On management risk and price in the mutual fund industry: style and performance distribution analysis

This study shows how investing in mutual funds involves an additional risk, which we call management risk as a consequence of the uncertainty in the results of active management. To address this issue, we analyze a sample of 2539 US equity mutual funds. For comparative purposes, we differentiate among index funds and actively managed mutual funds with different investment styles. We observe that performance distribution shows negative mean, negative skewness, and excess kurtosis. Results also show that management risk is not rewarded with higher abnormal performance. Moreover, higher active management prices are linked to funds with higher management risk and negative asymmetry. Therefore, investors seem to be risk-seeking since they are paying more to participate in high asymmetric bets. Finally, we attempt to solve this puzzle from the behavioral finance perspective.


Introduction
Mutual funds are a type of financial intermediation which allows investors to participate in financial markets through a professionally managed portfolio and in exchange, investors pay fees and expenses. Although some mutual funds may provide added value for investors, the overall evidence (Carhart 1997;Ferreira et al. 2012;Elton and Gruber 2013;Leite and Cortez 2020, among others) shows that funds do not add value in the aggregate and can even destroy it, since their results do not offset the expenses associated with such management. Despite this evidence, the volume of assets managed by the mutual fund industry has generally continued to grow (see, for instance, Tchamyou et al. 2018;and Qureshi et al. 2019), standing at 25.7 trillion dollars in the US in 2019 and representing 47% of the total worldwide. 1 Gruber (1996) analyzed the puzzle of why these funds continue to grow in spite of their underperformance, and suggested that mutual funds attract a different type of investor: those who invest mainly in underperforming funds form a disadvantaged clientele. In the same spirit, Pástor and Stambaugh (2012) provide some theoretical explanations to justify the rise in popularity of active management, despite its poor track record. In turn, Malkiel (2013) explored the issue of why investors continue to pay excessive fees for financial services of such questionable value. Marekwica and Steininger (2014) also suggest that the fees actively managed mutual funds charge are, in general, too high. In the same vein, Wasik (2013) considers that mutual funds should reduce their fees to values similar to index funds, given the adverse impact they have on fund performance (Mansor et al. 2015).
In this framework, we contribute by considering an additional component. Specifically, we propose that delegated asset management involves an additional risk, which we will call management risk. The risk of the mutual fund therefore consists of two components. The first relates to the risk from passive asset allocation, while the second refers to the management risk due to the uncertainty of the results of managers' activity and its interaction with passive allocation. In this schema, management expenses--mainly due to fees--can be defined as the price paid by investors for bearing the management risk, with the expectation of obtaining a positive abnormal performance. We carry out an empirical study to analyze the relationship between the components of this framework: performance, management risk, and its price or management expenses.
In the empirical part of the study, and following the literature (e.g., Jiang and Verardo 2018;and Dong et al. 2019) we use the widely applied model developed by Carhart (1997) to estimate the abnormal performance of a sample of US domestic equity mutual funds. Firstly, in line with previous studies, the results show that the performance is negative in the aggregate, with a value (-1.10% annual) that is very close to the average expense ratio of the funds (-1.17% annual). Next, the relationship between performance, management risk and price is developed at two levels: first at the individual level within each style fund; and second at the industry level by analyzing the cross-sectional distribution of abnormal performance.
The relationship between performance and expenses has garnered attention in the literature. Elton et al. (1993) found that the most expensive mutual funds show lower gross returns. In the same line, Carhart (1997) found that expense ratios are negatively related to performance. Also Barber et al. (2005) showed how, in aggregate, more expensive mutual funds achieve worse abnormal performance. Gil-Bazo and Ruiz-Verdú (2008) also proposed a model in which worse-performing funds set fees that are greater or equal to those set by better-performing funds. As Parida and Tang (2018) have demonstrated, the huge development of the mutual fund industry during the last decades, together with the subsequent rise in competition in the industry, may have not been sufficient to reduce their management and operating expenses, regardless of their pervasive implications on fund performance and performance persistence. In addition, Chang et al. (2019) report evidence on the detrimental effect of portfolio fees on fund performance, since funds bearing low levels of expenses achieved significantly higher risk-adjusted funds than other comparable funds. This negative performance-fee relationship is also shown in Chuprinin et al. (2019) and El Ghoul and Karoui (2019), among others. Our results are in line with this literature: in aggregate we found a negative and significant relationship between price and performance, so more expensive mutual funds achieve worse abnormal performances. However our analysis of this relationship makes a further contribution by considering mutual funds separately according to styles, finding that the relationship is mainly driven by the effect of large and Index mutual funds.
