Correlation Neglect in Financial Decision-Making
Correlation Neglect in Financial Decision-Making
Erik Eyster and Georg Weizs?cker
November21,2010
Abstract
Good decision-making often requires people to perceive and handle a myriad of statistical correlations.Notably,optimal portfolio theory depends upon a sophisticated understanding of
the correlation between…nancial assets.In this paper,we examine people’s understanding of
correlation using a sequence of portfolio-allocation problems and…nd it to be strongly imperfect.
Our experiment uses pairs of portfolio-choice problems that have the same asset span—identical
sets of attainable returns—and di¤er only in the assets’correlation.While any outcome-based
theory of choice makes the same prediction across paired problems,the subjects behave very
di¤erently across pairs.We…nd evidence for correlation neglect—people treat correlated vari-
ables as uncorrelated—as well as for a form of“1=n heuristic”—people invest half of wealth in
each of two available assets.(JEL B49)
Keywords:portfolio choice,correlation neglect,1=n heuristic,biases in beliefs
Eyster:Department of Economics,London School of Economics(email: e.eyster@https://www.sodocs.net/doc/3313343173.html,);Weizs?cker:DIW Berlin&Department of Economics,University College London&Department of Economics,London School of Economics(email:gweizsaecker@diw.de).We are grateful to Xavier Gabaix,Jacob King,Matthew Rabin,Sven Rady,Tobias Schmidt,Paul Viefers and audiences at Berkeley,DIW Berlin,Bonn,East Anglia,Innsbruck,Toulouse and UCL,as well as the Second Behavioral Economics Annual Meeting(BEAM)at Cornell for their comments and to Jacob King and Wei Min Wang for superb research assistance.Both authors thank the STICERD research center at LSE for funding the experiments,Technical University Berlin for access to the decision laboratory,and the Schweizerische Nationalbank for funding the research opportunities at the ESSET2008conference.Weizs?cker thanks the ELSE Centre at UCL for…nancial support,and Eyster thanks the NIH for travel support.
1Introduction
Financial decision-makers face a panoply of correlations across di¤erent asset returns.Yet people have limited attention and…nd it cognitively challenging to work with joint distributions of multiple random variables.Even if in principle the typical investor could analyse…nancial variables’co-movement adequately,she still might fail to account for correlation properly at the moment of allocating her…nancial portfolio.As a consequence,investors may hold portfolios that contain undesirable and avoidable risks.For example,many household investors invest disproportionately in stock of their own employers(Benartzi and Thaler2001)or hold only a handful of positively-correlated assets(Polkovnichenko2005).Following the recent…nancial-markets crisis,numerous commentators have asked whether both households and institutional investors relied on de…cient risk modelling.1
In this paper,we explore people’s tendency to neglect correlation.Although such“correlation neglect”may play an important role in numerous economic settings,we focus entirely on its con-sequences for…nancial decision-making.Many of the investment behaviours described above are broadly consistent with correlation neglect—but also with a multitude of other forces.To eliminate such confounds,we design and run a series of controlled experiments that test people’s attention to correlation.Our experiment studies the standard textbook model of portfolio choice with state-dependent returns using a framing variation in which each participant faces two versions of the same portfolio-choice problem.Across the two framing variations,we switch asset correlation on and o¤.This design thus ensures that under the null hypothesis that people correctly perceive the covariance structure,the framing variation does not a¤ect their behavior.Nevertheless,we …nd that behavior changes strongly and the data analysis supports two alternative hypotheses, both irresponsive to correlation.First,people tend to ignore correlation and treat correlated assets as independent.Second,people tend to follow a simple"1/n"heuristic that prescribes investing equal shares of a…nancial portfolio into all available assets(as in Benartzi and Thaler(2001)). Our experimental data support these theories despite the fact that,through an understanding test, we ensure that the participants understand the payo¤structure,including the co-movement of the 1Brunnermeier(2009)and Hellwig(2009)discuss erroneous perceptions of systemic risks during the crisis.
asset returns.Consistent with limited attention,subjects appear to omit these considerations when choosing their portfolios.
