Global Shapes of Preference Scaling Functions

Abstract

In the expected utility framework, concavity (or convexity) of the preference scaling function corresponds to risk-aversion (or risk-seeking) preferences. Friedman and Savage (1948) and Markowitz (1952) indicate that individuals may be risk-averse at some wealth levels but risk-seeking at others. Prospect theory (Kahneman & Tversky, 1979) proposes an S-shaped utility function exhibiting risk-aversion in the domain of gain and risk-seeking in the loss domain. In this study, we use lottery experiments to infer global shapes of the preference scaling functions that determine risk attitudes in interaction with uncertainty. The global shapes that emerge are—fully concave, fully convex, linear, Reverse-S and S-shaped preference scaling functions. Using annual income as a proxy for wealth, we examine the relationship between global shapes and wealth. The data largely supports the Friedman–Savage hypothesis; we find that at low levels of wealth people are concave, followed by Reverse-S and then a second upper concave segment at higher levels of wealth.

JEL: C91, D81

Keywords

Experiments decision making under risk and uncertainty risk attitudes individual behaviour

Introduction

Individual choices are often made in contexts where information is limited and knowledge is uncertain. The basic elements of decisions under uncertainty include actions available to the individual, states of nature, and consequences associated with combinations of acts and states. The expected utility (EU) approach (Neumann & Morgenstern, 1947) integrates these elements and provides a basis for individual decision making under uncertainty. In this approach, the individual combines preferences for outcomes with a probability function describing beliefs about the likelihood of occurrence of states of nature to arrive at expected utility that enables an ordering of acts. Preferences for consequences or outcomes are usually described by a preference scaling function and the curvature of this function in interaction with the probability function describes the individual’s attitude to risk. Concave, linear and convex preference scaling functions represent diminishing, constant and increasing marginal utility respectively and the expected utility derived from these in the presence of uncertainty represents risk-aversion, risk-neutrality, and risk-seeking attitudes respectively. In the expected utility framework, the curvature of the preference scaling functions determines the magnitude of local absolute and relative risk attitudes and allows for representation of constant, increasing, and decreasing magnitudes of risk attitudes over the domain of outcomes.

Another perspective on decisions under uncertainty is provided in prospect theory (Kahneman & Tversky, 1979) which suggests that individuals evaluate outcomes in terms of gains and losses rather than final states of wealth as assumed in the EU approach. Here, preferences are represented by S-shaped functions where people are risk-averse in gains and risk-seeking in losses, as well as loss averse, that is, they are considerably more sensitive to losses than to gains. Also, critics of the expected utility approach see the assumptions of rationality, cognitive ability as unrealistic and do not accurately describe how people make decisions in many situations (Camerer, 1992). Within the EU approach, while risk-aversion is considered to be the normal case, individuals may have differing risk attitudes across circumstances and situations. One such instance is that of a risk-averse individual accepting unfavourable bets when gambling in casinos and this could be explained by considering the recreational utility derived from gambling (Hirschleifer & Riley, 1992). It is also possible that individuals are risk-averse in some ranges of income and risk-seeking in other ranges and this would imply preference scaling functions that are inflected with curvature changing from concave to convex and vice versa over the domain of outcomes. In such cases, the global shape of the preference scaling function is of interest. Friedman and Savage (1948) proposed a doubly inflected preference scaling function and Markowitz (1952) proposed a preference scaling function with three inflection points over wealth. Such preference scaling functions suggest that there may be levels of wealth associated with inflection points where curvature changes and individuals exhibit different attitudes on either side of these points. The global shape of the preference scaling function has been the focus in some recent studies in experimental economics (Pennings & Smidts, 2003; Schunk & Betsch, 2006). Pennings and Garcia (2009) suggest that the global shape of the utility function may contain information that predicts investment decisions among portfolio managers. Schunk and Betsch (2006) suggest that the global shape of utility functions is related to deliberative vs. intuitive modes of thinking and decision making.

In this article, the focus is on the global shape of the preference scaling function and its relationship with wealth levels using annual income as a proxy for wealth. In an experiment, certainty equivalents of lotteries are elicited and used to infer the local curvature of respondents’ preference scaling functions. The respondents are then classified according to the global shape and the relationship of the same with income level is assessed. The global shapes that emerge are strictly concave, strictly convex, Reverse-S and S-shaped and find preliminary support for the global shape proposed in Friedman and Savage (1948). In the next section, the expected utility and prospect theory perspectives are discussed with a particular focus on the Friedman–Savage model of preferences. In the third section, the experiment is described and respondent choices are analyzed under the assumption that choices maximize expected utility. In the fourth section, the relationship between the shape of the preference scaling function and income levels is examined and conclusion with a brief discussion is presented in the fifth section.

