periods.

These are the costs of socially conscious investing with Vanguard during the period. In our tabulations of indexed versus managed funds, we exclude the FTSE social index, as it is not a managed fund or an index fund that one invests in for the purposes of raising return or controlling risk.

Performance Results: A Mixed Score

One useful set of findings is the top row in each set in Figures 1 and 2. There, we combined all of the managed funds into a single portfolio, using the portfolio weights at the beginning of 2003 and assuming continual monthly rebalancing. This managed portfolio beat its tracking index by 0.98 percent per year for the period from January 2003–December 2004 (a statistically significant difference), but underperformed by 0.04 percent per year for January 2005–January 2006 (not a statistically significant difference). On the whole, this gave the managed portfolios an average outperformance of 0.46 percent per year, which was not statistically significant.

Much of this differential is explained by the extraordinary performance of the Vanguard International Explorer, Capital Opportunity and PRIMECAP funds.12

Individual Funds Have Statistically Significant α's … After Style Adjustment

Figure 1 shows that, over the whole four-year period, the Vanguard Capital Opportunity, International Explorer and PRIMECAP funds have arithmetic α's that are statistically greater than zero, while the Vanguard Equity Income, U.S.

Value and International Growth funds have α's that are statistically less than zero. That is, the probability of any of these funds showing such an extreme value for α but representing no improvement over the tracking basket if the experiment were repeated indefinitely is at most 0.05. However, in the 2005-2006 subperiod, the only significant α is for the Vanguard International Explorer fund.

Of the 22 funds, the α's are large and positive (greater than one) for four funds; small and positive for four; and negative for 14.

Figure 2

An Important Discovery

Figure 3 illustrates an important discovery. Figure 3 graphs the average short-period geometric α against R2, the measure of how well the monthly managed returns fit the monthly returns of the corresponding tracking index. We see that managed funds that do not track closely to their index, i.e., that have low R2's, tend to beat those tracking indexes, whereas managed funds that are tracked closely tend to underperform their tracking indexes. Thus, investors would be well-advised to buy index funds instead of managed funds in situations where those funds closely track their tracking index fund baskets. However, when managed fund performance is not well explained by a tracking index fund basket, investors may look toward the managed funds.

Why might this be true? Perhaps only competent managers dare to deviate from the indexes. Alternatively, perhaps risk-averse executives at Vanguard are willing to give fund managers room to deviate from the indexes only when they have lots of confidence in the managers, or when a style is not captured by its index funds. Then again, perhaps we are making an apples-to-oranges comparison and we should disregard these results because we failed to find a good tracking index.

Figure 3 suggests that we further explore this issue by dividing our sample into closely tracked managed funds, with high R2s; and loosely tracked managed funds, with low R2s. We do this in Figure 4.
Figure 4 presents long- and short-period geometric α's. The 13 managed funds which closely track their tracking indexes (i.e., have high R2s) have an average short-period α of -1.28 percent per year. Meanwhile, the seven managed funds that loosely track their tracking indexes have an average short-period α of +1.87 percent per year. Thus, the closely tracked funds lose to their tracking indexes, while the loosely tracked funds beat them.13

Broad-Based Results

Overall, the average geometric α is positive, while the median geometric α is negative. Sixty percent of the short-period calculations show managed funds to have negative α's. Thus, we can't identify either managed or index funds overall as the winning class.

The managed funds on average have expense ratios that are 0.26 percent greater than their corresponding tracking index. They also have turnover rates that are 33 percent greater than the tracking index. If we follow Bogle [2006a] and estimate that each 100 percent increase in turnover reduces return by 1 percent per year, we would expect the managed funds to underperform their tracking indexes on average overall by 0.59 percent per year.

