Publisher: Now Publishers, Inc. Document Type: Article. Length: 3, words. Lexile Measure: L. Translate Article. Set Interface Language. Decrease font size. Increase font size. Display options. Default More Most. Back to Default Settings Done. It is well known that for a given stock return distribution and for a given riskless interest rate, the investment proportion in the stock decreases with an increase in the degree of risk aversion.
The explanation for the fact that the optimal proportion is almost constant, even when revisions are not possible, is as follows.
First, note that the return distributions in Task 2 are just the distributions obtained by investing in Task 1 for two periods, assuming i. In this case, his rate of return in the first period is 0. Thus, after two periods the portfolio grows by 1.
This is a little lower than the return of As the table reveals, the return distributions are not exactly identical, but they are very close.
As the return distributions are very close, this implies that the optimal asset allocation is very similar whether portfolio revisions are possible or not. Indeed, Table 3 reveals that the optimal investment proportion in the stock with no revisions is almost constant across the three tasks. This result is quite general and does not hinge on the specific distributions given in our experiment: it holds for different return distributions, and for horizons much longer than the 3-period horizon employed in our experiment.
The figure in S3 Appendix shows the optimal asset allocation with and without revisions, for alphas in the range 0. As most empirical and experimental evidence suggest that the risk aversion parameter is around 1—2 see [ 37 ] and references within , the figure reveals that in the relevant range of risk aversion the optimal weight in the risky asset with and without revisions is almost identical.
Thus, the optimal asset allocation of a CRRA investor is almost the same whether he invests for 1 year or for 10 years, even if portfolio revisions are not allowed. In the context of our experiment, this result yields a clear-cut prediction for the asset allocation behavior predicted for CRRA individuals: the investment proportion in the stock should be constant or almost constant across all three tasks.
The experiment was conducted in a classroom setting with pen and paper. It is composed of three main choice tasks. In each task there is a risk-free asset and a risky stock, with a given return distribution. In each task the subject is asked to allocate his investment between these two assets.
The complete questionnaire is provided in S4 Appendix. The instructions that appear before each one of the three tasks are:. The possible outcomes at the end of the investment period are as given below. The three tasks are given in Table 1. Note that Task 2 and Task 3 are in fact the 2-period and 3-period versions of Task 1, under the assumption of i.
Similarly, the risk-free rate in Task 2 is the 2-period return of the risk-free asset of Task 1. In the same manner, Task 3 is the 3-period version of Task 1. These three tasks allow us to investigate asset allocation choices, and their dependence on the horizon. The tasks are not framed in terms of a 1-period, 2-period and 3-period choices, but rather as three different stand-alone tasks.
Specifically, in Tasks 2 and 3 the subjects have no opportunity to revise their allocations after each period as in [ 5 , 6 ]. This is important, because as shown in [ 33 ], subjects are strongly affected by the information presented to them: if the actual investment horizon is 3 periods, but the information provided to the subjects is for the 1-period returns, subjects tend to treat the investment as a 1-period investment.
Thus, to obtain preferences for a 3-period horizon, the 3-period return distributions should be presented, which is the setting of our experiment, as well as in the experiments in [ 5 , 6 ].
This approach also avoids possible biases that investors may have in translating 1-period returns into longer-period returns [ 38 ]. In addition, the cash flows of the various choices are obtained immediately, hence there is no need to discount the returns and the subjects do not face anxiety corresponding to immediate and far-away risks. The experimental setup employed corresponds to the framework in which the theoretical predictions the optimal asset allocation of PT and CRRA investors are derived—the optimal allocation is based on the t-period return distributions, which are the distributions presented to the subjects.
The experiment is designed to resemble a situation of large-stake investments, such as investment of life-long saving for retirement. This has obvious pros and cons. The advantage of the large-stake setting is that it allows us to employ large potential gains and losses, which resemble decisions about life-long savings involving payoffs in the magnitude of hundreds of thousands of dollars.