Decision-making Approaches and Shapes of Utility Curves

Expected Utility Approach (EUT)

The development of the concept of utility is credited to Daniel Bernoulli (1738) who stated that ‘…the value of an item must not be based on its price, but rather on the utility that it yields’. The prevailing perspective at that time was that the value of a lottery should be equal to its mathematical expectation and hence identical to all people. However Bernoulli justified his ideas in many ways, such as the famous St Petersburg Paradox, to show that two people facing the same lottery may value it differently and the value of a lottery is not equal to its mathematical expectation. Bernoulli introduced the notion of decreasing marginal utility and proposed the concave shape of the preference scaling function. Hence, a gain of ₹1000 was not necessarily worth twice as much as a gain of ₹500. This relationship between utility and wealth is characterized by a preference scaling function v, which for every wealth level x tells us the level of ‘satisfaction’ or ‘utility’ v(x) attained by the decision maker. Two centuries later, von Neumann and Morgenstern (1944) developed an axiomatic theory of utility and redefined utility in monetary terms. The central notion is that an uncertain prospect should be valued according to its expected utility which is the weighted sum of the utilities from payoffs. The preference scaling function v(x) represents preferences for wealth and expected utility of $E [v (\tilde{x})]$ describes preferences for uncertain prospects $\tilde{χ}$ . The expectation is over the probability distribution associated with $\tilde{χ}$ and the expectation of concave (convex) function of a random variable is more (less) than the function evaluated at the expectation of the random variable (Jensen’s inequality). Risk-aversion implies that the expected value of a gamble is preferred to the gamble and risk-seeking implies that the gamble is preferred to its expected value. Thus, risk attitudes are related to the curvature of the preference scaling function: Risk-Aversion: v″(x) < 0 and E[v(x) < v[E(x)] Risk-Seeking: v″(x) > 0 and E[v(x) > v[E(x)] Risk-Neutral: v″(x) = 0 and E[v(x) = v[E(x)]

Attitude to risk is then an interaction of two factors working together; the individual’s marginal utility described by the curvature of the preference scaling function and uncertainty (Hirshleifer & Riley, 1992).

The crucial axioms in expected utility models are transitivity, dominance and invariance. Transitivity assumes that if option A is preferred to option B and B is preferred to C, then A is preferred to C as well. Dominance argues that if one option is better on at least one aspect, and at least as good on all other aspects, it will be preferred to lesser options. Invariance posits that a preference should remain unchanged regardless of order or method of presentation. In the EU approach, empirical determination of an individuals’ attitude towards risk begins with the specification of a preference scaling function. By assuming an appropriate preference scaling functions, one can represent risk-averse, risk-seeking or risk-neutral individuals.

Pratt (1964) and Arrow (1971) proposed a local measure of risk-aversion based on the curvature of v(x) known as the Arrow–Pratt coefficient of absolute risk aversion (ARA), the negative of the ration of the second derivative to the first derivative.

ARA= - \frac{v^{″} (x)}{v^{'} (x)}

A higher positive ARA implies more risk-aversion and higher negative ARA implies higher risk-seeking, at 0 it is risk-neutral. This measure is useful as it is invariant to an affine transformation of the utility function, and such transformation does not affect the preferences expressed by v(x). The advantage of this measure is that it can be used to compare individuals with different utility functions to draw conclusions about differences in local risk-aversion across people. However, it is conceptualized as unidimensional construct and locally applicable over consistent risk preferences. Another important notion is that of the certainty equivalent (CE) measure (Hershey & Schoemaker, 1985); the individual is indifferent between the certain value and the gamble, that is, $v (C E_{\tilde{χ}}) = E [v (\tilde{χ})] .$ . Risk-aversion (seeking), associated with concave (convex) preference scaling functions would mean, the expected value of the gamble is greater (lesser) than the certainty equivalent.

Friedman and Savage (FS) Hypothesis

The classical approach assumes marginal utility is decreasing in wealth, indicating a general aversion to risk. However, it is observed that some individuals exhibit risk-aversion in some settings and risk-seeking in others. One explanation for such behaviour is provided by Friedman and Savage (1948) who suggest that it is not necessarily true that an individual’s preference scaling function has the same kind of curvature everywhere; there may be levels of wealth or income associated with risk-seeking and others with risk-aversion.

