Subjective Measures of Risk: Seminar Notes

Transcription

1 Subjective Measures of Risk: Seminar Notes Eduardo Zambrano y First version: December, 2007 This version: May, 2008 Abstract The risk of an asset is identi ed in most economic applications with either some statistical measure of dispersion or with some percentile value of its underlying probability distribution. These notions neither capture what most people have in mind when they think about risk nor have economic, decision theoretic foundations (in particular, they are not monotone with respect to stochastic dominance). In this paper I build on Aumann and Serrano (2008) and Foster and Hart (2008) and develop subjective measures of the riskiness of an asset that are both based on decision theory and that respect the attitudes toward risk of speci c decision makers. As a bonus, the theory yields an operational interpretation of Aumann and Serrano s economic index of riskiness. In the paper I also provide examples of how how to apply these measures to standard microeconomics and nance problems. I would like to thank participants at a seminar at the University of Notre Dame for their very useful comments to an early version of this research. y Department of Economics, Cal Poly, 1 Grand Ave., San Luis Obispo, CA ezambran@calpoly.edu. Website: 1

2 Risk 1: Possibility of loss or injury. 2: Someone or something that creates or suggests a hazard. 3 a: The chance of loss or the perils to the subject matter of an insurance contract. b: A person or thing that is a specified hazard to an insurer. c: An insurance hazard from a specified cause or source. 4: The chance that an investment (as a stock or commodity) will lose value. -Merriam Webster Dictionary 1 Introduction It goes without question that, in a world plagued with uncertainties, an accurate measurement of risk is essential for the proper allocation of resources at the individual and the societal level. In order to accomplish this task economists and other social scientists have used distinct measures of dispersion developed by statisticians over the years. A problem with this approach is that the most widely used measures of statistical dispersion (such as the standard deviation, the variance, the expected absolute deviation and the interquartile range) su er from the inconvenient of not being monotone with respect to stochastic dominance and hence may be inadequate for aiding the decision making process of risk averse individuals. To see what the problem is consider an asset g that pays $120 and -$100 with equal probablity. One could ask: what is the risk in accepting g? This asset has standard deviation $110 and it is a common practice to identify the risk of g with this number. Now, consider the asset h that pays $185 with probability.8 and -$90 with probability.2. The chances of losing money from owning h are much smaller than those from owning g. Moreover, one faces smaller loses, when losing, from owning h rather than g: A casual application of any of the de nitions of risk o ered by the Merriam Webster dictionary (as in the quote above) would rank h as less risky than g yet the standard deviation criterion ranks h to be as risky as g since the standard deviation of h is also $ To address this problem Aumann and Serrano (2008) and Foster and Hart (2008) have recently developed measures of riskiness that are meant to be universal (apply to any asset) and objective (apply to any decision maker) and that are based on economic and decision theoretic considerations (in particular, that they are monotone with respect to stochastic dominance). According to Aumann and Serrano (AS), if g is a risky prospect and E is the expectation operator one can measure the risk of g as the number R that solves Ee g=r 1 = 0: 1 Technically, h rst order stochastically dominates g and this lower degree of risk of h relative to g is not detected by the standard deviation of those assets, as they are the same. 2