Figure 3

This predicted underperformance is greater than any of the observed underperformance figures in Figure 4 for the short- or long-run average. However, for the managed funds that closely track their index basket, the reverse is true: The managed funds underperform the tracking indexes by more than can be accounted for by the excess expense ratios and turnover of managed funds: from -1.28 percent per year to -0.68 percent per year. This implies no stock-picking prescience over our two-year periods for the closely tracked managed funds, net of the cash drag inherent to most managed funds. These funds are to some degree closet indexers, and they would have served their clients well by adhering even more closely to the indexes. Using turnover and expense ratio differentials between the managed fund and the index basket to predict α's is tricky, because Vanguard tells us 17 out of the 20 managed funds we consider have expense ratios which reward performance of the fund.

Managed Funds Are More Risky

From the last row of Figure 4, we see that managed funds have an average standard deviation of returns that is 0.11- 0.12 percent per month higher than that of the corresponding tracking index.14 Hence, managed funds are more risky.

This is surprising in view of the fact that managed funds tend to be less fully invested in equity than the index funds. We think managed funds tend to be less well-diversified than index funds, making them more risky. Windsor II, which strives to improve diversification by hiring multiple managers, is one of the few funds that has a lower risk than its tracking basket.

α's Are Not Generally Predictable

When we regressed the second period geometric α on the first period geometric α for 2003-2006, we found a positive relationship. Thus, from the bull market of 2003 through 2006, outperformance is predictable. This predictability might be due to autocorrelation in expense ratios, turnover, the share of assets held out of the stock market, the stockpicking genius of managers or the returns of the style of the mutual funds that are not captured by the styles of the indexes. As examples of the last point, Vanguard has no international value index, international growth index or international small-cap index fund. Overall, the correlation between the first- and second-half period geometric α's is 0.74. The correlation between the arithmetic α's is a similar 0.75.

However, when we look only at the 17 funds that are closely tracked over the four-year period (with R2's greater than 0.9), the correlation falls to 0.18. Moreover, when we break the eight-year period (1999-2006 inclusive) into four two-year periods (with the two funds discussed in footnote 12 missing), the correlation between α's in one period and in the subsequent period is negative. Thus, we don't see that picking mutual funds on the basis of past α's is profitable.
These results are consistent with Bogle (2007, p.xvii) who writes, ''Fund investors are confident that they can easily select superior fund managers. They are wrong.''

Figure 4

Mutual Funds Don't Make Prescient Style Adjustments

If mutual funds adjust their styles and correctly anticipate the winning styles, we would expect α's for the whole period to exceed the average α's for the shorter periods. In fact the whole-period average α's exceed the average of the short-period α's by 0.55 percent per year for the closely tracked funds, and by 0.05 percent per year for the loosely tracked funds, for an average of 0.38 percent per year and a median of 0.14 percent per year. Thus, in the last four years, mutual funds did make prescient style adjustments, and this adjustment is more marked for the closely tracked funds.

However, when we do the same analysis for the last eight years for all funds in existence during the entire period, we find average α's exceed the average of the short-period α's by -0.11 percent per year for the closely tracked funds and by +0.16 percent per year for the loosely tracked funds, for an average of -0.01 percent per year, where R2=0.85 is the cutoff between close and loose tracking.

Thus, there is no evidence that on average fund managers make prescient style choices. Individuals may wish to adjust their styles in accordance with anticipated differential returns to different styles if they are able to correctly predict differential style returns.

Are Vanguard's Managed Funds Better?

Dan Wiener, editor of The Independent Advisor for Vanguard Investors, in the quote at the beginning of this paper, denigrates the idea of investing in broad-based indexes. Both Bogle and Swensen [2005], however, among others, strongly advocate investing in broad-based indexes.

Who is right? It is instructive to see how Wiener's growth portfolio has performed. We draw on data from The Hulbert Financial Digest from the inception of Wiener's portfolios through December 31, 2006.
Wiener's Growth Portfolio from January 1992 (the first January after its inception) through December 2006 returned 2.27 percent per year more than the Wilshire 5000, with less risk. The probability that the Growth Portfolio could have outperformed by such a wide margin because of luck rather than skill is only 13.4 percent.15 The outperformance over the last nine years is 5.44 percent per year, and in each of those years, the Growth Portfolio has outperformed the Wilshire 5000.