A small set of previous experiments examining people’s responses to correlation uncovers evi-dence consistent with neglect.Kroll,Levy and Rapoport(1988)and Kallir and Sonsino(2009)…nd that changing the correlation structure of a portfolio-choice problem leads to little or no change in participants’decision-making,even when many expected-utility preferences common in the eco-nomics and…nance literatures predict signi…cant change.Correlation neglect predicts no change in participants’behavior,consistent with the data.2Our design reverses the previous ones:whereas Kroll et al.(1988)and Kallir and Sonsino(2009)vary the decision problems and…nd behavior unchanged,we keep the decision problems economically unchanged and…nd that behavior changes. These two approaches complement each other and together paint a consistent picture of correlation neglect.3
Our experimental design sharpens the…ndings of this past work because it allows us to test the null hypothesis that people correctly appreciate correlation without making any ancillary assump-tions about subjects’utility functions.Chie?y,we need not assume that subjects are risk averse to test the null hypothesis that subjects correctly appreciate correlation.The reasons for this is that any fully rational agent will choose the same distribution of earnings in both isomorphic investment problems because the set of available portfolios is identical between them.This isomorphy arises through a straightforward manipulation:the assets in the correlated frame are linear combinations of the assets in the uncorrelated frame and thus span the exact same set of earnings distributions.4 Section2present the main experiment,which involves four such pairs of problems.In two of the problems,the correlated frame involves a positive correlation across assets,and in the other 2Kroll et al.(1988)also report evidence that choices respond remarkably little to feedback on realized returns. Kallir and Sonsino(2009)shed additional light on the cognitive nature of the bias that is consistent with the interpretation of correlation neglect as deriving from limited attention:when asked to predict one asset’s return conditional upon the other’s return,subjects demonstrate that they do perceive correlation correctly despite making investment choices that do not incorporate this understanding.
3Further closely related evidence is provided by Gubaydullina and Spiwoks(2009),whose subjects fail to minimize portfolio variance in a problem with correlated assets,but not in a di¤erent problem without correlation.
4This statement is modulo a quali…er about necessary short-sales constraints,which we clarify in Section2.
two problems it involves a negative correlation.The set of problems with correlated assets thus o¤ers a nontrivial range of hedging opportunities,which may or may not be appreciated by the participants.5
Section3describes the theoretical predictions for these choice problems,based on a sequence of assumptions.For the standard,“rational”benchmark we…rst impose the most basic assumption that people have rational preferences over portfolios that depend only upon the distributions of their monetary payo¤s.This consequentialist assumption—encompassing the entire set of expected-utility preferences as well as many generalizations in the literature—implies that if the decision-maker were to make only one choice in our experiment,it would not matter which of the two correlation frames she faced.To invoke this implication in the data analysis,we make a second mild assumption ensuring that the decision-maker considers her choices in isolation:we assume that she either obeys the independence axiom or that she brackets narrowly between her choices (as de…ned in Section3).Jointly,the stated assumptions in our experimental design enable us to reject of the null hypothesis that subjects correctly perceive correlation,regardless of their risk attitudes.A further(and standard)assumption that we use for some purposes is that the decision-maker is risk averse.Risk aversion su￠ces to make a unique prediction in three of our four pairs of problems that have a unique portfolio that second-order stochastically dominates all others.For the remaining pair,we use the stronger assumption that the(risk-averse)decision-maker is close to being risk neutral to make a unique prediction.
The same full set of of assumptions su￠ces to make a unique prediction in each choice problem for a decision-maker who fully neglects correlation and treats all random variables as independent. To model correlation neglect,we begin with both assets’marginal distribution over payo¤s and construct their product distribution over payo¤s;someone who neglects correlation misperceives payo¤s as deriving from this product distribution—where,by construction,the assets’payo¤s are uncorrelated—rather than from the true conditional distribution.Finally,Section3also discusses the third behavioral model that we consider,a decision-maker who simply invests equal proportions 5A notable feature of our design is its minimality:switching correlation between non-degenerate random variables on and o¤requires at least as many assets(two)states of nature(four)and distinct monetary prizes(three)as we employ.In this sense,correlation neglect may appear in the simplest possible set-up.
of her wealth in each available asset.As will become clear,this may be viewed as an extreme version of“variance neglect”,where the decision-maker ignores di¤erences in the asset variances.