3 In turn, Foster and Hart (FH) would have us measure the risk of g as the number R that solves Elog(1 + g=r) = 0: Foster and Hart s measure can be thought of as a reservation wealth level, a level of wealth below which one risks bankruptcy by repeatedly accepting g, above which one guarantees avoiding bankruptcy with probability one even as one repeatedly accepts g. Aumann and Serrano s measure is harder to interpret along operational lines (and it is indeed one of the purposes of this paper to provide such operational interpretation). To better understand the AS and FH measures I decided to see what would happen if, instead of trying to come up with an objective measure of riskiness, we fully allow our measurements to be a ected by the particular attitudes towards risk of the decision makers. I develop such measures here and then compare them with the objective measures proposed by AS and FH. We learn the following: on the one hand, the subjective measures include the AS and FH measures as special cases; on the other, the subjective measures can be derived from the objective ones in the same way that wind chill temperature factors can be derived from the objectively measured air temperature. In this speci c sense the research presented in this paper strongly complements the work by AS and FH. Let u i be an arbitrary Bernoulli utility function and let w i (R) be the wealth that makes this individual have an absolute risk aversion coe cient of 1=R (for the moment assume this can always be done). To measure the riskiness of g for this decision maker I propose that we nd the number R that solves Eu i (w i (R) + g) u i (w i (R)) = 0: We thus obtain one such number R i (g) for each asset g, for each decision maker i. As the standard deviation, any such measure will be measured in the same units in which g is measured but, unlike the standard deviation, the subjective measures will be monotone with respect to rst and second order stochastic dominance. In the paper I establish some other properties satis ed by R i (g) : In particular, I provide a behavioral interpretation of those measures: they can be thought of as a reservation slope of the demand for the asset, a measure of how sensitive the individual demand for the risky asset has to be to a drop in price for the individual to want to hold the asset in his or her portfolio. This interpretation holds for all such measures. In particular it will hold for the AS and the FH measures as well, as they can be derived from a suitable 3

4 application of the principles developed above to individuals with CARA and log preferences, respectively. Once we have a way to measure the riskiness of a portfolio that is tailored for a decision maker one can include those measures as part of more elaborate decision theoretic exercises. To illustrate this I show in the paper how changes in the price of an asset will a ect the riskiness of the optimal portfolio for a decision maker and how to decompose the e ect of a price change on the demand for the asset into a risk e ect (the change in demand due to the price change keeping the riskiness of the portfolio constant) and a more familiar wealth (or income) e ect. I also illustrate how one would use the measures in applications by computing the riskiness of some popular zero cost portfolios, such as those studied by Fama and French (1993). Other important questions that have yet to be studied include the proper way of accounting, in the riskiness measures, for multiple assets and their comovements and what, if anything, all of this has to contribute to the di erent theories of asset pricing. All that is left for future work. 2 Background An asset g is a random variable with real values some of which are negative, and that has a positive expectation. Random variables are denoted in bold letters. I consider expected utility maximizers with Bernoulli utility functions u i de ned over the entire real line that are twice continuously di erentiable, strictly increasing, and concave. Let w be a real number, interpreted as a wealth level. The coe cient of (absolute) risk tolerance of a decision maker i is de ned as the reciprocal of the Arrow-Pratt coe cient of absolute risk aversion, that is: R i (w) = Individual i accepts g at w if Eu i (w + g) > u i (w): u0 (w) u 00 (w) : 2.1 Aumann, Serrano Individual i is more risk averse than individual j if for all possible wealth levels w i and w j for i and j, j accepts at w j all the investments than i accepts at w i, but not the other way around. Consider investments g and h such that whenever a less risk averse decision maker rejects g, a more risk averse decision maker rejects h. Aumann and Serrano will then call asset h more risky than asset g. An index of riskiness Q(g) is homogeneous of degree one if Q(g) = tq(tg). One then has the following: Theorem 1 (AS, 2008) For each asset g there is a unique positive number R(g) with Ee g=r(g) = 1. Then the index R AS thus de ned satis es the riskiness order and is homogeneous of degree one. Moreover, any index satisfying these two principles is a positive multiple of R AS. 4

