Draft: 5/10/05 Comments Welcome Biases in allocation under risk and uncertainty: Partition dependence, unit dependence, and procedure dependence

Previous studies of employee investment in retirement plans suggest that people typically "naively diversify" their investment funds, tending to allocate 1/n of the total to each of n available instruments (Benartzi & Thaler, 2001). In this paper we provide experimental evidence that this bias extends to allocation among simple chance prospects and demonstrate three new violations of rational choice theory implied by use of the naïve diversification strategy. Study 1 demonstrates “partition dependence” in which participants' allocations among a fixed set of investments varies with the hierarchical structure of the option set (e.g., by vendor and instrument). Study 2 demonstrates “unit dependence” in which participants' preferred allocations vary with the metric in which the investment is reported (dollars versus number of shares). Study 3 demonstrates “procedure dependence” in which the bias toward even allocation disappears if participants are asked to choose from a menu of possible portfolios. We show that these results extend to sophisticated participants, simple well-specified gambles and incentivecompatible payoffs. We close with a discussion of theoretical and prescriptive implications. Running Head: Dependencies in Naïve Diversification Address Correspondence to: Thomas Langer University of Muenster Universitaetsstr. 14-16 48143 Muenster, Germany [email protected] The authors gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) and the SFB 504 at the University of Mannheim. The present work was also supported by grant SES-00-99209 from the National Science Foundation to the second author. Finally, we acknowledge the support of the Fuqua School of Business, Duke University, who supported the authors when these studies were initially conceived. This paper has benefited from discussions with Shlomo Benartzi. We are grateful for helpful comments on previous drafts of this paper from David Bardolet and Brett Rogers. Partition, Unit, and Procedure Dependence page 1 Biases in allocation under risk and uncertainty: Partition dependence, unit dependence, and procedure dependence Organizations and individuals must often decide how to allocate resources among projects whose consequences are risky or uncertain. Firms must periodically choose how to divide capital and human resources among various research and development projects; venture capitalists must decide how to distribute funds among start-up companies; law firms must decide how they will invest time and money among a set of cases whose payout is uncertain. Among the most important financial decisions that individuals make is how to allocate their savings among potential investments. In the last few decades, American employers have rapidly shifted from offering defined retirement benefit plans to offering defined contribution retirement saving plans in which employees choose how to allocate their money among a fixed set of possibilities. As of 1999 about 85 percent of private retirement savings contributions went to accounts in which individuals decided how to invest plan assets (Poterba, Venti & Wise, 2001), and as of the end of 2003 there were an estimated 42 million participants and $1.9 trillion in total assets invested in 401k plans (Employer Benefit Research Institute, 2004). In light of the importance and prevalence of risky and uncertain allocation, there should be great interest in understanding how people make such decisions and how they might be improved. To date, however, there has been relatively little behavioral research on how people allocate funds over a fixed set of risky or uncertain prospects. Especially noteworthy is a groundbreaking investigation of personal savings decisions in which Benartzi and Thaler (2001; hereafter “BT”) argued that people typically employ “naïve diversification” strategies, allocating 1/n of their funds to each of n investment prospects available to them with little regard to the nature of these investments. For instance, in a survey of UCLA employees, these investigators found that a substantial proportion of participants allocated half of their (hypothetical) retirement savings to each of two investments with little regard to whether the two options were a stock fund and a bond fund, a stock fund and a balanced stock/bond fund, or a bond fund and a balanced stock/bond fund. In another survey UCLA employees offered four fixed-income funds and one equity fund allocated a median 40% of their retirement savings to equities, whereas employees offered Partition, Unit, and Procedure Dependence page 2 one fixed-income fund and four equity funds allocated a median 75% of their retirement savings to equities. BT found further evidence of naïve diversification in their analysis of actual investments in 170 retirement savings plans obtained in a database from Money Market Directories. The results showed that employees allocated more of their savings to equities if their company offered a greater proportion of equity investment options. For instance, University of California employees, who allocated their retirement savings among a stock fund and four bond funds, invested 34 percent of their money in stocks, whereas TWA employees, who allocated their retirement savings among five stock funds and one bond fund, invested 75 percent of their money in stocks. BT’s results suggest that naïve diversification is a strong and robust phenomenon in individual decisions concerning retirement investment, but it remains to be seen to what extent this phenomenon would be observed in other domains of decision under uncertainty. For instance, it could be that naïve diversification is a prescriptive norm that people apply only to personal investment decisions. Or it could be that people only apply naïve diversification in situations where the probability distribution over outcomes or the correlation among returns of potential investments is unclear. The first purpose of the present paper is to investigate whether naïve diversification is exhibited by sophisticated respondents making allocations among a small number of simple well-defined lotteries with incentive-compatible payoffs and transparent correlations. The second and more central purpose of this paper is to characterize and test new implications of naïve diversification. In particular we conjecture that this