A Prospect Theoretical Approach to Investing

Transcription

1 A Prospect Theoretical Approach to Investing D I S S E R T A T I O N of the University of St. Gallen School of Management, Economics, Law, Social Sciences and International Affairs to obtain the title of Doctor of Philosophy in Economics and Finance submitted by Tatiana Dvinyaninova from Russia Approved on the application of Prof. Dr. Enrico De Giorgi and Prof. Dr. Thierry Post Dissertation no Difo Druck GmbH, Bamberg 2016

2 The University of St. Gallen, School of Management, Economics, Law, Social Sciences and International Affairs, hereby consents to the printing of the present dissertation, without hereby expressing any opinion on the views herein expressed. St. Gallen, November 10, 2015 The President: Prof. Dr. Thomas Bieger

3 I Table of contents Abstract II Introduction... 1 I. Mutual fund performance decomposition: a behavioural approach II. Garbage in, garbage out? III. Momentum in a prospect theory framework 74

4 II Abstract Traditionally, financial economists have been evaluating investment decisions in a mean-variance framework introduced by Markowitz (1952). However, experiments have shown prospect theory (PT), introduced by Kahneman and Tversky (1979), to provide a better description of how investors make choices under uncertainty. Overall, my thesis highlights the importance of incorporating investor preferences into investment decisions. While banks and asset managers typically base their recommendations on investors general degrees of risk aversion and investment horizons only, their heterogeneity with respect to loss aversions or reference points are largely neglected in practice. My results suggest that both investment advisors and academics might be better off devoting more time and resources to efficiently extracting investors true preferences rather than providing investment advice based on performance measures that do not adequately fit investors risk profiles. Abstract in German Traditionell haben Finanz-Ökonomen Anlageentscheidungen im sogenannten Mean- Variance-Framework von Markowitz (1952) evaluiert. Experimente haben jedoch gezeigt, dass die Prospect Theory von Kahneman and Tversky (1979) eine bessere Beschreibung der Entscheidungen von Anlegern unter Untersicherheit liefert. Meine Dissertation zeigt die Relevanz die Präferenzen von Anlegern realistisch in die Anlageentscheide einzubeziehen. Während Banken und Vermögensverwalter ihre Anlageempfehlungen typischer Weise basiert auf der generellen Risikoaversion und dem Anlagehorizont der Investoren machen, wird ihre Heterogenität bezüglich Verlustaversion und Referenzpunkten in der Praxis häufig vernachlässigt. Meine Resultate zeigen, dass sowohl Praktiker als auch Akademiker mehr Zeit und Ressourcen in die Analysen der Präferenzen von Anlegern investieren sollten, anstatt Empfehlungen auf Basis für den Anleger nicht adäquater Performance Masse zu geben.

5 Introduction 1 Introduction Traditionally, financial economists have been evaluating investment decisions in a mean-variance framework introduced by Markowitz (1952). However, despite of its simplicity and wide acceptance among both academics and practitioners, the meanvariance approach does not properly account for important information contained in the higher moments of the return distribution. In this context, numerous findings in the field of behavioral finance have shown investors to have more complex preferences and utility functions that are not captured by mean and variance alone. More specifically, experiments have shown prospect theory (PT), introduced by Kahneman and Tversky (1979), to provide a better description of how investors make choices under uncertainty. While many studies nowadays rely on PT to more realistically model investors preferences, several important topics in finance have not yet been thoroughly investigated in the PT framework. One of the topics most extensively researched in the recent financial literature is the performance of mutual funds. In order to gain more insights into the drivers of fund performance, a large body of literature is looking at various observable fund-specific attributes that might be related to performance. Traditionally, these studies have been using intercepts of factor regression models, so-called alphas, to measure mutual fund performance. However, the higher moments of the return distribution captured in the PT framework raise the natural question of how the most important findings on mutual fund performance change in this behavioural setting. In my first paper, I therefore decompose the performance of US equity mutual funds expressed in terms of PT utilities into contributions of different fund-specific attributes. I find that behavioural decision makers are significantly better off investing in well-diversified, older funds with a good track record that belong to large fund families. Moreover, I provide evidence that funds charging higher fees and engaging in more trading activity are unfavourable for behavioural investors. My results also imply the existence of persistence in funds loss taking behaviour. Importantly, the obtained results are largely driven by the loss aversion inherent to PT utilities. This high relevance of investors preference profile for the relation between fund characteristics and performance suggests that the selection of mutual funds might have to