5 Aumann and Serrano show, among other things, that this index is measured in the same unit as g is measured and that it is positive with respect to rst and second order stochastic dominance. They characterized this index in terms of CARA preferences in their Theorem B as follows. Theorem 2 (AS, 2008) R AS (g) is the coe cient of absolute risk aversion of a CARA decision maker who is indi erent between accepting and rejecting g. Example 1 The asset g discussed in the introduction that pays $120 and -$100 with equal probablity has an AS-riskiness given by R AS (g) = 601:66: In turn, the asset h that pays $185 with probability.8 and -$90 with probability.2 has AS-riskiness given by R AS (h) = 57:04 and so according to the AS criterion h is less risky than g. While R AS (g) is the riskiness of g one may wonder what operational interpretation, if any, one may give to such measure. Foster and Hart s research started as an attempt to provide such an interpretation, and their work led to a di erent measure of riskiness. 2.2 Foster, Hart Consider again the asset g that pays $120 and -$100 with equal probablity. What is the risk in accepting g? Foster and Hart argue that the risk of g clearly depends on the wealth of the individual who owns the asset. After all, if one s wealth is $100 one risks bankruptcy by accepting g. On the other hand, if one s wealth is, say, $1,000,000, the risk of g is very low indeed. Moreover, in this case one s wealth would grow arbitrarily large if one was exposed repeatedly to g (as it has a positive expected value). This line of reasoning led Foster and Hart to the following: Theorem 3 (FH, 2008) For each asset g there is a unique positive number R F H (g) with Elog(1 + g=r F H (g)) = 0 such that: To guarantee no-bankruptcy, when one s wealth is w, one must reject all assets g for which R(g) > w. Example 2 The asset g that pays $120 and -$100 with equal probablity has an FH-riskiness given by R F H (g) = 600: In turn, the asset h that pays $185 with probability.8 and -$90 with probability.2 has FH-riskiness given by R F H (h) = 91:08 and so according to the FH criterion h is less risky than g. Foster and Hart show, among other things, that this index is measured in the same units as g is measured, is monotone with respect to rst and second order stochastic dominance and that it can be characterized in terms of log preferences as follows. Theorem 4 (FH, 2008) R F H (g) is the wealth level that makes a log decision maker indi erent between accepting and rejecting g: A version of Aumann and Serrano s Theorem B immediately follows: 5

6 Corollary 5 R AS (g) is the coe cient of risk tolerance of a log decision maker who is indi erent between accepting and rejecting g. All proofs are relegated to the Appendix. We now have two measures of riskiness that are both meant to be objective and universal and that share many desirable properties among which prominently stands the fact that both are monotone with respect to stochastic dominance. The Foster and Hart measure, however, has the advantage that it has a clear operational interpretation, whereas such interpretation had not, until now, been obtained for the Aumann and Serrano measure. It turns out that to develop such interpretation it is convenient to rst de ne measures of risk that are subjective and universal, namely, that take into account the individual perceptions of risk of di erent decision makers. 3 Subjective measures of risk We noticed above that the AS and FH measures are related to CARA and log preferences in a similar way, via Aumann and Serrano s Theorem B and Corollary 5 in this paper. One can then use the property identi ed in the Theorem to de ne, for an arbitrary decision maker, the riskiness of g. This idea has antecedents in Serrano and Palacios-Huerta (2006) and Zambrano (2008). Serrano and Palacios-Huerta noticed that when decision makers consider an asset too risky (in that they would never accept it) this implies a lower bound for the absolute risk aversion for the individual, and hence such bound could be thought to contain information about the riskiness of the asset. Zambrano (2008) in turn noticed that if one knows the decision maker s certainty equivalent CE for the asset and his tolerance for risk, then one could derive an upper bound to the individual s subjective beliefs over tail events, namely, to how pessimistic the decision maker must have been when he deemed the asset to be worth CE. It takes but a step from these notions to identify, for these decision makers, the riskiness of an asset with the risk tolerance level that would make their certainty equivalent for the asset equal to zero. Consider some asset g and a decision maker i with Bernoulli utility function u i. One could argue that g is too risky for i if Eu i (w + g) < u i (w) for all w and not risky at all for i if Eu i (w + g) > u i (w) for all w. The remaining case is one in which there is a wealth level that would make this individual indi erent between accepting and rejecting g, namely one in which there is w i (g) such that Eu i (w (g) + g) = u i (w (g)) for this decision maker. 2 Now de ne the riskiness R i (g) of asset g for individual i in this case to be equal to R i (w i (g)) ; that is, equal to the level of risk tolerance associated with the individual being indi erent between accepting and rejecting g. To summarize, de ne R i (g) as follows: 2 This number w i (g) is not to be confused with the certainty equivalent of g. The certainty equivalent of g; at wealth level w; is de ned in this context as the number CE i (g;w) such that Eu i (w + g) u i (w + CE i (g;w)): From this it follows that w i (g) is the number that makes the certainty equivalent of g; at wealth level w i (g) ; equal to zero. 6