strategy will give rise to three systematic dependencies that cannot easily be reconciled with rational choice theory and pose a substantial challenge for decision analysis. First, if decision makers rely on naive diversification to even a modest extent, the final distribution will be affected by the particular grouping of options over which resources are allocated. For instance, participants in retirement plans who are offered a choice of vendors followed by a choice of 1 BT did note that naive diversification may be an instance of a more general diversification heuristic that also includes the tendency to seek variety when making multiple choices among snacks and other consumer products (e.g., Simonson, 1990; Read & Loewenstein, 1995; Fox, Ratner & Lieb, 2005), but we are aware of no studies of naïve diversification in allocation among risky or uncertain prospects outside of an investment context. Partition, Unit, and Procedure Dependence page 3 instruments within vendors (as is the case with the typical 403b plan offered by not-for-profit organizations) might first allocate evenly across vendors, then evenly across instruments within vendors. Participants in a retirement plan who are offered these same instruments in parallel (as is the case with the typical 401k plan offered by for-profit organizations) would instead allocate evenly across instruments with little regard to their corresponding vendor. We refer to the sensitivity of allocations to the way in which prospects happen to be grouped as partition dependence. Second, if decision makers are biased toward allocating resources evenly over potential investments, the final distribution will depend on what, specifically, is being allocated (i.e., the unit of measurement). For instance, brokerage accounts typically require investors to indicate the number of shares of common stocks that they wish to purchase, whereas they typically require investors to indicate the number of dollars worth of mutual funds that they wish to purchase. Thus, if share prices of investment instruments differ then the strategy of allocating money evenly among instruments will yield a different portfolio than the strategy of purchasing an equal number of shares of each instrument. To illustrate, suppose a person wishes to invest $3,000 in two stocks: stock A, currently trading at $10 per share, and stock B, currently trading at $20 per share. In this case, a purchase of 100 shares of stock A and 100 shares of stock B may seem more attractive than a purchase of 150 shares of stock A and 75 shares of stock B. However, the opposite may be true when the same portfolios are expressed in terms of dollars invested: $1000 worth of stock A and $2000 worth of stock B may seem less attractive than $1500 worth of stock A and $1500 worth of stock B. We refer to the tendency for portfolio choice to be affected by the particular unit in which allocations are expressed as unit dependence. Finally, if people rely on naïve diversification when allocating funds among a fixed set of prospects, the question arises to what extent they will do so when choosing among possible mixtures. Previous work in judgment and decision making has found that response strategies are influenced by the compatibility between the strategy and the response mode (Slovic, Griffin & Tversky, 1990; Fischer & 2 Indeed, when the second author originally signed up for the 403(b) plan at his first tenure-track job, he asked the university benefits counselor how most people allocated among the three available vendors, to which the counselor responded, “most people allocate 1/3 to each.” Partition, Unit, and Procedure Dependence page 4 Hawkins, 1993). In this case we predict that the proportion of resources invested in each prospect should be more salient to decision makers when they are called on to make an explicit allocation than when they are asked to choose between prespecified portfolios. Moreover, the heuristic to distribute equally will be more accessible when allocating whereas other choice heuristics such as maximizing the minimum possible return or maximizing expected value will be more accessible when choosing a portfolio, especially if this information is made more transparent. Thus we predict that explicit allocation will lead to more even distributions than an equivalent choice among prespecified portfolios that are mixtures of the same underlying assets. We refer to the tendency of different elicitation procedures to give rise to different portfolio choices as procedure dependence. In this paper we provide an experimental investigation of naïve diversification in allocation under risk and uncertainty that extends previous work by BT. First, we seek to replicate the basic result that people are biased toward allocating 1/n of their budget to each of n investment prospects, with insufficient regard to the nature of those prospects. We depart from previous work by using not only simple investments but also well-defined chance lotteries with incentive-compatible payoffs. Second, we explore the extent to which participants will exhibit the three hypothesized dependencies under such conditions. In Study 1 we seek evidence of partition dependence for both hypothetical investments and chance lotteries. In Study 2 we test for unit dependence for both hypothetical investments and chance lotteries. In Study 3 we seek evidence of procedure dependence in a within-subject experiment involving chance lotteries. Study 1: Partition Dependence In this study we investigate whether the allocation of funds is affected by the grouping of lotteries or investment instruments. As mentioned earlier, many retirement savings plans call on employees to first allocate funds by vendor then by specific instruments within each vendor. Many firms offer a focal investment such as company stock versus a set of other instruments, in which case employees may Partition, Unit, and Procedure Dependence page 5 allocate first between this focal investment and other investments, then proceed to allocate among remaining investments. Preliminary evidence of partition dependence can be found in an analysis of field data reported in BT. They find that among a wide range of pension plans that do not include the option of company stock, employees allocate their retirement savings roughly evenly among stocks and bonds (49 percent to equities), whereas among pension plans that do offer company stock as an option, employees invest 42 percent