The data summary of Section4shows that very few participants choose equivalent portfolios in paired choice problems.Only one out of146participants in the main experiment behaves fully consistently,choosing four equivalent portfolios in the four pairs of choices.Of the remaining participants,a majority(60%)does not choose even a single isomorphic pair of portfolios in any of the four pairs of choices.A surprising result appears regarding the relative predictive value of the three basic models(rational,perceived independence,1/n).In linear regressions,the rational model adds no explanatory power to the other two.Regressing subjects’choices in the correlated problems on the predictions of the…rst two models,the rational model has a point estimate with the wrong sign.
Section5further examines the patterns in subjects’deviations from rationality.There again we exploit the fact that all assets pay the same expected return to decompose subjects’preferences over portfolios into preferences over the variance in payo¤s.Thus retaining the assumption of mean-variance preferences,we allow for a wider set of erroneous perceptions of the covariance structure.6 First,participants’perceptions of correlation may be biased towards zero.Second,they may underestimate the magnitude of di¤erences between the entries of the variance-covariance matrix by using a concave transformation of variance.We classify subjects into types according to two parameters,one measuring correlation neglect and the other variance neglect.The single type that best…ts the entire subject pool is one that entirely neglects correlation and exponentiates variance to the power0:43.The two types that best…t the subject pool are one covering ninety-one percent of subjects that essentially coincides with that just described and a second,rational type best…tting the remaining nine percent of subjects.Adding additional types does not signi…cantly improve the statistical…t.These estimations indicate that substantial proportions of the participants follow the extreme cases of these two biases—they either neglect all covariance or all variance di¤erences—or show combinations of the biases.Under the simplifying assumption that all participants either correctly appreciate the covariance structure or show one of the two extreme biases,we…nd that 6The result in Eyster(2010)clari…es that mild risk aversion leads to approximate mean-variance preferences.
an estimated10%of the participants show the correct appreciation of the distribution.
Our study also includes a few more experimental demonstrations of correlation neglect,which we present in Section6.In one,we o¤er the participants two portfolios,the…rst riskier than the second,with the property that if one mistakenly ignores the correlation between the underlying assets the…rst portfolio appears to…rst-order stochastically dominate the second portfolio.Indeed, almost all subjects choose the apparently dominant option.But in a control treatment,where the the same two portfolios are o¤ered but their true distributions are explicitly shown to the subjects, about half of the subjects choose the second option.
The…nal demonstration of correlation neglect is that participants can be made to violate ar-bitrage freeness.In a separate task we present the participants with three assets,where one asset is state-wise dominated by appropriate combinations of the other two.Any investment in the dominated asset is an arbitrage loss:spreading that investment appropriately across the other two assets would yield more money in every state of the world.We…nd that more than three quarters of our subjects indeed fall prey to this arbitrage.Section7concludes.
2Experimental Design and Procedure
The following excerpt from the instructions shows one of the decision problems,labelled Choice3.
3.Invest each of your60points in either Asset E or Asset F,as given below.
[1][2][3][4]
E12241224
F12122424
Let =f1;2;3;4g be the four equi-probable possible states of the world,denoted by columns in the table above.Each row’s label X denotes an asset that pays out X(!)in state!2 .In Choice 3,for instance,Asset E pays out E(1)=12in state1,E(2)=24in state2,etc.Each entry states how many Euros a subject who invested her entire portfolio in that asset would earn.The state!is chosen through a random draw with equal probabilities across the four states in .Each participant faces N=11or N=12choice problems in this format,without feedback between choices.Of
these,ten problems were presented in the…rst part of the experiment.The additional one or two choices,described in Section6,followed after a brief additional instruction.Although the content of this additional instruction varied among participants,its placement after the…rst ten problems prevented it from in?uencing the initial ten choices.We focus on eight of the initial ten choices in this section,which constitute the main experiment;the remaining two choices are reported in Appendix B.7Only one of the N choices of the experiment is paid out for each subject,following another random draw that each subject makes at the conclusion of the experiment.8Given this random payment procedure,participants can be assumed to make each choice in isolation given preferences that satisfy the Independence Axiom or given that she“brackets narrowly”.We return to this issue in Section3.