In addition, this may make subjects perceive their decisions as more substantial, relative to the typical experimental setup, where the magnitude of outcomes are only a few dollars.
As [ 39 ] argues, individuals may behave very differently when large stakes are involved, compared to situations with small or moderate stakes. The drawback of the hypothetical large-stake setup is that one may argue that subjects are not sufficiently incentivized when payoffs are hypothetical.
The control task is the last task of the experiment, Task 4 see S4 Appendix. In this control task, the subject is asked to choose between two alternative investments, where one alternative investment G dominates the other investment F by First-order Stochastic Dominance FSD. This means that any rational individual any expected utility maximizer as well as any cumulative PT expected value maximizer should choose the dominating investment G for more information on First-order Stochastic Dominance, see [ 40 ].
This provides a strong indication that the vast majority of subjects did give careful consideration to the experimental tasks, and that they understood the instructions. We report below only results for those subjects who answered the control task correctly, i.
The experiment was conducted with a hard-copy questionnaire filled out by pen. Written instructions were given in the questionnaire, as well as provided verbally. There was no show-up fee. The original questionnaire composed of the four tasks discussed above is provided in S4 Appendix. We have four groups of subjects, from three different countries:. There are a total of subjects. Another two subjects failed to complete the entire questionnaire.
Thus, we are left with subjects who completed the questionnaire and answered the control task correctly. The choices across the groups of subjects from various cultures and with different experience in the capital market are quite similar, which increases the reliability of our results.
Below we report the results combined across all four groups. S5 Appendix provides the results for each subject group separately, and the individual level data, including group affiliation. We find the following average investment proportions in the stock in the three tasks across all subjects : where the sub-indices refer to the task number.
The average proportion increases somewhat from Task 1 to Task 2, and slightly decreases from Task 2 to Task 3. While the increase in the average proportion from Task 1 to Task 2 is statistically significant matched-pair t-value 2. It is interesting to note that [ 6 ] observe exactly the same pattern for the average proportions: a significant increase in the average proportion allocated to the stock from the short horizon to the medium horizon, and a slight decrease from the medium horizon to the long horizon.
The results are compared in more detail in the next section. In general, the average proportions do not change much across tasks. These average proportions may be consistent with most subjects having CRRA preferences, where the small variation in the proportions across tasks is due to a small group of subjects with preferences different than CRRA.
However, these average proportions may also be consistent with many other possible models, including a scenario where some subjects have PT preferences. Thus, in order to reach more definitive conclusions about preferences one cannot rely only on the average proportions, and we therefore turn to analyze the results at the individual level, which is our main experimental focus.
The preceding analysis shows that CRRA preferences predict investment proportions that are practically constant across all tasks, i. Table 5 summarizes the choice patterns predicted by the different models, and the number of subjects with investment allocation choices conforming to each pattern.
Pattern 1, where the investment proportions are exactly equal across all three tasks, is consistent with CRRA preferences.
While there are slight differences in the theoretically optimal proportions across tasks because there are no revisions, see Table 3 , these differences are in the order of 0. As reported in the table, 59 subjects However, we do not consider this choice pattern as supporting PT for two reasons. This does not seem reasonable. Note that the 59 subjects in pattern 1 are a subgroup of the 80 subjects in pattern 2, i.
We turn now to the choice patterns predicted by PT. Pattern 3 with weights 0, 0, 0 , pattern 4 with weights 0, 0, 1 , and pattern 5 with weights 0, 1, 1 of Table 5 correspond to these three cases. Pattern 3 is predicted for all PT investors with a reference point at the future value of wealth.
In aggregate, only 6 subjects 3. Thus, even if we allow for a wide range of PT parameters, there is very little support for PT with a reference point at the future value of wealth. In order to allow for some noise, or bounded-rationality errors as we allow for approximate CRRA in pattern 2 , we allow for deviations from these values and define pattern 6 as: 0. Notice that PT subjects with a reference point at current wealth should follow this pattern whether they employ the objective probabilities or the CPT decision weights see the table in S2 Appendix.
