Thomas Langer
email: thomas.langer@wiwi.uni-muenster.de

Martin Weber
email: weber@bank.bwl.uni-mannheim.de

Does Commitment or Feedback Influence Myopic Loss Aversion? An Experimental Analysis

Keywords: intertemporal decision making, myopic loss aversion, feedback frequency, length of commitment, evaluation period

Empirical research has demonstrated that a lower feedback frequency combined with a longer period of commitment decreases myopia and thereby increases the willingness to invest in a risky asset. In an experimental study, we disentangle the intertwined manipulation of feedback frequency and commitment to analyze how each individual variable contributes to the change in myopia and how they interact. We find that the period of commitment exerts a substantial impact and the feedback frequency a far less pronounced impact. There is a strong interaction between both variables. The results have significant implications for real world intertemporal decision making.

Introduction

This paper investigates two important features of intertemporal decision making. First, it considers how decisions are influenced by the number of periods a decision maker is committed to a decision, that is, how long he is bound to his initial choice. Consider the stock market as an example of intertemporal decision making. Suppose you own some cash and want to make an investment in some risky asset, given some fixed planning horizon (e.g. your year of retirement). Does the amount you invest in the risky asset depend on the time you are committed to your investment? For example, does the decision depend on whether you can change the amount invested each month or only each year? The second feature studied is the feedback frequency. Using the stock market context again, the question arises as to whether the frequency of feedback influences the amount invested in the risky asset. Take, for example, a (risky) investment fund that provides information about its current value each year versus another fund that provides similar information each quarter. Does the difference in frequency make people invest more or less into such funds? In addition to the impact of each of these individual variables, one should consider whether there is any interaction between length of commitment and feedback frequency. Knowing the answers to the above questions is essential. The sheer volume of long-term investments in the stock market makes it worth studying factors that influence these investments.

In addition to the practical relevance of our study, our results are important for interpreting previous work. It will be shown that one cannot interpret previous results effectively without understanding the independent and joint influence of feedback frequency and commitment. Finally, knowing the influence of both variables might help us to design better investment decision support tools.

The concept of myopic loss aversion (MLA), introduced by Benartzi and Thaler (1995), can answer some of the above questions. The key idea behind MLA is that, because of loss aversion, a sequence of risky investments looks less attractive in a myopic evaluation.

This argument was originally confirmed in the controlled environment of two experimental studies. Gneezy and Potters (1997) examined which portion of a riskless endowment participants were willing to invest in a risky asset, whereas Thaler et al. (1997) asked individuals to split their money between two assets with different levels of risk. In both studies, a manipulation of the degree of myopia systematically influenced the willingness to invest in the riskier alternative. If participants received less frequent feedback and were forced to make a binding multi-period decision, they evaluated the assets less myopically and were more willing to accept the risk.

Subsequently, these original results on the impact of myopia on risk taking have been confirmed and extended in many ways. Haigh and List (2005) replicated the study of Gneezy and Potters (1997) with traders from the Chicago Board of Trade and found even stronger effects of MLA. Professional experience thus does not seem to weaken the bias. Gneezy et al. Kapteyn and Potters (2003) demonstrated the effect in an experimental market setting. Market prices for risky assets were significantly higher if feedback was provided less frequently and decisions were binding for several periods. Markets thus do not seem to eliminate MLA. Langer and Weber (2005) build on Prospect Theory (Tversky and Kahneman 1992) to derive refined hypotheses about the impact of myopia on risk taking. They confirmed their predictions in an experimental setting similar to that of Gneezy and Potters. Summing up, all the evidence on MLA suggests that the willingness to invest in the risky asset is influenced by a simultaneous manipulation of feedback frequency and period of commitment. 1 However, we 1 A different approach is used in Benartzi and Thaler's (1999) analysis of MLA. They manipulate the presentation format of the lottery sequence and exclude a myopic evaluation by displaying the aggregated distribution of the sequence instead of the repeated trial format.

know almost nothing about each factor's individual influence or about a possible interaction effect. 2

In our study, we find three main results. First, binding decisions cause people to be less myopic, perhaps because they must to think through the implications of a longer time horizon.

Participants invest more in the risky asset in the binding condition than in the non-binding condition. This tendency does not disappear over time. Second, providing less frequent feedback seems to help people learn over time that it is better to go with the risky prospect (i.e. to be less myopic). Third, there is no simple main effect from combining commitment and feedback, but an interaction between these two variables. The strongest effect is observed when participants are committed to their decisions, but receive frequent feedback. The effect is even stronger than the combination of a long period of commitment and less frequent feedback. It seems that if people are committed to their decisions, more frequent feedback is helpful because over time it becomes more salient that occasional losses are outweighed by ultimate gains.

In Section 2, we will present some conceptual thoughts on MLA and introduce both hypotheses and our study design. The results are presented in Section 3. The paper concludes with a short discussion (Section 4).

2 Some General Thoughts on MLA and Experimental Design

Myopic Loss Aversion

MLA combines two important behavioral concepts: loss aversion and myopic evaluation.

Loss aversion refers to the fact that in evaluating a risky alternative, individuals tend to 2 In a recent study, however, Bellemare et al. ( 2005) address similar questions. Their findings are different from our results, which might be due the fact that we implement a multiplicative scenario instead of the additive approach commonly used in the literature. This issue will be discussed later.

weight losses more heavily than gains.3 Myopia is the short-sightedness that induces a decision maker to evaluate each alternative within a sequence independently, whereas a rational decision maker would evaluate the sequence as a whole. Loss aversion implies that a myopic decision maker will invest too little in the risky asset (i.e. myopia leads to decreased willingness to take risk). The question addressed in this paper is to what degree myopia (and thus the attitude to risk) is influenced by commitment and by feedback, and how these two variables interact.

Most experimental research on MLA uses the feedback frequency and the period of commitment simultaneously to manipulate myopia. Less frequent feedback delivers distributional information at a more aggregated level and a longer period of commitment induces investors to think in terms of several periods into the future. The experimental finding that these manipulations lead to a higher willingness to invest in a lottery sequence (Gneezy and Potters 1997, Haigh and List 2005, Gneezy et al. 2003) shows that the feedback frequency and the period of commitment in fact influence the degree of myopia in the evaluation. The results obtained by Langer and Weber, though questioning the robustness of MLA, provide additional support for this mechanism. The study extends the argument from myopic loss aversion to myopic prospect theory (MPT), considering not only loss aversion, but also diminishing value sensitivity. MPT implies that myopia results in stronger risk aversion with respect to most sequences of alternatives but results in lower risk aversion for sequences based on alternatives with small loss probabilities and high loss magnitudes. They find experimental support for their refined predictions, demonstrating, on the one hand, that the effect of myopia on risk taking is less general than suggested by MLA, but showing at the same time that the basic mechanism of inducing different degrees of myopia through the feedback frequency and the period of commitment seems to be robust.

In this paper, we are interested only in the phenomenon of inducing different degrees of myopia. To avoid potential ambiguity in the results due to the divergences between MLA and MPT, our experimental analysis concentrates on alternatives for which predictions about the impact of myopia on the attractiveness of the gambling sequence are unique. For a prospect that offers a 40% chance of gaining 7% on the invested amount and a 60% chance of losing 3%, as used in our experiment, a less myopic evaluation of multiple plays should generally (i.e. in MLA and MPT) be more attractive.

In addition to the separate consideration of commitment and feedback, our design differs from most used in the literature (e.g. Gneezy and Potters 1997) in one important respect. Most studies use an additive approach (i.e. the decision maker faces identical investment opportunities in each period, and the aggregated outcome of the decision sequence is the sum of all single decision outcomes):

[ ] ∑ = α - + + ⋅ α ⋅ = T 1 t )] t ( 1 [ )] t ( r 1 [ ) t ( X FW , ( 1
)
where FW is the final wealth, X is the identical (new) endowment in each period, α(t) is the proportion of endowment invested into the risky asset in period t with 0 ≤ α(t) ≤ 1, r(t) is the return on the risky asset in period t, and T the planning horizon. 4 We use a multiplicative approach instead, in which the returns of the periods are compounded. Investors receive an initial endowment that is transferred from period to period, altered by the outcomes of the investment decisions:

[ ] ∏ = α - + + ⋅ α ⋅ = T 1 t )] t ( 1 [ )] t ( r 1 [ ) t ( Y ' FW , (2)
where FW' is the final wealth, Y is the initial endowment, α(t) is the proportion of current wealth invested in the risky asset in period t, r(t) is the return on the investment in period t, and T the planning horizon.

Clearly, the multiplicative case is more realistic than the additive case as it more closely resembles the accrual of returns in real asset markets. On the other hand, it might lead to weaker results in terms of MLA. Myopia must be expected to be less extreme, since the obvious relevance of early round decisions for later round endowments might induce investors to think less myopically, even in conditions that support myopic thinking in the additive case. It is useful to examine how strong the effects of MLA remain in this more realistic multiplicative setting.

In the following section we present the hypotheses and then explain which design is used to test them.

Hypotheses

This paper deals with the effect of commitment and feedback on myopia, which itself influences the amount invested in the risky asset. As described in formula (2), α(t) is the percentage of current wealth a person invests in the risky asset in period t. Thus α(t), or just α if the specific period is of no relevance, is the key variable for which hypotheses should be derived.

We use the notation α* for the average percentage invested over the full planning horizon.