In the Friedman and Savage model, risk preferences vary with wealth and individuals could accept some fair gambles while reject others. In this model, preferences are represented by a doubly inflected preference scaling function (Figure 1) where v(z) is concave up until inflection point B and then becomes convex until inflection C after which it becomes concave again. Thus, at very low wealth levels (between the origin and z_B) and at high levels of wealth (above z_C) people exhibit risk-averse behaviour. The expected value E(z) of a lottery z = {z_A, z_B} is preferred to the lottery. However between the inflection points B and C, people are risk-seeking; the lottery z′ = {z_B, z_C} is preferred to the expected value E(z′) of the lottery. An individual with very low wealth (less than z_A) would be risk-averse. Those with wealth between z_A and z_B may be risk-averse in the face of small gambles but may take risk to cross a threshold level z_B of wealth. At wealth levels between z_B and z_C risk-seeking behaviour is exhibited but closer to z_C individuals may trade-off gains against large losses if the probability associated with loss is low. The very rich with wealth levels exceeding z_C may not be interested in risky prospects at all. Thus, this model explains why people may take low probability high-payoff risks such as lottery tickets, while at the same insuring against mild risks with mild payoffs. From the perspective of empirical verification, this model assumes that concave preference scaling functions are associated with very low or very high levels of wealth. As wealth increases from very low levels, the likely shape of the preference scaling function is Reverse-S followed by convex and S-shapes. Finally, at very high levels, the shape is likely to be concave again.

Prospect Theory

Prospect theory is a descriptive model of choice under risk and uncertainty (Camerer, 2001) and explains frequently observed behavioural tendencies. Prospect theory and cumulative prospect theory developed by Kahneman & Tversky (1979, 1992) describe choice as a two-stage process. In the first phase, alternatives are edited and values are attached to outcomes and weights to probabilities. In the second phase, the edited alternatives are evaluated. Prospect theory suggests that decision makers exhibit consistent violations of the assumptions of utility theory in this two-staged process. First, they consider choices as adjustments to their current wealth from a personal reference point rather than final wealth. They tend to be risk-averse towards outcomes seen as gains and risk-seeking towards outcomes seen as losses from this reference point. Second, decision makers tend to overweight unlikely events and underweight likely events when assigning probabilities. Finally, the manner in which alternatives are presented can influence the choices made known as framing effects. Thus, Prospect theory seeks to accommodate several important behavioural tendencies observed empirically:

Reference Dependence—Individuals make decisions by evaluating gains and losses relative to a reference point rather than evaluating expected final wealth. Reference points are flexible such as the status quo (Kahneman & Tversky, 1979), aspiration level (Siegel, 1957; Tversky & Kahneman, 1991) or past observations (Baucells et al., 2011). Koszegi and Rabin (2006, 2007) argue that expectations about the future form the most natural reference point for valuing realized outcomes.

S-shaped Value Function—Prospect theory shows people process outcomes as gains/losses using a value function that is concave for gains and convex for losses. This S-shaped value function captures individuals’ risk-aversion over gains and risk-seeking behaviour over losses. Figure 2 depicts the Prospect theory S-shaped value function.

Loss Aversion—People with preferences consistent with prospect theory are willing to take on additional risk in order to avoid feeling a loss. This implies individuals weigh losses more heavily than gains and this aspect of prospect theory has been termed ‘loss aversion’.

Diminishing Sensitivity—Sensitivity to marginal changes in the outcome reduces as the agent moves further away from the reference point, making the curve of the value function S-shaped.

Probability Weighting—Finally, prospect theory preferences use a weighting function that over-weights small probabilities and under-weights large probability since individuals have been shown to be more sensitive to small gains/losses relative to larger ones. This effect is modelled with a Reverse-S-shaped probability weighting function.

The section above describes alternative perspectives on consumer preferences and shapes of preference scaling functions. Within the expected utility approach, the Friedman–Savage hypothesis accommodates concave, Reverse-S, convex and S-shaped preferences across various levels of wealth and Prospect theory describes an S-shaped value function which is convex in the loss domain and a concave in the gain domain. An empirical analysis, by eliciting risk preferences of investors, could help in assessing which of these perspectives prevail in explaining different shapes of utility curves.

Survey Design and Analysis Plan

We use the reference lottery technique to elicit risk preferences of investors and classify investors based on global shapes of utility curves. In designing the lottery task, the main source of bias arises when the assessment does not match the respondents’ real decision situation (Pennings & Smidts, 2003). An important decision for an investor is to analyze his savings, consumption and financial goals while making investment plans. The lottery task fits this decision context as it asks the respondent to identify an amount (in ₹) that he would like to invest for a year, considering his savings and expenditure. The dimensions of the lottery in terms of money are different for each investor and computed depending on what he reports as his investment amount for the year. The basic premise underlying this design is that it is necessary to create a lottery that matches the subject’s financial consideration. Binary lotteries with equal probability for the two states are presented to the respondents.

A spreadsheet macro is used to generate the lotteries. The respondents are informed that there are no right or wrong answers and the program begins with a question, ‘How much would you like invest (₹) for a year? Kindly take into account your savings and expenditure for the year’. Once the respondent identifies an amount, two options are presented in terms of a risk-free asset which has a fixed return and a risky asset with a variable return both constructed around the investment level indicated.

