Bracketing the Portfolio Problem

By substituting equation 6.17 into equation 6.16, we can rewrite the broad bracketing optimization problem as

If we ignore the kink in the value function (in other words, use a single smooth function to represent gains and losses), the solution can be written as

Where v″ is the slope of the marginal utility function (negative over gains, positive over losses). Here, σij = E zi − μizj − μj is the variance of investment i when i = j and the covariance of investment i and j when they are unequal. The covariance is a measure of how random variables are related. A positive covariance indicates that the random variables are positively related (when one is higher, the other is likely to be higher, like height and weight), and a negative covariance indicates a negative relationship, like speed and weight. The interested reader can turn to the Advanced Concept box, Bracketing the Portfolio Problem, to read the derivation of this solution.

If instead, we suppose that decisions are narrowly bracketed, then substituting equation 6.17 into equation 6.18 yields

which has a solution given by

Notably, this is very similar to equation 6.20, except that all of the covariance terms are omitted. This means that the investor will fail to recognize when the return of investment options are related, allowing diversification to reduce the risk. In particular, negatively correlated investments allow one to create portfolios that should reduce the variance of the outcome. If risks are negatively correlated, equation 6.20 suggests that a broadbracket decision maker will increase her level of investment in that given asset. Alternatively, if risks are positively correlated, this too will be ignored, leading the investor to take a much larger risk than she might otherwise by overinvesting in stocks that all move together. Loss aversion can amplify this effect.

More than the Sum of Its Parts

By nature, narrow bracketing leaves the person worse off than broad bracketing if one decision will affect the choices available in other decisions. An example of how this might happen is if there are properties of groups of decisions that are not apparent or present in single decisions. For example, it is not possible to diversify a portfolio of investments with only a single stock. To diversify, one must be able to make assessments about the covariance of returns among stocks, a property that would not be assessed when examining a single stock by itself. Whenever an attractive property is present in groups of choices but not individual choices, narrow bracketing leads the decision maker to make suboptimal choices.

The Utility Function and Risk Aversion

Expected utility theory supposes that all risk behavior is due to changes in the marginal utility of wealth or the concavity or convexity of the utility function. Whenever the function is concave, the decision maker behaves so as to avoid risks. Whenever it is convex, the decision maker seeks risk. Typically, economists assume that people are risk averse, displaying diminishing marginal utility of wealth. Matthew Rabin made an important observation regarding the curvature of utility of wealth functions. One of the basic principles upon which calculus is based is that any continuous and smooth function is approximately a line over very small ranges of the input variable. Thus, even though the sine function, sin x, wiggles up and down when plotted over 0, 2π, it is closely approximated by a straight line if we plot it over 0, 0.0001. Thus, a standard concave utility function is also approximated by a straight line over small changes in wealth. Utility functions that are straight lines display risk neutrality, leading to choices that maximize expected payouts without regard for variance of the gamble. Thus, when gambles involve very small amounts of money, people who maximize expected utility should behave approximately as if they are risk neutral.

Matthew Rabin turns this principle on its head. If people are risk averse over small gambles, this tells us something of how concave their utility function must be for somewhat larger gambles. More explicitly, suppose someone was offered a gamble that yielded x1 < 0 with 0.50 probability and x2 > 0 with 0.50 probability. Further, suppose that the gamble had a positive expected value but that we observed the person reject the gamble. This possibility is depicted in Figure 6.4. In this case, we know that u0 > Eux. Then we know that u0 > 0.5ux1 + 0.5ux2. This means that the utility function must be at least as concave as the bolded line segments in Figure 6.4. Knowing this, we can use these line segments to find other gambles for larger amounts that must also be turned down.

Therefore, suppose that for all w, Uw is strictly increasing and weakly concave. Suppose there exists g > l > 0, such that for all w, the person would reject a 0.50 probability of receiving g and a 0.50 probability of losing l. Then, the person would also turn down a bet with 0.50 probability of gaining mg and a 0.50 probability of losing 2kl, where k is any positive integer, and m < mk, where

that participants were instructed to treat the decisions concurrently. Thus, choosing A and D is choosing a compound gamble A and D over every other combination of gamble. The compound gamble A and D yields a 75 percent chance of losing $760 and a 25 percent chance of gaining $240.

Consider the alternative choice (chosen by less than 13 percent of participants) of the compound gamble B and C. This gamble yields a 75 percent chance to lose $750, and a 25 percent chance to win $250. The compound gamble B and C clearly dominates A and D, yielding more money for the same probability. Clearly, if the gambles had been presented as combined, people would have chosen B and C instead of A and D.

Narrow bracketing was perhaps induced by the difficulty in calculating the compound gambles from the separate gambles, thus leading to a suboptimal decision. In this case, the lack of transparency in the choice led to narrow bracketing.

EXAMPLE 6.4 A Taste for Diversity—or Not

Suppose it was Halloween and you were trick-or-treating. You visit three houses that happen to offer exactly the same three types of candy bars. At the first house, you are allowed to take one candy bar, and you choose your favorite. At the second house, you are also allowed to choose only one, and you decide to take your favorite. At the third house, you are allowed to choose two, and you decide to choose one of your favorite and one of the others. Daniel Read and George Loewenstein set up three such houses on Halloween and observed the choices by individual trick-or-treaters. At the houses allowing only one choice, people tended to choose the same candy. At the house allowing children to take two, trick-or-treaters overwhelmingly chose two different treats.

A majority of the time, we make decisions in a sequence, one at a time. For example, someone might purchase her lunch each day, deciding each day what to eat independently. Sometimes, however, we make a single decision that determines our consumption over several periods. For example, someone who brings a bag lunch might buy the materials for her lunches only once a week or even less often. In this case, it is natural to think that the sequential choices will be narrowly bracketed and the single choice will be broadly bracketed.

Itamar Simonson asked students in his class to choose among six possible snacks: peanuts, tortilla chips, milk chocolate with almonds, a Snickers bar, Oreo cookies, or crackers and cheese. They were to receive these snacks once a week at the end of class for three consecutive weeks. Students in some of his classes were asked to choose in class on the day they would receive the snacks. Students in other classes were asked to choose before the first distribution of snacks what they would receive all three weeks. A total of 362 students participated in either the sequential or the joint choice conditions. In the sequential choice condition, 9 percent of students chose a different snack each week. In the joint choice condition, 45 percent chose a different snack each week. Read and Loewenstein replicated this experiment, but they asked the students in the simultaneous choice condition if they wanted to change their mind on the days they were to receive their second and third snacks. A little less than half of the participants in these sessions wanted to revise their choices, and those who wanted to change desired less diversity in their selection.