in company stock, 29 percent in other equities, and 29 percent in fixed-income investments. This pattern could arise from hierarchical partitioning in which participants who are offered company stock first divide their savings roughly equally between company stock and all other instruments, then divide equally among remaining instruments. We seek more direct evidence of partition dependence in two studies in which we hold a set of risky or uncertain prospects constant, but vary the subjective grouping of these prospects in various ways. In Study 1A we test for partition dependence using three simple generic investment instruments that are assigned to one of two vendors; participants are first asked to allocate funds among vendors then among instruments offered by a common vendor. In Study 1B we explore whether partition dependence might be observed in the case of simple chance lotteries with incentive-compatible payoffs. Study 1A Method We recruited 184 MBA students at Duke University during an orientation session to complete a brief survey in exchange for a donation to charity. Participants were asked to suppose that they had an opportunity to invest a portion of their income, with a matching contribution from their employer, in taxdeferred investments for retirement. Next, they were asked how they would divide this benefit among three asset classes: bonds (long-term U.S. treasury notes), equities (an S&P 500 stock index fund), and real estate (a nationally diversified Real Estate Investment Trust). Assets were grouped so that two were offered by one fictitious vendor (“Fiduciary”) and one was offered by a second fictitious vendor (“Integrity”). The mapping from vendor to asset was fully counterbalanced (so that there were three meaningful variations, one with each asset mapped to the singleton vendor) and the order of presentation Partition, Unit, and Procedure Dependence page 6 was also counterbalanced. After all of the vendors and assets were described, the allocation decision was presented as a two-stage process. In the first stage, participants were asked to allocate their endowment (in percentage terms) among the two vendors. In the second stage, participants were asked to subdivide the money allocated to the two-asset-vendor among its two assets. For one group of participants (n = 61) the one-asset vendor offered bonds (treatment B); for a second group of participants (n = 63) the one-asset vendor offered stocks (treatment S); for a third group of participants (n = 60) the one-asset vendor offered real estate (treatment R). We hypothesized that people would be biased toward even allocation among the available vendors, and also biased toward even allocation among assets within vendors. Thus, participants who employ perfect naïve diversification should allocate half of their savings to each vendor, and half to each instrument within the vendor that offers two instruments. For example, in treatment R perfect naïve diversifiers would allocate 50% to real estate, and 25% each to stocks and bonds. More generally, we expected a bias in this direction so that allocations are biased toward 50% for singleton assets and 25% for non-singleton assets. Hypothesis 1A: Each asset will be allocated more funds when it is the only asset assigned to a vendor than when it is one of two assets assigned to a vendor. Results Table 1 displays mean and median allocations to the three assets for the treatments B, S and R; means are also depicted visually in Figure 1. For instance, the mean allocation into bonds was 35% when bonds were the singleton (treatment B), compared to a 21% allocation, on average, when paired with another instrument (treatments S and R, see Table 1). The difference between the bond allocation in treatment B and the two other treatments is highly significant (p<0.0001 by Mann-Whitney). Even higher differences are observed in the respective analysis of stocks (59% in treatment S vs. 38% in the other treatments) and real estate (40% in treatment R vs. 23% in the other treatments). These differences are also highly significant (p<0.0001). 3 We further asked subjects to rate their own knowledge concerning such investments on a scale from 0 (no knowledge) to 10 (a great deal of knowledge) and found that self-rated knowledge does not influence the strength of the effect. The only significant effect of knowledge is a generally higher allocation into stocks. Partition, Unit, and Procedure Dependence page 7 It is worth pointing out that although participants were strongly influenced by the grouping of investments, they apparently did distinguish among them: Median allocations to stocks were above the proportions implied by pure two-step naïve diversification in all three experimental conditions (i.e., above 25%, 50%, and 25% in treatments, B, S, and R, respectively) and median allocations to bonds and real estate were below proportions implied by pure naïve diversification in all three conditions. 0% 10% 20% 30% 40% 50% 60% 70% mean allocation to bonds mean allocation to stocks mean allocation to real estate treatment B treatment S treatment R Figure 1: Results of Study 1A. % allocation to bonds % allocation to stocks % allocation to real estate mean (median) mean (median) mean (median) treatment B n=61 35.2 (30.0) 39.0 (35.0) 25.8 (24.0) treatment S n=63 20.1 (15.0) 59.1 (65.0) 20.8 (20.1) treatment R n=60 22.4 (20.0) 37.9 (38.3) 39.7 (37.5) Table 1: Mean and median allocations to bonds, stocks and real estate in Study 1A. Study 1B We wished to test the robustness of partition dependence through a replication. Study 1B differed from Study 1A in three important respects. First, participants were asked to allocate their endowment among chance lotteries with explicit probability distributions over outcomes rather than more abstract assets with more ambiguous (i.e. vaguer) distributions. Second, we introduced incentive-compatible payments. Third, in order to obtain a neutral benchmark of allocation preferences we included a nonhierarchical treatment in which assets were not grouped by vendors. Partition, Unit, and Procedure Dependence page 8 Method We recruited 171 Masters degree students specializing in Banking and Finance at the University of Mannheim. Participants received a questionnaire in which they were asked to divide an endowment of €30,000 between three assets. These assets were presented as simple two-outcome lotteries and had the following form: Lottery P Lottery G Lottery L