Introduction:
Risk preferences are integral to many fields of economic study. They are the central focus of literature on decision making under uncertainty, also playing a key role in financial and insurance economics. Risk preferences are also major drivers in consumption models, investment and asset pricing in macroeconomics. As risk preferences play such a key role in many economic fields, it follows that the ability to accurately estimate these preferences is of major importance.
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Thinking about agent’s risk aversion or lack thereof started in 1738 with Daniel Bernoulli’s paper. This predates economics and psychology as distinct subsets of inquiry. Bernoulli points out that the expected value of payoffs is not a sufficient description of agent’s behaviour as fails to consider the risks associated with these payoffs. His experiment consisted of a repeated coin flip, with no payoff occurring until heads appears. The payoff is doubled after every tails. Expected payoff is an infinite sum of ones so you would expect agents would always pay for the chance to receive infinite reward. However, this is not the case and this experiment is now known as the ‘St Petersburg paradox’. Bernoulli concluded that the utility function is nonlinear and concave, meaning higher payoffs are underweighted regardless of risk. This could be considered the first economic experiment.
The main article I will be looking at in this essay is ‘Assessment and Estimation of Risk Preferences’ by Holt and Laura (2014). This article looks at literature pertaining to experimental methods available to measure risk attitudes.
Assessment approaches:
Binary choice:
Nearly 200 years after Bernoulli’s solution to the ‘St Petersburg paradox’, little progress had been made in the field of decision making under uncertainty. This remained the case until Markowitz (1952) proposed an experiment, constructed as follows: Markowitz asked colleagues if they would certainly lose a penny or gamble, with the gamble being a 1/10 chance to lose 10 cents. Almost all colleagues exhibited risk aversion (preferring the sure loss of a penny). Markowitz scaled this up to see if people’s risk preferences changed and indeed they did, with people preferring a 1/10 chance of losing $10,000 compared to a sure loss of $1000. The inverse however was true when it came to gains. In the case of small gains people tended to prefer the gamble of 1/10 chance of gaining $10 compared to a sure dollar. This pattern did not continue in large gambles where people almost always preferred a sure $1million compared with 1/10 chance of gaining $10million.
Continuing from Markowitz’s seminal paper, Kahneman and Tversky (1979) conducted a similar experiment. Given a choice between a sure $3 and a 0.8 chance of $4, 80% of subjects prefer the safe choice. This indicates a degree of risk aversion as the safe payoff is below the expected payoff of 0.8(0.4) = $3.20. When the gains were flipped into losses only 8% of people preferred the 0.8 chance of losing $4. This intuits that losses are avoided more than gains are sought. To further test this loss aversion, they scaled down the probabilities in the gain cases. The sure $3 became a 0.25 chance of $3 and the 0.8 chance of $4 became 0.2 chance of $4. An expected utility maximiser that chose the sure $3 in the first task was expected to choose 0.25 chance of $3 in the second as probabilities and therefore expected utilities had been quartered. However, this was not observed, an example of the ‘Allais paradox’ where people do not always maximise their utility as expected. This led Tversky and Kahneman to believe that it was certainty that led people to choose the certain option in gains and gamble in losses.
In Kahneman and Tversky’s paper (1992) this loss aversion was quantified and found that median coefficient on loss aversion was 2.25 meaning losses hurt 2.25 times more than their equivalent gains.
Investment menus:
In Kahneman and Tversky’s (1992) papers anomalies were sometimes explained in terms of non-linear perceptions of probabilities where probabilities near 0 are overweighted and probabilities near 1 underweighted. To examine whether anomalies are truly caused by nonlinear perception, high, real payoffs can be used, demonstrated in Binswanger’s 1981 paper. This approach contrasted with the low hypothetical payoffs considered up to this point. These payoffs were possible to implement as the experiment was set in rural Bangladesh with participants being farmers, where the highest outcome for a decision exceeded typical monthly wages. The following gambles were proposed:
The results are ordered from highest EV to lowest and riskiest to safest. Gamble 0, the lowest expected payoff and the least risky (analogous to putting all your assets in the safe asset). Gamble 2* and 3* are inefficient versions of gambles 2 and 3 respectively. They are inefficient as give no extra EV only extra risk, so only risk seeking agents would choose these. Paradoxically gamble 2* and 3* are not the highest EV gambles. This further enforces Allais’ result. This simple approach boils down to an investment menu where participants are asked to invest in risky or safe options.
Binswanger calculated the midpoints of risk aversion coefficients for the choice of each gamble as shown in the CRRA midpoint column. The higher the CRRA value the more risk averse a person. When the payoff scaled up from 0.5 rupees to 50 rupees, the percentage that gamble 5 fell from 19% to 2%. Conversely the 100x payoff increase doubled (20% to 40%). This led to Binswanger rejecting the hypothesis of risk aversion being constant over payoff scales and implied risk aversion increased when payoffs increased.