There will be two conditions for commitment: "c" for single period commitment and "C" for multiple period commitment, and two conditions for feedback: "f" for frequent feedback which will be given each period and "F" for less frequent feedback. Thus, the variable α Cf (t) denotes, for example, the percentage of wealth invested in the risky asset in period t, where the decision maker is committed to his investment decision for multiple periods, but receives feedback about outcomes after each period. 5

The first hypothesis states that we can extend the results presented in the literature for the additive case to the multiplicative case: A longer period of commitment and simultaneously less feedback will increase the amount invested in the risky asset. 6

Hypothesis 1:

α cf * < α CF *.
The second hypothesis states that both commitment and feedback frequency will individually influence the amount invested in the risky asset. Section 2.1 has demonstrated that the degree of myopia influences α. Therefore, both more commitment and a lower feedback frequency should reduce myopia. This is the rationale behind Hypothesis 2.

Hypothesis 2:

α c• *< α C• * and α •f * < α •F *.
Ex ante, we see no reason to hypothesize any interaction effect between both variables (even if we did in fact find one subsequently).

Thus far, we have considered the average allocation over time. Next, we will present some refined hypotheses with respect to allocation patterns over time. Previous experimental research has shown that allocations to risky assets will generally increase over time, independent of the specific treatment. Langer and Weber found such an effect for an additive allocation scenario. Similar empirical evidence can be found in Weber and Camerer (1992),

where individuals played an investment game. We believe that learning plays an important role in these asset allocation experiments. Over time, as subjects learn to cope with the risky situation in the experiments, ambiguity about the process as a whole diminishes. This results in increased risk taking. 5 We will denote α •f (t) and α •F (t) if we consider the average amount invested for "f" and "F" for both commitment conditions. Note that αC• (t) cannot change for those periods in which the decision is binding.

6 It should be clear that decreasing the decision making flexibility is not just a framing issue, but a marginal change in the decision problem even from a normative point of view. As Gneezy and Potters (1997) argue, however, the effect on the risk taking behavior should be minor.

Hypothesis 3: For all treatments, the proportion of wealth α(t) invested in the risky asset increases over time.

Thus far, we have not considered a possible difference between the effect of commitment and that of feedback. Clearly, it should make a difference at the beginning of the experiment.

Commitment should have an immediate effect, as if comitted subjects immediately become less myopic. By contrast, feedback should influence subjects only over time (and not have any effect in period 1).

Hypothesis 4:

α •f (1) = α •F (1) and α c• (1) < α C• (1).

Experimental Design

In our computerized experiment, participants were confronted with 30 independent draws of the same gamble. At the beginning of the experiment, each individual received an initial endowment of 25 € that could be invested totally or partially in a lottery L = (+7 %, .4; 3 %, .6) that increased the invested amount by 7 % or decreased it by 3 % with the stated probabilities.

This gamble has an expected overall return of +1 %. Following the multiplicative case, the endowment in period t+1 was equal to the outcome of the investment in period t plus the amount transferred, thus not invested in period t:

Y(t + 1) = Y (t) [α (t) [1 + r(t) ] + [1-α (t) ]].
Each allocation decision had to be stated in percentage terms, so we explicitly asked for the proportion α, not the absolute investment amount. 7 Participants were fully informed about the return distribution of the asset and the fact that asset returns of different periods were stochastically independent.

7 Percentage allocations seem to be more natural for real world stock/bond allocation decisions. It should be noted that Gneezy and Potters asked for amounts and not for percentages in their original work in the additive framework. However, since Langer and Weber replicated the results of Gneezy and Potters while asking for percentages, it does not look like this design difference has a major impact on the results.

Two factors, the period of commitment and the feedback frequency, were varied within the experiment, resulting in a 2x2 design. Short term commitment (condition c) was for one period; longer term commitment was for three periods (condition C). Frequent feedback was provided in each period (condition f), and infrequent feedback was after each third period (condition F) .

In design cf, subjects were asked in each period to make a new allocation decision (i.e. to state which portion of their current endowment they wanted to invest in the risky asset). Following the decision, the lottery was played out and the change in wealth calculated and displayed. Figure 1 shows a typical screen from experimental condition cf. On the top left side of the screen, current wealth (here 28.62 €) is displayed. Below, the allocation decision is made either by using a slider or typing in the number (here 91 %). The feedback box at the bottom appears after the draw and presents the outcome of the gamble (here -3 %) as well as the calculation of gains or losses and the new level of wealth (here 27.84 €).

In design Cf, participants' allocation decisions were binding for three rounds. Therefore, although the gambles were played out and feedback was presented in each round, participants could only adjust their allocation after each third round. In design cF, feedback about the outcome of the investments was presented at a more aggregated level. After each third round, participants received information about the total change of wealth since the last provision of feedback. Nevertheless, allocations could be adjusted in each individual round. 8 In design CF, participants made binding decisions for three rounds and obtained feedback information only about the aggregate change of wealth over the three periods.

The experimental subjects were masters students from Mannheim University, recruited in an advanced finance class. Overall, 107 students took part in the experiment, 26 in treatment CF and 27 in each of the other treatments. 9 They entered the computer lab individually and were randomly assigned to one of the treatments by the computer. They read the instructions on the screen, were given the opportunity to ask questions, and then started the experiment independently. 10 The outcomes of the gambles were generated randomly for each individual participant by the computer, according to the specified return distribution. On average, the experiment took about 30 minutes, 15 minutes for reading the instructions and 15 minutes for the 10 (in treatments Cf and CF) or 30 (in treatments cf and cF) allocation decisions. All 8 Undoubtedly, this is a rather strange investment condition. We included it nevertheless to cover the complete 2x2 design.

9 Somewhat reflecting the typical gender distribution in advanced finance classes in Germany, there were only 14 female students taking part in the study (13%). That explains why we cannot present any second order results on gender differences in the following. 10 The complete instructions can be downloaded from the journal webpage.

participants received a flat fee of € 3 for showing up and had a 10 % chance of being selected by the computer for real payment according to their final level of wealth in the experiment.

The twelve selected individuals earned an additional € 30.19 on average. The payment procedure was made known to participants ex ante.

Results 11

The average allocation for the different conditions are given in Table 1 and in Figure 2.

Tab. 1: Average proportion of wealth invested in risky asset lottery over all 30 rounds

Table 1 indicates that we found a main effect for commitment (α c• * < α C• *, p < 0.01, Mann-Whitney one-tailed), but no effect for feedback. Thus Hypothesis 2 can only be partly confirmed. This is surprising, given the fact that in most other studies presented in the literature so far, the feedback frequency is suggested to be the driving force. Figure 2 helps to explain the surprising result. It clearly demonstrates that there is a significant interaction effect (ANOVA, p < .05) between both the commitment and feedback frequency variables. Either manipulation on its own increases the percentage invested in the risky asset (α cf * < α Cf *, p <

.01 and α cf * < α cF *, p < .05) as the dotted lines illustrate. However, the joint effect is reversed (α Cf * > α CF *, n.s) or non-existent (α cF * ≈ α CF *) at best. The data indicate that the effect of a longer period of commitment is stronger when the feedback frequency is greater. We will discuss this finding in more detail below. Nevertheless, we can confirm hypothesis 1 and thus replicate the earlier findings in a multiplicative setting (α cf * < α CF *, p < .05). For a first simple test of Hypothesis 3, we compute the average allocation for both individual in the second half (periods 16-30) and in the first half (periods 1-15) of the experiment and define the 'trend' to be the difference between these numbers. As shown in Table 2, the median trend turns out to be positive in three of the four treatments. It is highest in treatment cF (11%) and lowest in treatment CF (0%). Using a Wilcoxon signed rank test, the trend is significantly positive at a 1% level for treatments Cf and cF and on a 5% level for treatment cf.

α c• * α C• * α •f * α •F * n=54 n=53
In treatment CF, the effect is insignificant; we even observe more participants with a negative (11) than a positive (9) trend.

Tab. 2: Change in average allocation in first half (rounds 1-15) to average allocation in second half (rounds 16-30) of the experiment.

Hypothesis 4 states that the allocation in Period 1 is the same both for low and high feedback frequencies and that it is greater in the case where subjects make binding decisions for three periods. Calculating the medians (means) we obtain α •f (1) = 50.0% (49.3%) , α •F (1) = 50.0% (49.8%) and

α c• (1) = 36.5% (41.5%), α C• (1) = 50.0% (57.6%) .
The difference between α •f (1) and α •F (1) is insignificant, whereas the difference between α c• (1) and α C• (1) is significant on a 1% level. The results clearly support Hypothesis 4.

In our examination of Hypotheses 1-3, we considered average allocations over the complete course of the experiment. We now investigate how much subjects will allocate to the risky investment in the long run. This seems especially important in light of the applications mentioned in the introduction. Based on the allocations during the experiment, we esti- mate a process that converges to some limit allocation α(∞), or α ∞ for short, for t →∞. More explicitly, we consider a partial adjustment model,

α'(t) = b+c⋅α'(t-1), (3)
and estimate the parameters b, c, and α'(1) to minimize the quadratic deviation of α' from α. 12 Because of the interfering end effects in the data (see Figure 3), we exclude the allocations of the final three rounds from the fitting procedure. 13 We further impose the natural restriction on the allocation function α'(t) to remain within the domain [0, 1] for all t. This is achieved by restricting the estimated parameters to α'(1)∈ [0, 1], c∈ (0, 1), and b∈ [0, 1-c]. The function then monotonically converges to the long run allocation

) 1 ( c b - ∞ = α ∈[0, 1] for t →∞.
To provide a general impression of the results, Figure 4 presents such fitted functions α'(t) for the average allocations within each treatment. It can be seen that α'(t) increases in all four treatments (providing further support for hypothesis 3). The values of b, c, α'(1), and 3. Again, the interaction effect is shown in the data as α Cf (∞) and α cF (∞) are both larger than α CF (∞). In the long run, the combined manipulation of feedback frequency and period of commitment thus seems to have less impact on the willingness to invest in the risky asset than each manipulation alone.