Alternative A: Fixed sum available for certain (C).

Alternative B: A 50/50 chance of receiving a relatively high amount (x₁) or a relatively low amount of money (x₀). The program computes x₁ as (+50%) and x₀ as (–50%) of the investment amount indicated by the respondent and presents the lottery {x₀, x₁} where both outcomes are equally likely.

Further, referring to the lottery generated, the programme asks the respondent; ‘If you had to choose the risky asset option, what would be the certain amount of money you would view as equally desirable as the risky asset?’ The response to this question is the certainty equivalent of the presented lottery. It represents that amount for which the decision maker is indifferent between the lottery and the amount for certain. Designating the response as CE0.5, _0.5(1)₀₁

where v represents the preference scaling function, w denotes the individual’s average annual income (used as a proxy for wealth) measured in ₹ and p is the probability of realizing x₀. The expected wealth from accepting the gamble is calculated as 0.5 [(w + x₀) + (w + x₁)] and its certainty equivalent is (w + CE0.5). In the expected utility framework, positive linear transformations of utility functions are strategically equivalent. Hence, v(w + x₀) and v(w + x₁) can be set to 0 and 1 respectively so that the utility at the response to this question is 0.5 and hence the response is designated as CE0.5.

Once the respondent identifies CE0.5, it becomes the input amount for generating the 0.25 lottery and then the respondent identifies CE0.25 and that in turn becomes the input for the 0.125 lottery and the process goes on. Six other lotteries are generated corresponding to utilities 0.25, 0.125, 0.375, 0.75 0.625 and 0.875 in that order. Thus, the assessment of certainty equivalents is an iterative process and seven lotteries are generated; the process took about 20 minutes for each subject. Annexure 1 presents the elicitation process for a respondent. Given the certainty equivalents corresponding to these utilities, the preference scaling function v(.) for the respondent can be constructed using non linear regression. Further, the risk premium (π) of the respondent for each lottery, the difference between the mathematical expectation of the lottery and the elicited certainty equivalent, can be computed. It is the amount the decision maker is willing to give up in order to get rid of the risks associated with the lottery.

(2) π = E(w + x) – CE(w + x)

If π is negative, it identifies the subject as risk-seeking; if π is positive, the subject is risk-averse and if 0, the subject is risk-neutral. Risk premium, which indicates the local curvature and magnitude of the risk attitude, can be computed for each of the elicited CEs. The sequence of CEs or risk premiums defines the global shape of the preference scaling function.

If prospect theory were applicable, individuals would consider gains and losses with respect to a reference. If the reference point were the initial wealth, the utility at the response to the first question would be equal to the expected utility from the lottery {–x0, + x₁} with the loss and gain being equally likely. It is also possible that the reference is arrived such that it maximizes the consequent expected utility. In this case, the response is the reference and value of wealth at which the expected utility from the lottery of loss and gain (both evaluated from the reference) is maximized. In either case, the remaining responses can be split into loss and gain domains and assessed separately for curvature.

It should be noted that the method of analyzing the relationship between risk attitudes and possible antecedents such as income, age, etc., would depend on the global shapes that emerge. If the shapes are limited to globally concave and globally convex types, it is possible to use the responses to fit preference scaling functions using non linear regression. The parameters of the preference scaling function could then be used to derive a measure of magnitude of risk such as absolute risk-aversion/seeking. The degree of local risk-aversion/seeking computed at the midpoint could serve as a representative measure of the magnitude of the individual’s risk attitude. Since this would be a continuous measure on the real line, the relationship between risk attitude and possible correlates such as income, age, etc., could be examined in a simple regression framework. However, if the shapes that emerge include inflected types, local measures are not comparable and the analysis would require examining the relationship between category membership and the possible correlates.

Data, Results and Discussion

National Institute of Securities Markets (NISM) periodically conducts workshops to aid financial literacy in India. Since the experiment requires subjects to have some knowledge about investments, investors who attend these workshops were chosen as respondents for the study. The data sample consisted of housewives, students, retired people, industry professionals and market participants who take investments decisions and are seeking to enhance their understanding about decisions under uncertainty. The purposive sampling (Patton, 1990) approach, in which subjects are selected because of certain specific characteristics, was used. By means of a computer aided interview, data was collected from 90 individual investors. In the experiment, respondents were presented with seven lottery options which represented a risky asset and they were asked to indicate certainty equivalents for each lottery option. The raw data needed cleaning as some responses were incomplete and therefore considered invalid leading to valid data from 69 respondents. The survey was administered in person, to explain the context to the respondents and prevent non-sampling errors like response differences, definitional difficulties and differing respondent interpretations.