Regarding the existence of management risk, we first analyze its relationship with abnormal performance, finding no significant evidence. This result holds at both the individual and the industry levels. Therefore, investors that bear more management risk are not rewarded with greater abnormal performance. Secondly we study the relationship between management risk and expenses, finding that investors are paying a higher (lower) price in funds with higher (lower) management risk. This result suggests that investors show risk-seeking behavior because they pay more to participate in high bets.
Furthermore, this paper analyzes the distribution of performance in the different fund categories according to their style in the Morningstar 9-style box. Kraus and Litzenberger (1976) note that investors have an aversion to variance and a preference for positive skewness. Moreover, as Dittmar (2002) points out, kurtosis captures the probability of extreme outcomes. Therefore, as well as utility-based arguments such as aversion to standard deviation (s.d.), investors would be averse to kurtosis. In this line, part of the asset pricing literature focuses on the importance of the higher moments of return distribution of financial assets (e.g. Prakash et al. 2003;Doan et al. 2010;Neuberger 2012 andAmaya et al. 2015, among others). Our empirical evidence demonstrates that, in general, performance distributions show negative skewness and excess kurtosis, (i.e. adverse characteristics for investors). Negative asymmetry implies that active management is more likely to provide poor than excellent results; to quote Baumeister et al. (2001), "Bad is stronger than good" Also at the mutual fund industry level, mutual fund expenses are found to be higher (lower) for distributions with higher (lower) dispersion. Thus, mutual fund styles with higher management risk at the industry level are associated with a higher active management price.
Therefore, the evidence indicates that retail investors in mutual funds are surprisingly risk-seeking in relation to active management. That is, they habitually pay a certain price for bearing the management risk, which is very similar to the negative average abnormal performance they experience. Moreover, the higher the price, the higher the management risk. In this regard, the last section of the paper aims to explain the behavior of mutual fund investors from the perspective of behavioral finance. Consequently, and from the perspective of an efficient market (Fama 1970) and the arithmetic of active management (Sharpe 1991), the active management of portfolios would not be justified or, at least, not at such a high price (Wasik 2013;Malkiel 2013;Chang et al. 2019). Thus, the recent growth of index funds is justified (Khan 2018;Leippold and Rueegg 2020), since these funds involve lower prices and lower management risks.
The rest of the paper is organized as follows. The next section defines management risk and its price. The performance methodology and the data used in the empirical part of the paper are then described. Next we comment on the results obtained in the study. The paper ends with a summary of the main conclusions and their relationship with behavioral finance and the mutual fund industry.

The price of management risk
The return of investor i (without considering their cash flows) in mutual fund p, R i,p,t , can be expressed by (1), where R p,p,t is the return on the passive asset allocation or style benchmark linked to the mutual fund portfolio. It can be defined by a single benchmark, factor, or a set of these benchmarks or factors. R a,p,t is the return on the active management of the mutual fund, and f p,t is the expense ratio from its fees and operational costs. In (2) we define α p,t , as the value added for the mutual fund investor: As mentioned in the introduction, we analyze the management risk at two levels: individual and industry. Regarding the individual level, to define the management risk from the investment of investor i in mutual fund p we apply the traditional mean-variance framework considering expressions (1) and (2). Hence, the risk of the mutual fund investor's investment may be defined by (3), which can be expanded to expression (4): where 2 R p,p,t represents the risk from passive asset allocation, whereas the rest of the terms in (4) are linked to the decision to invest through a mutual fund. In the latter case, the investor assumes an additional risk, which we call management risk, determined by 2 p,t and 2 R p,p,t , p,t . Both terms represent the risk due to managers' activity, arising from the uncertainty of the results eventually derived from their investment decisions. The second term depends on the interaction between passive and active management, and includes the effects that active management has when carried out on different asset classes or different market situations. For example, active management might not have the same effect in the bond market as in the stock market, and the effect could also be different in a bearish or a bullish market. 2 Our proposal is in line with previous studies in the literature. For instance, Berk and Green (2004) proposed a model to explain the relationship between mutual fund flows and performance in which a variable alpha is also defined but under the assumption of normal distribution. Our approach can be linked to the proposal of Grinold and Kahn (1999), who pointed out that performance (measured by the information ratio) depends on the skill of the managers in their strategies and on the breadth or the number of strategies that they employ per year. Therefore, the higher the breadth and the less persistent the skill, the greater the management risk. Unlike other studies, we do not focus on the level of active management of the mutual funds (Cremers and Petajisto 2009;Amihud and Goyenko 2013;among others), but on the risk derived from the results of that active management.