The set of available portfolios in decision problem n2f1;:::;N g is characterized by the two available assets X n1and X n2.Any portfolio can be viewed as a lottery over wealth W that assigns wealth W(!)to state!:a decision maker who invests the fraction n1of her wealth (60 n1“points”)into asset X n1and the remaining fraction1 n1into asset X n2ends with wealth W(!)= n1X n1(!)+(1 n1)X n2(!)in state!.9 n1also must lie in some constraint set C n R,which precludes short sales of either asset,0 n1 1,and in some cases includes more stringent constraints.The following table shows our main8choice problems by reproducing the 8speci…cations of(X n1;X n2;C n)for n=1;:::8.The experiment presented the problems in two di¤erent sequences,used abstract labels for states,and varied the order in which states were presented.
7The two remaining choices exactly resemble those discussed in the main text and were designed to demonstrate that correlation neglect can lead to violations of blatantly obvious statewise dominance between assets.Appendix B reports a statistically signi…cant(but economically small)e¤ect of this nature.
8In addition,participants receive a show-up fee of5Euros.
9Although the participants were required to choose integer point allocations,putting n
onto a grid,we ignore
1
this complication for simplicity.
n f X n1(1);X n1(2);X n1(3);X n1(4)gf X n2(1);X n2(2);X n2(3);X n2(4)g C n
1f15;21;15;21gf12;12;24;24g0 11 1
2f18;30;6;18gf12;12;24;24g0 21 12
3f12;24;12;24gf12;12;24;24g0 31 1
4f12;24;12;24gf12;18;18;24g0 41 1
5f14;21;14;21gf14;14;21;21g0 51 1
6f14;21;14;21gf14;0;35;21g2
3 61 1
7f12;30;12;30gf12;12;30;30g0 71 1
8f12;30;12;30gf12;18;24;30g0 81 1
Table1:8portfolio-choice problems.
An important feature of the8problems is that half—those with odd-numbered n—involve only pairs of uncorrelated assets,whereas the other half involve non-zero correlations.In Section3,we explain how each even-numbered decision problem is isomorphic to its immediate predecessor in the table.
The participants for the eight decision problems described above were148students of Technical University Berlin,mostly undergraduate.Of these,two participants violated one of the contraints C n and their data were thus excluded from the analysis,leaving146complete observations.The eight experimental sessions were conducted in a paper-and-pencil format(with instructions trans-lated into German)following a…xed protocol and with the same experimenters present.Each session lasted about90minutes,including all payments.Before the main part of the experiment, the participants underwent an understanding test asking three questions about the payment rule, all of which subjects had to answer correctly before proceeding.About10%of the subjects needed help from the experimenters to pass the understanding test.Controlled variations of the eight decision problems were also used in four further sessions with96additional subjects(see Section 6).
3Predictions
In this section,we develop predictions in our experiments deriving from various di¤erent as-sumptions about the decision maker’s choice functions,moving from weakest to strongest.Let W n=W(X n1;X n2;C n)be the set of wealth lotteries feasible for the investor in decision problem n. Because subjects were only paid for a single choice chosen at random,we work with their prefer-ences over W,the feasible set of wealth lotteries across the entire experiment.10For any S W, let m (S)=f s2S:8s02S;s s0g,the investor’s set of preferred lotteries from S.
Assumption A.The investor makes choices that maximise a preference relation over W that is complete and transitive.Moreover,for each n,m (W n)is a singleton set.
Assumption A implies that investors’preferences over asset portfolios are rational;they depend only on the span of assets and not the correlation structure underlying it.When di¤erent assets pay di¤erent expected returns,then the risk-return tradeo¤normally produces non-singleton in-di¤erence curves.In our experiment,however,in each choice,each asset pays the same expected return.Because our design eliminates any risk-return tradeo¤,it makes sense to assume that in-di¤erence curves are degenerate.Nevertheless,it su￠ces to assume–and that we do assume—that the decision maker has a unique preference-maximising portfolio in each of the choice sets of the experiment.Whenever two portfolio-allocation problems have the same asset span,any decision maker who makes choices to maximise rational preferences over wealth lotteries must make the same choice across both problems.In particular,if a subject were to make only one choice in the experiment,then it would only depend upon the span of assets in that choice.