The behavioural finance literature suggests that demographics, socio-economic characteristics and psychological factors are correlates investor behaviour and risk preferences. Women are observed to be on average more risk-averse than men (Croson & Gneezy, 2009; Schubert et al., 1999; Eckel & Grossman, 2008). Experimental evidence suggests that women are more risk-averse towards gambles (Hershey & Schoemaker, 1980) and they behave more conservatively in familiar as well as in unfamiliar gambles, and in gain as well as in loss gambles (Powell & Ansic, 1997). Riley and Chow (1992) find that asset allocation and risk tolerance of investors are positively related to education, income and wealth levels. Schooley and Worden (1996) find that investors with post-secondary education and those who are married hold higher percentages of equity securities in their portfolios. Barnewall (1987) and Hartog et al. (2002) find that certain occupational groups like civil servants, lawyers, medical and dental non-surgeons are more risk-averse than entrepreneurs and private sector employees. Dohmen et al. (2011) find that gender, age, height and parenting have economically significant impact on willingness to take risks. Barber and Odean (2000) find that greater overconfidence leads to greater trading and to lower expected utility among overconfident traders than rational traders. Grinblatt and Keloharju (2009) analyze the role that two psychological attributes; sensation-seeking and overconfidence that influence the tendency of investors to trade risky stocks. People tend to be consistently more willing to take risks when certain losses are anticipated, and are more willing to settle for a sure gain when absolute gains are anticipated (Shefrin & Statman, 1985). Loewenstein et al. (2001) highlight the role of emotions in risky situations that often diverge from cognitive assessment of risks. Guiso and Paiella (2008) find that household attributes do not predict risk-aversion; however uncertainty or liquidity constraints in consumers’ environment influence greater risk-aversion. While our primary focus is on income and its relationship to global shapes and risk attitudes, we elicit information on basic demographics so as to control for these in our analysis. Thus in addition to the lottery tasks, respondents provided basic demographic information which includes annual income level.

Global Shapes

In this study, we categorize respondents according to global shape of the preference scaling functions over the total wealth domain. Global shape is defined here as the general shape, that is, concave, convex, S-shaped or Reverse-S-shaped. To identify the global shape of the utility, we elicit seven CE points, infer corresponding risk premiums (π) and respondents are classified as globally concave—if risk-averse in every lottery segment; fully convex—if risk-seeking in every lottery segment; Reverse-S—if they are concave to start with and then convex, S-shaped—if convex to start with and then concave; and globally linear. The prevalence of different shapes of utility curves in the data is presented in Table 1. Approximately 33 per cent of the subjects exhibit concave preferences, 38 per cent are Reverse-S and 17 per cent have completely risk-seeking preferences. The linear and S-shape categories are sparsely populated.

Table 1.

Global Shapes of Utility Curves

Shapes of Utility Curves	Number	Percentage
Fully concave	23	33%
Reverse-S	26	38%
Fully convex	12	17%
S-shaped	3	4%
Linear	5	7%
Total	69	100%

Source: Authors.

Empirical research attempting to examine the FS model has been sparse: Premus (1979) used the FS model to explain various phenomena such as migration; Brunk (1981) found that those who gamble do so primarily to increase their wealth; Hawley and Fujii (1993) found higher income individuals to be more tolerant of financial risks, and interpreted this as evidence against the FS hypothesis and Eisenhauer (2005) find that around 18–20 per cent of individuals in their data evaluate risks in a manner consistent with FS preferences.

The FS model hypothesizes concavity in low levels of wealth as well as very high levels of wealth and convexity between, leading to a doubly-inflected curve (Figure 2). As wealth increases, the shape of the preference scaling function changes in the following sequence: concave, Reverse-S, convex, S-shaped and again concave (Figure 1). We use annual income as a proxy for wealth and examine whether the data provides any support for the FS model. We first categorize shapes of utility curves across their corresponding income levels. We classify concave shapes into concave low and concave high since concavity is possible at very low and very high income levels (we realize this may lead to a circularity problem in the analysis). Within the concave category, respondents with income levels between 100 thousand and 1 million are classified as concave low and the rest (income level between 1 million and 2 million) are classified as concave high (Table 2). We do not consider the risk-neutral cases in this analysis.

Table 2.

Shapes of Utility Curves across Income Range

Income Category (₹)	Concave Low	Reverse-S	Convex	S-shaped	Concave High	Total
1 (< 100 thousand)	4	1	–	–	–	5
2 (100–500 thousand)	8	11	7	–	–	26
3 (500 thousand– 1 million)	2	5	4	1	–	12
4 (1 million– 1.5 million)	–	3		2	5	10
5 (> 1.5 million)	–	6	1		4	11
Total Shapes	14	26	12	3	9	64

Source: Authors.

Table 3.