6 2 Introduction follow different guidelines for different types of investors. In particular, while mutual fund rankings traditionally depend on alphas or past performance, the ranks might turn out to vary significantly with respect to the chosen selection or ranking measure. In my second paper, I therefore address the question of whether mutual fund selection needs to be tailored to behavioural preferences. In particular, considering several well-known fund selection measures, I evaluate whether relying on these measures when selecting funds translates into significant opportunity losses for PT investors. My results indicate that widely accepted measures fail to properly capture investors preferences and largely neglect their heterogeneity. Specifically, I reveal that correctly identifying the investors benchmarks and properly accounting for their risk profiles in general and aversion to losses in particular are crucial premises for sensible fund selection. Moreover, I propose a simple behavioural fund selection measure, created from readily available risk measures dynamically adjusted for loss aversion, that is closely related to investors true risk profiles and therefore allows to better capture their heterogeneity. Intuitively, the large discrepancies between fund ranks under different ranking measures also imply that investment strategies providing seemingly abnormal profits in the classical framework might lose profitability under PT preferences. While this has recently been shown for the popular momentum effect in a long-short framework, related studies at the same time suggest that momentum strategies might regain profitability when accounting for investors preferences in the strategy construction stage. Nevertheless, the question of whether PT preferences indeed explain momentum returns or simply require a tailor-made momentum strategy to exploit the momentum effect has remained unanswered in the literature. In our third paper, we therefore evaluate classical long-only momentum strategies in a PT framework and propose several modifications to behaviouralise classical momentum. Applying these modifications, we manage to create behavioural strategies that provide additional value to PT investors for a certain range of loss aversion, thereby reviving the sensibility of momentum investing for PT investors. Overall, this thesis highlights the importance of incorporating investor preferences into investment decisions. While banks and asset managers typically base their recommendations on investors general degrees of risk aversion and investment horizons only, their heterogeneity with respect to loss aversions or reference points are largely neglected in practice. Our results suggest that both investment advisors and academics might be better off devoting more time and resources to efficiently extracting investors

7 Introduction 3 true preferences rather than providing investment advice based on performance measures that do not adequately fit investors risk profiles.

8 4 Mutual fund performance decomposition: a behavioural approach I. Mutual fund performance decomposition: a behavioural approach Tatiana Dvinyaninova University of St.Gallen, School of Economics and Political Science, Bodanstrasse 8, 9000 St.Gallen, Switzerland This version: 27 July 2015 Abstract Using a dataset comprised of nearly US mutual funds, we analyse the relation between cumulative prospect theory utilities and various fund characteristics. We find that behavioural decision makers are significantly better off investing in well-diversified, older funds with a good track record that belong to large fund families. Moreover, we provide evidence that funds charging higher fees and engaging in more trading are unfavourable for behavioural investors. In general, our results imply the existence of persistence in funds behaviour, in particular in their attitudes towards relative drawdowns. Indeed, our sensitivity analyses indicate that the most natural explanation for our results is provided by the loss aversion inherent in CPT utilities. Key words Mutual fund performance Prospect theory Loss aversion JEL Classification G02, G11

9 Mutual fund performance decomposition: a behavioural approach 5 1. Introduction According to the Investment Company Institute (ICI), the global mutual fund industry s assets under management have grown more than sevenfold in the last two decades, from $4 trillion in 1993 to $30 trillion in 2014, generally reflecting investors ever increasing demand for professionally managed and well diversified products. The mutual fund industry remains very competitive and dynamic, with hundreds of funds opened, merged or liquidated every year, making it an increasingly complex task to identify those funds that can potentially provide superior performance. A natural way to facilitate the selection process is to rely on various observable fundspecific attributes that might be related to fund performance. In fact, a large body of recent literature has been dedicated to analysing the dependencies between popular performance evaluation measures and fund-specific managerial and organizational factors, including among others past performance, diversification, expenses, turnover, load fees, fund size, fund inflows, fund family size and fund age (e.g. Grinblatt and Titman, 1994; Carhart, 1997; Dahlquist et al., 2000; Chen et al., 2004; Prather et al., 2004; Kacperczyk et al., 2005; Pollet and Wilson, 2008; Cremers and Petajisto, 2009; Massa and Patgiri, 2009; Huang et al., 2011; Agnesens, 2013). Traditionally, this literature has been using intercepts of factor regression models, socalled alphas, to evaluate performance. Specifically, the existing literature relies on the simplest one-factor or Jensen s (1969) alpha obtained from the Capital Asset Pricing Model (CAPM), three-factor alpha resulting from Fama and French (1993) type regression extending CAPM by size and value factors, and currently the most commonly used Carhart s (1997) four-factor alpha which additionally captures Jegadeesh and Titman s (1993) momentum anomaly. Although evaluating performance in terms of alpha remains a widely accepted approach among both academics and practitioners, numerous findings in the field of behavioural finance (see Barberis and Thaler (2003) and Shefrin (2008) for an extensive overview) suggest that alpha fails to properly capture investors perception of funds returns in general and their negative and extreme returns in particular. Indeed, even popular proponents of traditional finance seem to acknowledge the importance of investor preferences with respect to factor models (e.g. Fama and French, 2007). Nevertheless, to