12 This process is used in market experiments to estimate equilibrium prices (see, e.g. Camerer et al., 1989). 13 Such end-effects are also found in other similar experiments such as Weber and Camerer. We believe that these experiments reflect the conventional wisdom that one should lower the exposure to risk at the end of the planning horizon. Tab. 3: Process α'(t) = b+c⋅α'(t-1) fitted on average allocations in each design.

To test the significance of the long run effects, we use the procedure described above to fit functions α'(t) to each individual's allocations and compare the resulting individual long run allocations α(∞) for the different treatments. The results are summarized in Interestingly, in conditions Cf and cF more than half of the participants have an α(∞)-value of 100% (i.e. in the long run their willingness to invest is only limited by the fact that at most the complete endowment can be invested). A Mann-Whitney test shows that α cf (∞) is significantly smaller than α Cf (∞) on a 1% level, whereas the difference between α cf (∞) and α cF (∞) is only marginally significant. 14 Thus, even in the long run, subjects invest more in the risky asset when they are committed to their decisions or feedback is given more frequently; no such effect is observed if both manipulations are present. 15 4 Discussion

In this paper, we investigate the effect of commitment and feedback frequency on myopic loss aversion and examine the robustness and determinants of the phenomenon. In contrast to previous studies, our experimental design is based on a multiplicative scenario that more closely resembles the investment process in real asset markets. We further disentangle the intertwined manipulation of feedback frequency and period of commitment, commonly examined in conjunction in previous research, to understand better how each aspect contributes to the change in myopia.

14 Marginal significance is also given for the difference between α Cf (∞) and α CF (∞).

15 Thus, the long run data is not in line with hypothesis 1 and with the findings of previous studies. It should be noted, however, that previous research mostly concentrates on average allocations over time and does not consider estimated long run allocations. Thus, our results are not directly comparable with those in the literature. We find that each isolated manipulation has an impact on myopia and thereby the willingness to invest, even in a multiplicative scenario. Surprisingly, we find a strong and persistent interaction effect. The effect of the feedback frequency is reversed for the long period of commitment. This is an interesting result, because the manipulation of feedback frequency is usually considered the main driving force behind myopic loss aversion effects in experimental studies. The underlying intuition is that a longer period of commitment induces the decision maker to consider the consequences of the investment decision in a more far-sighted manner, thereby limiting the negative effects of narrow framing (Kahneman and Lovallo 1993). This effect is especially strong if the decision makers receive frequent feedback (i.e. they inevitably learn over time that occasional losses are outweighed by larger gains). Limiting feedback helps people learn not to behave myopically, but this is less the case when decisions entail commitment.

α cf (∞) α Cf (∞) α cF (∞) α CF (
The documented effect of decision flexibility on risk taking behavior has obvious relevance for investment advice and the design of saving plans. Making investment decisions unchangeable for several periods could, somewhat counterintuitively, help to increase the investor's willingness to invest in risky assets. Stock funds should advertise the possibility to make, for example, a year long commitment, but should send statements more frequently during the year.

A number of questions remain. We found the effects described in the literature to persist in a multiplicative setting, whereas we do not know which of the two settings (additive or multiplicative) induces stronger effects. Interestingly, in similar research, but in an additive investment scenario, Bellemare et al. do not find the interaction between feedback frequency and period of commitment that we observe. This suggests that it might be worthwhile to examine further the impact of the investment scenario (additive vs. multiplicative) on the determinants of myopia. Furthermore, our explanation of the strong effects of frequent feedback and longer commitment is ex post. It would be worth running replicative studies to gain a greater level of understanding of this important interaction effect. Finally, it is not clear to what degree our results (as well as the results presented in the literature so far) depend on the specific parameters of the experiment. In this first study building on the multiplicative scenario, we intentionally chose a risk profile that was as simple as possible and seemed suitable to produce myopic loss aversion effects. 16 Of course, it can be questioned whether the percentage changes (and incentive sizes) were large enough to induce participants to think carefully about their decisions. Further experimental studies might be able to shed light on such questions.

Even more important, and surely also more challenging, would it be to look for effects of myopic loss aversion in the field. We are not aware of any existing empirical data set that would allow an immediate and clean test of the myopia effects discussed in this paper. 17 16 Our only other concern was that the risk profile should be somewhat similar to a stock market investment. In this case, the 1% expected return and about 5% standard deviation of returns make our lottery comparable to the monthly return distribution of a stock market index. 17 Gneezy et al. motivate their research on myopic loss aversion by referring to the Israelian Bank Hapoalim and its intention to send out less frequent feedback to their customers as this was supposed to increase the willingness to hold mutual funds. However, this has to be considered anecdotal evidence, of course, and no information on the customers' reaction to such changes is reported.

Fig. 1 :

 1 Fig. 1: Screenshot from condition"bf" (translated).

Fig. 4 :

 4 Fig.4: Average allocations in each treatment and fitted process α'(t) = b+c⋅α'(t-1).

  However, it might be possible to convince investment companies to provide slightly different investment environments to their customers (e.g. different feedback frequencies) to mimic the manipulations of the existing experimental studies. Finding similar effects of myopia in such real world data would not only eliminate the usual concerns about the insufficient incentives and artifical investment environment underlying the experimental findings, but it could also be extremely helpful to demonstrate to practitioners the relevance of behavioral research. 5 Literature Bellemare, C., Krause, M., Kröger, S., Zhang, C., 2005. Myopic loss aversion, information feedback vs. investment flexibility. Economics Letters 87, 319-324.

Table 4 .

 4

					Condition					cf	Cf	cF	CF
						α'(1)						38.0 %	61.7 %	39.1 %	53.8 %
						b						0.091	0.064	0.109	0.132
	α	(	∞	)	=	c ) -= 1 ( c b	lim t ∞ →	α	'	( t	)	0.836 55.8 %	0.923 82.8 %	0.849 72.1 %	0.804 67.7 %

For a review of loss aversion see[START_REF] Kahneman | The endowment effect, loss aversion, and status quo bias: anomalies[END_REF] and[START_REF] Tversky | Advances in prospect theory: cumulative representation of uncertainty[END_REF].

The fraction 1-α(t) is kept at hand and thus remains unchanged.

Note that all tests in the result section are nonparametric and are based on medians unless specified otherwise.


Vasyl Golosnoy
email: vgolosnoy@stat-econ.uni-kiel.de

Yarema Okhrin
email: okhrin@euv-frankfurt-o.de

General Uncertainty in Portfolio Selection: A Case-Based Decision Approach

Keywords: act similarity, case-based reasoning, general uncertainty, model belief degree. JEL classification: D81, G11, G14

Often a portfolio investor can hardly imagine all states of nature relevant to his investment problem, causing general uncertainty concerning an asset allocation model. We quantify general uncertainty as the weakness of an investor's belief in a conventional portfolio procedure, then we develop the case-based decision making approach for determining the optimal belief degree. The economic effect of the proposed case-based methodology is investigated in the empirical study. The empirical results suggest two successful patterns of case-based decisions that could be linked to the issue of market efficiency. Moreover, our case-based modeling reflects some behavioral phenomena observed on financial markets.

Introduction

The economic literature differentiates between several kinds of uncertainty (cf. Dequech 2006). According to [START_REF] Knight | Risk, Uncertainty and Profit[END_REF], uncertainty characterizes situations where the states of nature are known but their probabilities are not. The multi-prior maximin expected utility (EU) approach of Gilboa and Schmeidler (hereafter 'GS') (1989) allows us to model this type of uncertainty. However, such an approach is inappropriate for decision problems with general uncertainty where neither states of nature nor their probabilities are known. GS (1995GS ( , 2001a) ) develop the case-based decision theory (CBDT) for decision-making in situations with general uncertainty. The CBDT is applied to economic problems, among others, by GS (1997bGS ( , 2001b)), [START_REF] Blonski | Social learning with case-based decisions[END_REF], and [START_REF] Jahnke | Coordinating service-sensitive demand and capacity by adaptive decision making: an application of case-based decision theory[END_REF].

The portfolio investor encounters general uncertainty because it is often hard to imagine all states of nature relevant for the asset allocation problem. In particular, general uncertainty can be seen as uncertainty whether the pre-selected asset allocation model is correct or/and the underlying assumptions are valid. To our knowledge there is no literature dealing with this type of uncertainty in portfolio context. This paper aims is to fill this gap and to assess general uncertainty in portfolio selection with the tools of the case-based reasoning. We adopt the CBDT for this purpose and investigate which case-based strategies lead to portfolio performance improvement.

We quantify general uncertainty as a degree of the investor's distrust in the possibility of achieving positive utility effects from investing according to the pre-selected formal model for asset allocation. The time-varying model belief degree is considered as a proxy for general uncertainty. The dynamic choice of the model belief degree is implemented using a case-based approach in a two stage procedure. In the first stage the investor applies a conventional portfolio rule, while in the second stage he uses case-based reasoning for determining his degree of model belief. The CBDT investor makes decisions solely using his experience, past performance and similarity of past situations to the present. The similarity function quantifies the distance between two situations; its form depends on the value of the indifference parameter. The decision to choose should be close to decisions in situations with favorable and far from those in situations with unfavorable past outcomes. The aspiration level distinguishes between favorable and unfavorable outcomes. It is adjusted using incoming information.

