# «Edwin J. Elton* Martin J. Gruber* Jeffrey A. Busse* October 2002 Abstract Financial theory is often based on the belief that the actions of rational ...»

Although we have studied the association over a three-year period, the question remains whether similar results hold over a shorter one-year time horizon. Since Gruber (1996) and Zheng (1999) show that investors react to short-term performance, we wish to establish whether the same variables that are associated over longer horizons are also associated in the short term. Panel A of Table 3 shows that the results for the one-year horizon closely parallel the three-year results.7 As we move from three years to one year, the R2 across all relationships decreases slightly. An interesting point to note is that, unlike the results for the three year horizon, lagged expenses have a higher association with future differential return (and alpha) than lagged values of either of these variables.8 The reason for this is that the short-term return contains more random noise as a predictor of the future than long-term return, while expenses are much more constant over time.

The R2 of the regression of expenses in one year on expenses in the following year is 0.97.

Panel A of Table 2 shows that using both expenses and pre-expense differential return does not increase the R2 above its value when we only include post-expense differential return. The coefficient of determination does not change, and if we examine the coefficient on expenses and the coefficient on differential return plus expenses, we see that the net weight on expenses is approximately zero.

We use Fama McBeth techniques for calculating t-statistics because it is a convenient way to summarize numbers. However, in every case where we discuss average significance, each of the coefficients in the yearly regressions has the same sign each year, and the coefficient in question is statistically significant at the one percent level each year.

In the one-year case, using both pre-expense differential return (or alpha) and fund expenses together leads to higher association with future performance than using past performance or expenses separately.

Since we will be examining the impact of the past differential return and alpha on subsequent cash flow, it is worthwhile to examine the subsequent performance of funds chosen on the basis of past predictive variables. Tables 4 and 5 show this for three- and one-year periods. For each holding period we rank funds on some predictive variable and then divide them into deciles.

First, consider the three-year results shown in Table 4. Note that buying the ten percent of funds with the highest past return rather than the ten percent with the lowest past return gives an extra return in the next period of 0.97 percent per year. Buying the single fund with the highest past return rather than the single fund with the worst past return leads to a difference of 1.40 percent per year in future return. Furthermore, performance decreases as we move down the deciles, with a highly significant rank correlation between past differential return and future differential return.

For alpha, the results are analogous: funds ranked in the top ten percent according to alpha have an alpha that is 1.07 percent per year higher than funds in the bottom ten percent, and the best fund outperforms the worst fund by 1.43 percent. Once again, the rank correlation between the deciles formed on past alpha and the subsequent realized alpha is highly significant. Clearly, these results are economically significant and indicate that investors who are interested in good future return should buy funds with higher past returns.

What if we buy funds based on expenses? Buying the ten percent of funds with the lowest expenses rather than the ten percent with the highest expenses results in an extra differential return of 0.92 percent per year, while the lowest expense fund outperforms the highest expense fund by 1.52 percent. Likewise, the difference in next year’s alpha from buying the lower expense decile of funds rather than the higher is 0.943 percent per year, while the difference between funds at the extreme of expenses is 1.54 percent. Once again, all rank correlations are highly significant.

When we examine Table 5 for the one-year results, we find that they are very similar to the three-year results. All of the measures work well and with even more predictive power.

B. Predictability of Management Skill In this section, we examine how well managers do, as opposed to investors, relative to holding a passive portfolio at their chosen risk levels. To measure this we add expenses back to return. If mutual funds did not charge expenses, their performance would still likely be different from the return on an index for a number of reasons. First, by purchasing and selling securities, they incur a transaction cost that reduces their return below that of an index. Second, index funds need cash management policies to handle inflows and outflows from investors and policies regarding the timing of the reinvestment of dividends. Funds can freely choose their policies, while index returns are calculated based on a mechanical rule for reinvesting dividends and assuming no inflows or outflows. A fund’s cash management practices can enhance or decrease performance relative to an index. Third, funds can freely choose how they handle sales and purchases caused by changes in the companies contained in the S&P index. Again, these changes are handled mechanically when calculating a return on an index. Fourth, index funds need to have policies on how to handle tender offers and mergers while these are handled mechanically in index construction. Finally, index funds can lend securities and earn a return on the securities that are lent; the index return cannot do so.

Do managers add value before expenses, and if they do, is this value predictable?

We look at this in two ways. First, we examine alpha plus expenses. This is the fund’s risk-adjusted return if there were no expenses. Table 1 shows that, on average, management adds 3.4 basis points (T-value 3.69) in value per year. A more naive way to examine management skill is to compare a performance measure that is not risk-adjusted to a benchmark. Examining differential return plus expenses, management underperforms by 4.1 basis points per year. Management skill in the future is related to past skill over long periods. Panel B of Table 2 shows that, for a three-year holding period, alpha plus expenses in the last three years regressed on alpha plus expenses in the first three years has a slope of 0.665 with a t-statistic of 4.558. Similarly, differential return plus expenses in the second three years regressed on the same variable in the first three years has a slope of 0.265 with a t-statistic of 2.328. In both cases there is regression toward the mean so that the best estimate of future management skill is a fraction of its prior value.

However, in both cases there is a statistically significant association between past and future management performance. Ranking funds in the first three years on the basis of alpha plus expenses and selecting the top decile rather than the bottom decile leads to differential management performance of 16.2 basis points per year. Likewise, ranking funds by differential return plus expenses in the first three years and selecting the top ten percent rather than bottom ten percent leads to differential management performance of

19.5 basis points per year. Management performance is not only predictable in the tails, the rank correlations across deciles are also statistically significant for both measures of performance.