10 6 Mutual fund performance decomposition: a behavioural approach our best knowledge, no effort has yet been made to integrate these behavioural biases into the mutual fund performance decomposition literature. This study provides a first attempt to analyse mutual fund performance in a more realistic setting, taking investor sentiment into account. In particular, we base our evaluations on diverse specifications of Tversky and Kahneman s (1992) cumulative prospect theory (CPT) utility functions. 1 Specifically, using a dataset comprised of nearly US mutual funds, we decompose fund performance expressed in terms of certainty equivalents (CEs) corresponding to CPT utilities computed for diverse reference point specifications, and examine contributions of different fundspecific attributes to these measures. 2 We additionally inspect the sensitivity of our results to changes in important behavioural parameters, namely risk aversion, loss aversion and probability distortions, thereby providing extensive insights on the actual drivers of mutual fund performance in the behavioural framework. To our best knowledge, decompositions of funds CEs corresponding to CPT utilities into contributions of fund characteristics are novel to the literature, thereby providing new important insights on mutual fund selection in a more realistic, behavioural framework. Several interesting results emerge from our analysis. First of all, we observe a significant persistence in fund utility, thereby suggesting that past performance is a strong indicator of future performance in a behavioural framework. Likewise, in contrast to analyses based on traditional performance measures, we find a significantly positive impact of diversification on performance, driven by the increased relevance of (negative) fluctuations in the behavioural framework. Our results further indicate that fund family size has a significantly positive effect on utilities, providing support for the popular hypothesis of scale economies existing within a fund family. We also provide evidence that older funds might have risk characteristics that are favourable to behavioural investors. On the contrary, we observe management expenses, as well as marketing and distribution fees, to have a significantly negative impact on fund performance in the behavioural setting, implying that funds charging higher management and 12b-1 fees cannot compensate their clients with an appropriate performance. Similarly, our findings 1 The cumulative prospect theory is an established standard in the behavioural literature, thereby being a natural candidate for such applications. 2 Throughout this article, we refer to certainty equivalents corresponding to CPT utility values as behavioural performance evaluation measures.

11 Mutual fund performance decomposition: a behavioural approach 7 show that funds with a relatively high trading activity provide lower utilities. Finally, we find fund size and fund flows to have no significant effect on performance. Importantly, our sensitivity analyses reveal a strong relation between the significance of our results and changes in the loss aversion coefficient. Indeed, our findings indicate that the most natural explanation for the relations between fund performance and fund characteristics in our setting have their roots in the loss aversion inherent to the functional form of CPT utilities. Among others, this effect is remarkable for the impact of past performance, implying the existence of persistence not only in performance, but also in funds risk-taking behaviour in general and their attitudes towards large relative drawdowns in particular. Likewise, the impact of diversification on funds performance can be largely attributed to loss aversion, therefore illustrating the importance of diversification for loss averse investors. The significantly negative effect of funds turnover on funds utility also increases with loss aversion, lending support to our proposition that fund managers engaging in more trading activity might be exposed to higher drawdowns that are captured in the behavioural setting. Our sensitivity analysis furthermore supports our hypothesis that older funds might more frequently manage to avoid large intermediate drawdowns, thereby making them more attractive for loss averse investors. The remainder of this article is organized as follows. In Sections 2 and 3, we provide an overview on the dataset and general methodological approach applied throughout the paper. In Section 4 we discuss our empirical results and their implications. Section 5 describes our sensitivity analyses and robustness checks. Section 6 discusses potential implications for future research, while Section 7 concludes. 2. Data Consistent with the existing literature, we create an extensive dataset, covering a 15- year time period from January 1999 to December 2013, comprised of a survivorship-bias free sample of US equity mutual funds, obtained from the Centre for Research in Security Prices (CRSP) database. More specifically, our sample contains daily data on fund returns and quarterly data on popular fund characteristics considered by previous studies. In particular, these fund-specific attributes include past performance, diversification 3, 3 The data on past performance and diversification is not readily available on CRSP.

12 8 Mutual fund performance decomposition: a behavioural approach marketing and distribution costs (12b-1 fees), management fees 4, turnover ratio measured as the minimum of aggregated sales or purchases of securities divided by the average 12- month Total Net Assets (TNA) and indicating fund s trading activity, front loads and rear loads which are charged at the time of investment initiation and redemption respectively, fund size measured in terms of total net assets under management (TNA), fund flows accounting for asset purchases (inflows) and withdrawals (outflows) as a share of the TNA, fund family size computed as a sum of TNAs of all other funds belonging to the same family, and fund age. Guided by popular studies (Carhart, 1997; Wermers, 2000; Prather et al., 2004; Pollet and Wilson, 2008; Agnesens, 2013) in processing our dataset, we restrict our sample to include diversified equity funds only. Therefore, excluded from our study are closed-end funds, hybrid funds, international funds, bond funds, money market funds, real estate funds, commodity funds, as well as low turnover or passive funds, and those funds whose equity holdings are below 50% (e.g. Prather et al., 2004; Pollet and Wilson, 2008; Agnesens, 2013). Moreover, for the funds that offer different share classes, we combine the CRSP net returns and characteristics of all share classes into a single portfolio, whereby each variable is weighted by the TNA-share corresponding to that share class (e.g. Pollet and Wilson, 2008; Agnesens, 2013). Our final dataset is comprised of funds and quarterly observations. The corresponding summary statistics are presented in Table I Table I about here The median fund in our dataset has an unexplained variation (1 RR 2 ) of around 6% of its total risk. It charges 0.18% and 0.74% of its TNA for 12b-1 and management fees, respectively, and has a turnover ratio of approximately 72%. It imposes no front load commissions, but charges 1% rear load fees. TNA of the median fund in our sample are worth around 165 million USD with about 1% of the managed assets withdrawn from the fund every quarter. It belongs to a fund family managing another 2 billion USD in US diversified equities and is about 10 years old. 4 Note that, according to CRSP, these fees can be offset by reimbursements, which can lead to negative management fee values.