The case-based approach amends the EU reasoning in situations without information about possible states of nature (GS 2001a, p. 27ff). The linkage between the CBDT and EU rules is shown by [START_REF] Matsui | Expected utility and case-based reasoning[END_REF]. The CBDT is grounded on analogical thinking where current preferences depend on past experience. It implies bounded rationality in the sense of [START_REF] March | Organizations[END_REF]. Deviations from rationality influence the formation of general equilibrium due to limits to arbitrage. Behavioral phenomena arise due to biases in people's belief formation and/or due to preferences concerning decision-making based on given beliefs (cf. [START_REF] Barberis | A survey of behavioral finance[END_REF]. Alternatively to the EU approach, the prospect theory of [START_REF] Kahneman | Prospect theory: an analysis of decision under risk[END_REF] and its successors provides a successful description of economic agent preferences. On the contrary, the CBDT is not an alternative, but an amendment to the EU and prospect theories. Our case-based investor may exhibit the representativeness bias (too much weight on recent patterns in the data), conservatism (too much weight on prior beliefs), anchoring (too much importance on the starting value) or availability biases. However, we see the focus of this paper not on explaining behaviorial effects but on exploring which patterns of case-based decisions could be of interest for the portfolio investor.

The agent considered in the paper is an uninformed price-taker, maximizing his EU for a given risk A c c e p t e d M a n u s c r i p t aversion coefficient. He is not a representative investor, but merely a person willing to account for general uncertainty by means of the case-based approach. The indifference and memory parameter values determine his case-based strategy. The empirical study explores which case-based strategies bring the best investment results. Pure EU maximization strategies and simple portfolio rules as well as modified EU approaches are used with and without case-based amendment. The CBDT investor with a small risk aversion achieves the best EU results for good memory and high indifference degree. Such an investor may exhibit anchoring and conservatism biases. His aspiration level and model belief degree would be gradually adjusted to new information. This corresponds to the fundamental trading strategy oriented on predictability of the future portfolio performance by the past one. On the contrary, the investor with a large risk aversion mostly benefits when his CBDT strategy is characterized by poor memory and a low indifference degree. This implies quick adjustments of the aspiration level and model belief degree. Such an investor may exhibit availability and representativeness biases. This is a variation of the active trading strategy, implying improvement of arbitrage on financial markets.

The empirical evidence provides insights about the link between case-based reasoning and market efficiency. Active trading should be profitable on markets with a high efficiency degree, while fundamental trading is more appropriate for non-efficient markets, characterized by a higher degree of predictability. Investigating the portfolio of Dow Jones Industrial stocks, we find that benefits of the fundamental strategy reduce over time, while the positive effects of following the active trading case-based strategy increase. This supports the hypothesis of increasing US market efficiency. The active case-based trader quickly reacts to newcoming information. Such investment strategy improves market efficiency and may survive in financial markets in the long run. These findings shed light on the consequences of applying CBDT to model uncertainty (MU) assessment in portfolio selection.

The rest of the paper is organized as follows. Section 2 discusses the issues of risk and uncertainty in portfolio selection. Our case-based methodology to assessment of general uncertainty in portfolio problems is introduced in Section 3. Section 4 provides the empirical study, while Section 5 concludes the paper. The technical details are provided in the Appendix and in the supplement available on the JEBO website.

Risk and Uncertainty in Portfolio Selection

Asset Allocation under Uncertainty

The conventional wealth allocation approach suggests choosing portfolio weights by maximizing the investor's EU. In the [START_REF] Markowitz | Portfolio selection[END_REF] framework, the EU is maximized in the mean-variance procedure by finding the optimal trade-off between the expected portfolio return and risk associated with future investment outcomes. The unknown model parameters cause estimation risk, its importance recognized since [START_REF] Klein | The effect of estimation risk on optimal portfolio choice[END_REF]. Estimation errors, especially those in expected returns, seriously hamper portfolio performance [START_REF] Best | On the sensitivity of mean-variance-efficient portfolios to changes in asset means[END_REF]. Bayesian [START_REF] Jorion | Bayes-Stein estimation for portfolio analysis[END_REF][START_REF] Polson | Bayesian portfolio selection: an empirical analysis of the S&P index 1970-1996[END_REF][START_REF] Kumar | Learning about investment risk: The effect of structural uncertainty on dynamic investment and consumption[END_REF] or frequentistic (Ledoit and Wolf 2003[START_REF] Ter Horst | Incorporating estimation risk in optimal portfolios[END_REF][START_REF] Kan | Optimal estimation for economics gains: portfolio choice with parameter uncertainty[END_REF][START_REF] Golosnoy | Multivariate shrinkage for optimal portfolio weights[END_REF] methods reduce of the estimation risk in portfolio selection. The portfolio rule should be adjusted using one or a combination of the aforementioned methods.

However, the EU-based portfolio strategies often perform unsatisfactorily even after adjusting for estimation risk [START_REF] Michaud | Efficient Asset Management[END_REF]. They may be unable to beat simple benchmark approaches in terms of the out-of-sample EU. Constraining portfolio weights [START_REF] Frost | For better performance: constrain portfolio weights[END_REF][START_REF] Grauer | Do constraints improve portfolio performance[END_REF][START_REF] Jagannathan | Risk reduction in large portfolios: why imposing the wrong constraints helps[END_REF] or different modifications to equal weight portfolio strategies [START_REF] Black | Global portfolio optimization[END_REF]Litterman, 1992, DeMiguel et al., 2008) frequently outperform Markowitz-based rules. The non-satisfactory performance points at possible misspecification of the pre-selected models. The investor remains unaware of the true model; thus accounting for MU can mitigate the negative effects of false model assumptions.

Economic literature differentiates uncertainty types depending on the amount of information about states of nature and their probabilities. According to Knight, the issue of risk refers to situations where the states of nature as well as their probabilities are given, while uncertainty refers to situations where the states of nature are given but the probabilities are unknown. Moreover, there exist situations in which the decision-maker faces general uncertainty or structural ignorance (cf. GS 2001a), where even states of nature are neither known nor can be easily constructed. The EU framework is appropriate for decision making under risk. GS (1989) propose the maximin EU approach for situations with unknown probabilities and known states of nature. However, the EU rule is inappropriate for situations with unknown probabilities and states of nature. GS (1995GS ( , 2001a) ) develop the CBDT for decision-making in such situations. The case-based economic agent makes decisions using his experience and the similarity of encountered situations to the current one. Next we discuss the assessment of Knightian and general uncertainty in portfolio selection.

Assessment of Knightian Uncertainty

Decision making under Knightian uncertainty is grounded on the papers of GS (1989) and [START_REF] Chamberlain | Econometric applications of maximin expected utility[END_REF]. Their probabilistic maximin approach implies the choice of the EUs over the set of alternative prior distributions. The application of maximin to MU is based on choosing the most favorable variant from the set of least favorable alternatives. The maximin-based multiprior approaches are widely used for uncertainty modeling in economic applications (cf. [START_REF] Hansen | Robust control and model uncertainty[END_REF][START_REF] Kogan | A simple theory of asset pricing under model uncertainty[END_REF][START_REF] Aiolfi | Model uncertainty, thick modeling and the predictability of stock returns[END_REF].

An elegant way to quantify the degree of Knightian uncertainty in portfolio context in a Bayesian framework is developed by [START_REF] Pastor | Portfolio selection and asset pricing models[END_REF] and [START_REF] Pastor | Comparing asset pricing models: an investment perspective[END_REF]. The prior variance of the intercept in asset-pricing factor models serves as a proxy for the uncertainty degree. The size of the variance determines the trade-off between the influence of the prior density and of the likelihood function on the posterior density. In the case of low uncertainty, the prior plays a more important role; otherwise the investor gives more weight to the likelihood (i.e. to the data). [START_REF] Avramov | Stock predictability and model uncertainty[END_REF] assesses MU by averaging alternative forecasting models, extending the approach of Pastor. [START_REF] Tu | Data-generating process uncertainty: what difference does it make in portfolio decisions[END_REF] additionally consider the uncertain data-generating process (DGP) by means of Bayesian updating of prior beliefs about the true DGP.

The quantification of Knightian uncertainty in portfolio selection is combined with decision making in a multiprior framework by [START_REF] Wang | A shrinkage approach to model uncertainty and asset allocation[END_REF] and [START_REF] Garlappi | Portfolio selection with parameter and model uncertainty[END_REF]. Wang extends the approach of Pastor by considering an investor with a maximin portfolio problem, max

w min ω EU(w, ω), ( 1
)
where the EU is maximized with respect to the portfolio weights w and minimized with respect to the MU degree ω. The approach of [START_REF] Wang | A shrinkage approach to model uncertainty and asset allocation[END_REF] is Bayesian; the prior variance of ω determines the degree of uncertainty, which could be seen as a degree of trust in the factor model. Garlappi et al. propose a frequentistic multiprior approach to MU. They impose additional constraints on model parameters, allowing a wide range of possible alternative distributions for the uncertain values. The size of the constraints influences asset allocation and represents the degree of Knightian uncertainty. This approach is based on selecting from a set of different models; unfortunately it is limited by a relatively small number of alternatives.

However, it remains unclear why MU should be Knightian (i.e. limited by any clearly restricted set of alternatives). In real economic situations there is uncertainty about the underlying distribution of asset returns or whether this distribution will be unchanged in the future. The investor may be uncertain whether his investment opportunities are representative enough as well as about systematic risk of his investments. Hence, there are very different uncertainty types, so it may be difficult to imagine all states of nature, not even considering assigning utility outcomes to each possible state. The EU reasoning is inappropriate for this type of modeling, where analogy-based methods should be used (GS 2001a, p. 45ff).

Assessment of General Uncertainty

The portfolio investor unable to imagine possible states of nature faces general uncertainty. GS (2001a) suggest using the tools of the CBDT for making decisions in these situations. The case-based reasoning amends the EU rule and is grounded not on probabilistic but on psychological aspects of decision-making. Next we propose a way of adjusting conventional portfolio rules by introducing a degree of general uncertainty.

