Interval risk aversion

Abstract

The conventional measures of absolute and relative risk aversion are appropriate for measuring preferences locally, but because they rely on differential calculus, they cannot accurately capture attitudes towards high-stakes risks involving potentially large changes in wealth. Eisenhauer (2006) has recently proposed an alternative approach which avoids the use of calculus. The present article extends that work in two ways. First, the Pratt–Arrow coefficient of absolute risk aversion is generalized into a measure of interval risk aversion, suitable for analysing preferences over risks of any magnitude, and a corresponding interval measure of relative risk aversion is constructed from preference and risk parameters, without explicit reference to initial wealth or income. Second, the new measures are applied to survey data from the Bank of Italy, to illustrate their empirical applicability to large-scale risk.

Acknowledgements

We thank Marco Lippi and an anonymous referee for helpful comments; any errors are ours.

Notes

¹ Pratt (1964) did observe that a utility function exhibiting risk aversion in the small at all wealth levels could be considered as displaying risk aversion in the large. Rabin (2000) takes this further, arguing that aversion to small risks at every wealth level would imply an implausibly exaggerated aversion to large risks.

² The lira was the unit of currency in Italy prior to its conversion to the euro on 1 January 2002. At the time of the conversion, 10 million lire were worth approximately 5000 euros.

³ As with any such survey, one might question the respondents' veracity, or more generally, the consistency between their answers to purely hypothetical questions and their actual behaviour outside the interview setting. This may be especially pertinent if the hypothetical scenario does not seem plausible to the respondents, in which case they may not take it seriously. See Warneryd (1996) for a careful analysis of this subject. Although important, this issue is neither more nor less applicable to the estimation of interval risk aversion than it is to the estimation of conventional models of small-scale risk aversion using the SHIW or similar survey data, as in the studies by Barsky et al. (1997), Brunello (2002), Hartog et al. (2002), Guiso and Paiella (2003) and others.

⁴ Guiso and Paiella (2003), for example, multiply absolute risk aversion by consumption, whereas Eisenhauer and Ventura (2003) multiply by income. See Meyer and Meyer (2006) for a further discussion of this issue.

⁵ Further discrepancies arise in studies that interpret w as wealth, over whether to exclude liabilities or measure net worth, how to measure human capital and so forth; see, for example, Siegel and Hoban (1982). Meyer and Meyer (2005) show that estimates based on different wealth measurements can vary substantially, in terms of both magnitudes and slopes.

⁶ Eisenhauer and Ventura (2003), for example, found that on average, men had greater relative risk aversion than women, but that was essentially an artefact of higher male incomes; on average, absolute risk version was slightly higher among women. Similarly, differences in absolute risk aversion were outweighed by differences in income across education levels, employment status, geographic locations and other categories.

⁷ Indeed, by the year 2000, a less ambiguous version of the question read, 'You are offered the opportunity of buying shares which, tomorrow, with equal probability, will be worth either 10 million [lire] or nothing. How much would you be prepared to pay (maximum amount) to buy these shares?' The later version more clearly indicates a gross gain in the favourable state. However, we deliberately exploit the ambiguity of the earlier version to illustrate an anomaly associated with the empirical estimation of local risk aversion.

⁸ In practice, some researchers using the 1995 SHIW, such as Guiso and Paiella (2003), have adopted the net-gain interpretation and measured absolute risk aversion according to Equation 4; others, including Brunello (2002), have employed the gross-gain interpretation and measured absolute risk aversion according to Equation 5.

⁹ The weights are given by the magnitudes of gains and losses involved in the two states of nature. Because DUC ₂ measures the change in utility over a gain of k units and DUC ₁ measures the change in utility over a loss of z units, the discrete utility changes in the favourable and unfavourable states are weighted by k and z, respectively.

¹⁰ For infinitesimal changes in wealth, it makes little difference whether the second derivative is divided by a right-hand first derivative (as in Equation 6) or a left-hand derivative, since both tend to U'(w)in the limit. When the changes in wealth are large, however, the magnitude of the resulting ratio is sensitive to division by a wealth gain or a wealth loss. Thus, for concave utility functions, -SDUC/DUC ₁ < -SDUC/DUC_A < SDUC/DUC ₂. So as not to unduly exaggerate the coefficient of interval risk aversion in either direction, we use the average value of the discrete utility change, DUC_A , as the denominator.