13 Mutual fund performance decomposition: a behavioural approach 9 Additionally, we use standard data available on the Kenneth French web-page 5, including daily data on the risk-free return as measured by the Treasury bill rate, the market excess return (MMMMMM) calculated as an average of all NYSE, AMEX and NASDAQ stocks less the risk-free return, as well as returns for the small minus big (SSSSSS), high minus low (HHHHHH) and momentum (MMMMMM) factors. 6 Note further, that we compute the diversification measure 7 for each portfolio jj as the coefficient of determination RR 2 resulting from the Carhart s (1997) type regression estimated as follows: rr jj,tt rr rrrr tt = αα 4FF jj,qq + ββ MMMMMM jj,qq MMMMMM tt + ββ SSSSSS jj,qq SSSSSS tt + ββ HHHHHH jj,qq HHHHHH tt + ββ MMMMMM jj,qq MMMMMM tt + εε jj,tt, where rr jj,tt and rr tt rrrr are the time tt returns of fund jj and the risk free rate, respectively, and the time tt returns of the risk factors, MMMMMM tt, SSSSSS tt, HHHHHH tt and MMMMMM tt defined above. Intuitively, RR 2 reflects the percentage of a fund s excess return movements that can be attributed to movements in the four risk factors included in the regression. Hence, the higher this RR 2, the less idiosyncratic risk the portfolio has, implying a higher level of diversification. 3. Methodology 3.1. Fund utilities We consider a decision maker who evaluates her investment according to the cumulative prospect theory of Tversky and Kahneman (1992). Thereby, in our setting we incorporate four important features of investors behaviour, namely investment assessment in terms of gains and losses relative to a certain reference point, risk aversion, loss aversion and probability weighting. 5 We would like to thank Kenneth French for making the data available. 6 In our analysis, we compute performance measures quarterly to match the frequency of fund characteristics data in the decomposition exercise, whereby one quarter contains approximately 63 daily observations. 7 As compared to other fund characteristics employed in this study, analysing the impact of diversification on performance is relatively new to the literature, and therefore a commonly accepted diversification measure has not yet been developed. In our opinion, previously used diversification measures have a number of drawbacks. Prather et al. (2004) define diversification as a share of the fund s total assets invested in its top 10 holdings and thereby neglect all remaining stocks, as well as the diversification within the top 10 holdings. Pollet and Wilson (2008) and Cremers and Petajisto (2009) use measures related to the number of stocks held by the fund and thereby fail to take into account weights assigned to each stock in the portfolio. Agnesens (2013) uses a sum of squared portfolio weights, thereby neglecting the correlation between the individual assets. We consider the diversification measure used in our study to be more sensible for at least two reasons. First, it features none of the above-mentioned issues. Second, it does not require any information on funds holdings and therefore allows keeping also those funds and time periods for which holdings data is not readily available.

14 10 Mutual fund performance decomposition: a behavioural approach More specifically, consider the following three statements advocated by the CPT and their implications for our analysis. First, investors do not derive utility from absolute wealth positions, but rather consider changes in their wealth relative to a reference point. Therefore, for each fund jj 1,2,, NN we consider a vector of relative daily tt 1,2,, TT RRRR returns xx jj,tt = rr jj,tt rr tt as compared to a reference point RRRR, whereby its elements xx jj,tt 0 and xx jj,tt < 0 are defined as gains and losses, respectively. 8 Second, investors are risk and loss averse. Typically, empirical estimates of loss aversion, denoted as λλ, are around two (e.g. Kahneman and Tversky, 1979; Pennings and Smidts, 2003; Booij and Van de Kuilen, 2009), meaning that a decision maker is twice as sensitive to losses as to gains. The loss aversion feature is reflected in the following value function representation: vv xx jj,tt = xx jj,tt αα, xx jj,tt 0 λλ xx jj,tt ββ (1), xx jj,tt < 0. The piece-wise power function vv( ) is usually concave in the domain of gains and convex in the domain of losses, with risk-aversion parameters αα and ββ determining its curvature. Third, investors tend to overestimate the likelihood of extreme events. As a consequence, they weight outcomes by so-called decision weights computed using a nonlinear function of objective probabilities. Importantly, in the CPT framework, this weighting function is applied to cumulative probabilities. 9 Specifically, denoting decision weights by ππ jj,tt, and defining the number of gains and losses for fund jj in quarter qq by GG jj,qq and LL jj,qq, respectively, we get: ππ + jj,tt = ww + pp jj,tt + + pp jj,ggjj,qq ww + pp jj,tt pp jj,ggjj,qq ππ jj,tt = ww pp jj, LLjj,qq + + pp jj,tt ww pp jj, LLjj,qq + + pp jj,tt 1. (2) In this study, we further follow Tversky and Kahneman (1992) in assuming a weighting function of the following form for gains and losses, respectively: 8 The exact specifications of reference points we employ in our analysis are explained in the Section Intuitively, decision weights ππ are computed as the weighted probability of obtaining a gain (loss) which is at least as big (at most as small) as xx tt minus the weighted probability of obtaining a gain (loss) which is strictly bigger (smaller) than xx tt, where the probability weighting is defined in Eq. (4).