Dan Wiener's belief that offering index funds is more profitable for Vanguard than offering managed funds recalls Paul Krugman's 1998 argument. Krugman asked, when he drives from Boston to New York and faces a headwind, does he drive more slowly? No, he depresses the accelerator further and maintains his speed. Likewise, he argues, imports from abroad do not cause unemployment. This economic headwind signals the Federal Reserve to sop up the incipient unemployment by depressing the interest rate accelerator, which stimulates investment and leaves employment where it was before. Similarly, when investors invest in a low-cost, low-turnover, broad-based index fund, Vanguard is able to sop up the incipient profits by depressing the expense ratio it charges. That Vanguard is unusual in doing this is the thrust of many of John Bogle's speeches. Bogle also champions low-cost competition in the mutual fund sector, and such competition should strengthen this mechanism.

Finally, if Vanguard's investors in indexed funds are less well-treated than those who invest in Vanguard's managed funds with the same style, we would expect α's for the managed funds that are closely tracked by their index baskets to be positive.

This is not the case: The average α is -1.28 percent per year. It may be that individual investors can pick styles presciently. If so, that argues for slice-and-dice indexing combined with managed funds for parts of the market not covered by index funds; it does not argue for the exclusive use of managed funds.

Is there any relationship between our α's and Wiener's buy, hold and sell recommendations? Looking at the February 2007 edition of his newsletter, Wiener [2007b], we find that those managed funds that he rates buy, hold and sell have average geometric α's for the four-year period of +1.39 percent, -0.83 percent and -1.66 percent per year, respectively. Thus, his recommendations are consistent with our α's.

Conclusion

We find that Vanguard's index and managed equity funds are comparable. There seems to be little reason to build a portfolio solely out of one or the other if one wishes to overweight some style categories, especially since Vanguard's managed equity funds are able to overweight patches of the market not overweighted by its index funds, like international value, international growth and international small cap.16

References

Bogle, John C. The Little Book of Common Sense Investing, Wiley 2007.
Bogle, John C. ''What's Ahead for Stocks and Bonds – And How to Earn Your Fair Share,'' Keynote speech, ''The Money Show,'' Las Vegas, Nev., May 15, 2006a.
_____ Interview with Paul Merriman on ''Sound Investing.'' August 4, 2006b, http://www.fundadvice.com.
Kizer, Jared. ''Index Fundamentalism Revisited: Redux.'' The Journal of Portfolio Management (Winter 2005), pp. 112-119.
Kritzman, Mark P. Puzzles of Finance, Wiley 2002.
Krugman, Paul, The Accidental Theorist and Other Dispatches from The Dismal Science, Norton, New York, 1998.
Reinker, Kenneth S., and Edward Tower, ''Index Fundamentalism Revisited,'' The Journal of Portfolio Management, Summer 2004, pp. 37-50.
Reinker, Kenneth S., Edward Tower and Wei Zheng, Barron's letter to the editor, March 14, 2005, p. 38.
Reinker, Kenneth S., and Edward Tower. ''Are Vanguard's Managers Good Stock-Pickers or Style-Pickers?'' The Journal of Portfolio Management (Spring 2005), pp. 109-111.
Rekenthaler, John, Michele Gambera and Joshua Charlson. ''Estimating Portfolio Style in U.S. Equity Funds: A Comparative Study of Portfolio-Based Fundamental Style Analysis and Returns-Based Style Analysis, Journal of Investing (Fall 2006), pp. 25-33.
Sharpe, William F. ''Asset Allocation: Management Style and performance Measurement.'' Journal of Portfolio Management (Winter 1992), pp. 7-19.
Swensen, David. Unconventional Success: A Fundamental Approach to Personal Investment, Simon and Schuster, Inc., New York, 2005.
Tower, Edward, ''Are GMO's Predictions Prescient?'' Duke Economics Department Working Paper, 2007. http://www. econ.duke.edu/Papers/PDF/GMO_Predictions1.pdf
Tower, Edward and Cheng-Ying Yang, ''DFA Versus Vanguard: Has DFA Outperformed Vanguard by Enough to Justify its Advisor Fees?'' Duke Economics Department Working Paper, 2007, http://www.econ.duke.edu/Papers/PDF/ Vanguard_Versus_DFA_30%20july_2007.pdf
Wiener, Dan, ''Action Plan for Vanguard Investors: A Publication of The Independent Advisor for Vanguard Investors, 2007a. Wiener, Dan, The Independent Advisor for Vanguard Investors, February 2007b.