7 8 < 0 if Eu i (w 0 + g) > Eu i (w 0 ) for all w 0 R i (g) = 1 if Eu i (w 0 + g) < Eu i (w 0 ) for all w 0 : R i (w i (g)) for w i (g) such that Eu i (w (g) + g) = Eu i (w (g)) It is important to keep track of the di erence between R i (g) and R i (w): R i (w) is a measure of the tolerance for risk of individual i at wealth w and R i (g) is the riskiness of g for individual i: how high the risk tolerance of the individual has to be for the decision maker to want to hold g. The measure R i (g) shares many of the properties that R AS (g) and R F H (g) have. For example, R i (g) is measured in the same units as g is measured and Theorem 6 R i (g) is monotone with respect to rst and second order stochastic dominace. Moreover, it is straightforward to show that R i (g) = R AS (g) for the CARA decision maker with risk tolerance coe cient R AS (g) and R i (g) = R AS (g) for a log decision maker. De ning the riskiness of an asset in terms of the critical risk tolerance that would lead the individual to accept this asset in his portfolio changes somewhat the answer to a question originally posed by Pratt (1964, p.126): The local risk aversion function r i (w) [:= 1=R i (w)] associated with any utility function u i contains all essential information about u i while eliminating everything arbitrary about u i. Pratt wondered if, because of this, one could eliminate u i from consideration in favor of r i : Pratt s answer to this question went as follows: Decisions about ordinary (as opposed to small ) risks are determined by r i only through u i (...), so it is not convenient entirely to eliminate u i from consideration in favor of r i. I note here that, once R i (g) is computed for every asset g one could, in principle, dispense with u i ; as decisions about ordinary risks can be determined now by a direct comparison between R i (w) and R i (g): 3 The following Theorem records this fact. Theorem 7 Individual i accepts g at w if and only if If R i (w) > R i (g): Example 3 Consider a decision maker with constant relative risk aversion, with relative risk aversion coe cient equal to 3 (call this individual 3 ) For the asset g that pays $120 and -$100 with equal probablity R (3) (g) = 600:37: For the asset h that pays $185 with probability.8 and -$90 with probability.2 3 Of course, it will still not be convenient to eliminate u i from consideration as R i (g) is determined through u i (in the same way that the certainty equivalent and the probability premium of g are also determined through u i ). : 7

