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1-11Actim n/R517rt n5353.1 iprrn5 T3inn THE CENTER FOR AGRICULTURAL ECONOMIC RESEARCH Mean-Gini, Portfolio Theory and the Pricing of Risky Assets by Haim Shalit and Shlomo Yitzhaki Rehovc:'.1, Israe!, P.O.B. 12 12 .1.11 The working papers in this series are preliminary rmi on lt n-rptn -ipmon nroNn and circulated for the purpose of discussion. The nwrn .nvwn n52p) )1)15 »lwx-1 views expressed in the papers do not ref:ea those nx of the Center for Agricultural Economic Research. .n,x5pn n5D5D2 -Ipm35 111017V/3 13,1•4 0n2 nwainn iD-Inn nw-r -. Mean-Gini, Portfolio Theory and the Haim Shalit and Shlomo Yitzhaki Rehovot, February 1982 MEAN-GIN:, PORTFOLIO THEORY, AND THE PRICII\,IG . OF RISKY . ASSETS . by Haim Shalit:.and ShloMo * Yitzhaki The Hebrew University of Jerusalem • ABSTRACT ' This paper presents the Mean-Gini (MG) approach to analyze risky prospects and construct optimum portfolios. The method possesses the simplicity of the mean-variance model with the efficiency of stochastic dominance. Hence, Gini's mean difference is superior to the variance for evaluating the variability Of a prospect. The analysis is further extended with the concentration ratio that permits to classify different securities with respect to their relative riskiness. The MG approach is then applied to capital markets and the security valuation theorem is derived as a general relationship between average return and risk. This is further 'extended to include degree of 'risk aversion. that can be estimated from capital market data. This research was supported by a grant from the Government of Israel, the National Committee for Research and Development, Grant No. 5107. We ,are grateful to Haim Levy for useful errors are ours. comments. All the remaining MEAN-GINI, PORTFOLIO ANALYSIS, AND THE PRICING OF RISKY ASSETS The formal similarity between modelling decision-making under uncertainty and evaluating and interpreting income inequality has been noted and used by several economists. An example is Samuelson who used the same method to prove that it pays to diversify risky investment (1967), and that equal distribution of income among identical Benthamites.maximizes the sum of social utility (1966). dominance Atkinson (1970) showed that ,the rules for stochastic which were developed as criteria for evaluating risky invest- ments, can be translated into Lorenz-curve terms in evaluating income inequality. The purpose of our paper is to interpret some of the recent results on income inequality and apply them o portfolio analysis. In particular, we establish that Gini's mean difference, the Lorenz curve, and the concentration ratio, which are extensively used in the field of income inequality, can also be used to characterize risky prospectsand construct optimum portfolios. The superiority of Gini s mean difference over the variance as an index of variability has been demonstrated by Yitzhaki (1982). In parti- cular, it was shown that using the mean and Gini's mean difference as summary statistics of the distribution of a risky investment permits the user to derive necessary conditions for stochastic dominance, enabling him to discard from the efficient set prospects that are stochastically dominated by others. This property means that Gini's mean difference is a better candidate for evaluating the variability of a distribution. In the present and paper, we argue that Gini's mean difference can replace the variance ce needed that the concentration ratio based on it can replace the covarian in portfolio theory. .ance The mean-Gini portfolio selection rule is - superior to the mean-varl of selection rule in the Sense that the mean-Gini efficient set consists be portfolios that, according to stochastic dominance criteria, cannot ariance dominated by any other portfolio. . Furthermore, as with the mean-v of' risky model, the mean=Gini ranking can be. used to derive the pricing' assets in an equilibrium framework. But it is easier to compute efficient simpler portfolios by mean-Gini rules than by mean-variance rules and much than using stochastic dominance rules. Moreover, Gini's mean difference can from be extended into a family of coefficients of variability, differing each other by a parameter, that represents the investor's risk aversion. It turns out that each member of this family can be used in portfolio -) theory for constructing capital-asset pricing models. In the first section, we define Gini's mean difference and justify its use. d In the second, the properties of a portfolio which is specifie third section by its mean and its Gini coefficient are developed, while the presents some extensions of the Gini coefficient to portfolio theory, such as the concentration ratio (Kakwani, 1977, Pyatt Chen, and Fei, 1980). The fourth section is devoted to the capital-asset pricing model and contains a discussion of the properties of the approach, whereas the last section presents the extended Gini's mean difference (Yitzhaki, 1980) and applies it to portfolio analysis. Much of the discussion in this paper is based on a reinterpretation of results on income inequality, in the portfolio context. Thus, whenever possible we do not prove the propositions but refer the reader to the _original papers. GINI'S MEAN DIFFERENCE I. bility of a random variable. mean difference is an index of the varia of the absolute difference between every It is based on the expected value * ble. That is, let F(y) and f(y) pair of realizations of the random varia distribution and the density function respectively represent the cumulative and b < co such that exist a > 0O of prospect y and assume that there difference is defined as follows:1 F(a) = 0, F(b) = 1, then Gini's mean 1 r = 2 (1) fa , a y - xlf f(y)dxdy . e and one finds at least eight This definition is not easy to handl mean difference in the literature. different formulations of Gini t s with two of them. our purposes it will be useful to deal F(y)].:137 a For The first is a (2) [1 - F(Y)] Y rence in terms of the expected Equation (2) presents Gini s mean diffe cumulative distribution function, value of the distribution, p, and its F(y). 