Only 5 subjects 2. We should note that some cases of pattern 6 may also be consistent and are a special case of patterns 1 and 2. Thus, there is some overlap between the choice patterns, and such a subject would be included in patterns 1, 2, and 6. Of the 5 subjects following pattern 6, four are also consistent with approximate CRRA, i.
The other subjects who do not fall into any of the categories in Table 5 , display a variety of choice patterns. The complete results of individual-level choices are provided in S5 Appendix. Yet, the most dominant choice pattern is pattern 1, where all proportions are exactly identical.
We find evidence supporting CRRA preferences for a relatively large proportion of the subjects. This finding is consistent with previous empirical and experimental findings [ 44 ]: concludes that the relative risk aversion is almost constant, implying CRRA [ 45 ]. This may seem surprising, in light of the vast experimental support for PT.
How can this result be reconciled with the extensive literature showing support for PT? We believe that the most plausible explanation is that individuals behave differently when faced with small or modest bets as in most experimental studies supporting PT and when faced with large stakes, as in the present experimental setup.
The importance of the scale of the amount at stake on choices is best summarized by Rabin [ 39 ] who writes:. Expected utility theory may well be a useful model of the taste for very large scale insurance. Despite its usefulness , however , there are reasons why it is important for economists to recognize how miscalibrated expected utility is as an explanation of modest-scale risk aversion.
Another possible explanation is that much of the experimental support for PT comes from experiments with either positive or negative prospects, rather than with mixed prospects. Indeed [ 50 ], find support for PT when employing prospects in the positive domain or in the negative domain separately, but find that most subjects contradict PT when mixed prospects are employed.
Needless to say, most real-life decisions involve both potential gains and potential losses, and this is also the framework of the present study. It is also possible that PT is approximately consistent with aggregate results, while it does not provide a good description of choice at the individual level.
This is what [ 51 ] find. Both of these studies experimentally investigate the influence of the investment horizon on the average asset allocation. While there are some important differences between these two studies, they both find that when subjects are presented with longer horizons, on average they tend to increase their investment proportions in the stock.
This is consistent with myopic loss aversion—when individuals with PT preferences observe short-horizon returns they are confronted with a large probability of a loss, and therefore they tend to avoid the stock; as the horizon increases, the probability of a loss decreases, and individuals tend to increase their investment proportions in the stock [ 4 , 33 ].
At the aggregate level, this is similar to what we observe here: the average proportion in the stock increases from However, as discussed above, these aggregate results may also be consistent with most subjects having CRRA preferences, and the relatively small increase in the average investment proportion being driven by a potentially small group of investors with non-CRRA preferences.
This implies that individual behavior may be very different than the averages reported. Thus, only by looking at decisions at the individual level we can reach more definitive conclusions regarding preferences.
In the present study we examine not only whether the optimal investment allocation to the stock increases with the horizon, but also the exact way in which it changes. Thus, the two main distinctions between GP and TTKS and the current study is that we analyze choices at the individual level , and that we have very clear predictions about the exact asset allocation values implied by PT.
One key difference between GP and TTKS is that GP investigate asset allocation between a risky stock and a risk-free bond as in the present study , while TTKS study the asset allocation between a risky stock and a risky bond which is less volatile than the stock, but still involves risk. This has an important implication for the optimal asset allocation of a PT investor: when the bond is risk-free, a dramatic jump in the asset allocation is theoretically predicted.
When the bond is risky as in TTKS, and also in [ 4 ] the optimal asset allocation may change more gradually. This is an advantage of the GP setup, and the setup employed here: it yields very clear and testable theoretical predictions. It is well-known that this behavior is consistent with CRRA preferences when revisions are allowed.