8 R (3) (g) = 59:77: This illustrates the content of Theorem 6, as h rst order stochastically dominates g. Moreover, Table 1 below shows the tolerance for risk and the certainty equivalents of g and h for di erent wealth levels in a way that illustrates the content of Theorem 7: for wealth levels that lead to R (3) (w) < [>]R (3) (g) the individual rejects [accepts] g, and similarly for h. w R (3) (w) CE (3) (w; g) CE (3) (w; h) :81 54:18 179:31 59:77 70: :20 95: :11 600: : :98 126:93 Table 1: A CRRA(3) individual E(g) = 10, E(h) = 130, (g) = (h) = Comparisons across risk measures Consider again asset g as in the example from the Introduction. We determined previously that R AS (g) = 601:66, R F H (g) = 600 and R (3) (g) = 600:37. That these measures are all similar for this asset is no coincidence: Theorem 8 Assume that u i is C 1 and let T (g; u i ) be a function of the sum of third-and-higher order terms in a Taylor series expansion of Eu i (w (g) + g) P 1 1 u i (w (g)) around w (g) ; that is, T (g; u i ) = u 00 i (wi(g)) n=3 1 n! u(n) i (w i (g)) E (g n ). Assume that 0 < R i (g) < 1. Then R i (g) = 1 E g 2 2 E(g) + T (g; u i) : When those higher order terms are small R i (g) 1 E(g 2 ) 2 E(g) =: R 0 (g) : Indeed, for asset g in the example above we have that R 0 (g) = 610: An immediate corollary of this result is that all the measures of riskiness discussed in this paper can be derived from each other. Corollary 9 Let u i and u j be two C 1 Bernoulli utility functions. If 0 < R i (g) < 1 and 0 < R j (g) < 1 then R i (g) = R j (g) + [T (g; u i ) T (g; u j )] : Corollary 9 can be used in applications as follows: Suppose we have determined that R AS (g), R F H (g) (or another measure R j (g)) is the objective measure of riskiness. One can then use Corollary 9 to compute, out of the objective measure, the corresponding riskiness measures for di erent decision makers. 4 4 Notice that Corollary 9 is silent as to which is the objective measure of riskiness: it will work with any measure R j (g) as the benchmark. 8

9 Remark 1 In Theorem 8 the term R 0 (g) := 1 E(g 2 ) 2 E(g) occupies a prominent role as it is the common component in the computation of the riskiness measures for all decision makers. For this reason, it is tempting to dub it the objective component of any R i (g): Unfortunately, as Foster and Hart noted in their paper, R 0 (g) is not monotone with respect to rst order stochastic dominance. For example, the asset f that pays $200 and -$100 with equal probablity has R 0 (f) = 250:whereas the asset m that pays $350 and -$100 with equal probablity has R 0 (m) = 265 even though m rst order stochastically dominates f. Remark 2 While R 0 (g) may not be the best candidate for an objective measure of riskiness, it is not without interest. Consider a decision maker with quadratic Bernoulli utility function given by u q (w) = w w 2 : This function is increasing as long as w < 1 2 : To de ne the riskiness function R q for this decision maker it is important to notice that there will always be assets m for which Eu i (w 0 +m) > Eu i (w 0 ) whenever w 0 + m < 1 2 : In this case, as before, R q(m) = 0. For any other asset f, it turns out that R q (f) = R 0 (f) and, for as long as this is the case, the measure R 0 will be monotone with respect to stochastic dominance. Example 4 For the assets m and f de ned above we have that R q (f) = 250 and R q (m) = 0; which respects stochastic dominance. Also, for g from the Introduction and f as de ned above we have that R q (g) = R 0 (g) = 610 and R q (f) = R 0 (f) = 250, which also respects stochastic dominance. 4 An operational interpretation of R i (g) So far we know that the intended interpretation of R i (g) < R i (h) is that asset g is less risky for individual i than asset h. In the case of individuals with log utility we know, thanks to the work of Foster and Hart, that R F H (g) also stands for the reserve wealth level such that, if the individual s wealth is below R F H (g) then he should not accept this asset if he wishes to avoid any risk of bankruptcy. Then the question follows: is there any operational interpretation we may give to any of the other measures studied in this paper, including R AS (g)? To investigate this matter I formulate and solve a standard portfolio optimization problem as follows: Imagine that shares of a large asset g are o ered for sale to the public ( is the standard deviation of g). Let x be the number of shares of g the individual with Bernoulli utility function u i buys at a price p per share. As before, let R i (w) = is u0 (w) u 00 (w) max Eu i x : The decision problem for the individual g p w + x with solution x i (p; w) : Clearly, if those shares of g are o ered at expected value, the demand for shares will be zero. A question follows: what would the demand for shares be if the price were to drop slightly from expected value? = x=0 R i (w) 9