3 ng with stochastic dominance criteria. This equation is important when deali for analysing portfolios is The other Gini formula which is useful r = y[F y 2 a _ 2 f(y)dy •• that is, Gini's mean difference is twice the covariance of variable its cumulative distribution. y a For completeness, we state the following pro- perties of Gini's mean difference: 1. If The Gini coefficient is non-negative and bounded from above by y is a given constant, F = 0 . p - Furthermore, as is shown in Yitzhaki (1980 ), its maximum value is reached for the distribution 2. - The - Gini coefficient is sensitive to mean-preserving spreads Atkinson, 1970). ay where . Let where 5. Let , where variables; then . Let y = a//if 3- is a constant; then 2 is a constant, then y 1 and y . are two independently distributed + r be a normally distributed variable with p and a 2 then (Nair, 1936) •• Properties 2 to 5 are similar to those attributed to the standard deviation. Hence it is not surprising that the Gini coefficient can be used to derive the efficient set of uncertain prospects in the same way as is done using the mean-variance criterion. This feature is implied-by the behavior of investors. 'who rank uncertain 'prospects by their mean and the dispersion of returns. Efficient sets of uncertain prospects ,,re •• in the construeted such that no other feasible prospect will be included mean for set unless it has a lower dispersion for a given mean or a higher given dispersion. Usually the standard deviation is used as the measure . f dispersion; we propose to use instead Gini's, mean difference- Hence, obtained by the 'efficient set that answers the mean-Gini (MG). criterion is . finding, for each given mean that prospects xahich are not in the efficient set have at least larger or equallGini's mean difference. If combinations of I is obtained by uncertain prospects are allowed to be held, the efficient set the Gini constructing, for each given mean, a mix of. prospects that minimize coefficient of that portfolio. We advocate the use of the mean-Gini (MG) t' method first because, if prospects are normally distributed, the efficien iance set of the mean-Gini is identical to the efficient set of the mean-var (MV) Method (see property 6 above). Secondly, it is justified by the superiority s n f the mean-Gini over the mean-variance 'approach in ranking uncertai prospect according to stochastic dominance .(SD) rules as we will now bring forward. Let y l and y2 be two uncertain prospects. Proposition p 1 - r1 a p 2 - F 2 y is a.necessary condition for Lo first and second stochastic dominance rules 1 The condition to dominate Yitzhaki, 1982). y 2 according See Appendix for a proof. Define SmG the efficient set obtained by the MG method, the efficient set obeying the stochastic dominance first and second rules and obeying proposition 1, S is also a subset of we'assert the following: S is a subset o the set Scm and S MG .• • •••• Mg 'Hence, applyjng.propp,pitionl:to the efficient set constructed by the method enables. us obtain an .efficient set which.is a subset of theefficient.. set according to first and second SD rules. This subset is not liable to 'the criticism usually advanced against the efficient set constructed by the MV rule (See 'Rothschild and Stiglitz, 1970; of proposition 1 can be demonstrated by an example: uniformly distributed between between 2 and 4 . Both 0 and and yl 1 y2 to mean-variance and mean-Gini rules, and 1 2 = 3 r ' = 2 while y2 assume that is y1 is uniformly distributed are in the efficient set according with p 1 = 1 2 r 1 1 6 1 1 But clearly all investors prefer 1 3 The use Hanoch and Levy, 1969). y2 over • Applying proposition 1 to the set 'obtained by the mean-Gini criterion enables us to discard ' 1 from the efficient set. The superiority of the mean-Gini approach over the stochastic dominance (SD) criteria results from its similarity to the MV method. As far as we know, there is no method for constructing optimum portfolios by 'SD rules. Users of the mean-Gini method may minimize the Gini, coefficient of a linear combination of prospects subject to a given required mean return. the user to construct the efficient set Changing the mean permits corresponding to the mean-Gini criterion. This set canbe used for portfolio analysis and capital asset pricing equilibrium according to MG rules. Furthermore, by applying proposition 1 to that efficient set, a subset is obtained which is contained in the efficient set of portfolios according o SD rules. . PORTFOLIO ANALYSIS portfolio whose performance is summarized by the mean The properties of to those of. the regular meat-Standard. and. the Gini coefficient These, properties can be illustrated by the familiar .deviation model. textbook diagram in which r, the Gini coefficient of the return on the portfolio, is depicted on ,the horizontal axis, while, the mean of the - p one-period return, is