The neglect of correlation in allocation decisions.

AuthorKallir, Ido
  1. Introduction

    The observation that nonperfect correlation in asset returns may increase diversification possibilities is an essential component of efficient portfolio theory (Markowitz 1952). Finance textbooks commonly demonstrate the expansion of efficient frontier as the correlation in returns of underlying assets decreases. (1) This paper deals with an experimental investigation of the effect of changes in correlation on allocation decisions in practice. We design a simple two-asset investment problem where changes in correlation must affect optimal allocations in a predetermined, intuitive direction. More than 140 subjects with advanced backgrounds in finance are asked to predict future returns and allocate funds, in several distinct problems, for different levels of correlation. The experiments reveal that subjects consistently neglect the distributional data in their allocation decisions; although, predictions are adapted to changes in correlation. While the results for the prediction tasks demonstrate that subjects recognize differences in experimental correlation, the observed allocations suggest they fail to incorporate such variation into their investment decisions. Our simple two-asset design moreover allows for direct derivation of formal results regarding the optimal allocation patterns of rational investors. The experimental allocations consistently contradict formal theory; the inconsistency can be attributed to documented bias in allocation of funds (Benartzi and Thaler 2001) and perception of probability (Kahneman et al. 1982).

    In each experimental allocation problem, subjects are asked to distribute funds between two "virtual" assets, A and B, which admit only two levels of return: high or low. The marginal distribution of return on A always dominates the corresponding distribution on B, and the only rationale for allocating funds to the second asset (B) follows from the possibility that realized return on A would be low when the return on B is high; that is, from nonperfect correlation in returns. Different versions of this problem are presented to more than 140 master of business administration (MBA) and business undergraduate students in three distinct versions of the experiment. In the first two versions, subjects are asked to allocate funds in five different problems where the correlation in asset returns takes the values +2/3, +1/3, 0, -1/3, and -2/3. The high/low return levels on each asset are randomly drawn for every subject and each problem to examine the effect of changes in correlation in general for different combinations of underlying returns. The random drawing of returns for different problems is also intended to preclude schematic response to the distributional stimuli. Information on the joint distribution of returns is present in the form of empirical frequencies for the 12 preceding periods. Subjects are requested to predict returns for four additional periods under the assumption that future returns are sampled from the empirical distribution.

    The experiment was designed to investigate three specific issues/hypotheses: (i) test the rationality of predictions; (ii) examine if subjects increase the allocations to the dominated asset B as the correlation in returns decreases; and (iii) check the compatibility of allocations with rational models of choice, such as expected utility (EU) and rank-dependent utility (RDU). (2) The results reveal that subjects with advanced backgrounds in finance violate the rational hypotheses in all three domains: subjects try to replicate the empirical distributions when filling in their predictions, while they should fill in the most probable return combination in order to maximize the probability of correct forecasting. In this sense, subjects exhibit the well-known probability-matching, or "'law of small numbers," bias (Tversky and Kahneman 1971).

    The allocation to the dominated asset B does not increase significantly as the level of correlation with A decreases; although, the results for the prediction tasks suggest that subjects "recognize" shifts in correlation. Actual allocation decisions are mostly affected by the magnitude of returns on A and B, rather than being affected by correlation levels. In this sense, subjects neglect the joint distribution of returns in their allocation decisions.

    Finally, we prove that more than 50% of the subjects violate conditions required for consistency with EU or RDU maximization by allocating too much of the funds to the dominated asset B. The inconsistency is attributed to either a common bias in allocation patterns (naive diversification) or perception of probability (the overweighting of salient, representative events).

    To further examine the robustness of results we run a third version of the experiment where the joint distribution of returns was presented in a multicolored pie-chart (in addition to the numerical table), and the number of problems in each questionnaire was decreased from five to two. The experiment was run in advanced MBA seminars in finance. The allocations still did not vary with level of correlation. The neglect of distributional data therefore reappears in a focused "shorter" experiment with modified instructions.

    Kroll et al. (1988) ran an experimental examination of the Capital Asset Pricing Model (CAPM), where subjects are asked to allocate funds between three risky assets with normally distributed returns. Between-subject comparisons suggest that allocations are not affected by the correlation in returns (of two assets), even after 20 successive investment rounds. Anderson and Settle (1996) more recently demonstrate that subjects are insensitive to distributional characteristics when creating portfolios from field 1-year and 10-year stock data. Benzion et al. (2004) let subjects diversify between a bond, a stock, and an option and found that inefficient naive allocations persist even after 40 investment rounds where subjects receive feedback on their realized payoff at the end of each period. Our current experiments test the effect of correlation on investment, within-subject, in a simpler two-asset binary-return design where the effect of decrease in correlation on optimal allocation is intuitively transparent. Moreover, we run a multi-task experiment where subjects are asked to predict returns and allocate funds concurrently. The results demonstrate that even when subjects comprehend the difference between positive and negative correlations (differences are reflected in the prediction tasks), they fail to incorporate such variation into their allocation decisions.

    The nonprofessional financial media is frequently engaged with analyzing mean returns, trends, and cycles in the stock market. Data on correlation, volatility, and higher moments of historical returns are rarely examined. Our experiments may suggest that the media's emphasis on average data is complemented by low demand of potential investors for higher-level distributional information. Higher moments and joint-distribution statistics are harder to process or digest. The investors therefore naturally focus on "salient" mean return statistics while ignoring finer properties of underling distributions (Reyna 2004; discussed further in section 7).

    The paper is organized as follows: Section 2 presents the main experiment (Versions 1 and 2), the results for the prediction tasks are discussed in section 3, and the allocations are examined in section 4. Version 3 of the experiment is discussed in section 5, section 6 analyzes the allocation problem of an expected utility maximizer and reexamines the data in light of the model (a generalization to RDU is provided in the supplementary Appendix), and section 7 concludes.

  2. The Main Experiments

    Typical Problem Format

    The experiment consisted of five prediction/allocation problems. The format of a typical problem is illustrated in Appendix A. Subjects receive tabulated information on the empirical distribution of returns on virtual assets A and B in a sample of 12 observations. The returns on each virtual asset take only two possible values: high or low. The joint distribution table thus refers to four possible scenarios: the case where both assets earn their high return level, the case where both assets pay their low return level, and the two cases where one asset gains high and the second asset pays low. Subjects are told (see instructions in Appendix B) that the tabulated data describes the joint distribution of returns on A and B and are asked to assume that a sample of five additional observations (observations 13-17) was drawn from the tabulated distribution to compose the experimental tasks. The first experimental task dealt with predicting the return for one asset from the realized return on the second asset for observations 13-16 (see the prediction table in Appendix A). The asset with concealed return was randomly selected for each prediction task (and each questionnaire) and could vary across the four prediction periods. The prediction task was followed by an allocation task, where subjects allocated a fixed endowment between the two assets under the assumption that payoff will be determined by the realized returns in observation 17. (3) The instructions explained that at the end of the experiment one of the five problems and one prediction period would be randomly selected for each subject. Correct prediction in the selected prediction task paid the subject 20 New Israeli Shekel (N.I.S.). (4) In addition, subjects received a payoff that depended on the realized return of their portfolio in the selected problem (see details in subsection below).

    The Allocation/Prediction Problems

    Let hh denote the realization where both assets score the high return level; ll will similarly denote the case where the level of return on both assets is low; hl (lh) will then describe the cases where asset A (B) scores the high level of return while the other asset...

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