In the preceding analysis we show that even when revisions are not allowed, as in the final decision in the TTKS setup, and as in our experiment, CRRA implies almost the same behavior. This study employs the setting of investments for different horizons to theoretically analyze the predictions of two main contending preferences: Expected utility with Constant Relative Risk Aversion CRRA preferences, and Prospect Theory PT preferences with and without decision-weights, and with various alternative reference points.
The special feature of this setup is that it yields very clear predictions for the individual-level behavior of CRRA and PT preferences. We show that for CRRA preferences, regardless of the risk aversion parameter, the asset allocation should remain virtually constant at different horizons, even when revisions are not allowed. It is important to note that this jump does not require a very long horizon, it already occurs at shifts from a one-year horizon to a two-year or a three-year horizon.
These theoretical results are quite general: they do not depend on the specific return distributions, the specific PT parameters and the functional form of the value function, and whether objective probabilities or CPT decision-weights are employed. It is important to emphasize that the jump in the asset allocation to the risky asset is a fundamental characteristic of PT and loss-aversion, and it occurs even if the horizon does not change, e.
Generally, we do not obtain such a jump in the expected utility framework. Like in [ 5 ] and in [ 6 ], in our experiment each subject faced three choices, not knowing that they correspond to various horizons or to a combination of several lotteries.
Thus, by the selected choices we can investigate the horizon effect on asset allocation, neutralizing the effect of the timing of cash flows see, for example, [ 52 , 53 ]. This very weak support for PT is surprising, given the large number of experimental studies supporting PT. We suspect that this difference may be partly due to the fact that many of these studies investigate the domain of gains and the domain of losses separately, while we employ mixed prospects.
More importantly, it is possible that loss aversion plays an important role in setups where gains and potential losses are modest e. Our findings have several key implications. The first is about the equity premium puzzle [ 37 ].
Benartzi and Thaler [ 4 ] suggest that this puzzle may be solved by myopic loss aversion: PT preferences and a short horizon or a high evaluation frequency. We must conclude, like [ 54 ], that the equity premium puzzle is yet to be solved. The second implication of our results is about the important role of heterogeneity. We find that subjects are very different not only along one dimension such as the degree of risk aversion , but rather they follow completely different patterns of the asset allocations as a function of the horizon some monotonically increasing the allocation to the stock with the horizon, some monotonically decreasing it, and some changing it non-monotonically.
As [ 17 — 22 ] and others have argued, one should be very careful about reaching conclusions about the aggregate behavior of very heterogeneous individuals. At the same time, though, there is evidence that human beings try to avoid risk in both physical and financial pursuits. The same person who puts his life at risk climbing mountains may refuse to drive a car without his seat belt on or to invest in stocks, because he considers them to be too risky.
As we will see in the next chapter, some people are risk takers on small bets but become more risk averse on bets with larger economic consequences, and risk-taking behavior can change as people age, become wealthier and have families. In general, understanding what risk is and how we deal with it is the first step to effectively managing that risk. While we can talk intuitively about risk and how human beings react to it, economists have used utility functions to capture how we react to at least economic risk.
Individuals, they argue, make choices to maximize not wealth but expected utility. We can disagree with some of the assumptions underlying this view of risk, but it is as good a staring point as any for the analysis of risk. In this section, we will begin by presenting the origins of expected utility theory in a famous experiment and then consider possible special cases and issues that arise out of the theory. Consider a simple experiment. I will flip a coin once and will pay you a dollar if the coin came up tails on the first flip; the experiment will stop if it came up heads.
If you win the dollar on the first flip, though, you will be offered a second flip where you could double your winnings if the coin came up tails again. The game will thus continue, with the prize doubling at each stage, until you come up heads.
How much would you be willing to pay to partake in this gamble? This is the experiment that Nicholas Bernoulli proposed almost three hundred years ago, and he did so for a reason. This gamble, called the St.
Petersburg Paradox , has an expected value of infinity but most of us would pay only a few dollars to play this game. It was to resolve this paradox that his cousin, Daniel Bernoulli, proposed the following distinction between price and utility: [1].