Observation1.Under Assumption A,if W(X01;X02;C0)=W(X1;X2;C),then m (W(X01;X02;C0))= m (W(X1;X2;C)).
This results holds not only for expected-utility preferences but for all rational preference relations, regardless of whether they are continuous or satisfy the independence axiom.In our experiment, twinned problems are constructed to have identical asset spans.For example,in Choices3and4 10Since n
lies on a grid in each choice problem n,the experiment o¤ers only a…nite number of achievable wealth 1
levels.The set of probability distributions over these wealth levels contains W.
presented above in the text,Asset G is identical to Asset E,and Asset H equals1
2E+1
2
F.Someone
who invests^ 1
2
in Asset E of Choice3can achieve the same portfolio with =2^ 1in Asset G of Choice4.This holds because the remaining1 =1 (2^ 1)=2(1 ^ )invested in Asset
H,itself comprised of1
2E+1
2
F,gives1
2
2(1 ^ )=(1 ^ )invested in Asset F,just like in Choice
3.Hence,any portfolio produced using^ 1
2
in Choice3can be reproduced through a suitably
constructed portfolio in Choice4,and vice versa.Although this argument breaks down for^ <1
2
, the symmetry of Choice3across assets and states,as well as our experiment’s random presentation of states(columns are randomly permuted),suggests that anyone who wishes to invest less than one-half of her portfolio in Asset E,essentially betting on State3over the symmetric State2, should be equally willing to invest more than one-half in Asset F.Similarly,the other three pairs of twinned problems have been constructed so that every feasible portfolio in the uncorrelated problem can be replicated in the correlated problem,and vice versa.
Observation1implies that if each subject in our experiment were to make only a single portfolio-allocation choice,then she would take positions in twinned problems that result in the same state-contingent wealth lottery.However,each subject in our experiment does not make a single choice but instead a sequence of choices,with only a single choice randomly chosen to be paid out.In this case,we must also assume that the decision maker adheres to the independence axiom in order to conclude that she chooses the same portfolio across twinned problems.This limits the scope of the result to expected-utility preferences,be they risk-averse,risk-loving,or neither.
Yet Tversky and Kahneman(1981)and Rabin and Weizs?cker(2009),among others,demon-strate that subjects in experiments do not makes choices on individual problems that take into account the entire set of problems they have to solve—even when explicitly told to do so—but instead make each choice in isolation,neglecting all remaining problems.We capture this idea by a de…nition similar to that in Rabin and Weizs?cker(2009).
De…nition1.The decision maker brackets narrowly if she makes each portfolio-allocation choice as if it were her only one:she applies applies her preference relation to the choice set given by (X n1;X n2;C n).
A subject who“brackets narrowly”in this sense chooses the same portfolio across twinned
problems to conform to Observation1regardless of whether she obeys the Independence Axiom.
Assumption B.The decision maker brackets narrowly or makes choices to maximise preferences that satisfy the Independence Axiom.
Together Assumptions A and B imply Observation1in our experiment where only a single choice is actually paid out to subjects.
A standard assumption about preferences under uncertainty is that people dislike risk. Assumption C.The decision maker is risk averse.11
Because each asset in our experiment pays out a positive amount in each state of the world,a subject who maximised Kahneman and Tversky(1979)loss-averse preferences over lab winnings—using a zero reference point—satis…es Assumptions A-C.
In three out of four pairs of twinned problems(Choices3-8),Assumptions A,B and C su￠ce to make unique predictions about subjects’choices.For instance,consider Choice3,described in the last section,together with Choice4.
4.Invest each of your60points in either Asset G or Asset H,as given below.
[1][2][3][4]
G1*******
H12181824
In both Choices3and4,any portfolio leads to a payo¤of12in state1and24in state4.In Problem 3,investing E in Asset E and1 E in Asset F gives payo¤s24 E+12(1 E)=12+12 E in state2and12 E+24(1 E)=24 12 E in state3.Since states2and3occur with the same probability,any risk averter prefers to the expected value of her money payo¤across the two states,18,in each state to any lottery.This can be achieved by choosing E=1
.Each of Choices
2
3through8has the feature that in two states the decision maker can do nothing to hedge her risk while in the remaining two she can perfectly hedge her risk just as in this case.Since each twinned 11Formally,for every lottery L,the decision maker weakly prefers the degenerate lottery paying E[L]with certainty to L.