Within our framework, a mutual fund could be understood as a risky asset, in which there are two components: first, the passive allocation; and second, the active management, which implies an additional risk. The most important element of expenses is management fees and it is usually possible to know the amount beforehand when they are defined as a percentage of assets under management. Thus the variable f p,t can be understood as the price paid by investors for this second component (i.e. for bearing the management risk with the expectation of a positive abnormal performance). Managers' decisions are not usually observable (Berk and Green 2004), so although investors know that the manager can have market timing and stock picking abilities, they do not usually know much more about them, and only sometimes do they know the past mutual fund performance. Therefore if there is no persistence in such performance (Carhart 1997;Fama and French 2010;and Matallín-Sáez et al. 2016; among others) or investors are not able to identify skilled fund managers (Jiang and Yuksel 2017), investing in mutual funds could be understood as a casino in which f p,t is the price to participate and the results, negative or positive, are uncertain under the management risk. In this case, we define the management risk at the industry level, according to the characteristics of the crosssectional distribution of the abnormal performance for the whole set of mutual funds and also considering their different styles.

Performance methodology and data
In order to estimate the alpha or performance as defined in (2), we apply Carhart's (1997) multifactor model. This model has been widely applied in the previous literature addressing mutual fund performance (e.g., Ammann et al. 2012;Alves 2014;Lee et al. 2015;Choi et al. 2017;Alda 2019). It is specified as follows: where α p,t measures abnormal performance for month t. For day s, r p,s is the excess of the return (net of fees and other expenses) of mutual fund p over the daily riskfree return (r f,s ); r m,s is the excess return of the stock market; r smb,s and r hml,s are, respectively, the returns on the size and book-to-market factors defined by Fama and French (1993); and r wml,s is the momentum factor by Carhart (1997).
Data on mutual funds such as daily net returns, the annual expense ratio, assets under management and investment styles are taken from the Morningstar Direct database, while data on risk-free and daily factor returns used in model (5) are taken from Professor K. French's website. 3 The sample period analyzed covers almost 29 years, from January 1992 to October 2020. The scope analyzed is the US domestic equity mutual funds. Initially the sample was made up of 17,773 share-class funds. This sample is free from survivorship bias because we consider both the surviving and non-surviving mutual funds during the period analyzed. After aggregating the share-classes that belong to the same mutual fund, we obtain a sample of 5,251 different mutual funds. From that sample, and following a usual practice in the literature (Elton et al. 2001;Chong et al. 2020; and Stark 2019), we require funds to manage at least USD 15 million of assets to avoid upward bias in performance results. Nonetheless, Evans (2010) argues that a size filter is not enough to remove incubation bias in returns. Hence, and similar to other studies (Yan 2008;Muñoz and Vicente 2018), we also exclude funds with less than two years of maturity. The final sample contains 2,539 mutual funds. Table 1 shows some statistics of the sample mutual funds grouped by style. On the one hand, we observe that the most numerous groups correspond to large funds. Specifically, large funds represent 49.43% of the number of funds and 56.08% of the assets under management. Considering also Index funds, these categories amount to 57.94% of the funds and 79.95% of the assets. On the other hand, these categories are, on average, those with the lowest expense ratio, ranging between 1.12% and 1.20% for large funds and 0.60% for Index funds. Conversely, mid-cap and small represent only 20.05% of the assets and require more expensive management, although their average return is higher. In addition, growth funds show the highest expense ratio, average return and annualized standard deviation (s.d.). In contrast, value funds are generally those with the lowest expense ratio and standard deviation. Regarding the aggregate, the average expense ratio for all funds is 1.17% per year and the average annualized return is 10.58%.

