ریسک صندوق سرمایه گذاری مشترک و جریان وجوه سهم تنظیم شده بازار

Several papers use a fractional specification (net inflow/ assets under management) to infer a convex relation between flow and past performance. However, heterogeneous linear response functions combined with the pooled analysis commonly used in these studies can yield false convexity estimates. We show that such heterogeneity obtains in practice. Along these same lines, the paper also finds that several previously unexamined implications of a convex flow-performance relation fail to hold. Moreover, convexity with fractional flows (which we confirm) largely disappears in a conditional analysis that controls for heterogeneity. Market shares offer an alternative specification for flow that is more resilient to heterogeneity. Using this alternative specification, we again find no evidence of convexity in the flow-performance relation. We conclude that the widely held belief that the flow response function is convex is due solely to misspecification of the empirical model. The flow-return relation is linear.

Numerous studies show a convex relation between mutual fund flows and past returns, including those by Chevalier and Ellison (1997), Sirri and Tufano (1998), Fant and O’Neal (2000), and Huang, Wei, and Yan (2007). These findings have since been used as a foundation for theoretical models of inter-fund competition (Carpenter, 2000, Lynch and Musto, 2003, Taylor, 2003 and Basak and Makarov, 2012). Convexity plus Jensen's inequality naturally leads these articles to conclude that increasing a fund's risk increases its expected capital inflows. While these papers differ in many ways, they share a common element: Returns are assumed to map directly into flows divided by assets under management (AUM). While this fractional flow specification may appear innocuous, it is not. If these models are properly specified, then aggregate flows should be linked to the cross-sectional distribution of fund returns. Our tests, however, indicate otherwise, suggesting that the standard fractional flow model is misspecified. Market shares offer an alternative, more resilient specification and show no evidence of convexity in the fund growth-performance relation. As the paper demonstrates, misspecification can fully account for the previously empirically documented convex flow-return relation. In reality, the relation appears to be linear. A simple example shows how the fractional flow model links the distribution of individual fund returns to aggregate flows. Consider an economy with two funds: one has $100 under management, the other, $10. The fractional flow model states that the better performing fund will see an inflow of 10%, the other, 0%. If the large fund does better, the aggregate flow equals $10, and if the small one does better, the aggregate flow is $1. This relation between fund returns across different size groups and aggregate flows is, in fact, a general implication of the standard fractional flow model: When large funds do relatively well, aggregate flows should be larger. However, our tests yield little evidence that this is the case. Aggregate flows are seemingly determined by economy-wide events, such as the overall market return, not by whether large or small funds have recently done relatively well. A model based on market share changes provides a simple way around this problem. By construction, market share changes add to zero. Thus, in the absence of additional specification, a market share model does not link aggregate flows to the distribution of individual fund returns. Any empirical specification is necessarily misspecified to some degree. What is important is its robustness to such errors—in the case of mutual funds, an ability to handle unaccounted for heterogeneity in the cross section and across time. In practice, this can be difficult to accomplish with a fractional flow-return model as the (nonexhaustive) example depicted in Fig. 1 illustrates.In Fig. 1, the dashed blue lines represent the relation between period t flows (ft) and period t−1 returns (rt−1) within a period for two different fund types. They are labeled hot and cold money funds. The picture has three important elements. First, heterogeneity exists in the cross section of functions mapping flows to returns, which the empirical model has not fully controlled for. (In this example, the researcher does not estimate separate functions for the hot and cold money funds.) Second, when a fund's inflows are poised to be relatively high (meaning a relatively high cross on the y-axis), the flow-return relation steepens. Third, hot money funds have more volatile returns than cold money funds. As a result, the cold funds have relatively more observations close to the y-axis and the hot ones relatively more further away, as the red ovals indicate. Importantly, in all cases the relation between flows and returns is linear, implying volatility will not affect a fund's expected asset accumulation. Nevertheless, a regression with fractional flows as the dependent variable and returns as the explanatory variable will produce the convex green line. The convexity is entirely due to misspecification error. In this case, the curve tries to minimize the sum of squared errors by moving closer to the relatively densely populated areas, i.e., closer to the red ovals. To see how Fig. 1 might come about in actual data, consider a scenario in which one fund is growing (hot) while another has essentially stabilized at its current asset level (cold). To fix the general idea, the stable fund