In Choices1and2,risk aversion alone does not su￠ce to identify subjects’optimal choice.In this case,adding Assumption D su￠ces to make a unique prediction.
Assumption D.The decision maker is arbitrarily close to risk neutral.12
Observation2.Under Assumptions A-D,for each n,m (W)minimises the variance in portfolio earnings in each choice W n.
Eyster(2010)contains a more general theorem along these lines and its proof.13Intuitively,uniform arbitrary closeness to risk neutrality implies arbitrary closeness to constant-absolute risk aversion (CARA),which collapses to lexicographic mean-variance preferences as the decision maker ap-proaches risk neutrality.
An alternative hypothesis is that our decision maker neglects the correlation in assets’returns, treating each asset as independent.This violates Assumption A because such a decision maker does not maximise rational preferences over state-contingent payo¤s.We model a hexed decision maker as one who maximises rational preferences over a transformed,…ctitious asset space where assets are uncorrelated.To construct this…ctitious asset space,let
b =f1:1;1:2;1:3;1:4;2:1;2:2;2:3;2:4;3:1;3:2;3:3;3:4;4:1;4:2;4:3;4:4g
and suppose that in state a:b,the…rst asset pays X1(a)and the second X2(b).Whereas X1;X2: !R,these new assets b X1;b X2:b !R.Most importantly,whatever the correlation between X1and X2,b X1is independent of b X2by construction:whatever the payout of b X1,b X2pays out b X2(1);b X2(2);b X2(3)and b X2(4)with equal probability.
Let c W n=c W(b X n1;b X n2;C n)be the set of wealth lotteries feasible for the investor when the assets are b X n1;b X n2given the constraints C n.Let c W be the set of lotteries over wealth levels across the entire asset with the transformed assets(b X1;b X2).
12Formally,consider a sequence of expected utility maximisers with C2,concave Bernouilli utility functions(u n)
n2N and their associated Arrow-Pratt coe￠cient-of-absolute-risk aversion functions(r n)n2N.Let n denote the portfolio that maximises the expectation of u n.When(r n)n2N converges uniformly to zero,and the decision maker chooses a portfolio that belongs to the limit of( n2N),we say that the decision maker is arbitrarily close to risk neutral.
13Formally,( n)
converges to the portfolio that maximises lexicographic preferences in the moments,with n2N
alternating sign.In the context of our experiment where assets have equal means and there is at most one portfolio with any given variance,this coincides with minimising portfolio variance.
Assumption E.The decision maker makes choices that maximise a preference relation over c W that is complete and transitive.Moreover,for each n;m (c W n)is a singleton set. When X1and X2are uncorrelated,W is equivalent to c W,which implies that rational and hexed investors make the same choices.
Observation3.When X1is independent of X2,Assumptions A and B produce the same choices as do Assumptions B and E.
When assets are correlated,Assumptions B,C,D,and E su￠ce to make a unique prediction for exactly the reasons that Assumptions A,B,C,and D do above.In particular,in each choice the investor allocates her portfolio to minimise her perceived portfolio variance in the original problem, which equals the actual variance in the hatted problem.
When X1is positively correlated with X2,a hexed investor underestimates the variance in her portfolio.Let 12be the covariance between X1and X2.
Proposition1.When 12>0,Assumptions B,C,D,and E give a(weakly)more equal split of portfolio across assets than Assumptions A,B,C,and D.When 12<0,Assumptions B,C,D,and E give a(weakly)more unequal split of portfolio across assets than Assumptions A,B,C,and D.
When assets are positively correlated,a hexed investor overestimates diversi…cation bene…t of moving his portfolio from a low-variance asset to a high variance one,which leads her to take a more highly diversi…ed portfolio than a rational decision maker,a form of false diversi…cation e¤ect.When assets are negatively correlated,a hexer underestimates the diversi…cation bene…t of moving his portfolio from a low-variance asset to a high variance one,which leads her to take a less diversi…ed portfolio than a rational decision maker,a form of hedging neglect.