Structured binary choice:
Another method to elicit risk preferences is structured binary choices, this method provides a list of binary choices between safe and risky options. Risk aversion is inferred from the amount of risky options respondents choose or when they move from risky to safe options. For example a participant is given information about a drug trial and must decide whether to participate with a 10% success, then 20%, then 30% etc., from this it can be seen at which success rate they changed from not participating to participating, implying that the later someone switches from not participating to participating the more risk averse.
This experimental structure can be run with money payoffs, pioneered by Murnighan et al. (1988). This experiment provided participants with a menu of choices where they would choose between option A (a sure $5) and option B (p% chance of $10 and 1-p% of $4) where p is initialized at 60% reducing by 10% each subsequent choice. If a participant chooses A in the initial choice, where B has an EV of $7.60 then they are extremely risk averse. Only when p= 10% does expected value of B drop below 5(EV of A). Given this we deduce that only at this point would risk neutral agents switch from B to A, therefore the level of risk aversion can be measured by how early they switch.
As in Binswangers paper, payoffs were increased to measure scale effects. As expected higher real payoffs led to higher risk aversion. No statistically significant change was observed in hypothetical payoffs. Implying it is very important to incentivize subjects properly to elicit true preferences as there is a large disparity in the experiments with the only difference being real or hypothetical payoff.
This method can be seen as groups of different investment portfolios and is perhaps a more thorough test of risk attitudes than investment menus as approximates real life decisions better as you rarely only own one security or the risk-free instrument.
Price based assessments of a certainty equivalent.
Subjects provide the price at which they are willing to sell the gamble and a buying price generated at random. If buying price is above selling price, then they will receive the buying price and not take the gamble, otherwise they take the gamble. Subjects should give their certainty equivalent when giving a selling price because they will guarantee at least obtaining what they value the gamble at if the buying price is above their selling price. If buying price is lower, then they value gamble more than amount offered and would much rather have the gamble determine the outcome. In this experiment, most participants submitted selling prices below the EV of the gamble and are thus risk averse. The larger the difference between EV and selling price the more risk averse a subject is. Other studies showed increasing relative risk aversion still holds with certainty equivalents becoming relatively further away from the EV of the gamble, the higher the payoffs became.
In a Canadian study by Kachelmeier and Shehata , subjects were asked to give a selling and buying price for a 50/50 gamble between $0 and $20. Average selling price was $11 for a gamble with EV of $10, which indicates risk seeking tendencies, whereas average buying price was $5.50 indicating quite severe risk aversion. A possible reason for this is the endowment effect of selling the gamble. It is impossible to lose but by buying the gamble, you are at risk of losing money as it has had to be paid for. A way to avoid biases in buying and selling is asking subjects for a certainty equivalent for a gamble and not reference buying or selling to determine whether the player gets their certainty equivalent or plays the gamble. Another way is offering multiple price lists with several opportunities to move from a gamble to an increasing certain amount; the point at which the subject moves to the certain amount can be defined as their certainty equivalent.
Methodological Issues:
As was shown previously, risk preferences change as payoffs increase, becoming increasingly risk averse. However, in general, people are still risk averse with low payoffs. Demonstrated in the first experiment where with even low payoffs, two thirds of participants were risk averse, so it is acceptable to use just low payoffs to establish general attitudes towards risk but also not sufficient in establishing true sense of an individual’s risk preferences.
Incentive Effects:
Using hypothetical payoffs is an easier way to run experiments because payoffs can be scaled up endlessly to demonstrate how people’s attitudes change. The problem with this is people may not respond as they would if gambles and payoffs were real. The obvious alternative to this is using real payoffs and thus people’s responses would be genuine. This causes an increased experimental cost and is not practical when large payoffs are involved. Experiments conducted with and without incentives showed that risk aversion increased in most cases with incentives and thus they seem necessary to reduce ‘noise’. Overall, if incentivised participants act as they would in similar real-world situations, know how they would behave and have no special reason to behave disingenuously, the ‘noise’ is fairly small and results should be accurate.
Order Effects:
Most experiments on this subject featuring low and high payoffs proposed the lower payoff gamble to participants first. To test whether results vary if this were not the case, Harrison et al. (2005) ran an experiment whereby half of subjects would choose their options from a gamble and then choose again with the payoffs scaled 10x, and the other half only chose options for the 10x gamble. The results showed more risk aversion amongst those who did the 10x gamble second, implying previous gambles may influence people’s behaviour and prevent them from acting as they would if the 10x scenario was presented independently. This should cause researchers to randomise treatment orders between real and hypothetical to avoid such bias.
Structure of Choice Menus:
Most crossover points in choice menus are between points 5 and 7 (up to 10), away from the endpoints of the possible choices. Harrison analysed the position of where most people crossed over in regard to middle of the table by changing tables around so the same points that would be between 5 and 7 were now nearer on end of the menu. It was found that people tended to weight themselves towards the middle even if that makes them more risk averse or risk-seeking than before. Another similar experiment where the top 3 rows of the menu were removed showed similar effects. Crossover points should come between 2 and 4 but this was not the case and people remained near the middle and were more risk averse. Conversely, removing the bottom 3 rows made people riskier.