might be old, large, and housed within a well-established mutual fund family. While the cold fund's many long-term investors might not look to it as the place to put their money when seeking out the latest and greatest, neither are they likely to pull out if recent returns are subpar. This could arise from simple inertia (Choi et al., 2002, Choi et al., 2004a, Choi et al., 2004b, Duflo and Saez, 2002 and Choi et al., 2009) or if the fund's assets derive largely from essentially automated 401(k) retirement flows (Choi et al., 2002, Choi et al., 2004b and Mitchell et al., 2006). These and other possible factors produce for the older (cold money) fund the flow-return relation described by the lower blue short dashed line. The high long dashed blue line in Fig. 1 can be thought of as potentially representing young (hot money) funds with relatively few assets under management. Due to their size, these hot money funds can easily hold portfolios that, compared with large cold money ones, are relatively undiversified. Consequently, the hot money funds have a higher return volatility than the cold money funds. The important point is that the red ovals, indicating where data for the cold money funds are relatively dense, lie closer to the origin than those for the hot money funds. Finally, consider the impact of the age of the hot and cold money funds. If a hot money fund's investors have been with it for only a short period of time, its flow will likely vary dramatically in response to its recent returns, at least relative to the cold money fund. Combining the impact of heterogeneity in the unconditional flow, return volatility, and investor responsiveness to past returns yields the flow-return relation displayed in Fig. 1. A typical flow-return model will now produce the convex solid green line. Can controls fix the problem? Yes, if one knows how to properly divide up the funds. In this paper, we show that separating out funds that are both young and small from the rest yields a pair of linear relations. These young-small funds have all the relative properties depicted in Fig. 1: volatile returns, high relative unconditional growth, and a particularly steep flow-return relation. Thus, even accounting for just this one source of cross-sectional heterogeneity, the convexity found by running a pooled regression largely disappears. While we show that the empirical problems that lead to false convexity can be mitigated, it requires knowing exante what cross-sectional controls are needed. But any number of scenarios also yield the pattern depicted in Fig. 1, so there is no guarantee that any single set will be sufficient. Consider that the relation depicted in Fig. 1 could arise dynamically. High aggregate market returns induce high aggregate flows (Warther, 1995; Edelen and Warner, 2001; Goetzmann and Massa, 2003; Boyer and Zheng, 2009; and, using Israeli data, Ben-Rephael, Kandel, and Whol, 2011)). During these high aggregate flow periods, one might expect hot money funds to experience particularly high unconditional flows and an increase in the slope of the flow-return relation. The relative number of hot and cold fund types may also be impacted. High aggregate inflows encourage entry leading to an abundance of hot funds in such periods. (We test for and find these intertemporal patterns within the data and show that they explain a sizable fraction of the estimated convex fractional flow-return relation in standard models.) The important point, however, is that whatever the underlying causality, a nonlinear panel data regression of fractional flows given returns will generate the convex green curve in the absence of a full set of controls. What happens if one uses market share changes instead of flows as the dependent variable? While market shares are not impervious to misspecification, they are more robust. This is illustrated via the simulation results in Fig. 2.Fig. 2 derives from four thousand simulations of a single period economy. In each simulation, there are one thousand funds split evenly between hot and cold money. The parameter values have been selected to match the simulated data with some of the moments in the actual data for monthly fund flows (after trimming the top and bottom 5%) and returns. In both the actual data and the simulation, the flows have a mean value of 4.7%, with a standard deviation of 1.35%, and fund returns net of the market have a standard deviation of 5.4%. The three upper lines display the results using fractional flows and essentially reproduce the idealized lines depicted in Fig. 1. The three lower ones represent the results using market share changes. The upper orange long dashed line displays the hot fund flows for a given return; the lower short dashed one, the cold fund flows. The solid green line in the middle is the estimated impact of relative returns on market share changes using the pooled data. To make the comparison between fractional flow and market share change estimates easier to visualize, Fig. 2 scales the two measures so that at the origin the estimated response functions (the two solid lines) have the same slope. Even though the pooled regressions have identical slopes around the origin, the lines representing the flows to hot and cold funds are closer to each other when market shares, instead of fractional flows, are used. This greatly attenuates the distorting effect of uncontrolled for heterogeneity. The result holds whenever flows are an affine function of returns; market shares produce a smaller angle between the response lines, which reduces the