¹¹ Consider the two utility values U(w + k) and U(w) as approximated by symmetric, first-order Taylor expansions of utility around the point (w + 0.5k). In particular, let U(w + k) = U[(w + 0.5k) + 0.5k]≅U(w + 0.5k) + 0.5kU'(w + 0.5k) and U(w) = U[(w + 0.5k) - 0.5k]≅U(w + 0.5k) - 0.5kU'(w + 0.5k). Subtracting the second from the first and rearranging, U'(w + 0.5k)≅[U(w + k) - U(w)]/k = DUC ₂. The same argument can be applied to DUC ₁.

¹² For infinitesimal risks, however, it indeed reduces to the standard measure of relative risk aversion, evaluated at a level of wealth equal to (k + z)/2.

¹³ The discrepancy between local and global risk aversion lies at the heart of a recent dispute concerning the possibility of a risk averter voluntarily accepting an unfair wager; see Sounderpandian (1999).

¹⁴ The Friedman–Savage utility function is not only theoretically possible, it is also empirically important. In a Dutch survey, Eisenhauer (2005) finds that at least 18% of respondents have utility functions with 1 or more inflection points. In an experiment involving 21 undergraduates, Bosch-Domenech and Silvestre (1999) found 11 with utility functions having at least 1 inflection point.

¹⁵ For 5 < z < 10, IA_G < 0, implying a love of risk. IA_G is undefined for z = 10, and z > 10 can be deemed irrational by any reasonable standard.

¹⁶ Viewed another way, a reservation price of z = 0 implies a zero tolerance for risk. The inverse of interval absolute risk aversion correctly measures this as 1/IA = 0, while the inverse of absolute risk aversion in the small does not, since 1/a(w) = 5.

¹⁷ Given the ambiguities in the definition and measurement of an initial value of w as discussed above, we will not attempt to estimate local relative risk aversion. For earlier efforts to do so using the SHIW, see Guiso and Paiella (2003) or Eisenhauer and Ventura (2003).

¹⁸ Note that these median and modal values are close to the values often assumed for relative risk aversion in analytical studies. For example, Friedman and Warshawsky (1988) assume relative risk aversion to have a value of 4, and Skinner (1988) assumes values between 1 and 6. They are also within the range (4.2 to 5.4) simulated by Azar (2006) for local relative risk aversion.

¹⁹ As Beetsma and Schotman (2001, p. 821) note, 'Despite the importance of risk preferences, we still have very little idea about the answer to empirical issues like the average risk aversion of individuals, the determinants of risk attitudes and the heterogeneity of risk preferences among individuals.' For example, estimates of the magnitude of r(w) have ranged from less than 1 to more than 40, and studies have variously found r(w) to be increasing, constant, decreasing, or even (as in Barsky et al., 1997) parabolic in w. Meyer and Meyer (2005) demonstrate that consistent measurement of the outcome variable w removes some but not all of the discrepancy. Halek and Eisenhauer (2001) show that attitudes towards speculative risks, in which either gains or losses are possible, differ from attitudes towards pure risks, in which the outcome is either a loss or the absence of a loss (but not a gain). Because the present study measures aversion to a speculative risk, it should be compared primarily to other speculative risk studies such as those by Hartog et al. (2002) and Guiso and Paiella (2003).

²⁰ Because we would anticipate that those who are most risk averse in general would be most inclined to purchase insurance policies, ceteris paribus, our finding suggests that those who already possess insurance against pure risks are significantly less averse than others to undertaking the proposed speculative risk.

²¹ It remains an open question whether those who pursue higher education are initially endowed with greater risk tolerance than others, become more risk tolerant as a result of education, or both.

²² Northern Italy includes Piemonte, Valle d'Aosta, Lombardia, Liguria, Trentino Alto Adige, Veneto, Fruili Venezia Giulia and Emilia Romagna. Central Italy includes Toscana, Umbria, Marche and Lazio. Southern Italy includes Abruzzo, Molise, Campania, Puglia, Basilicata, Calabria, Sicilia and Sardegna.