Summary:
As we have seen, there are several issues with each experimental method and most pertain to more than one method. Despite this, if I were to run an experiment to analyse risk preferences, I would attempt to combine these methods by attempting to discover people’s certainty equivalent using a list of choices in a menu. To mitigate some issues involved, I would use real payoffs at smaller stakes and hypothetical payoffs at higher scales. I would also present menus in different orders to different groups, so order bias balances. Additionally, I would present participants with several varying versions of the same menu, where each point moves up or down in menu position so they are not drawn towards the middle, regardless of what they represent. I believe this would be a fitting experiment, with lots of bias to consider between groups, in terms of order effects and position of the crossover points in each menu.
Appendix B:
The value of r that maximises the likelihood (-18.93759) is 0 meaning I’m risk neutral with CRRA utilityThe value of r that maximises the likelihood (-19.92446) is 0.01 meaning I’m almost risk neutral with CARA utilityThe value of r that maximises the likelihood(-24.03395) is 0.2 meaning I’m almost risk neutral with expo utility
The utility function that characterizes my risk preferences the best is CRRA utility which means I will always invest the same proportion of my wealth in risky assets regardless of wealth.
data=read.csv(“data_risk.csv”)
pa1 = data$pa1
pa2 = data$pa2
pb1 = data$pb1
pb2 = data$pb2
xa1 = data$a1
xa2 = data$a2
xb1 = data$b1
xb2 = data$b2
yy=data$choice
R=seq(0,0.99,0.01)
LL_all = c()
for (r in R){
eua = pa1*crra(xa1,r)+pa2*crra(xa2,r)
eub = pb1*crra(xb1,r)+pb2*crra(xb2,r)
probA = eua/(eua+eub)
total = ifelse(yy==1, probA, 1-probA)
LL=log(prod(total))
LL_all=c(LL_all,LL)
}
crra= function(x,r){
u=x^(1-r)/(1-r)
return(u)
}
cara= function(x,r){
u=x^(1-r)/(1-r)
return(u)
}
expo=function(x,r){
u=1-exp(-x^r)
return(u)
}
max(LL_all)
key= which.max(LL_all)
R[key]
task
pa1
a1
pa2
a2
pb1
b1
pb2
b2
choice
1
0.34
24
0.66
59
0.42
47
0.58
64
0
2
0.88
79
0.12
82
0.2
57
0.8
94
0
3
0.74
62
0.26
0
0.44
23
0.56
31
1
4
0.05
56
0.95
72
0.95
68
0.05
95
1
5
0.25
84
0.75
43
0.43
7
0.57
97
0
6
0.28
7
0.72
74
0.71
55
0.29
63
0
7
0.09
56
0.91
19
0.76
13
0.24
90
0
8
0.63
41
0.37
18
0.98
56
0.02
8
0
9
0.88
72
0.12
29
0.39
67
0.61
63
1
10
0.61
37
0.39
50
0.6
6
0.4
45
1
11
0.08
54
0.92
31
0.15
44
0.85
29
1
12
0.92
63
0.08
5
0.63
43
0.37
53
1
13
0.78
32
0.22
99
0.32
39
0.68
56
0
14
0.16
66
0.84
23
0.79
15
0.21
29
1
15
0.12
52
0.88
73
0.98
92
0.02
19
0
16
0.29
88
0.71
78
0.29
53
0.71
91
1
17
0.31
39
0.69
51
0.84
16
0.16
91
1
18
0.17
70
0.83
65
0.35
100
0.65
50
0
19
0.91
80
0.09
19
0.64
37
0.36
65
1
20
0.09
83
0.91
67
0.48
77
0.52
6
1
21
0.44
14
0.56
72
0.21
9
0.79
31
1
22
0.68
41
0.32
65
0.85
100
0.15
2
0
23
0.38
40
0.62
55
0.14
26
0.86
96
0
24
0.62
1
0.38
83
0.41
37
0.59
24
1
25
0.49
15
0.51
50
0.94
64
0.06
14
0
26
0.1
40
0.9
32
0.1
77
0.9
2
1
27
0.2
40
0.8
32
0.2
77
0.8
2
1
28
0.3
40
0.7
32
0.3
77
0.7
2
1
29
0.4
40
0.6
32
0.4
77
0.6
2
1
30
0.5
40
0.5
32
0.5
77
0.5
2
0
31
0.6
40
0.4
32
0.6
77
0.4
2
0
32
0.7
40
0.3
32
0.7
77
0.3
2
0
33
0.8
40
0.2
32
0.8
77
0.2
2
0
34
0.9
40
0.1
32
0.9
77
0.1
2
0
35
1
40
0
32
1
77
0
2
0
References
Holt, C. and Laury, S. (2014). Assessment and Estimation of Risk Preferences. Handbook of the Economics of Risk and Uncertainty, pp.135-201.
Tversky, A. and Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), pp.297-323.
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