impact of misspecification error. At the same time, if the data exhibit true flow convexity, our power tests (based on the estimated flow-performance relation) indicate that with very high probability the market share model will yield a statistically significant convex relation as well. One can adjust the flow-return estimator to account for cross-sectional and intertemporal heterogeniety. However, market shares provide a simple alternative. Relative to flows, market shares mitigate many of the problems associated with intertemporal variation in the fund-by-fund cross section of investor responses to past returns without needing to know what they are exante. This paper concludes that the widely held belief that increasing a fund's risk will help it grow is due to misspecification error and is not reflected in the data either in the short or long term. Nevertheless, this is a separate issue from whether funds vary their risk over time. It is true that the risk-leads-to-growth idea has often motivated the empirical time-varying risk literature. However, it is not needed to accept the empirical conclusion that such variation is occurring. Managers may be systematically varying their fund's risk (Brown et al., 1996, Chevalier and Ellison, 1997, Taylor, 2003, Busse, 2001, Qiu, 2003 and Goriaev et al., 2005) but this does not appear to be motivated by some sort of fund flow tournament, as is typically hypothesized. If the underlying flow tournament which motivated many studies of mutual funds, does not exist, what then is responsible for the observed results? A number of other potential forces can provide an explanation. For example, mutual fund managers may vary their fund's risk in response to tournament-like incentives from forces other than fund flows. Career issues can play a role, as in Qiu (2003), who attributes the time-varying fund volatility in his data set to the risk of termination. Undoubtedly, other compensation factors such as bonus payments and the like could provide similar incentives. The paper is organized as follows. Section 2 looks at some long-term patterns that seem difficult to reconcile with a convex flow-performance relation. Section 3 discusses the mathematical implications of various empirical specifications of the fund return-flow relation. Section 4 introduces a market share based model as an alternative empirical specification. Section 5 discusses the data used. Section 6 presents a statistical summary and compares the level of observed convexity in the raw data across the fractional flow and market share-based models. Section 7 tests whether or not aggregate flows have patterns consistent with the embedded implications from empirical models that regress fractional fund flows on performance. Section 8 shows how the observed convex fractional flow-return relation can potentially be reconciled with the linear market share change-return relation found here. Section 9 concludes.

An extensive literature indicates that current fund flows are convex in past performance. This linkage has been established within an empirical model that regresses individual fund flows divided by AUM on past performance rankings and controls. However, the fractional flow model specification is not without its own economic and empirical implications. If fractional flows increase in performance, then aggregate flows should be larger when large funds do well relative to small ones. If fractional flows are also convex in performance, then aggregate flows should be larger when large funds populate the performance distribution tails. The empirical tests conducted here do not lend support to either hypothesis. The relation between aggregate flows and relative fund performance by size implied by the fractional flow model appears to be absent from the data. A power analysis shows that a lack of data is not behind this non-result, implying that the standard fractional flow model is misspecified in some way. Part of the problem with a fractional flow model is that it is sensitive to uncontrolled-for heterogeneity in the way investors respond to past returns either in the cross section of funds or over time. While in principle it is possible to correct for this, it requires knowing exante the full list of necessary controls. Absent that, simulations show that a fractional flow model can easily yield a spuriously estimated flow-performance relation, like that seen in the mutual fund data. Our analysis shows that allowing for various types of typically unaccounted-for heterogeneity greatly reduces the estimated convexity from a fractional flow model. An alternative to the fractional flow model is one based upon market shares. As our study shows both theoretically and empirically, it is considerably more robust to unaccounted-for heterogeneity within the data. Furthermore, our tests indicate that investors first decide how much to invest and then determine how to split it up—a sequence more in line with a market share than a fractional flow model. In contrast, a fractional flow model assumes investors first decide how much to invest based on each fund's individual performance in isolation and then generate the aggregate flow necessary to yield the required amount. Using market share changes instead of fractional flows as the dependent variable produces no evidence that increasing a fund's volatility will help it grow. This is in line with the long-term data showing that investors are not, in the aggregate, moving their investments into more volatile mutual funds. If anything, the opposite seems to be occurring.