Benartzi and Thaler(2001)have suggested that investors facing a menu of n di¤erent mutual funds often use the simple heuristic of investing the fraction1
of her portfolio in each fund.In
n
the context of our experiment,investing half the portfolio in each of the two choices would lead an investor to make di¤erent portfolios in paired problems.
We have seen that Assumptions A-D as well as B-E imply lexicographic mean-variance prefer-ences,albeit with di¤erent perceived covariance matrices.In settings where people are close to risk
neutral such that they maximise mean-variance preferences,we can consider a simple parametric alternative that incorportates diminishing sensitivity to variance of any kind.While much research has demonstrated people’s aversion to risk—even at its smallest scale—that does not imply that they dislike increased risk as much as the independence axiom implies.Like Fechner and Weber fa-mously demonstrated with physical stimuli,people’s perception of or distaste for risk may increase slower than linearly with variance.We capture this possibility by positing that people maximise mean-variance preferences using a transformed covariance matrix as follows:
V=0@ 21 l k sgn( 12)j 12j l k sgn( 12)j 12j l 22 l 1A;
where k;l2[0;1].
Using this transformed covariance matrix allows us to nest our several hypotheses about subject behaviour.
Rat:Subjects satisfy Assumptions A-D,i.e.,k=l=1.
Hex:Subjects satisfy Assumptions B-E,i.e.,k=0and l=1.
1/n:Subjects use the1
N
heuristic,i.e.,l=0.
Finally,when k=0and l<1,behaviour conforms to a hexer who also exhibits diminishing sensitivity to variance.
4Evidence of Correlation Neglect:Raw Data and Analysis of Means
This section shows the raw data of the experiment and tests whether Assumptions A and B are satis…ed.The observed data patterns show evidence of both directions of correlation neglect(Hex and1/n).We will examine these patterns further with a richer set of estimations in Section5.
Figures1to4depict all choices by the participants.Each…gure has the146participants’choices in two paired problems,f n 1;n g for n even,indicated by the small markers.(The large markers
are the predictions of the di¤erent behavioral assumptions,explained below.)The horizontal axis
in each…gure measures a participant’s investment in the…rst asset, n1,of the even-numbered choice problem—the one with correlated assets.The vertical axis uses the investor’s choice in the
is the twinned problem n 1and expresses this choice in the action space of problem n:b n 1
1
share that the participant would need to invest in problem n in order to hold the same distribution
in problem n 1.By construction of the of earnings that she chooses via her actual choice n 1
1
twinned-problem design,this share is uniquely determined for each feasible investment in problem
n 1.Formally,for each n 1
12C n 1there exists a unique b n 112C n such that
8x2R:Pr(W n 1(!)=x j n 11)=Pr(W n(!)=x j n1)
and thus each risk that the investor takes on in choice problem n 1can be generated by a possible action in choice problem n.By Observation1,the null hypothesis that participants correctly perceive covariance gives a simple prediction for the…gure:under Assumptions A and B(which
and n1are identical implicitly prescribe that correlation is fully understood),the two shares b n 1
1
and thus the data lie on the45-degree line.The data are also summarized in terms of mean and standard deviation in Table2,second column.14
14For Choice6,the prediction of the1/n model violates the a lower bound of40points for the…rst-listed asset. In the…gures,in Table2and all of the subsequent analysis,we therefore set the1/n prediction to40points.
Figure1:Distribution of investments in Choices1and2.Horizontal axis: 21.Vertical axis:b 11. Large{red,green,blue,x}markers represent{Rat,Hex,1/n,parametric}predictions.
Figure2:Distribution of investments in Choices3and4.Horizontal axis: 41.Vertical axis:b 31. Large{red,green,blue,x}markers represent{Rat,Hex,1/n,parametric}predictions.
Figure3:Distribution of investments in Choices5and6.Horizontal axis: 61.Vertical axis:b 51.
predictions.