10 Thus, R i (w) gives us exactly the slope (at the origin) of the demand function for shares of g for individual i and wealth w. In other words, it tells us how much exposure to g the individual would want, at the margin, if g wasn t priced exactly at fair value. This suggests the following operational interpretation for R i (g): how sensitive the demand for g must be to a small drop in price (from expected value) for the individual to want to hold the entire issue of g (at the current price). In other words, it tells us exactly the required marginal exposure to g (at the origin) that would make the individual to want to own (all of) g. Measure R i (g) is thus a reservation slope of the demand for g. This interpretation makes sense especially in light of Theorem 7 as one can say that R i (g) = the marginal exposure to g at the origin that would make you want to own all of g. R i (w) = how much marginal exposure to g at the origin you actually want. If R i (w) {z } desired marginal exposure < R i (g), reject the investment {z } required marginal exposure The polar cases R i (g) = 1 and R i (g) = 0 also t the interpretation: R i (g) = 1 means the demand for the asset has to be in nitely elastic for the individual to want to hold g. R i (g) = 0 means that any price drop below expected value will make the individual want to hold all of g. To summarize, make R i (g) operational as follows: 1. Identify the smallest wealth level that makes the individual indi erent between accepting and rejecting g (If the individual always wants the asset, let R i (g) = 0; if the individual never likes the asset, let R i (g) = 1;and go to step 3). 2. At that wealth level, determine how much exposure to g (in units of ) the individual would want if the price of g dropped slightly from expected value. 3. Call the resulting number the riskiness R i (g) of g for individual i. This procedure will yield all the riskiness measures developed in this paper and, in particular, will yield the measures R AS (g) and R F H (g) for the CARA and log utilities, respectively. 5 Application: Wealth and Risk e ects of a price change Consider now portfolios of the form w + x (g p). The interpretation is that the individual has wealth w and that he can buy x units (or shares) of asset g that are sold at a price of p per share. Let x i (p; w) be the individual demand 10

11 for g that maximizes the individual s expected utility. Now consider a drop in the price of g from p 1 to p 2. It follows that the riskiness of the portfolio will also drop, as (g p 2 ) rst order stochastically dominates (g p 1 ) : This will entice the individual to take more risks. At the same time, a drop in the price of g makes the individual wealthier (in the standard manner in which a price drop has an income e ect), and this may or may not entice the individual to take more risks. The implication is that one can decompose the change in quantity demanded due to a change in price of g into a wealth e ect and a risk e ect, as in the Slutsky equation. Details follow. For simplicity I work in this Section with individuals that have risk tolerance functions R i (w) that are invertible, with inverse given by w i (R): The non-invertible cases (which include individuals with CARA preferences) can be handled analogously. Let D be the domain of w i (R). The total e ect of a drop in the price of g is standard and is : To decompose this e ect into it s wealth and risk e ects recall how to measure the riskiness of the optimal portfolio [x i (p; w) (g p)]: nd R that satis es Eu i [w i (R) + x i (p; w) (g p)] = Eu i (w i (R)); and call R the riskiness of [x i (p; w) (g p)] as before. Conversely, given R 2 D, one can nd the portfolio for this decision maker that would have riskiness R. Such portfolio [h i (p; R) (g p)] is de ned by u i (w i (R)) Eu i [w i (R) + h i (p; R) (g and it is the optimal choice for a decision maker with wealth w (p; R) ; as de ned by x i (p; w (p; R)) h i (p; R): It follows that The interpretation is as intended: the total e ect on the quantity demanded of g from a drop in p is and this e ect can be decomposed into a risk e ect and a wealth e ect. The risk e ect arises because a price drop will decrease the riskiness of owning units of (g p) in the sense of rst order stochastic dominance and this will allow the individual, if he wished, to hold more units of g without increasing the riskiness of the portfolio. The change in the demand for g that would keep the riskiness of the portfolio constant is given by and it can be shown that this derivative is always negative. The wealth e ect is more standard: it combines the e ect of wealth changes on the demand for @w ; with a of how much wealthier the individual appears to be (net of the risk e ect) as a consequence of the drop in p. 5 5 Notice that this wealth e ect is not entirely 6= x i as in the Slutsky equation. This is so because w (p; R) is not obtained as the solution to an expenditure minimization problem and so the Envelope Theorem does not apply. p)] 11