We relate general uncertainty to the strength of an investor's belief in the chosen model. Uncertainty is quantified as a degree of disbelief in the possibility of achieving some EU gains by investing in risky assets compared to the risk-free alternative. The degree of model belief at time point t is denoted by θ t ∈ [0; 1] where θ t = 0 stands for complete disbelief while θ t = 1 indicates complete belief. The interval is restricted from zero to unity in order to escape model "overconfidence" θ t > 1 and "negative belief" θ t < 0. These issues remain beyond the scope of the paper, but may occur in practical situations. Accordingly, the degree of disbelief is quantified by 1θ t .

The portfolio strategy with general uncertainty is described as follows. The optimal portfolio weights ŵt at time point t are estimated using one of the portfolio selection models. We suggest using the model belief degree for the linear weighting of wealth proportions chosen with a non-CBDT rule. The optimal weights of risky assets adjusted to the model belief degree are given by vt = θ t ŵt .

(2)

This way of modeling originates from the idea of adjusting portfolio composition by moving along the capital market line (CML). If the degree of model belief θ t decreases, the investor moves on the CML towards the risk-free rate. This is equivalent to choosing a larger coefficient of risk aversion, similar to the idea of ter Horst et al.

A c c e p t e d M a n u s c r i p t 3 Case-Based Approach to General Uncertainty

Now we introduce the CBDT approach for determining the model belief degree θ t . First we describe the essence of the case-based approach, then we adopt the CBDT for our purposes. Finally, we discuss the role of case-based reasoning in portfolio selection.

The Essence of Case-Based Approach

The CBDT grounds decision making on previous experience and on the similarity of previous situations to the current one. The economic agent chooses current action by maximizing utilities of previous outcomes weighted by a similarity function. The similarity function provides the distance between the current and previously encountered problem-act pairs. GS (1997a) introduce similarity of acts for situations where the same problem should be solved many times and there is a large number of different acts to choose from.

To formalize the approach let Θ denote the discrete set of possible/experienced acts. The CBDT selects the decision θ * t ∈ Θ at the time t maximizing the objective function Q(θ t )

θ * t = arg max θt∈Θ Q(θ t ) with Q(θ t ) = θ∈Θ (U(θ) -H t )s(θ, θ t ).
The objective function is a weighted sum of the excess utilities, calculated as the difference between realized utilities of possible outcomes θ ∈ Θ and the aspiration level H t , measuring the decision maker's satisfaction. The aspiration H t differentiates between favorable and unfavorable outcomes and is updated with incoming information. The similarity function s(θ, θ t ) penalizes the distance between decision θ * t and possible outcomes θ ∈ Θ. The current decision (act) θ * t should be chosen "close" to decisions in situations with favorable and "far" from decisions in situations with unfavorable outcomes. The distance depends on the indifference parameter; low indifference indicates large distance even between similar situations. The CBDT decision maker assumes that acts favorable for a current problem in the past will also be favorable in the future. Thus the chosen θ * t can be seen as a CBDT forecast for the optimal model belief degree in the next period.

The implementation of case-based procedure depends primarily on the choice of the similarity function and the way the aspiration level is updated. Extensive discussion of these issues for various economic situations is provided by GS (2001a).

Selecting Degree of Uncertainty

The act similarity, proposed by GS (1997a), is appropriate for our problem because it remains the same for all periods. The belief degree θ t specifies the act and reflects investor's uncertainty. We consider a continuous set of possible acts and determine the optimal degree of model belief θ * t ∈ [0; 1] by maximizing the case-based objective function Q(θ t ):

θ * t = arg max θt∈[0,1] Q(θ t ), with Q(θ t ) = 1 0 ∆U t (θ)s(θ t , θ)dθ, ( 3
)
where s(•) is a similarity function for acts, ∆U t (θ) = U t (θ) -H t is a hypothetical excess utility for a fixed θ, and H t denotes the aspiration level. In the classical CBDT (e.g. by visiting different restaurants), a person chooses just one and remains unaware of realizations of other alternatives. Our investor can easily determine the results of all hypothetically possible choices of θ at any point of time. Thus we calculate the hypothetical excess utilities ∆U t (θ) for each choice of θ ∈ [0; 1] and then select the optimal θ * t by maximizing Q(θ t ). [START_REF] Gilboa | Utility in case-based decision theory[END_REF] axiomatize hypothetical reasoning. The hypothetical CBDT returns r p,i (θ) at time i = 1, . . . , t for any θ are given by

r p,i (θ) = θ ŵ i-1 r i + (1 -θ ŵ i-1 1)r f = θ(r p,i -r f ) + r f , ( 4
)
where ŵi-1 denotes the portfolio composition at time i -1 and r p,i denotes the realized portfolio return from portfolio rules without general uncertainty. Thus for each θ we get a time series of portfolio returns {r p,i (θ)} of length t. Then we calculate the hypothetical utility function by

U(θ) = θE(r p ) + (1 -θ)r f - γθ 2 2 V (r p ). ( 5
)
The expectation E(r p ) and variance V (r p ) are estimated using the whole history of realized {r p,i }.

Here we neglect the variability in r f because it is tiny compared to those of risky assets.

The aspiration level is a "happiness point" of the investor. The acts outperforming the aspiration level are perceived as favorable, while underperforming acts are regarded as unfavorable. We do not consider exogenous [START_REF] Oechssler | Cooperation as a result of learning with aspiration levels[END_REF] or uncertain aspirations (MacLeod and Pingle, 2005), but use endogenous aspirations depending on individual payoff history only. Our endogenous aspiration level is updated with outcomes of the investor's acts. Because the portfolio problem stays the same, the aspiration updating corresponds to the investor's learning procedure, similar to game theory literature [START_REF] Boergers | Naive reinforcement learning with endogenous aspirations[END_REF]. We utilize the proposition of GS (2001a, p. 155ff) to update aspirations according to the exponential rule,

H t = (1 -λ)H t-1 + λu t , (6)
with a smoothing λ ∈ [0; 1] denoting the memory strength. The exponential rule is widely used for adjustments of the aspiration level for innovations (cf. [START_REF] Hanaki | Learning strategies[END_REF]. GS (2001a, p. 159ff) argue that the CBDT can also be used with some other aspiration updating rules. The aspiration innovations are defined by u t = r p,t (θ * t-1 )γr 2 p,t (θ * t-1 )/2. The aspiration level can be written as

H t = H 0 + λ t i=1 (1 -λ) t-i r p,i (θ * i-1 ) - γ 2 λ t i=1 (1 -λ) t-i r 2 p,i (θ * i-1 ), (7)
where H 0 denotes the initial level. Thus it is calculated by exponential smoothing of the realized r p,t (θ * t-1 ) and squared r 2 p,t (θ * t-1 ) CBDT returns. The choice of the similarity function is of great importance for the case-based reasoning. We consider the modulus similarity function:

s(θ i , θ j ) = 1 -|θ i -θ j | κ . ( 8
)
A c c e p t e d M a n u s c r i p t

Other similarity functions may also be of interest [START_REF] Billot | Probabilities as similarity-weighted frequencies[END_REF]. The indifference parameter κ ∈ (0, +∞) determines the form of the similarity function. Small values of κ lead to a large distance even between very similar acts. The modulus similarity function is linear for κ = 1, concave for κ > 1 and convex for κ < 1.

To simplify the problem (3), we provide an explicit expression for the objective function Q(θ t ) for the given form of hypothetical utility (5), aspiration adjustments (6) and similarity function (8). Proposition 1 is obtained by integrating out θ from Q(θ t ).

Proposition 1. For the modulus similarity function s(θ i , θ j ) = 1 -|θ iθ j | κ with κ ∈ (0, +∞), it holds that the case-based objective function Q(θ t ) from ( 3) is given by

Q(θ t ) = (r f -H t ) × c 1 + (E(r p ) -r f ) × c 2 - γ 2 V (r p ) × c 3 , with c 1 = 1 - θ κ+1 t + (1 -θ t ) κ+1 κ + 1 , c 2 = 1 2 - θ κ+2 t + (1 -θ t ) κ+1 (1 + κ + θ t ) (κ + 1)(κ + 2)
, and

c 3 = 1 3 - 2θ κ+3 t + (1 -θ t ) κ+1 [(1 + κ)(1 -θ t ) + (1 + κ + θ t )(1 + κ + 2θ t )] (κ + 1)(κ + 2)(κ + 3) .
The influence of the similarity function s(•) on the objective Q(θ t ) is complex. The investor perceives the similarity of different portfolio choices based on the indifference parameter κ, which ascertains how strongly the investor distinguishes between different θs. Moreover, the κ value determines the inclination of the investor to the sub-optimal choices. A value of κ close to zero would give the unit weight to the largest utility and zero weights for the utilities with other θs. In this case the optimal θ * t would be the value providing the best historical CBDT performance. For large κs the investor would hardly differentiate between utilities; thus the chosen θ * t may be far from the value of θ maximizing the historical EU. The Appendix provides more discussion on the similarity functions.

The Role of Case-Based Reasoning in Portfolio Selection

The purely EU-maximizing investor with a risk aversion γ has no need to be case-based, because following CBDT implies departures from Savage rationality. He would likely follow some of the conventional portfolio rules. Nevertheless, some investors may be concerned about MU and would like to account for it. Their decision process for uncertainty modeling can be formalized by case-based intuition. Our CBDT methodology quantifies MU in portfolio selection.

The case-based portfolio investor determines his model belief degree θ * t by maximizing his objective function Q(θ t ) for the given risk aversion γ. The optimal θ * t is calculated for the case-based strategy, characterized by parameters (κ, λ). Thus the case-based approach is not really an improvement in the portfolio selection procedure but a possible pattern of the investor's behavior in the presence of MU. The investigation of case-based decision patterns via perception of uncertainty in portfolio application is a core point of our study.