Mutual fund performance and management risk
We apply model (5) to each fund to estimate its performance with a non-overlapping monthly rolling window. From this time series we compute the annualized average of abnormal performance of each mutual fund as shown in the third column of Table 2. Additionally, the column with the average of the expense ratio from Table 1 is also included for comparative purposes. Recall that the performance was estimated with returns net of fees and expenses. The last row of the table presents the aggregate results for all the funds in the sample, with the average abnormal performance being negative and significant; specifically, it is an annualized -1.10%, which is close to 1.17%, the average annual expense ratio. Therefore, in aggregate, mutual funds are not able to provide added value to investors as established by Sharpe (1991) and in line with ample evidence in the literature (e.g., Carhart 1997;Elton and Gruber 2013;and Ferreira et al. 2012, among others). One might think that although the added value is, on average, negative, there could be differences among funds that justify the investors' interest. When analyzing the different subgroups, the average performance in all styles is always negative, and significant in practically all cases. In the comparison of style by size, the results of the small funds are generally somewhat better than the mid-cap funds, and these in turn are better than the large ones. Within each style according to value, blend funds have the worst performance.
Regarding the management risk defined in (4), the fifth column on the right in Table 2 shows the average of the ratio between the variance of the monthly abnormal performance and the variance of the monthly return of each mutual fund. For the whole sample of mutual funds, the variability of the alpha on average is 10.27% of the fund's risk. The highest value, 29.86%, is obtained by the other funds category and the lowest value, 3.82%, by Index funds. These values tell us how management risk is higher in mutual funds without a defined style and lower for the opposite case, that is, mutual funds closer to a market index. The following column shows the average of the ratio for each fund between the covariance of the stock market monthly return, R m,t , with the monthly alpha and mutual fund return volatility. As a proxy for passive asset allocation, we used the monthly return of the S&P 500 instead of the combination of factors and betas that define the fund style in (5), in order to avoid possible biases because the alpha would also have been estimated within the same model. The results show that in general this covariance represents a low value with respect to the mutual fund risk. Finally, the last column shows the correlation between the monthly market return and monthly alpha, taking on average small values, which indicates a scarce interaction between the two variables. The greatest negative correlation, -5.08%, is observed for Index funds, which implies that for higher (lower) market returns there is a certain tendency for the alpha of these funds to be somewhat lower (higher). Table 2 showed how the variance of the monthly abnormal performance is the main component of the management risk of mutual funds. Next we are interested in analyzing whether this component is related to performance. A positive (negative) relationship would indicate that those investors who invest in funds with greater management risk are compensated with a higher (worse) abnormal performance. With this aim, we ordered the mutual funds according to the value of this component and grouped them in decile portfolios, so the funds with lower (higher) management risk are in the first (last) decile portfolio. Table 3 shows the abnormal performance of the decile portfolios both for each style and for the whole set of mutual funds. The last columns show the difference between the annualized performance of high minus low decile portfolios and its significance. The difference takes both positive and negative values, and is only significant at the 10% level for Large Blend (-1.23%). Finally, the last row of the table shows that the difference in performance for the whole set of funds (-0.18%) is not significant. Therefore, in general, there is no clear evidence of the relationship between management risk and performance.

The price of management risk
In the previous section we discussed the results for the abnormal performance achieved by the funds and their management risk, within each style and also in aggregate. Next we analyze its relationship with the price of such management. To do this, we grouped mutual funds in decile portfolios according to their price. Thus, the first (last) decile portfolio groups the funds with the lowest (highest) expense ratio.