15 Mutual fund performance decomposition: a behavioural approach 11 ww + pp jj,tt = ww pp jj,tt = pp jj,tt δ pp jj,ttδ + 1 pp jj,tt δ 1 δ pp jj,tt γ pp γ jj,tt + 1 pp jj,tt γ 1 γγ (3) Note, that the probability distortion discussed above, occurs only for δ 1 and γ 1, with the effect being greater for lower levels of δ and γ. Next, for each fund jj 1,2,, NN and each quarter qq 1,2,, QQ we compute the quarter qq CPT utility uu jj,qq assigned to fund jj by summing up the corresponding value functions as obtained from Eq. (1) multiplied by the decision weights from Eq. (2) for all tt 1,2,, TT trading days in that quarter, that is: TT uu jj,qq = ππ jj,tt vv xx jj,tt tt=1 (4) As a starting point for our empirical analysis, we use the parameter specification elicited by Tversky and Kahneman (1992), namely α = β = 0.88, λ = 2.25, δ = 0.69 and γ = Section 5 provides extensive sensitivity analyses with respect to these parameter values. Finally, since CPT utilities do not have an intuitive interpretation, in our analysis we use certainty equivalent gains and losses (CEs16T16T) 10 corresponding to any specified utility value. In our setting, certainty equivalents are calculated as follows: uu jj,qq 1/αα, uu jj,tt 0 CCCC uu jj,qq = uu 1/ββ jj,qq λλ, uu jj,tt < 0. Importantly, for the sake of interpretability we scale certainty equivalents to quarterly values. (5) 10 Certainty equivalent for any gamble is the certain amount that is equally preferred to that gamble. Note that while the certainty equivalent gains and losses correspond to the certain gain or loss as compared to the reference point rather than the certain absolute return, for simplicity we refer to these certainty equivalent gains and losses as certainty equivalents throughout this paper.

16 12 Mutual fund performance decomposition: a behavioural approach 3.2. Decomposition of utilities In the next step we decompose performance measures, defined in Eq. (5) into the contributions of fund-specific attributes, namely past performance, diversification, expenses, turnover ratio, load fees, fund size, fund flows, fund family size and fund age. Note that past performance corresponding to our behavioural setting is just a one quarter lagged value resulting from Eq. (5). For funds jj = 1, 2,, NN and quarters qq 1,2,, QQ we perform a pooled regression of quarterly performance measures yy jj,qq on our set of fund characteristics zz ii,jj,qq (ii 1,2,, II) as follows: yy jj,qq = bb 0 + bb 1 zz 1,jj,qq bb II zz II,jj,qq 1 + ee jj,qq (6) To avoid potential endogeneity problems, we use one quarter lagged variables for all fund-specific attributes. Moreover, to account for a non-linear dependence, consistent with existing literature (e.g. Cremers and Petajisto, 2009; Massa and Patgiri, 2009; Huang et al., 2011) we take logarithmic values of some variables, namely fund size, fund inflows, fund family size and fund age. To ensure robustness in the presence of crosssectional dependence, we use Driscoll and Kraay (1998) standard errors as suggested in Agnesens (2013). 11 Whether or not a particular fund attribute has a significant impact on that fund s risk-adjusted performance is then judged based on the coefficient estimates and their respective t-statistics obtained from Eq. (6). Taking into account that a fund s CPT utility crucially depends on the selected parameter set, we conduct sensitivity analyses with respect to different reference point specifications and diverse parameter combinations. In particular, we employ six reference points in our study, namely zero, the risk-free rate, the market return, and the risk-free rate plus exposures to one (Jensen, 1969), three (Fama and French, 1993) and four (Carhart, 1997) factor models, computed as rr rrrr tt + ββ MMMMMM jj,qq MMMMMM tt, rr rrrr tt + ββ MMMMMM jj,qq MMMMMM tt + ββ SSSSSS jj,qq SSSSSS tt + ββ HHHHHH jj,qq HHHHHH tt and rr rrrr tt + ββ MMMMMM jj,qq MMMMMM tt + ββ SSSSSS jj,qq SSSSSS tt + ββ HHHHHH jj,qq HHHHHH tt + ββ MMOOMM jj,qq MMMMMM tt, respectively, which we refer to as one (1ff), three (3ff) and four (4ff) factor model exposures throughout the study. Intuitively, in our case, a decision maker therefore 11 Note that by estimating quarterly utilities using the returns of the respective quarter only, we do not lose the information on cross-sectional dependence in the first step of our two-step approach. Hence, we can employ standard errors that are robust to cross-sectional dependence while still relying on the two-step approach that is necessary to allow incorporating behavioural performance measures.