Endnotes

1 Morningstar's portfolio instant X-Ray is also useful. It breaks managed funds down into the nine style groups (from large-cap value through mid-cap blend and small-cap growth) by proportion. But it does not distinguish between domestic and foreign equity.

2 Throughout the paper we ignore taxes. Thus, the analysis applies to mutual funds held in a tax-sheltered account. We don't find mutual funds appealing as a saving vehicle in a taxable account, as there, we prefer to hold individual stocks, selling off losers for capital losses when needed and postponing taxable sales where possible.

3 Our study is a variant of the usual returns-based-style analysis. For a useful survey, see Rekenthaler et al [2006].

4 R is a general-purpose statistical language that provides enormous flexibility and has become the academic standard for statistics departments at most universities in the U.S. It is a freeware version of the S language, originally developed in the AT&T labs and currently commercialized by the Insightful Corporation under the S-PLUS® brand®. Alternatively, one can use the solver feature in Microsoft Excel to perform the constrained regression described here.

5 Although the regular theory of linear models is rendered invalid by the introduction of the constraints on the coefficients of the model, these values provide a good approximation. In particular, we think it is safe to assume that t-values over 3 in absolute value correspond to funds that have a statistically significant arithmetic α, even after adjusting for non-normality and multiple constraints. Our R2s are not corrected for degrees of freedom.

6 Let R1 and R2 be successive monthly returns, expressed as proportions. The average arithmetic return is (R1+R2)/2, the average return each period. The average geometric return is [(1+R1)*(1+R2)].5 – 1, the common return each period that would generate the observed return over the entire span.

7 The expected one-period return exceeds the expected long-period return where future returns are drawn evenly from past returns without replacement. For example, if the past annual returns were 0% and 300%, the expected one-year return is 0.5[0+3] = 150%. This is the past arithmetic rate of return. This is also the expected annual return over any number of years if we expect that the return in all future periods will be drawn randomly from past returns with replacement; so, for example, if after two years we expect to have returned sequences of 0% then 300%, 0% then 0%, 300% then 0% and 300% then 300%, each with probabilities of 0.25%, after two years, it turns one dollar into an expected $6.25=(2.5).5, for an expected annual return of 150%. However, the expected annual return over a two-year span, when returns are drawn evenly from past returns, is [(1+0)(1+3)].5-1= 100%. This is the past geometric rate of return. It is also the expected return over many periods when future returns are drawn evenly without replacement from past returns. The reason it is lower than the expected return is that in the replacement case, the magic of compounding marries big returns with big returns more frequently to raise the expected return beyond the geometric average return. We believe in regression to the mean, i.e., that big returns are more likely to be married with small returns in the future. Therefore, we think the sensible policy is to report only geometric average returns and use them to calculate geometric α. But the standard in the literature is the arithmetic α. We bow to this misleading convention by reporting the latter in combination with our preferred measure. This discussion is based on Kritzman [2002, chapter 4]. For our managed funds, the average arithmetic α exceeds the average geometric α by 26%. This is partially explained by the fact that the standard deviation of return for the managed funds is 4% higher than that of the index baskets, and arithmetic returns exceed geometric returns by a larger amount the higher the standard deviation of return.