Firstly, Table 4 shows the abnormal annualized performance of each decile portfolio as well as the difference between the performance of the last and the first deciles and its significance. A positive (negative) difference would indicate that those funds with the highest expenses achieve a better (worse) performance than the cheapest funds, which would imply that the added value of active management per unit of cost is higher (lower) in the most expensive funds.
Only for the Small Growth style is this difference positive and significant (at the 0.10 level), implying that when investing in this style it would be advisable to select these more expensive mutual funds. However, for large, mid-cap, Other and Index styles the difference in performance is negative, suggesting that it would be better to select cheaper funds when investing in these styles. However, only for Large Growth, Large Blend, Mid-Cap Growth and Index funds are the negative differences significant (at the 0.05 or 0.10 level). Therefore, it seems that managers' ability to provide added value per monetary unit of cost may vary according to the type of assets or markets in which the fund primarily invests. Indeed, for instance in the case Table 3 Performance according to management risk Mutual funds of a sample from January 1992 to October 2020 are grouped in decile portfolios within each style according to their management risk, measured by the variance of the monthly abnormal performance. The table reports (in percentages) the equally-weighted annualized abnormal performance of the decile portfolios and the difference between those with a higher and a lower management risk. Tests for significance of these differences were run by bootstrapping one-sided p of an index fund where theoretically no active management is expected, the main fund selection criterion could reasonably be the lowest value of the expense ratio. For the whole set of funds, the last row of the table shows that including index funds yields a weakly significant negative difference of -0.95% (p-value equal to 0.000). This result is in line with the evidence reported by Barber et al. (2005), who found that in aggregate, more expensive mutual funds achieve worse abnormal performance. As a contribution to this previous study, we also analyze this relationship within each style, finding different results. Therefore, to some extent the evidence suggests that, depending on the mutual fund's style, selecting more or less expensive funds would be advisable, although this is not a robust enough criterion to find significant differences in performance.
Second, Table 5 shows the relationship between expense ratio and management risk. Table 2 showed that the main component of this risk in the expression (4) was 2 p,t . Therefore, Table 5 shows the square root of this component in each decile portfolio and the difference between the last and first deciles and its significance. A positive (negative) difference indicates that more expensive funds show a higher (lower) management risk than cheaper funds.
In general, the differences are positive and significant in most fund styles, except for the Mid-Cap Value case. In addition, in the last row for the whole set of mutual funds, we observe that the value of management risk is monotonically increasing from the low (1.09%) to the high (2.12%) decile portfolio, the difference (1.03%) thus being positive and significant. These results imply that in aggregate and for most styles, investors pay a higher (lower) price in funds with higher (lower) management risk. In other words, the higher the spread of the bet, the higher the price Mutual funds of a sample from January 1992 to October 2020 are grouped in decile portfolios according to their expense ratio within each style. The table reports (in percentages) the average of the square root of the management risk from expression (4) of the decile portfolios and the difference between those with a higher and a lower expense ratio. Tests for significance of these differences were run by bootstrapping one-sided p values investors are paying for it. Since in general this higher price does not provide a better performance (in view of the results of Table 4), this result is compatible with investors' risk-seeking behavior.

Cross-sectional distribution of performance and industry management risk
Next we look at the cross-sectional distribution of performance, in order to analyze the risk of active management not only in a fund, but at the industry level. As pointed out previously, under a context of low persistence, the future and past performance of a fund would be poorly correlated. Consequently, the cross-sectional distribution of abnormal mutual fund performance in the industry could be used as a proxy for the risk of an investor who chose to invest in a fund from that industry. For the whole set of mutual funds, the last row of Table 6 shows us how the percentage of funds with negative alphas is higher (72.72%) than the funds with positive ones (27.28%). In addition, the worst annualized alpha, -44.11%, has a higher absolute value than the best one, 15.77%. In short, these results indicate the possible existence of an asymmetric distribution, as confirmed by the negative value, -2.96, of the skewness coefficient. Moreover, the distribution is leptokurtic (i.e. with fat tails). In fact, the Jarque-Bera and the Anderson-Darling tests clearly reject the null hypothesis of normal distribution for all mutual fund styles. Therefore, not only do the funds represent, on average, a loss for the investor which is similar to f pt , as Table 2 showed, but also that the distribution of their abnormal performances, far from being normal, presents an asymmetry to the left and fat tails, which implies adverse characteristics for the investor (Kraus and Litzenberger 1976;and Dittmar 2002). Thus it seems that in the mutual fund industry is more likely to achieve worse than better results.