17 Mutual fund performance decomposition: a behavioural approach 13 assesses her investment either in absolute terms, or relative to a risk-free investment, to the return provided by the market portfolio, or to a strategy with similar factor exposures 12, respectively. To check the sensitivity of our results to the five behavioural parameters introduced in Section 3.1, namely risk aversion on the side of gains (αα) and losses (ββ), loss aversion (λλ) and probability distortion for gains (δδ) and losses (γγ), we conduct a ceteris paribus analysis by assigning discrete values from a selected interval to one of the parameters while keeping all others constant. Specifically, we consider risk-aversion parameters in the range of 1.00 to 0.40, loss aversion parameters in the range of 1.00 to 3.00, and allow probability distortion parameters to vary discretely from 1.00 to Table II shows the summary statistics for the six performance measures employed throughout this article, namely certainty equivalents calculated for CPT utility values with reference points specified as zero, the risk-free return, the market return, and the one, three, and four factor exposures, respectively Table II about here For the average parameters elicited by Tversky and Kahneman (1992), for the median fund we obtain negative CPT certainty equivalents ranging from 13.17% to 3.63% depending on the corresponding reference point. Moreover, we observe that the pairwise differences in the median CEs between reference points specified as zero and risk-free rate, market and one factor benchmark, as well as three and four factor reference points are rather negligible. Therefore, in the rest of the paper we demonstrate and discuss our findings corresponding to four out of six reference-point specifications only, namely the risk-free return, as well as exposures to one, three and four factor models. 13 In this context, note that some of the obvious differences in results for changing reference points may be attributed to purely technical reasons rather than economic relations. In particular, since risk factors absorb considerable variation in returns (Carhart, 1997), factor adjusted models capture random fluctuations better than specifications that do not take factor 12 The factor-adjusted reference points provide the natural behavioural equivalent to the factor-adjusted performance evaluation measures employed in the traditional performance evaluation literature. In particular, the CPT CEs with factor-adjusted reference points are identical to alphas from the corresponding factor models if all CPT parameters are set to We include the reference point specified as exposure to three factors despite its similarity with the four-factor reference point to take into account the most popular factor models suggested by the existing literature.

18 14 Mutual fund performance decomposition: a behavioural approach exposures into account, leading to higher explanatory power of individual fund characteristics in the factor-adjusted setting. 4. Results Given the large body of literature on decompositions of mutual fund performance in traditional, alpha-based frameworks, and the lack of such literature in the behavioural setting, the goal of this section is to provide a thorough analysis of the relation between funds CPT utilities and their fund-specific attributes. Therefore, we estimate pooled regressions of CPT certainty equivalents on fund characteristics, including past performance, diversification, 12b-1 fees, management fees, turnover, front-end loads, rear loads, fund size, fund inflows, fund family size and fund age. Table III demonstrates our results. While the table reports the results for the average parameter values elicited by Tversky and Kahneman (1992), we perform extensive sensitivity analyses on these parameter values in Section Table III about here First of all, across all model specifications we observe a significant persistence in fund utility. Contradicting to the market efficiency hypothesis, a large body of traditional, alpha-based literature supports the conventional wisdom that manager s track record is an important indicator of fund s future performance (e.g. Grinblatt and Titman, 1992; Elton et al., 1993; Hendricks et al., 1993; Goetzmann and Ibbotson, 1994; Brown and Goetzmann, 1995; Elton et al., 1996; Davis, 2001; Agnesens, 2013). Similarly, our results suggest that past performance is a strong indicator of future performance in the behavioural framework. Our results further indicate a significantly positive impact of diversification on CPT certainty equivalents. With our diversification measure driven by the zero-mean fluctuations captured by the regression error term of a four-factor model, ceteris paribus a lower diversification (and hence higher fluctuations) has a negative effect on CPT CEs per definition. For the four-factor reference point, the significantly positive coefficients on diversification therefore imply that fund managers taking higher active exposures as compared to the reference point cannot compensate these negative effects with higher

19 Mutual fund performance decomposition: a behavioural approach 15 average returns. While the negative effect of our diversification measure on CEs is not given per definition for the other reference points, the positive impact on diversification nevertheless remains significant for the one-factor and three-factor reference points. For the risk-free reference point both the coefficient and its significance decline. This indicates that diversification as compared to a style-adjusted equity benchmark is not as relevant for an investor that focuses on absolute returns or that lower diversification is compensated by higher or more favourably distributed returns. In general, we observe management fees, as well as marketing and distribution (12b-1) expenses, to have a significantly negative impact on fund performance in the behavioural setting. Therefore, we cannot reject the hypothesis stemming from the traditional literature (e.g. Sharpe, 1966; Carhart, 1997; Dahlquist et al., 2000; Prather et al., 2004; Kacperczyk et al., 2005; Pollet and Wilson, 2008; Cremers and Petajisto, 2009; Huang et al., 2011) that funds charging higher management and 12b-1 fees cannot compensate their clients with an appropriate performance. An exception is provided by the risk-free reference point case, where despite a negative coefficient on management fees, we cannot reject the hypothesis that funds with higher fees can compensate these expenses by higher or more favourably distributed returns. Our findings furthermore show that funds with a relatively high trading activity provide lower CPT certainty equivalents, as indicated by a significantly negative coefficient on the turnover ratio for all four reference point specifications. This result has been also documented in the alpha-based literature (e.g. Carhart; 1997; Massa and Patgiri; 2009) and suggests that larger transaction costs associated with more trading activity cannot be justified since they are not offset by a higher performance. In our setting, however, this finding could further imply that funds engaging in higher trading activity are more likely to exhibit drawdowns, both in terms of absolute and relative (as compared to risk factor exposures) returns, as these are captured by the loss aversion in the behavioural framework. Overall, our results also suggest that both commissions charged at the time of initial investment and at the time of redemption have a significant impact on CEs. Interestingly, though, front-end and rear loads have an opposing effect on funds utilities. Specifically, funds that charge front-end fees provide significantly higher CEs at the 10%-significancelevel. While load fees are conceptually not directly related to a funds strategy or performance, this finding might suggest that funds with high TNA-weighted front-end