There is less predictability over the one-year horizon. We examine the one-year association by regressing the value of the variable during one year on its value during the prior year. Panel B of Table 3 shows the regression results. Although past differential return plus expenses is associated with its future value, past alpha plus expenses is not.

Since the difference between return and alpha is the beta adjustment, this suggests that the association of differential return with its past value comes from many funds maintaining a stable beta that is different than one over time.

When measured by differential return, there is a lot of predictability in management skill across funds. Table 5 shows the average results based on one-year horizon deciles. For differential return plus expenses, the rank correlation between past and future values is 0.915. The ten percent of funds with the highest management skill in period t outperforms the index by 5.6 basis points per year in the subsequent period.

Likewise, the ten percent of funds with the lowest management skill in period t underperforms the index by 5.6 basis points in the next period. When we examine alpha plus expenses, the rank correlation is insignificant, with no difference in the subsequent performance of funds in the top and bottom deciles.9 All of these results together suggest that management skill is predictable in the long run, but only weakly predictable in the short run.

C. Predictability of Risk The basic risk in purchasing an index fund is that the fund’s return pattern does not match the index. There are two reasons why the fund return can deviate from the index. First, the fund’s beta could systematically differ from one. This could occur if, for example, the fund held cash to service inflows and outflows and did not adjust the beta by using futures. We measure this type of tracking error with the absolute value of beta minus one. Second, less than perfect replication could cause a fund’s return to randomly deviate from the index even if the fund’s beta is one on average. We measure this type of tracking error with R 2.

Panel C of Table 2 shows that R 2 and the absolute value of the difference between beta and one computed over a three-year period are each associated with their values computed over the previous three years. In each case the slope of the regression of the value in the second three years on the value in the first three years is significant at the one percent level. However, Table 4 shows that the Spearman correlation between past and future levels of one of the two risk measures, R 2, is not significant at the five percent We examine funds that are only open for institutional sales separately from funds that are open to individuals. When we separate the analysis into institutional and retail, we observe the same relationship as in the combined sample.

level. There is less association when we examine a one-year horizon. The regressions in Table 3 show that neither measure is significantly related to its prior year value. In addition, in contrast to the three-year decile analysis, when we examine the predictability of the risk measures with the one-year deciles in Table 5, the absolute value of beta minus one is not predictable, but R 2 is predictable.

If we look across funds at the variations in R 2 (from 0.9991 to 1.0000) and beta (from 0.979 to 1.005), we see very little difference. Thus, risk differences are unlikely to explain a large amount of the differences in cash flows among funds.

D. Predictability of Tax Efficiency Index funds also differ in their tax efficiency. Dividends and realized capital gains affect tax payments. To examine the predictability of dividends and capital gains, we divide each variable by its mean value across funds for each one- and three-year period.10 Doing so allows us to pool the results. Both of these tax variables are highly predictable.

First, examine the three-year results. Panel D of Table 2 shows that the slope coefficient in the regressions of the latter three years on the first three years is highly significant for both dividends and capital gains. Table 4 shows that the three-year rank correlation is also significant. Tables 3 and 5 indicate that a similar pattern arises with a one-year horizon. In particular, the rank correlation indicates that past values predict future values with a high level of significance. However, selecting funds based on past dividends leads to only a small variation in future dividends, with the top decile 16 percent above average and the bottom decile six percent below average. Given an average dividend yield of 1.25 percent per year implies that dividend differences have very little Dividends paid to shareholders are dividends paid to the fund less expenses. The predictability of expenses has already been examined. Thus, in examining the predictability of dividends paid to shareholders, we concentrate on the component that we have not examined: dividends paid to the fund. An interesting implication is that for taxable investors a higher expense ratio results in lower taxes as well as lower dividends. Thus, the government pays for part of any increase in expenses.

effect on tax efficiency. However, an investor selecting funds based on capital gains can obtain a great deal of future variation. The top decile has capital gains that are 134 percent of the average, and the lowest decile has capital gains that are 38 percent of the average. Capital gains average 2.24 percent per year over this period. Thus, the difference in tax efficiency caused by selecting funds on the basis of past realized capital gains is substantial across index funds.

**There are two other characteristics of index funds that investors could care about:**

maximum 12b-1 fees and loads. The expense ratio includes actual 12b-1 fees, and these directly affect returns. The investor could care about maximum 12b-1 fees because they might predict future actual 12b-1 fees and future expenses. Investors could also care about maximum 12b-1 fees because high fees may indicate a lack of management concern with performance. Ten of the funds in our sample have 12b-1 fees in some year.

Of these, seven have them during all six years. Thus, the presence of 12b-1 fees is highly predictable. Furthermore, although four funds adjust the magnitude of their 12b-1 fees some time during the sample period, these adjustments are small, and once adjusted tend to remain at the same level. Thus, past maximum 12b-1 fees predict maximum fees in the future.

Nine of the funds have loads during the sample period, mostly of the same magnitude; six funds have loads in every year, two drop loads, and one adds a load. Two funds change the size of the load during the sample period, but the change is small enough not to affect its relative rank. Thus, both the presence of loads and their magnitude are highly predictable on the basis of their presence and size during the prior year.