Income Descriptives for Shapes of Utility Curves

Shapes of Utility Curves (Codes)	N	Mean Income (₹ ’000)	Std. Deviation
Concave low (1)	14	183,333	98473
Reverse-S (2)	26	800,000	630079
Convex (3)	12	791,666	620056
S-shaped (4)	3	1,083,333	288675
Concave high (5)	9	1,350,000	316227

Source: Authors.

Table 3 details descriptive statistics for income across categories; the mean income is increasing across groups in the order: concave low, Reverse-S, Convex, S-shaped and concave high.

We consider pairs of shapes in the order suggested by the FS model and test for each pair the null hypothesis that the mean incomes are the same, versus the alternate hypothesis that the difference in means is negative (Table 4). The null hypothesis is rejected for two pairs (concave low = Reverse-S and Reverse-S = concave high) while it is not for the other three pairs; for two pairs the mean income of the first shape is significantly less than that of the second. There is a significant increase in mean income across concave low, Reverse-S and concave high groups of responses.

Table 4.

Independent Sample Test of Means

Pairs	Null Hypothesis(Shapes Indicate Group Means)	t-test for Equality of Means
		t-test for Equality of Means	t	Sig.	Mean Diff.	Std. Error Diff.	95% Confidence Interval of the Difference
		Lower	t	Sig.	Mean Diff.	Std. Error Diff.	Upper
1	Concave Low = Reverse-S	–4.863	0.000	–616667	126796	–876592	–356741
2	Reverse-S = Convex	.038	0.485	8333	217505	–442979	459646
3	Convex = S-shaped	–1.193	0.135	–291667	244575	–862729	279395
4	S-shaped = Concave High	–1.372	0.125	–266667	194365	–831108	297774
5	Reverse-S = Concave High	–3.460	0.001	–550000	158963	–874105	–225895

Source: Authors.

Shapes of Utility Curves and Income

Descriptive characteristics of the respondents’ demographics are presented in Table 5. To analyze the relationship between shapes of utility curves and demographic variables including income, we regress shapes of utility curves; that is, concave low, Reverse-S, convex, S-shaped and concave-high, on demographics variables like gender, age, academic qualifications, dependents, marital status, occupation and annual income. Logistic regression is used to see whether the idea of categorization is invariant to including demographic variables such as age, gender, academic qualification, occupation, etc. We exclude risk-neutral preferences from the analysis and hence the data is limited to 64 subjects.

Table 5.

Descriptive Characteristics of the Subjects

Demographics	Coding	Total (69)	Percentage
Gender	Female (0)Male (1)	3039	4456
Age	Below 30 (1)30–40 (2)40–50 (3)50–60 (4)> 60 (5)	1925 610 6	2840 914 9
Academic qualification	School Final (1)Graduate (2)Post Graduate (3) Professional degree (4)	4 52334	6 73354
Dependents	None (1)0–2 (2)3–5 (3)> 6 (4)	123814 2	176020 3
Marital Status	Unmarried (0)Married (1)	2445	3565
Occupation	Salaried (0)Business (1)Retired (2)Professional (3)	43 4 814	57 61120
Individual Income	Below 1,00,000 (1)1,00,001–5,00,000 (2)5,00,001–10,00,000 (3)10,00,001–15,00,000 (4)Above 15,00,000 (5)	52911 813	742161619

Source: Authors.

Multinomial logistic regression is appropriate to model relationships where the dependent variable is polytomous, that is, more than two categories, and the predictor variables are either categorical (factors) or numerical (covariates). The dependent variable is group membership or global shape of the preference scaling function, and independent variables include gender, academic qualifications, occupation and marital status as factors; and age, number of dependents, and annual income as covariates for the regression analysis. The focus is on income as a predictor while demographics such as age, occupation, etc., have been included as control variables. We define group specific random index functions

_j(3)_j_k_jk_k_j

with group specific intercepts α_j, explanatory variables x_k, group specific weights β_jk for the explanatory variables x_k, and a group specific random term ε_j so that follows a type II extreme value distribution. The concave (low) shape group is set as the base group with zero as the deterministic component of its index function. The probability of group membership then takes the multinomial logit form

P r o b (j) = \frac{e^{α_{j} + Σ_{k} β_{j k} x_{k}}}{Σ_{i} e^{α_{i} + Σ_{k} β_{i k} x_{k}}}

(4)

and estimates of α_j and β_jk are obtained through maximum likelihood estimation. The presence of overall test of relationship between utility curves and combination of independent variables is based on the statistical significance of the final model chi-square associated with the likelihood ratio test (Table 6) which is 0.000 (< 0.05) which indicates that the specified model explains significantly more variation that an intercept only model. The variation explained by the model is indicated by the Pseudo R-Square measures; Cox and Snell—80 per cent, Nagelkerke—84.9 per cent and McFadden—56.5 per cent. Table 7 describes the classification accuracy—the overall accuracy is 76.6 per cent (Table 7) and accuracy rates are high except for group 3 where the accuracy is 25 per cent.