With respect to the number of funds with negative and positive alphas, in all the categories the number of funds with negative alphas is higher. Of the large and Index funds, which are the categories with the most assets under management, between 70.37% and 80.98% present a negative performance. In most cases the absolute value of the minimum alpha is greater than that of the maximum alpha. Accordingly, most of the categories also present a negative asymmetry. The Mid-Cap Blend is the style with the lowest skewness coefficient, -7.21. This style is precisely the one that presents the worst average annualized performance (-1.46%, in Table 2), the lowest minimum alpha (-44.11%), the highest kurtosis (67.47) and the highest cross-sectional dispersion in the distribution of the abnormal performance (4.50%). Conversely, the Index funds category stands out for having the highest minimum alpha (-3.71%), a positive asymmetry (0.44), a lower kurtosis (3.39) and the lowest cross-sectional dispersion (1.16%). As expected and in line with the results in Table 2, index funds obviously show a lower management risk at the industry level. These differences in distribution show how the effect of active management can vary depending on the passive asset allocation or style in which it operates. We think that these differences in the performance distribution are relevant information for the investor. Several statistics of the performance distribution were presented in Table 6. To complement this information, we make a non-parametric estimation of these distributions. We estimate the densities using Gaussian kernel smoothing (Silverman 1986). Figure 1 shows the results; firstly, plots 1.a to 1.c illustrate the densities for mutual funds grouped by style according to the Morningstar Style Box.
In general, the distributions are shifted to the left of zero, have a greater probability mass to the left, are leptokurtic and show negative skewness. These characteristics are more evident in plots 1.a and 1.b for large and mid-cap funds, respectively, i.e. the groups with the largest number of funds and managed assets. In plot 1.c, where the density of the small funds is represented, we can observe how the distributions are no longer as leptokurtic or as asymmetric. Lastly, plot 1.d illustrates the densities of Index funds and non-Index funds. Figure 2 compares the density of the mutual funds (excluding Index style) with the density of a normal distribution with the same mean (-1.16% in Table 2) and s.d. (2.67% in Table 6). Plot 2.a clearly shows that mutual fund distribution is leptokurtic and negatively skewed. To better observe this asymmetry, plots 2.b and 2.c are prepared from plot 2.a., but only showing the density for abnormal performance lower than -5% and higher than 5%, respectively. A comparison reveals that mutual funds show a higher probability of more negative abnormal

Annualized performance
All except Index Index a b c d Fig. 1 Non-parametric estimation of performance distribution. The figure shows the densities for the different mutual fund groups, calculated by a non-parametric estimation method using Gaussian kernel smoothing. The abnormal performance of the mutual funds sample from January 1992 to October 2020 is estimated using model (5) performance (plot 2.b) than of better performance (plot 2.c). Moreover, this fat tail on the left exceeds the density of the normal distribution.

The price of industry management risk
In the previous sections we analyzed performance distributions. Now, we study the relationship between the average expense ratio obtained for each style of funds and certain variables that characterize the cross-sectional distribution of performance within each style. As Table 7 shows, we propose two models, using explanatory variables related to performance distribution: the mean, the standard deviation (s.d.), the coefficient of asymmetry and the logarithm of the excess kurtosis with respect to the normal distribution. In model (a) all the variables are included; however, given that all of them are significant except the kurtosis, model (b) was estimated without this variable. In this model, the intercept indicates that the mean expense ratio is not significant and the characteristics of abnormal performance distribution explain 83.22% of the variance of the expense ratio across mutual fund styles. The mean of the annualized alpha is significant (p-value of 0.004) and negatively related (coefficient of -0.5568) to the expense ratio. This result is in line with the evidence  Fig. 2 Comparing non-parametric performance and normal distributions. The figure compares the density for the abnormal performance of all mutual funds except index funds and the density of the normal distribution with same mean and s.d. Density is estimated by a non-parametric estimation method using Gaussian kernel smoothing. The abnormal performance of the mutual funds sample from January 1992 to October 2020 is estimated using model (5) shown in Table 4, where, in aggregate, expensive funds achieve worst performance than cheaper funds. The s.d. of the abnormal performance distribution is significant (p-value of zero) and takes a positive value of 0.3094. Therefore, mutual fund styles with higher dispersion in abnormal performance are linked to more expensive active management (i.e. higher management risk at the industry level is linked to higher management risk price). In other words, the greater the bet, the higher the price of that bet. This result is in line with evidence in Table 5 at the individual mutual fund level, when within each style, more expensive (cheaper) funds showed higher (lower) management risk. The asymmetry of the distribution is also significant and takes a positive value of 0.0013. Thus, the higher (lower) the skewness coefficient, the higher (lower) the expense ratio. Therefore, mutual fund styles with positive asymmetry are linked to higher management risk price. In other words, the higher the probability of better abnormal performance, the higher the price of that bet.