The price of the item is dependent only on the thing itself and is equal for everyone; the utility, however, is dependent on the particular circumstances of the person making the estimate. Bernoulli had two insights that continue to animate how we think about risk today. First, he noted that the value attached to this gamble would vary across individuals, with some individuals willing to pay more than others, with the difference a function of their risk aversion.
He was making an argument that the marginal utility of wealth decreases as wealth increases, a view that is at the core of most conventional economic theory today. Technically, diminishing marginal utility implies that utility increases as wealth increases and at a declining rate. Petersburg paradox. While the argument for diminishing marginal utility seems eminently reasonable, it is possible that utility could increase in lock step with wealth constant marginal utility for some investors or even increase at an increasing rate increasing marginal utility for others.
The classic risk lover, used to illustrate bromides about the evils of gambling and speculation, would fall into the latter category. The relationship between utility and wealth lies at the heart of whether we should manage risk, and if so, how. After all, in a world of risk neutral individuals, there would be little demand for insurance, in particular, and risk hedging, in general.
It is precisely because investors are risk averse that they care about risk, and the choices they make will reflect their risk aversion. In the bets presented by Bernoulli and others, success and failure were equally likely though the outcomes varied, a reasonable assumption for a coin flip but not one that applies generally across all gambles.
They argued that the expected utility to individuals from a lottery can be specified in terms of both outcomes and the probabilities of those outcomes, and that individuals pick.
The Von-Neumann-Morgenstern arguments for utility are based upon what they called the basic axioms of choice. The first of these axioms, titled comparability or completeness , requires that the alternative gambles or choices be comparable and that individuals be able to specify their preferences for each one.
The third, referred to as the independence axiom specifies that the outcomes in each lottery or gamble are independent of each other. This is perhaps the most important and the most controversial of the choice axioms. Essentially, we are assuming that the preference between two lotteries will be unaffected, if they are combined in the same way with a third lottery.
In other words, if we prefer lottery A to lottery B, we are assuming that combining both lotteries with a third lottery C will not alter our preferences.
The fourth axiom, measurability , requires that the probability of different outcomes within each gamble be measurable with a probability. What these axioms allowed Von Neumann and Morgenstern to do was to derive expected utility functions for gambles that were linear functions of the probabilities of the expected utility of the individual outcomes.
Extending this approach, we can estimate the expected utility of any gamble, as long as we can specify the potential outcomes and the probabilities of each one.
As we will see later in this chapter, it is disagreements about the appropriateness of these axioms that have animated the discussion of risk aversion for the last few decades. The importance of what Von Neumann and Morgenstern did in advancing our understanding and analysis of risk cannot be under estimated. By extending the discussion from whether an individual should accept a gamble or not to how he or she should choose between different gambles, they laid the foundations for modern portfolio theory and risk management.
After all, investors have to choose between risky asset classes stocks versus real estate and assets within each risk class Google versus Coca Cola and the Von Neumann-Morgenstern approach allows for such choices. In the context of risk management, the expected utility proposition has allowed us to not only develop a theory of how individuals and businesses should deal with risk, but also to follow up by measuring the payoff to risk management.
Gambling, whether on long shots on the horse track or card tables at the casinos, cannot be easily reconciled with a world of risk averse individuals, such as those described by Bernoulli. Put another way, if the St. Petersburg Paradox can be explained by individuals being risk averse, those same individuals create another paradox when they go out and bet on horses at the track or play at the card table since they are giving up certain amounts of money for gambles with expected values that are lower in value.
Economists have tried to explain away gambling behavior with a variety of stories. The first argument is that it is a subset of strange human beings who gamble and that that they cannot be considered rational. This small risk-loving group, it is argued, will only become smaller over time, as they are parted from their money. While the story allows us to separate ourselves from this unexplainable behavior, it clearly loses its resonance when the vast majority of individuals indulge in gambling, as the evidence suggests that they do, at least sometimes.