8 One could follow Reinker and Tower [2004] and calculate a risk-adjusted α by combining a low-risk asset with the security (the managed fund or the tracking basket) with the higher standard deviation of return, until the standard deviations of returns for the two portfolios were equal, and then comparing returns. But the result depends on the low-risk asset chosen. Moreover, the analysis may be misleading. Suppose fund A returns more each period than does fund B, but while fund B has a zero standard deviation of return, fund A's return is irregular. Then risk-adjusting fund A (using a low-return, low-risk asset) will bring its return below that of fund B, yet no investor would choose fund B over fund A.

9 This paper parallels Sharpe [1992]. Sharpe's paper is useful: clearly constructed, not technical and available online. Our R2 is the same as his. He compares a mutual fund with 12 component asset classes including bills, government bonds, corporate bonds and mortgage-related securities, and various stock indexes. We are interested in how managed equity funds compare with a collection of equity index funds, so that is what we use as our comparison group. His regressions span five years. Ours span four- and two-year periods. Our α is the sum of his ''tracking error.'' He found that Fidelity Magellan dramatically reduced its exposure to small-cap stocks as the fund got larger during the 1980s. He writes, ''style analysis provides measures that reflect how returns act, rather than a simplistic concept of what the portfolios include.''

10 Each figure is the estimate of the population standard deviation based on the standard deviation of the sample.

11 We restricted our study to the four most recent years in order to include Strategic Equity and U.S. Value in our sample and to learn about performance that reflects current management. We feel that the two-year period is the shortest period we could use and still have confidence that we have enough data points to produce a reliable regression. However, we do discuss some eight-year calculations.

12 We use the Wilcoxon sign-rank test to test the hypothesis that the ''median'' managed fund produces no improvement over its index basket. This test assumes that the values of α obtained from independent regressions represent a random sample from a common population, but it does not assume any parametric form for its distribution. A slight variation of this test can be used to compare the variances of the managed funds versus its corresponding tracking basket. This test shows that there is no statistically significant evidence that the median fund has a different from zero (p-value 0.5661).

13 This is true for a longer period as well. Over the eight-year period from 1999 to 2006, for the funds that existed during the whole eight years, the low R2 funds for the eight-year regression (R2<0.85) have an average short-period (two-year) α of 2.18, while the high R2 funds (R2>0.85) have an average two-year α of -0.14. When we use the funds in existence for the entire eight-year period and regress two-year geometric alpha on the R2 from the previous two-year period (as well as dummies for the two-year periods), our estimate is that a 10 percentage point decrease in R2 increases the alpha by 1.37 percent per year.

14 Also, from the Wilcoxon sign-rank test, we get significant evidence that the distribution of standard deviations of the tracking baskets has a significantly lower median than the distribution of standard deviations of the managed funds (p-value 0.001097, with a two-sided confidence band for the median of the difference equal to (-1.425 -0.210)).

15 These are real, continuously compounded returns. The test used is Microsoft Excel's t-test: paired two sample for means. More precisely, if the returns of the two series are samples drawn from populations, the population of the Wiener series has a greater mean than that of the Wilshire series with a probability of 0.87%. However, according to Hulbert, Wiener's Growth Portfolio performed better over the 10 years ending in December 2006, on both a risk-adjusted and a non-risk-adjusted basis, than his other three portfolios. The odds that one of his portfolios would perform well due to luck are greater than the odds that one particular portfolio will perform well, so arguably our 13.4 percent figure in the text overstates his portfolio-picking prowess. The March 2007 edition of Hulbert lists the Wiener newsletter fifth out of 24 mutual fund newsletters on the basis of total return and tenth out of 24 on the basis of risk-adjusted return.

16 Vanguard's new Midcap Growth Index and Midcap Value Index funds capture two more parts of the market for its index fund portfolio.