The fee price of mutual fund management is usually known previously and implies a reduction in the final return of the fund. Investors assume the cost of this reduction, expecting that this price will provide a higher return. On the other hand, from a financial utility perspective, investors should be adverse to variance (Markowitz 1952) and kurtosis (Dittmar 2002), and exhibit a preference for a positive mean outcome (Markowitz 1952) and a positive skewness (Kraus and Litzenberger 1976). Then, considering the alpha's distribution as the outcome from the mutual fund industry, results from Tables 2 and 6 show how the management risk is defined by adverse characteristics, namely dispersion, kurtosis and negative asymmetry. These results are therefore consistent with investors who are risk seeking rather than risk adverse, because they are paying a price set in advance to bear the  Tables 3 and 4 for the individual mutual fund analyses. Moreover, results from Table 7 show that investors are risk seeking (the higher the price of the management risk, the greater the dispersion of the outcome). The retail investor's behavior does not, therefore, fit with the classic financial utility perspective. In the next section we will attempt to answer this puzzling question from the perspective of behavioral finance.

Conclusions and discussion
This study has shown how investing in mutual funds involves an additional risk, which we call management risk, linked to the uncertainty regarding the results of active management. Our empirical results show that for the almost 29 years covering January 1992 to October 2020, the aggregate of mutual funds did not provide added value to investors, which is in line with the evidence found in previous literature. In fact, it meant an average loss of value (-1.10% per year), which is very similar to the value of the average price of active management (1.17%). However, these are average values and there could be differences when mutual fund styles or different levels of management risk and price are considered. To explore this issue further, we considered two perspectives: on the one hand, the analyses were performed within each fund style at the individual mutual fund level; on the other hand, the analyses focused on the industry level, considering the cross-sectional distribution of performance. Some interesting results arise from the individual level analysis. Firstly, and in general, no relationship was found between the levels of management risk and the abnormal performance of the mutual funds. Therefore, investors that bear more management risk are not rewarded with higher abnormal performance. Secondly, the relationship between the price of active management and the abnormal performance differs depending on the mutual fund style, so investors should show preference for cheaper or more expensive funds according to style. However there is also a lack of significance in many styles. In aggregate we found a negative and significant relationship between price and performance, but mainly driven by the effect of large and Index mutual funds; thus more expensive mutual funds achieve worse abnormal performance. Finally, the third individual level analysis, which explored the relationship between price and management risk, revealed risk-seeking investor behavior as investors pay a higher (lower) price in funds with higher (lower) management risk.
In turn, the main results from the industry level analysis are as follows. In general, the cross-sectional distributions of mutual fund performance show negative asymmetry and excess kurtosis, which implies adverse characteristics for investors. Negative asymmetry means that very poorly performing funds are more likely than funds with excellent results. From the perspective of a technical efficiency analysis, it is assumed that it is more difficult to overcome a certain level of efficiency and easier to be far from the efficient frontier. This asymmetry means that managers' decisions can lead to greater destruction than gain in value. This result could even be considered natural, if in any human decision process the likelihood of performing very badly is greater than performing very well, or as Baumeister et al. (2001) put it, "bad is stronger than good". Also, the relation between the price of active management and the characteristics of the performance distribution was analyzed. We found that price is positively related to dispersion (s.d.) and skewness. The dispersion result is in line with that achieved at the individual level. Thus, higher management risk at the industry level is linked to higher management risk price.