A c c e p t e d M a n u s c r i p t

Considering the patterns of case-based decisions, we aim to answer two questions. The first one is which patterns of case-based decision making are successful. This is of importance because investors with given γ following successful strategies (κ, λ) would stay in the market. On the contrary, the non successful CBDT investors would lose the money and abandon the market in the long run. This intuitively shows that the prevailing type of CBDT investors would potentially follow some successful (κ, λ) strategies. The second question is on the interpretation of these successful case-based strategies. Empirical study in Section 4 provides answers to both questions.

Empirical Study

The empirical study has the following structure. First we introduce the conventional non-CBDT portfolio rules. Then we describe the data and methodological details of the CBDT approach in portfolio selection. Finally, we provide and discuss the empirical evidence.

Conventional Asset Allocation Procedures

We consider the economy with k risky and one risk-free assets. Our investor chooses the portfolio weights w by maximizing the mean-variance objective function for the portfolio return r p :

max w E(r p ) - γ 2 V (r p ) with r p = w r + (1 -w 1)r f , (9)
where γ denotes a risk aversion coefficient, r is a k-dimensional vector of risky asset returns with E(r) = µ, V (r) = Σ. The problem (9) corresponds to the maximization of the expected quadratic utility. Now we introduce several approaches for selecting the portfolio weights w.

Markowitz-Based Procedures

The classical Markowitz optimal weights depend on the unknown true distribution parameters (µ, Σ).

We use the classical sample estimators ( μ, Σ) based on n past returns. The true w and estimated ŵ Markowitz optimal weights are given by

w = 1 γ Σ -1 (µ -r f 1), and ŵ = 1 γ Σ-1 ( μ -r f 1). ( 10
)
The practical application of Markowitz weights (10) often leads to unsatisfactory results primarily due to estimation errors in μ and Σ (cf. Michaud). Estimation risk could be mitigated by various methods, discussed in Section 2.1. For its reduction we combine the advantages of Jorion's Bayesian approach and the shrinkage methodology of Ledoit and Wolf. The shrunk covariance matrix is estimated according to

ΣLW = η Σ + (1 -η)F, ( 11
)
where Σ is a sample covariance matrix, F is a covariance matrix from the one-factor model, and η is a shrinkage coefficient defined by Ledoit and Wolf. The shrunk mean returns are computed according to the Jorion procedure:

μJ = (1 -φ j ) μ + φ j µ G 1, (12)
where μ denotes the sample mean, µ G is the mean of the global minimum variance portfolio, and φ j is a shrinkage coefficient defined by Jorion. The estimated optimal weights ŵJ,LW are obtained by substituting the shrinkage parameters μJ and ΣLW into the EU maximization problem (9).

Simple Rules for Portfolio Selection

Simple investment rules are not justified by the EU approach but yield good results in practical applications. The recent overview of these methods is given in DeMiguel et al. Here we consider the two simple equal-weight approaches widely used in practice. The first alternative is to choose equal weights for all k risky and the risk-free assets. The weights of risky assets and the fraction of wealth invested riskily are given by

w i = 1 k + 1
, and

k i=1 w i = k k + 1 . ( 13
)
The second alternative is based on the capital market line (CML) consideration and is developed in spirit of Black and Litterman. The investor chooses equal weights for all risky assets and constructs the composite asset with return r ct = (r 1t + ... + r kt )/k. Then the portfolio problem is reduced to allocation between the composite and risk-free assets. We use the sample estimator for the composite asset variance σ2 c and estimate the mean μc along the CML by

μ * c = r f + σc σ M (µ M -r f ), (14)
where (µ M , σ 2 M ) are the long-term parameters of the market portfolio proxy. Then the fraction of wealth invested in the composite risky asset is given by

w c = 1 γ μ * c -r f σ2 c , ( 15
)
so the weight of each risky asset is given by w i = w c /k for i = 1, ..., k. This approach combines the advantages of both Markowitz and CAPM procedures.

Constrained Markowitz-based Portfolio Selection

Constraining portfolio weights improves Markowitz-based procedures. Introducing constraints is advocated among others by [START_REF] Frost | For better performance: constrain portfolio weights[END_REF], [START_REF] Grauer | Do constraints improve portfolio performance[END_REF], and [START_REF] Jagannathan | Risk reduction in large portfolios: why imposing the wrong constraints helps[END_REF]. It avoids extreme portfolio positions while preserving the advantages of Markowitz rules. These procedures provide a good out-of-sample portfolio performance in practical applications,

A c c e p t e d M a n u s c r i p t see DeMiguel et al. Note that analytical solutions for constrained optimization problems are often unavailable; consequently numerical methods should be use. We implement the constraints by solving the problem (9) with the portfolio weights of risky assets restricted by 0 ≤ w i ≤ 1/a for i = 1, ..., k.

(16)

We impose equal constraints for all risky assets. The choice a = k guarantees w 1 ≤ 1, implying no risk-free borrowing or lending.

Data and Methodological Issues

In our study the US investor allocates his wealth to risky assets listed in the Dow Jones Industrial Average Index (DJIA). The US 3-month T-Bill rate serves as a proxy for the risk-free rate. The asset returns are based on weekly prices for the period from 01.01.1973 to 06.10.2005, in total 1709 observations, for k = 24 stocks permanently listed in the DJIA. The October 1987 crash has induced improvements in financial market regulation. In order to investigate whether our findings are valid for recent history, we report the results for k = 30 Dow Jones stocks for the subperiod from 01.01.87 to 06.10.2005, in total 980 observations. All data is taken from DataStream.

The CBDT approach is applied to three groups of portfolio rules. The first group consists of pure EU-maximization approaches, namely simple Markowitz strategy (10), the Jorion approach (12) and the combined Jorion and Ledoit & Wolf approach (11,12). The equal weight approach (13) and equal weight strategy based on the CML consideration (15) constitute the second group. The annualized parameters of the market portfolio proxy are taken to be µ M = 0.13 and σ M = 0.21 as in Farrell (1997, p.46). The third group consists of the adjusted EU-approaches, namely constrained procedures from the first group. We prohibit short selling and restrict the portfolio weights to [0, 1/k] for all risky assets. The risk-free rate serves as a basic benchmark.

The expected utilities (9) are reported for all non-CBDT and CBDT approaches for the investor with given γ = {2, 10, 25} and different case-based strategies (λ, κ). The chosen values of the memory parameter λ ∈ {0, 0.1, 0.5, 0.9, 1.0} correspond to perfect, good, average, poor and no memory, respectively. The indifference parameter of the similarity function takes the values κ ∈ {0.2, 1, 3.0}, defining convex, linear and concave functional forms. The moments in (9) are computed from the whole sample of realized portfolio returns. The non-CBDT portfolio weights are calculated as in Section 4.1. The respective CBDT portfolio weights are estimated by (2) with θ computed from (3) with the modulus similarity function. Other similarity functions do not alter the results significantly.

As in Ledoit et al. (2003), we estimate the parameters (µ, Σ, σ 2 c ) with n = 104 weekly returns. The choice of other estimation periods (n = 78, 130) does not change the reported evidence. The history of portfolio returns is needed for estimating portfolio expectation and variance for constructing the CBDT objective function Q(θ t ). Thus we introduce a learning period for computing E(r p ) and V (r p ). The first case-based decision is conducted at the end of the learning period at t = 0. The initial aspiration H 0 equals the average risk-free rate over the learning period. The one year (52 weekly observations) learning period is sufficient for getting reliable estimators of E(r p ), V (r p ). Their precision is constantly improved by taking all newly incoming r p s for estimation purposes. The casebased investor with a given γ chooses the optimal θ * 0 by maximizing Q(θ 0 ) for all κ values. The next period aspiration H 1 is obtained according to (6) updating for all λs.

A c c e p t e d M a n u s c r i p t

Empirical Results: Description

We report the effect of introducing case-based reasoning on portfolio performance. The best CBDT strategies are identified and interpreted from an economic viewpoint. Finally, a graphical illustration highlights the intuition behind case-based decision making. Thee EUs for both non-CBDT and CBDT strategies based on the out-of-sample portfolio returns are presented in Table 1 for the period 1973-2005 (24 risky assets) and in Table 2 for the period 1987-2005 (30 risky assets) for given γs and different constellations of (κ, λ).

The non-CBDT investor with risk aversion γ = 2 obtains the best overall results for the CML approach. The CBDT-CML approach is of interest for the investor with good memory λ 0 and the concave similarity function with indifference κ > 1, which implies a high degree of similarity between different acts. However, the advantages of the CBDT-CML approach decrease in the period 1987-2005 compared to the whole sample 1973-2005. The non-CBDT investor with γ = {10, 25} gets the best results with the CML approach, too. However, the CBDT-adjusted Markowitz rules are superior. They provide the best performance for the convex similarity function with a large distance even between similar acts. The memory λ ≈ 0.5 is of advantage for γ = 10, while for γ = 25 the best results are obtained for the weak memory λ ≈ 0.9, implying quick aspiration adjustments. These findings are even more pronounced in the recent history of financial markets from 1987 to 2005. The purely Markowitz-based approaches underperform the risk-free one. For this class of strategies, the CBDT investor would avoid risky assets and invest almost exclusively in the risk-free one. This evidence holds for all γs and is robust with respect to the sub-sample selection.

Thus we identify two CBDT strategies with the best performance for the considered portfolio rules. The investor with small risk aversion should prefer the strategy with large indifference parameter κ > 1 and nearly perfect memory λ 0. Such an investor hardly differentiates between acts with favorable and unfavorable outcomes and heavily relies on historical information. He can be characterized as a fundamental investor, following a variation of passive portfolio strategies. This approach implies gradual changes of model belief degree and the presence of anchoring and conservatism behavioral biases. Alternatively, the investor with large risk aversion should select a strategy with small κ < 1 and large λ 1. Then the model belief degree would quickly react to recent incoming information. Such an actively trading investor may exhibit representativeness and availability biases.