The second argument is that an individual may be risk averse over some segments of wealth, become risk loving over other and revert back to being risk averse again. Friedman and Savage, for instance, argued that individuals can be risk-loving and risk-averse at the same time, over different choices and for different segments of wealth: In effect, it is not irrational for an individual to buy insurance against certain types of risk on any given day and to go to the race track on the same day.
Why we would go through bouts of such pronounced risk loving behavior over some segments of wealth, while being risk averse at others, is not addressed. The third argument is that gambling cannot be compared to other wealth seeking behavior because individuals enjoy gambling for its own sake and that they are willing to accept the loss in wealth for the excitement that comes from rolling the dice.
Here again, we have to give pause. Why would individuals not feel the same excitement when buying stock in a risky company or bonds in a distressed firm? If they do, should the utility of a risky investment always be written as a function of both the wealth change it creates and the excitement quotient? The final and most plausible argument is grounded in behavioral quirks that seem to be systematic. To provide one example, individuals seem to routinely over estimate their own skills and the probabilities of success when playing risky games.
As a consequence, gambles with negative expected values can be perceived wrongly to have positive expected value. Thus, gambling is less a manifestation of risk loving than it is of over confidence. We will return to this topic in more detail later in this chapter and the next one. While much of the discussion about this topic has been restricted to individuals gambling at casinos and race tracks, it clearly has relevance to risk management.
Rather than going through intellectual contortions trying to explain such phenomena in rational terms, we should accept the reality that such behavior is neither new nor unexpected in a world where some individuals, for whatever reason, are pre-disposed to risk seeking. Which one would you pick? With conventional expected utility theory, where investors are risk averse and the utility function is concave, the answer is clear. If you would reject the first gamble, you should reject the second one as well.
Samuelson argued that rejecting the individual bet while accepting the aggregated bet was inconsistent with expected utility theory and that the error probably occurred because his colleague had mistakenly assumed that the variance of a repeated series of bets was lower than the variance of one bet.
In a series of papers, Rabin challenged this view of the world. He showed that an individual who showed even mild risk aversion on small bets would need to be offered huge amounts of money with larger bets, if one concave utility function relating utility to wealth covered all ranges of his wealth. The conclusion he drew was that individuals have to be close to risk neutral with small gambles for the risk aversion that we observe with larger gambles to be even feasible, which would imply that there are different expected utility functions for different segments of wealth rather than one utility function for all wealth levels.
His view is consistent with the behavioral view of utility in prospect theory, which we will touch upon later in this chapter and return to in the next one.
There are important implications for risk management. If individuals are less risk averse with small risks as opposed to large risks, whether they hedge risks or not and the tools they use to manage those risks should depend upon the consequences. Large companies may choose not to hedge risks that smaller companies protect themselves against, and the same business may hedge against risks with large potential impact while letting smaller risks pass through to their investors.
It may also follow that there can be no unified theory of risk management, since how we deal with risk will depend upon how large we perceive the impact of the risk to be. Measuring risk aversion in specific terms becomes the first step in analyzing and dealing with risk in both portfolio and business contexts. In this section, we examine different ways of measuring risk aversion, starting with the widely used but still effective technique of offering gambles and observing what people choose to do and then moving on to more complex measures.
As we noted earlier, a risk-neutral individual will be willing to accept a fair bet. The flip side of this statement is that if we can observe what someone is willing to pay for this bet or any other where the expected value can be computed , we can draw inferences about their views on risk. In technical terms, the price that an individual is willing to pay for a bet where there is uncertainty and an expected value is called the certainty equivalent value. We can relate certainty equivalents back to utility functions.
Assume that you as an individual are offered a choice between two risky outcomes, A and B, and that you can estimate the expected value across the two outcomes, based upon the probabilities, p and 1-p , of each occurring:. Furthermore, assume that you know how much utility you will derive from each of these outcomes and label them U A and U B. If you are risk neutral, you will in effect derive the same utility from obtaining V with certainty as you would if you were offered the risky outcomes with an expected value of V:.
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