20 16 Mutual fund performance decomposition: a behavioural approach load fees (and hence a presumably high share of retail investors) are more frequently targeted to investors with high sensitivity to relative drawdowns. A more intuitive explanation comes to mind for the significantly negative effect of rear loads. In particular, as investors in funds with rear loads are forced to be more patient regarding these funds intermediate performance given their high exit costs, managers of those funds have much less of an incentive to avoid large intermediate drawdown, which is immediately captured by loss aversion in the behavioural setting. We find fund size to have a barely significant influence depending on the model specification. One popular hypothesis concerning the sign of the TNA impact, advocated by the traditional literature, suggests that larger funds may suffer from insufficient liquidity on their best investment ideas, thereby being forced to invest some of their assets into non-preferred investment ideas and hence generating lower performance (Berk and Green, 2004). Although our coefficients of fund size have a negative and partly significant sign, we cannot provide strong evidence to support the hypothesis of Berk and Green (2004) in the behavioural framework. Likewise, the evidence on the relation between fund flows and CPT CEs is mixed, but rather insignificant. In the traditional setting, e.g. Dahlquist et al. (2000), Chen et al. (2004), Sapp and Tiwari (2004), Cremers and Petajisto (2009) and Agnesens (2013) provide evidence against the hypothesis of investors being able to determine the future top performing funds (smart money). Similarly, we do not support the hypothesis that money inflows or outflows provide any indication of fund s future utility. On the contrary, fund family size generally has a significantly positive effect on behavioural performance, providing support for the popular hypothesis of scale economies within a fund family widely used in alpha-based studies (e.g. Chen et al., 2004; Massa and Patgiri, 2009; Huang et al., 2011; Agnesens, 2013). Finally, for three out of four reference point specifications, we find fund age to have a significantly positive impact in the behavioural framework. One possible explanation could be older funds having enough expertise to avoid large intermediate drawdowns which are undesirable in the CPT setting. Alternatively, older funds might enjoy an established competitive position based on an extensive track record and, therefore, have no need to engage in tournament behaviour that frequently involves high risk investments. Hence, older funds would have risk characteristics that are favourable to behavioural investors.

21 Mutual fund performance decomposition: a behavioural approach 17 To conclude, the decomposition exercise conducted in this section provides some interesting insights on the relation between behavioural fund performance measures and fund specific characteristics. To further elaborate on this issue, we provide a thorough analysis of our results sensitivity to changes in behavioural parameters characterizing CPT preferences. 5. Sensitivity analyses and robustness checks In this section, we discuss the sensitivity of t-statistics 14 resulting from the regression in Eq. (6) to changes in behavioural parameters, namely risk aversion, loss aversion and probability distortions. We again demonstrate and discuss our findings corresponding to four benchmark reference point specifications, namely the risk-free return, and one, three and four factors, respectively. First, we analyse how relations between fund characteristics and fund performance discussed in the previous section change when all behavioural parameters are simultaneously set to one. Intuitively, for the risk-free reference point this case corresponds to risk-neutral expected utility preferences, while for the three remaining reference points we obtain intercepts of popular factor models, αα 1FF jj,qq, αα 3FF jj,qq, and αα 4FF jj,qq : rr jj,tt rr rrrr tt = αα 1FF jj,qq + ββ MMMMMM jj,qq MMMMMM tt + εε jj,tt, (7) rr jj,tt rr rrrr tt = αα 3FF jj,qq + ββ MMMMMM jj,qq MMMMMM tt + ββ SSSSSS jj,qq SSSSSS tt + ββ HHHHHH jj,qq HHHHHH tt + εε jj,tt, (8) rr jj,tt rr rrrr tt = αα 4FF jj,qq + ββ MMMMMM jj,qq MMMMMM tt + ββ SSSSSS jj,qq SSSSSS tt + ββ HHHHHH jj,qq HHHHHH tt + ββ MMMMMM jj,qq MMMMMM tt + εε jj,tt, (9) where rr jj,tt and rr tt rrrr are the time tt returns of fund jj and the risk free rate, respectively, and the time tt returns of the risk factors, MMMMMM tt, SSSSSS tt, HHHHHH tt and MMMMMM tt are defined as in Section Table IV about here Note that we discuss sensitivity in t-statistics rather than in coefficients, as the magnitude of coefficients may vary not only due to changes in the relation of certain fund characteristics to performance, but also due to different variation of the CEs in general under changing parameter specifications. In contrast to the coefficient estimates, the t- statistics takes the change in the variation of CEs into account as well, thereby providing the more relevant measure for our purposes.