The above results show how, in general, the distributions of the abnormal performance of the mutual funds have negative mean and negative skewness. Despite this, investors pay a price to invest in mutual funds and bear the management risk, and moreover, this price is higher, the higher the management risk. This behavior is consistent with risk-seeking rather than risk-averse investors according to the classic financial utility perspective. We use the behavioral finance framework in an attempt to untangle this puzzle, considering the following situations and emotional biases: (1) The investor does not have all the information on fund operations, the implicit expenses that they entail nor previous evidence regarding the evaluation of their results, among others. Alternatively, investors might be affected by cognitive dissonance bias (Goetzmann and Peles 1997;Pompian 2006), given that they may not accept new evidence or information which conflicts with pre-existing understandings since it causes psychological discomfort.
(2) The amount of the fee is not mentally valued in a reasonable way (mental accounting bias, Pompian 2006), either because it is assigned a mental value or utility different from what would rationally correspond to it (Kahneman and Tversky 1979), or due to the way the information is presented (framing bias, Pompian 2006) as an annual percentage. Indeed, investors may perceive a value of 1.17% (the average of the expense ratio in our sample) as negligible. However, the valuation of expenses could be different if they were calculated in the long term, for example in 10 years, assets would lose 12.34% of their value; or if the fee were valued in absolute terms, for example an investment of $ 100,000 would incur a management fee of $ 12,340. In this line, Bogle (2010) pointed out that most investors still do not understand the devastating impact high fees have on long-term investment results. Also, Malkiel (2013) proposed that investors should be informed of the percentage of the fund's long-run returns that have been consumed by fees. (3) Taffler (2018) presents an original perspective in which investors consider managers and mutual funds as phantastic objects. Investors unconsciously need to believe that managers are omnipotent agents who are capable of consistently outperforming over time. Taffler pointed out that the mutual fund industry is based on a dual state of mind in which the underlying reality (average underperformance and lack of persistence) is surpassed by the unrealistic expectations of investors. In this line, Malkiel also indicated that 'advertising by the fund industry is geared to promote the idea that investing is very complicated, that "experts" are required to help, and that actively managed funds are really worth the high prices that are charged' (2013, p 106). On the other hand, even though they are aware that fund managers inevitably fall short when meeting their impossible performance demands, managers also play a parallel psycho-logical role in helping investors to alleviate the emotional anxiety derived from the uncertainty of the financial investments. In this line, following Pompian's (2006) definition of behavioral biases, the above described situations could be attributed to two biases: i) the regret aversion bias, since delegating the management of a portfolio to another person means that the errors committed in such management will not be ours but someone else's (Chang et al. 2016); and ii) the optimism bias, by which investors tend to be over-optimistic about the fund managers' ability to achieve a positive performance. Malkiel (2013) also points out that overconfidence bias plays an important role in explaining investors' behavior to the extent that they may believe they are able to select the best managed funds.
Finally, if we consider that markets are efficient according to the strong definition (Fama 1970), results of active management would be merely random and, in accordance with the arithmetic of active management (Sharpe 1991), would not add value to the investor in aggregate terms. In that case, a rational investor should not pay a price for participating in a game (management risk) in which, on average, the price becomes a loss and there is also negative asymmetry. In fact, in line with our results, Malkiel (2013) indicates that the greatest inefficiency in the stock market is in industry for investment advice due to the puzzle of why investors continue to pay excessive fees in mutual funds. Consequently, investors should move away from actively managed funds and opt to invest in index mutual funds in order to reduce expenses and management risk. Wasik (2013), for instance, also considers that mutual fund fees are too high. He points out that only index exchange-traded funds have low fees, approximately 0.10%, and that this value should be a benchmark for anyone looking for a fair annual fee. In this respect, Khan (2018) claims that there is a current trend in the industry for index funds to increase their assets to the detriment of active funds.