Figure 1 illustrates the time evolution of the model belief degree θ * t for the CML and constrained Markowitz-Jorion strategies. The CBDT parameters are κ = 3.0, λ = 0.1 for γ = 2 and κ = 0.2, λ = 0.9 for γ = 10. An increase in λ and decrease in κ accelerate the model belief adjustment. The time evolution of θ * t is characterized by three domains of attraction θ d = {0, 0.5, 1}. They denote the states of complete disbelief, average and complete belief. The optimal θ * t strongly depends on the past performance of the underlying non-CBDT strategies and of the risk-free rate evolution. The optimal θ * t from the interval [0.5, 1], characterizing strong belief degree, can hardly switch to the interval [0, 0.5], characterizing weak belief, and vice versa. Previous investment failures increase the disappointment of investor. On the contrary, past successes increase the investor's confidence, stimulating him to invest more in risky assets. The technical motivation for this effect is provided in the Appendix. The aspiration level on Figure 2 visualizes the evolution of the investor's satisfaction. It varies around the risk-free rate and exhibits clusters coinciding with the periods of low and high market volatility. This agrees with March and Simon's proposition: "Over time, the aspiration level A c c e p t e d M a n u s c r i p t tends to adjust to the level of achievement ... the level of satisfactory performance is likely to be very close to the actually achieved level of recent performance" (pp. 182-183).

Empirical Results: Discussion

The CBDT investor allows departures from rationality. Two successful case-based patterns premise completely different behavior. The fundamental trading approach is based on perfect memory, with low differentiation between acts. On the contrary, active trading exploits recent information and profits from high differentiation between favorable and unfavorable outcomes. Consequently, fundamental and active traders exhibit different behavioral biases. Now we provide an economic explanation for why these case-based strategies are of interest.

Fundamental CBDT trading implies making decisions relying on good memory about acts and outcomes. This would bring the highest benefit in situations with predictable returns and/or their second moments. Predictability on asset markets points to departures from efficiency. Thus the fundamental strategy may be more successful in markets, which are rather far from efficiency due to limits to arbitrage. In our empirical study we investigate the success of CBDT on the established US market. We consider the entire sample from 1973 to 2005 as well as two subsamples, from 1973 to 1986 and from 1987 to 2005. The predictability of the utility function could be separated into predictability of portfolio return and predictability of portfolio variance. We illustrate the predictability by providing the autocorrelations for the first and second moments of the equal weight portfolio return for all periods, considered in the study.

Table 3 reports a drastic decrease of autocorrelations in the subperiod 1987-2005 compared to whole period 1973-1986. This suggests that the US market updates more precisely upon new information. This indicates a decrease of predictability and increase of efficiency. Comparing the case-based results for the period 1973-2005 with the results for the period 1987-2005 confirms the evidence that the US market is becoming more efficient. Benefits from the fundamental strategy have reduced over time. This can be interpreted as support for the hypothesis that acceleration of information transfer improves arbitrage in the market. From this viewpoint the fundamental CBDT investors would abandon efficient markets in the long run. On the contrary, the benefits from case-based active trading have increased in the period 1987-2005. Active trading exploits the latest information and can be interpreted as arbitrage. Thus the active CBDT strategy serves to improve market efficiency. Consequently, such traders would survive in the long run. This empirical evidence supports the CBDT application for the US market investor. The best identified case-based strategies may mimic widely observed behavioral biases. The success of fundamental and active trading CBDT strategies is linked to the degree of market efficiency. This stresses the importance of case-based decision making for general uncertainty assessment.

Summary

In this paper we quantify general uncertainty in portfolio selection using the case-based reasoning of Gilboa and Schmeidler (2001a). The situations with general uncertainty are characterized by an absence of information both about possible states of nature and their probabilities. Case-based The indifference parameter κ is set to 3 and the memory parameter λ to 0.1 for the strategies with γ = 2 and to 0.2 and to 0.9 for the strategies with γ = 25, respectively.

A c c e p t e d M a n u s c r i p t A c c e p t e d M a n u s c r i p t decision theory (CBDT) amends the EU approach for such situations. We define general uncertainty as the degree of the investor's belief in the possibility of benefit from conventional investment rules. The CBDT approach is suggested for selecting the degree of model belief. The major aim of the study is to identify and interpret successful patterns of case-based decisions.

The case-based investor determines his model belief degree to maximize the objective function, defined as the sum of excess utilities for all hypothetical acts weighted by similarity of these acts to the act of choice. Hypothetical reasoning is appropriate because the investor can easily construct the consequences of all possible past choices. The indifference parameter determines the shape of the similarity function and serves to measure the distance between acts. The excess utility is calculated as the difference of the hypothetical utility and aspiration level, serving as a level of investor satisfaction. The memory strength determines the speed of aspiration adjustment. Indifference and memory parameters formalize the case-based strategy for a given investor's risk aversion. The model belief degree is used for linear adjustment of conventional portfolio rules.

The proposed case-based methodology is investigated in the empirical study. Based on our findings, we identify two successful case-based strategies. The case-based investor with a small risk aversion should choose a strategy implying gradual adjustments of model belief degree. This strategy provides the best results in a non-efficient market with high predictability degree. On the contrary, the case-based investor with medium and large risk aversion benefits from a strategy with quick adjustments. This implies active trading with immediate reaction on newly incoming information and market efficiency improvement. Thus the successful case-based approaches provide motivation for various investment strategies. They incorporate some behavioral phenomena observed on financial markets and could be seen as a type of decision-making rules under bounded rationality.

Figure 1 :

 1 Figure 1: The dynamics of the optimal degree of model belief θ * t a

Figure 2 :a

 2 Figure 2: The dynamics of the aspiration level H t and risk-free rate r f a

A

  c c e p t e d M a n u s c r i p t

A

  c c e p t e d M a n u s c r i p t

Table 1 :

 1 Expected utilities with and without CBDT reasoning for the period 1973-2005 a For each value of γ the first line contains the non-CBDT expected utilities; the rest of the block contains the CBDTutilities with the given values of memory λ and indifference parameters κ. Mark denotes the classical Markowitz approach, MJ the Markowitz approach with the mean estimated as in Jorion, MJLW the Markowitz approach with the mean estimated as in Jorion and the covariance matrix as in Ledoit and Wolf, EQ the strategy with equal portfolio weights, CM L the approach based on capital market line considerations, Mc constrained Markowitz approach, MJc constrained MJ strategy, and MJLW c constrained MJLW strategy. The CBDT strategies with significantly higher EUs than the non-CBDT strategy are marked with * for 1% significance level, * * for 5% and * * * for 10%. The number of assets in the portfolio is equal to 24.

			RF	Mark	MJ	MJLW	EQ	CML	Mc	MJc	MJLWc
	γ = 2	λ = 0.0 λ = 0.1 λ = 0.5 λ = 0.9 λ = 1.0	1.159 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0	-133.938 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 *	-33.256 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 *	-22.726 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 * 1.159 *	1.488 1.391 1.405 1.422 1.484 1.502 1.476 1.376 1.385 1.398 1.312 1.304 1.304 1.255 1.246 1.233	1.811 1.662 1.652 1.664 1.898 1.903 1.895 1.673 1.686 1.696 1.502 1.488 1.483 1.481 1.469 1.452	1.406 1.287 1.298 1.302 1.251 1.251 1.269 1.254 1.237 1.230 1.266 1.269 1.288 1.269 1.264 1.257	1.202 1.173 1.187 1.184 1.088 1.098 1.092 1.149 1.140 1.151 1.104 1.099 1.062 1.121 1.129 1.126	1.205 1.170 1.185 1.182 1.089 1.106 1.097 1.144 1.132 1.163 1.116 1.132 1.083 1.127 1.166 1.132
	γ = 10	λ = 0.0 λ = 0.1 λ = 0.5 λ = 0.9 λ = 1.0	1.157 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0	-25.873 1.157 * 1.157 * 1.157 * 1.157 * 1.157 * 1.157 * 1.157 * 1.157 * 1.157 * 1.157 * 1.157 * 1.157 * 1.157 * 1.157 * 1.157 *	-5.733 1.157 * 1.157 * 1.157 * 1.157 * 1.157 * 1.157 * 1.156 * 1.157 * 1.157 * 1.156 * 1.157 * 1.157 * 1.156 * 1.157 * 1.157 *	-3.618 1.153 * 1.157 * 1.157 * 1.155 * 1.157 * 1.157 * 1.156 * 1.157 * 1.157 * 1.168 * 1.157 * 1.157 * 1.164 * 1.157 * 1.157 *	-0.473 1.081 * 1.054 * 1.114 * 1.132 * 1.079 * 1.061 * 1.144 * 1.129 * 1.031 * 1.228 * 1.164 * 0.969 * 1.205 * 1.108 * 1.051 *	1.288 1.249 1.249 1.247 1.305 1.305 1.307 1.252 1.253 1.255 1.239 1.239 1.240 1.236 1.236 1.236	0.721 1.131 * * 1.117 * * 1.081 * * 1.069 1.223 * * 1.182 * * * 1.236 * * 1.303 * * 1.256 * * 1.197 * * 1.205 * * 1.153 * * * 1.226 * * 1.253 * * 1.218 * *	0.703 1.137 * * 1.124 * * 1.097 * * * 1.193 * * 1.031 * * * 1.087 * * * 1.236 * * 1.257 * * 1.157 * * 1.187 * * 1.205 * * 1.133 * * 1.214 * * 1.199 * * 1.150 * *	0.720 1.138 * * 1.128 * * 1.104 * * 1.192 * * 1.138 * * 1.209 * * * 1.197 * * 1.194 * * 1.211 * * 1.197 * * 1.202 * * 1.146 * * 1.205 * * 1.179 * * 1.152 * *
	γ = 25	λ = 0.0 λ = 0.1 λ = 0.5 λ = 0.9 λ = 1.0	1.154 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0	-9.666 1.154 * 1.154 * 1.154 * 1.154 * 1.154 * 1.154 * 1.154 * 1.154 * 1.154 * 1.154 * 1.154 * 1.154 * 1.154 * 1.154 * 1.154 *	-1.607 1.154 * 1.154 * 1.154 * 1.150 * 1.153 * 1.154 * 1.159 * 1.149 * 1.154 * 1.168 * 1.150 * 1.154 * 1.174 * 1.152 * 1.154 *	-0.761 1.145 * 1.154 * 1.154 * 1.148 * 1.154 * 1.154 * 1.178 * 1.158 * 1.154 * 1.183 * 1.156 * 1.154 * 1.185 * 1.150 * 1.154 *	-4.149 1.147 * 1.154 * 1.154 * 1.153 * 1.153 * 1.154 * 1.099 * 1.154 * 1.154 * 1.070 * 1.154 * 1.154 * 1.042 * 1.156 * 1.154 *	1.207 1.189 1.189 1.188 1.202 1.204 1.205 1.188 1.190 1.191 1.182 1.182 1.183 1.181 1.180 1.179	0.020 1.118 * 1.091 * 1.091 * 1.190 * 1.073 * 1.149 * 1.222 * 1.198 * 1.166 * 1.141 * 1.110 * 1.134 * 1.156 * 1.116 * 1.129 *	0.224 1.112 * 1.094 * 1.093 * 1.171 * 1.156 * 1.137 * 1.186 * 1.179 * 1.135 * 1.168 * 1.143 * 1.117 * 1.148 * 1.126 * 1.041 *	0.290 1.104 * 1.081 * 1.082 * 1.171 * 1.169 * 1.137 * 1.194 * 1.234 * 1.155 * 1.160 * 1.153 * 1.108 * 1.142 * 1.132 * 1.046 *
	a