Table 6.

Model Fitting Information

Model	Model Fitting Criteria	Likelihood Ratio Tests
Model	–2 Log Likelihood	Chi-Square	df	Sig.
Intercept Only	176.629
Final	73.780	102.849	44	.000

Source: Authors.

Note: Pseudo R-Square measures are Cox and Snell (0.80), Nagelkerke (0.849) and Mcfadden (0.565).

Table 7.

Classification

Observed	Predicted
Observed	1.00	2.00	3.00	4.00	5.00	Percentage Correct
1.00	14	1	0	0	0	93.3%
2.00	1	23	1	1	0	88.5%
3.00	1	7	3	1	0	25.0%
4.00	0	0	0	3	0	100.0%
5.00	0	0	1	1	6	75.0%
Overall Percentage	25.0%	48.4%	7.8%	9.4%	9.4%	76.6%

Source: Authors.

The Likelihood ratio test (Table 8) shows the contribution of each independent variable to the model. Age, number of dependents, annual income, academic qualification, and occupation contribute to the overall model fit while gender and marital status are not significant. The parameter estimates of the significant variables in the model are presented in Table 9. As mentioned earlier, Concave-low is treated as the base group and index functions are estimated for Reverse-S, convex, S-shaped and concave-high relative to concave-low. The parameter estimates in the model are relative to the base group; hence for a unit change in the predictor variable, the probability of group membership relative to the base group is proportional to the estimate of the weight parameter associated with the predictor variable, other variables in the model being held constant.

Table 8.

Likelihood Ratio Tests

Effect	Model Fitting Criteria	Likelihood Ratio Tests
Effect	–2 Log Likelihood of Reduced Model	Chi-Square	df	Sig.
Intercept	73.780a	.000	0	–
Age	82.921	9.141	4	.058
Dependents	88.110	14.330	4	.006
Annual income	117.771	43.991	4	.000
Gender	79.029	5.249	4	.263
Acad. qualifications	101.877	28.097	12	.005
Marital Status	78.248	4.468	4	.346
Occupation	94.266	20.486	12	.058

Source: Authors.

Notes: The chi-square statistic is the difference in –2 log-likelihoods between the final model and a reduced model. The reduced model is formed by omitting an effect from the final model. The null hypothesis is that all parameters of that effect are 0.

a This reduced model is equivalent to the final model because omitting the effect does not increase the degrees of freedom.

Table 9 presents the significant parameter estimates of the regression. The S-shape group does not have any significant parameters so that the index function for this group is not different from zero and it is not distinguished from the base concave (low) group. Intercepts for all groups are not different from zero suggesting that the inherent probability of belonging to any group is the same. Annual income, post graduate degree and business as occupation are significant in distinguishing between shapes of utility curves. Respondents with post graduate education rather than professional degree are more likely to be part of Reverse-S group relative to the concave-low group. Salaried people are more likely to belong to the concave (low) group relative to professionals. These findings are consistent with Schoemaker (1993) who suggests that depending on education and occupation people may perceive options differently in their life and apply different norms to them.

Table 9.

Parameter Estimates

Shapes of Utility Curves	Demographic Variables	B	Std. Error	Wald	Sig.	Exp(B)
2.00—Reverse-S	Annual income	8.270	4.203	3.870	.049	3903.908
	Acad. qual.= Post Grad	3.712	1.825	4.138	.042	40.952
	Acad. qual.–Professional degree (Base variable)	0b
	Occup. = Salaried	–6.810	3.562	3.655	.056	.001
	Occup.–Professional (Base Variable)	0b
3.00—Convex	Annual income	8.433	4.221	3.993	.046	4598.138
5.00—Concave-high	Intercept	–35.286	20.046	3.098	.078
	Annual income	11.496	4.592	6.267	.012	98343.121

Source: Authors.

Notes: a. The reference category is: Concave-low (1).

b. This parameter is set to zero because it is redundant.

c. Floating point overflow occurred while computing this statistic. Its value is therefore set to system missing.

Controlling for other demographic variables, as annual income increases, subjects are more likely to belong first to the Reverse-S, second to the convex, and then concave-high groups, since the income parameter estimates are significant and increase in this order. Thus, these results provide some support for the Friedman–Savage hypothesis that as wealth increases the utility curve changes from concave, to Reverse-S, convex and then concave again at higher levels of wealth.

Overall, the data is consistent with the Friedman–Savage model of preferences across the wealth domain and the results indicate that individuals may be risk-averse at some wealth levels but risk-seeking at others.