12 Remark 3 As is known, the demand for shares of g is zero when the shares are priced at their expected value, regardless of wealth is also zero at that price, and thus the demand that follows a slight decrease in price from expected value can be thought to be entirely driven by the risk e ect of a price change. 5.1 Application: The risks behind some popular zero cost porftolios In this Section I illustrate how one would go about computing the riskiness of the three zero cost portfolios known as the three Fama-French factors according to the riskiness measures discused in this paper. The portfolio [R m R f ] borrows at the risk free rate (a Treasury bill rate) and invest the proceeds on all NYSE, AMEX, and NASDAQ stocks. The other two portfolios are formed from subsamples of this universe of stocks: Portfolio SMB shorts large companies and invest the proceeds on small companies. The portfolio HML shorts growth companies and invests the proceeds on value companies. See Fama-French (1993) for a complete description of the portfolios. I use annual data from the period , obtained from Ken French s website, to compute R AS, R F H and R (3) for each of these portfolios. I use the empirical distribution of payo s for these assets to compute the expectations that de ne the measures in question. For comparison purposes I also present some traditional measures of riskiness for these assets, such as the sample standard deviation, the sample variance, the interquartile range, etc., alongside with the common factor R 0, which is presented so that the reader can see how all the subjective riskiness measures are related to one another through R 0. Table 2 presents those results. From the examination of the table it becomes clear that di erent measures will rank the zero cost portfolios in di erent ways. R AS and R (3) agree with the standard deviation and the variance criteria in ranking R m R f as most risky and HML as least risky. On the other hand, R F H agrees with the V ar(5%); the interquartile range, Q 3 Q 1 ; and the expected absolute deviation, Ejg Egj; in ranking R m R f as most risky and SMB as least risky. The inverse Sharpe ratios, = and 2 = and are in a category of their own in that they do not agree with any other measure, not even with each other. 12

13 R m R f SMB HML = = M in V ar(5%) Q 3 Q Ejg Egj R R AS R F H R (3) Table 2: The riskiness of the Fama-French zero-cost portfolios. Returns computed at an annual frequency for the period Remark 4 It would be interesting to combine the approach to risk measurement developed in this paper with a judicious estimation of the tails of the probability distributions that generate those returns in light of the fact that (a) those tails tend to be poorly represented in the empirical distribution, and (b) small changes in the estimation of the left tail of those distributions are bound to have important e ects on the riskiness measures, as those measures are most sensitive to large losses. That is also left for future work. 6 Concluding comments Much remains to be done in the way of characterizing and testing, in applications, which riskiness measures among the ones discussed in this paper are the most sensible. One thing is clear, however: the argument in favor of using statistical dispersion measures (such as the standard deviation) as the prime measures of risk for lack of better alternatives is no longer tenable. 7 References Aumann R., and R. Serrano, An Economic Index of Riskiness, working paper, March, Fama, E. and K. French, Common Risk Factors in the Returns on Stocks and Bonds, Journal of Financial Economics,1993. Foster, D. and S. Hart, An Operational Measure of Riskiness, working paper, May, Pratt, J., Risk Aversion in the Small and in the Large, Econometrica,

14 Serrano R., and I. Palacios-Huerta, Rejecting Small Gambles Under Expected Utility, Economics Letters, Zambrano, E. Expected Utility Inequalities, Economic Theory, July,