22 increasing 18 Mutual fund performance decomposition: a behavioural approach Table IV demonstrates our results for the case αα = ββ = λλ = δδ = γγ = We observe that the relation between fund characteristics and performance crucially depends on the functional form for all reference point specifications. In general, while some relations between fund characteristics and fund performance are robust to the elimination of all behavioural effects (i.e. setting all parameters to 1.00), many fund characteristics interact differently with utilities than they do with non-behavioural performance measures in terms of sign, significance and magnitude. Table IV therefore suggests that the observed differences can be attributed to functional form implied by CPT preferences. Hence, we continue our analysis by considering the impact of each behavioural parameter individually. Figures I to IV demonstrate our findings for the risk aversion coefficients (αα = ββ) varying discretely in the interval from 1.00 (risk neutrality) to 0.40 (high risk aversion), while keeping other parameters equal to their average values as specified by Tversky and Kahneman (1992), namely λλ = 2.25, δδ = 0.69 and γγ = Figures I, II, III and IV about here In general, our results are rather insensitive to changes in risk aversion, with t-statistics for all fund characteristics and across all model specifications changing only marginally with increasing risk aversion. While intuitively some of relations between performance measures and characteristics could have been driven by different processing of funds risk in general, the sensitivity analysis suggests that risk aversion alone does not seem to explain our findings. Figures V to VIII report the sensitivity of our results to loss aversion λλ16t from 1.00 (no loss aversion) to 3.00 (high loss aversion), with αα = ββ =0.88, δδ = 0.69 and γγ = Figures V, VI, VII and VIII about here In sharp contrast to the sensitivity analysis for risk aversion, our findings appear to be extremely sensitive to changes in loss aversion. In particular, as the t-statistics obtained for the case of λλ = 1.00, αα = ββ =0.88, δδ = 0.69 and γγ =0.61 closely correspond to the

23 Mutual fund performance decomposition: a behavioural approach 19 values demonstrated in Table IV, the sensitivity analysis suggests that our results in Section 4 can be predominantly attributed to loss aversion. A striking example for this pattern is provided by past performance. Specifically, while the effect of past performance is insignificant or only marginally significant for some reference points for the case of loss aversion set to 1.00, it is becoming highly significant for increasing levels of loss aversion in all considered model specifications, implying the existence of certain persistence in funds behaviour, especially in their attitude towards relative drawdowns. Likewise, the impact of diversification on funds performance can be largely attributed to loss aversion. Specifically, the t-statistics corresponding to the diversification measure change from being negatively significant for the case of no loss aversion to being positively significant already for the lower than average levels of loss aversion. While the role of diversification does not seem to be very sensitive to investors risk aversion, its high sensitivity to loss aversion at a given level of risk aversion indicates that better diversified funds are favourable for behavioural investors in particular because of their ability to avoid large losses relative to the reference point. The negative effect of management fees is also boosted by an increase in loss aversion. Specifically, while the effect of management fees is insignificant or only marginally significant, depending on the chosen reference point, for the case of no loss aversion, it is becoming highly significant for increasing levels of loss aversion. On the contrary, we observe the effect of 12b-1 fees losing its significance with growing loss aversion. Similar to management fees, the significantly negative effect of the turnover on funds utility also becomes more prominent with increasing loss aversion, supporting our hypothesis that fund managers engaging in more trading activity might be taking higher risk captured in the behavioural setting. Front and rear loads are found to have no significant impact on performance for values of loss aversion set to one. However, with increasing loss aversion coefficient, the effect of front loads on performance becomes positively significant, whereas the impact of rear loads becomes negatively significant. While the intuition for this finding corresponds to our reasoning in the previous section, the sensitivity of this result on loss aversion again strengthens the role of loss aversion for investors investment decisions. Our sensitivity analysis also supports our hypothesis that older funds either have enough expertise to avoid large intermediate drawdowns or do not engage in high risk investments, making them more attractive for loss averse investors. Specifically, the

24 20 Mutual fund performance decomposition: a behavioural approach significantly positive effect of fund age on fund performance increases with the growing loss aversion. Finally, Figures IX to XII show our findings for the probability distortion coefficients (δδ = γγ) changing discretely from 1.00 (no probability distortion) to 0.10 (high probability distortion), while keeping λλ = 2.25 and αα = ββ = Figures IX, X, XI and XII about here Our results further suggest that overweighting extreme events slightly reinforces the strong effects created by loss aversion. Specifically, the sensitivity of our results to probability distortions δδ and γγ is especially prominent for those variables whose impact is related to negative fluctuations. In particular, our results indicate that diversification plays an increasingly important role for loss averse investors who additionally exaggerate the probability of extreme events. The significantly negative effect of turnover is also strengthened by a higher probability distortion, since the probability of high intermediate losses associated with more trading activity becomes overweighted. To summarize our sensitivity analyses, we observe fund flows, fund size and fund family size to be rather insensitive to changes in behavioural parameters, while all other fund characteristics seem to crucially depend on loss aversion. Moreover, in addition to their dependence on loss aversion, diversification and turnover are also sensitive to the chosen parameters of probability weighting. Therefore, the results of this section suggest that relying on observable fund-specific characteristics when choosing appropriate funds requires a thorough consideration of the investor s preference profile. 6. Implications for future research While both academic literature and practical applications invest large amounts of resources into analysing and selection mutual funds based on various traditional performance measures, our results show the output of these analyses to crucially depend on the investors preference profile. This suggests at least two directions for future research. One straightforward extension to our study would be to relate fund characteristics to cumulative prospect theory through the moments of the return distribution, e.g. mean,