Table 2 :

 2 Expected utilities with and without CBDT reasoning for the period 1987-2005 a For each value of γ the first line contains the non-CBDT expected utilities; the rest of the block contains the CBDTutilities with the given values of memory λ and indifference κ parameters. Mark denotes the classical Markowitz approach, MJ the Markowitz approach with the mean estimated as in Jorion, MJLW the Markowitz approach with the mean estimated as in Jorion and the covariance matrix as in Ledoit and Wolf, EQ the strategy with equal portfolio weights, CM L the approach based on capital market line considerations, Mc constrained Markowitz approach, MJc constrained MJ strategy, and MJLW c constrained MJLW strategy. The CBDT strategies with significantly higher EUs than the non-CBDT strategy are marked with * for 1% significance level, * * for 5% and * * * for 10%. The number of assets in the portfolio is equal to 30.

			RF	Mark	MJ	MJLW	EQ	CML	Mc	MJc	MJLWc
	γ = 2	λ = 0.0 λ = 0.1 λ = 0.5 λ = 0.9 λ = 1.0	0.778 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0	-217.379 0.636 * 0.778 * 0.778 * 0.732 * 0.778 * 0.778 * 0.877 * 0.778 * 0.778 * 0.906 * 0.778 * 0.778 * 0.427 * 0.778 * 0.778 *	-46.681 0.236 * 0.348 * -0.245 * 0.473 * 0.575 * -0.027 * -1.080 * 0.001 * 0.284 * 0.459 * 0.371 * -0.071 * 0.819 * 0.866 * 0.347 *	-28.363 -0.057 * 0.181 * 0.663 * -1.284 * -0.611 * -0.194 * -0.936 * -0.024 * -0.098 * -1.069 * -1.898 * -1.000 * -1.867 * -1.329 * -1.236 *	1.452 1.277 1.243 1.353 1.405 1.394 1.384 1.200 1.187 1.159 1.126 1.111 1.085 1.083 1.075 1.039	1.655 1.581 1.598 1.561 1.486 1.516 1.477 1.223 1.151 1.124 1.124 1.096 1.044 1.019 0.984 0.955	1.457 1.298 1.314 1.337 1.330 1.337 1.335 1.179 1.167 1.149 1.131 1.128 1.094 1.127 1.126 1.125	1.386 1.304 1.323 1.301 1.168 1.180 1.201 1.055 1.047 1.051 0.905 0.902 0.884 0.896 0.880 0.863	1.413 1.322 1.354 1.334 1.191 1.197 1.210 1.075 1.069 1.063 0.936 0.938 0.916 0.923 0.910 0.891
	γ = 10	λ = 0.0 λ = 0.1 λ = 0.5 λ = 0.9 λ = 1.0	0.778 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0	-42.857 0.752 * 0.778 * 0.778 * 0.787 * 0.778 * 0.778 * 0.797 * 0.778 * 0.778 * 0.803 * 0.778 * 0.778 * 0.721 * 0.778 * 0.778 *	-8.717 0.664 * 0.687 * 0.570 * 0.774 * 0.730 * 0.710 * 0.289 * 0.624 * 0.648 * 0.697 * 0.703 * 0.417 * 0.712 * 0.796 * 0.612 *	-5.042 0.609 * 0.668 * 0.752 * 0.387 * 0.487 * 0.585 * 0.431 * 0.447 * 0.599 * 0.317 * 0.268 * 0.426 * 0.241 * 0.272 * 0.379 *	-0.464 0.779 * * 0.606 * * 0.542 * * 0.814 * 0.704 * 0.628 * * 0.274 * * * 0.288 * * * 0.297 * * * 0.441 * * 0.403 * * 0.229 * * * 0.422 * * 0.384 * * 0.320 * * *	0.953 0.932 0.932 0.933 0.932 0.937 0.935 0.859 0.849 0.857 0.859 0.856 0.859 0.829 0.827 0.826	0.715 0.488 0.483 0.502 0.794 0.789 0.771 0.618 0.618 0.579 0.487 0.465 0.428 0.476 0.480 0.452	0.771 0.563 0.545 0.548 0.676 0.661 0.656 0.637 0.609 0.630 0.491 0.500 0.478 0.481 0.468 0.456	0.773 0.608 0.602 0.609 0.671 0.682 0.665 0.643 0.664 0.672 0.542 0.551 0.546 0.526 0.524 0.503
	γ = 25	λ = 0.0 λ = 0.1 λ = 0.5 λ = 0.9 λ = 1.0	0.777 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.5 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0 κ = 0.2 κ = 1.0 κ = 3.0	-16.679 0.766 * 0.777 * 0.777 * 0.767 * 0.777 * 0.777 * 0.788 * 0.777 * 0.777 * 0.792 * 0.777 * 0.777 * 0.754 * 0.777 * 0.777 *	-3.023 0.734 * 0.738 * 0.692 * 0.770 * 0.761 * 0.750 * 0.576 * 0.716 * 0.726 * 0.730 * 0.741 * 0.708 * 0.723 * 0.785 * 0.714 *	-1.556 0.710 * 0.738 * 0.762 * 0.602 * 0.707 * 0.680 * 0.625 * 0.662 * 0.708 * 0.602 * 0.588 * 0.554 * 0.531 * 0.513 * 0.601 *	-4.055 0.714 * 0.731 * 0.755 * 0.594 * 0.587 * 0.548 * 0.659 * 0.726 * 0.720 * 0.545 * 0.722 * 0.709 * 0.532 * 0.566 * 0.568 *	0.847 0.838 0.838 0.839 0.847 0.846 0.841 0.821 0.820 0.816 0.814 0.815 0.815 0.806 0.806 0.806	-0.146 0.677 * * 0.681 * * 0.695 * * 0.716 * * 0.623 * * 0.635 * * 0.639 * * 0.627 * * 0.577 * * 0.823 * 0.833 * 0.648 * * 0.841 * 0.837 * 0.753 *	-0.034 0.697 * * 0.698 * * 0.696 * * 0.677 * * 0.690 * * 0.686 * * 0.613 * * 0.699 * * 0.730 * * 0.803 * 0.802 * 0.688 * * 0.850 * 0.842 * 0.783 *	0.058 0.710 * * 0.710 * * 0.709 * * 0.701 * * 0.680 * * 0.569 * * 0.581 * * 0.627 * * 0.553 * * * 0.822 * 0.793 * 0.711 * * 0.847 * 0.838 * 0.758 * *
	a

Table 3 :

 3 Autocorrelations of the equal weight portfolio returns and squared returns for different investment periods. -0.0541 -0.0358 0.0220 1987-2005 -0.0204 0.0230 0.0221 -0.0258 -0.0405 0.0731 * * 1973-2005 -0.0250 0.0154 0.0525 * * -0.0351 -0.0367 0.0566 * * ACF of r 2 p 1973-1986 0.2865 * 0.1832 * 0.2156 * 0.2056 * 0.1643 * 0.1982 * 1987-2005 0.0878 * 0.0667 * * 0.0390 0.0515 0.0260 0.1650 * 1973-2005 0.1156 * 0.0835 * 0.0637 * 0.0736 * 0.0462 * * * 0.1706 *

	period	1	2	3	4	5	6
	1973-1986 -0.0342 0.0018	0.1065
	ACF of r p

*

Acknowledgements

We would like to express our gratitude to two anonymous referees for valuable comments. Special thanks are addressed to Thomas Lux for the helpful discussion of the paper. Any remaining errors are of our own responsibility.