Large{red,green,blue,x}markers represent{Rat,Hex,1/n,parametric}
Figure4:Distribution of investments in Choices7and8.Horizontal axis: 81.Vertical axis:b 71. Large{red,green,blue,x}markers represent{Rat,Hex,1/n,parametric}predictions.
n Data mean(std.dev.)Rat n Hex n(1/N)n
10:254(:113)0:40:40:25
20:381(:138)0:40:3330:5
30:126(:306)000
40:444(:234)00:3330:5
50:839(:053)0:8330:8330:833
60:778(:127)0:8330:9290:667
70:328(:225)0:250:250:25
80:545(:267)0:250:3570:5
Table2:Proportions of investment in…rst-listed asset.
Data and benchmark model predictions,separated
by choice problem
The…gures show strong and systematic deviations from the hypothesis that Assumptions A and B hold.Not only do only very few observations lie on the45-degree lines but there are also patterns in the data that cannot be driven by unsystematic disturbances.Corresponding statistical tests soundly reject the prediction of Assumptions A and B that behavior is unchanged within pairs of twinned problems.For each of the four pairs,Wilcoxon matched-pairs sign-rank tests reject the hypothesis of identical distributions of portfolios at a signi…cance level of0.001.This shows that the use of correlated versus uncorrelated asset returns has a signi…cant impact on choices.
As another measure of the accuracy of the null hypothesis of correct perception of correlation we ask how many participants make portfolio choices that are exactly identical between twinned problems.The answer is contained in Table3showing that the large majority of the participants never or almost never choose portfolios that are identical between twinned problems.97.2%of the participants choose a di¤erent set of portfolios between two twinned problems weakly more often than they choose the same portfolio.60.9%never choose the same portfolio twice.
#Freq.%Cum.%
08960:960:9
13725:386:2
21611:097:2
332:199:3
410:7100:0
Total146100:0
Table3:Frequencies of choosing identical portfolios in twinned
choice problems(out of4)
Of course,the deviations may be in?uenced by di¤erent sources and we must to be careful not to interpret random deviations from the45-degree line as evidence of correlation neglect.We therefore turn to statistical estimations that allow concluding that the deviations from the prediction of Assumptions A and B are indeed systematic in the way that we hypothesize.The…rst such estimation is an OLS regressions that summarizes the statistical connection between the data and extreme predictions Hex and(1/n).These predictions are indicated in the…gures—the green and blue marker,respectively—as well as in columns4and5of Table2.As a benchmark prediction, the…gures and Table2also contain the“rational”prediction Rat of a risk averter who understands the correlation structure(Assumption C).15The dependent variable in the regression is the vertical distance of the data points from the45-degree line in the…gure.The explanatory variables are, analogously,the vertical distances of the two predictions from the45-degree line,d Hex n 1 Hex n and d1/n n 1 (1/n)n,respectively.16
15Where needed to make a unique prediction,the degree of risk aversion is also assumed to be close to zero,as explained in Section3(Assumption D).
16The estimated model is
b n 11 n1= 2(d Hex n 1 Hex n)+ 3(d1/n n 1 (1/n)n)+ it
Since the model is di¤erenced we did not include a constant term.
b b b
d Hex n 1 Hex n0:85(:08) 0:18(:08)
d1/n n 1 (1/n)n 0:63(:05) 0:56(:04)
R20:210:350:36
#of obs.584584584
Table4:OLS regression of deviation from45-degree line on the predicted deviation
by two models Hex and1/n.Standard deviations in parentheses,clustered by subject.
The table shows that both extreme models of correlation neglect are predictive of the data means,as their coe￠cients are statistically signi…cant both individually(in univariate regressions) and when controlling for the respective other prediction.The1/n model has the larger predictive power.The regressions results are also consistent with the data feature that the deviations from the45-degree line are strong especially when Hex and1/n move together.This can be seen by inspection of the models’predictions in Table2:in Choices4and8,the predictions of Hex and1/n are both on the same side of the benchmark model Rat.In these two problems the data means also deviate from Rat in the same direction.More generally,the deviations in these problems from the 45-degree line are in the same direction as the deviations of the two stylized models.In contrast, in the two other choice problems,the two stylized models move in opposite directions away from Rat,and the data also conform much more to latter model’s prediction.
To see the partial predictive power of all three models,we now consider only correlated tasks (where the three model predictions di¤er),and regress on all three model predictions including Rat.