Conclusion

Empirical analysis of risk preferences (Donkers et al., 2001; Hartog et al., 2002; Holt & Laury, 2002; Pennings and Smidts, 2000; Wärneryd, 1996) typically assume a particular functional form for utility or simply measure a Pratt–Arrow coefficient of risk-aversion at a specific level of wealth, and neither of these approaches allow for FS type of preference scaling functions with multiple inflections. In the expected utility framework, the curvature of the utility function characterizes the risk attitude of the decision maker; concavity represents risk-aversion while convexity signifies risk-seeking preferences. As early as 1948, Friedman–Savage postulated a doubly–inflected utility function with a concave segment at low levels of wealth, followed by a convex segment and then a second upper concave segment at higher levels of wealth. Thus, risk-aversion and risk-seeking preferences can co-exist in an individual, supporting the empirical evidence that people buy insurance and lotteries simultaneously. Markowitz (1952) argued that simultaneous purchase of lotteries and insurance is not confined to individuals with low wealth, hence the utility function should accommodate change in wealth. In 1979, Kahneman and Tversky proposed the Prospect theory, an S-shaped value function, because people code outcomes as gains and losses and are risk-averse in gains and risk-seeking in losses.

These perspectives suggest that it could be useful to closely examine the global shape of preference scaling functions. Accordingly, in this study, the attempt is to calibrate risk attitudes of individual investors using the certainty equivalent method through a series of reference lotteries and arrive at the global shape of the underlying preference scaling function. The relationship of the shape with annual income is thereafter examined. The different shapes that emerge are—fully concave, fully convex, linear, S-shaped and Reverse-S-shaped utility curves. We find that the data supports the Friedman–Savage model where as wealth increases the utility curve changes from being concave, to convex and again concave, thereby allowing Reverse-S and S-shaped preferences in the interim. The primary limitation of the study is the small sample size owing to the difficulty in administering a series of reference lotteries. However, the results suggest that the global shape of preference scaling functions is an interesting focus area for further research.

Annexure 1

We present the response of one of the subjects in the survey. We use lottery tasks through an Excel spreadsheet and analyze attitudes to risk, we elicit certainty equivalents for 7 lotteries and evaluate them using the EU framework. The entire process took around 20 minutes for each respondent.

How much would you like to invest (₹) in a year. Kindly take into account your savings and expenditure for the year. For example, for ₹20 thousand enter as 20,000, Press Enter

₹100,000

2.	You can invest ₹100,000 in two options:
	Option A:	Option B:
	Risk-free Asset where you obtain 1.07 times (+7%) the invested amount for sure	Risky Asset with two possible outcomes
		equal chance of earning 1.5 times (+50%) of the invested amountORequal chance of earning 0.5 times (–50%) of the invested amount

3.	If you had to choose the risky asset option; what would be the certain amount of money you would view as equally desirable as the risky asset?
	I expect at least a certain amount of	A Risky Prospect with
		equal chance of gettingORequal chance of getting
Click on the arrow to enter → value and hit ENTER.	₹90,000	₹1,50,000₹50,000
	For any amount below this value, I will prefer to choose the risky prospect.	NEXT

Source: Authors.

In the above case, CE0.5 = ₹90,000

When the respondent identifies CE0.5 it becomes the input for establishing the 0.25 lottery and so on. Totally seven lotteries are generated corresponding to utilities 0.5, 0.25, 0.125, 0.375, 0.75, 0.625 and 0.875, in that order.

For the above respondent, the certainty equivalent responses for seven lotteries are: CE_0.5 = ₹90,000 (50,000,150,000; 0.5) CE_0.25 = ₹65,000 (50,000,90,000; 0.5) CE_0.125 = ₹55,000 (50,000,65,000; 0.5) CE_0.375 = ₹75,000 (65,000,90,000; 0.5) CE_0.75 = ₹140,000 (90,000,150,000; 0.5) CE_0.625 = ₹130,000 (90,000,140,000; 0.5) CE_0.875 = ₹147,000 (140,000,150,000; 0.5)

4. Evaluation of Certainty Equivalents:

The respondent choices are analysed under the assumption that choices maximize expected utility. The expected wealth from accepting the gamble is calculated as 0.5[(w + x0) + (w + x1)] and its certainty equivalent is (w + CE0.5).

As part of demographic variables, we elicit the range for annual income (₹) across five categories, that is, less than 100 thousand, 100 thousand–500 thousand, 500 thousand–1 million, 1 million– 1.5 million and above 1.5 million. The above respondent indicated his annual income to be in the range 500 thousand–1 million. Therefore we assume ₹750,000 as the average annual income which is a proxy for wealth (w).

We infer the risk attitude for every lottery segment from the elicited CEs and hence the global shape of the utility curve.

5. Global shape of the utility curve is Reverse-S for the above elicited certainty equivalents

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