on the vertical axis, as in Figure 1. The performance of two prospects, in the mean-Gini space. The return on portfolio combination .of: the return on is the share of is denoted by and B A and A wealth invested in Hence and and B is given by the convex y y A = aXA -17 (1 - a)X15-Where. and XR are the :one-period.. returns. As in the ordinary mean-standard deviation model the performance of depends also on the correlation between the portfolioY p on A and B and a . •To show this, consider the three special cases as drawn in Figure First, if prospects of correlation (p A AB) is and B are linearly dependent, the coefficient equal to unity and, following from the properties of Gini's mean difference, the line ACB represents all the possibilities of a portfolio mix composed of A . - 0), the curve pendent ( ' P AB y portfolio, performance. if A and B and B . ADB expresses the performance of the Second, if A and B are inde- showing intuitively that diversification improves that Furthermore, the portfolio returns would be much improved were negatively correlated as shown by the broken line for the extreme case of . 1 AFB 10 • The effect of the variability of a prospect on the variability of the portfolio can be presented much as it is in the mean-standard deviaijon. By equation (3), the Gini coefficient of a portfolio i model. y F (4) a = y Y is the cumulative distribution of the portfolio. F where -dF y) = 2cov p 2 for E a x. 1=1 i 1 0 i= is the return on prospect where . Since 1 , we Obtain • • E a.cov[x., F yji p 1 1 i=1 (6 that is, the risk ofthe portfolio can be decomposed into' a weighted sun) of the covariance between the variables x and the cumulative distribution i of the portfolio, p. It is worth mentioning that the variance of the portfolio can be written as (7) var Y P a cov x , y i=1 - 11 • and the difference in the decomposition of the nondiversifiable risk by the two methods is that i (6) the portfolio isrepresented by the cumu- lative distribution of its. retui-ns,. F', while in (7) it is represented by its returns By multiplying and dividing each component of (6) by where (8) where - 2covk -F.( ' F (x') is the cumulative distribution of prospect i, we obtain i N Z. .a.R.r. - iii R. = cov[xi, Fp(yp)J/cov[xi F (x)I which represents the ratio non-diversifiable risk tothe variability of prospect 7 r. = )] - 12 - THE CLASSIFICATION OF PROSPECTS BY RELATIVE RISK III. , F (y )] One can classify a security by its risk by using cov[xi p p as an whereas the diversiindex of the undiversified risk carried by a security i by including fied risk is the share of total risk that can be reduced in the portfolio security i - However, this classification is silent yp about the possibility that security I can be riskier than security in one situation and less risky in another. n curves ne way to improve the classification is' o'use concentratio on us to determine the security's risk according to the return that enable the portfolio.5 ' prospects prospect •I ine, for any two This classification enables us to determ within portfolio A and B A p , whether all investors agree that • is riskier than B with regard to portfolio relative riskiness of may disagree among them about the p A. or that they Assume two ed rates of return; securities with identical positive expect expectation of rate of return ( 9) g.(y ) as the conditional J P „ given the portfolio y ; that is, x Now define the function E (x ) = E (x ) = p J y) = E g(y) j jp JP We assume that > y P exists. If E[g.(y )] J P security j , g.(y). > 0 J P - and that the first derivative of g( ) tration curve of = P. , one can define the concen as the relationship between 13 - Y)f(Y)dY (10) f (y)dy . = F y P P P Concentration curves are plotted in Figure 2 below. seen that if x. = It can easily be y, equation (10) represents the Lorenz curve of port- .) folioyp (curve B in the figure). The relative riskiness of the securities in a portfolio can be compared according to the following proposition: Proposition 2:7 The concentration curve for the function above (below) the concentration curve for the function less (greater) than f r all, y ,) where!:n g (y ) will b P g (y ) ifn.(Y p) is i p is the elasticity of with reapect to Two corollaries follpw'from proposition corollaryl:neconcentrationcurveforthefunction-W gJ (below) the egalitarian line (45 line) if (y) will be above is less.(greater) than zero, for all .y . That is, in Figure 2, the 45degree line represents the risk free asset. Stocks which are always negatively correlated with the portfolio have a 0 concentration curve which is above the 45 -degree line. 14 ••• The second corollary permits comparison between the securities and, the portfolio they make up. Corollary 2: The concentration curve for the function (below) the Lorenz curve for the distribution (greater) than unity for all y F (y ) if P P .(y) lies above 1-1.(y ) is' less .3 P > 0' Corollary 2 permits us to distinguish between two kinds of securities in a , portfolio First, we observe aggressive securities, with concent- ration curves below the Lorenz curve of the portfolio, whose high.degree of responsiveness to the .portfolio leads to considerable instability. Second, we have defensive securities, with concentration curves above the Lorenz curve of the portfolio which reduce instability because they are less responsive.. These results are summarized Figure .2. Let OAB be the Lorenz. curve of the portfolio. The aggressivestock will b represented by OCA while the • defensive stock is represented by the concentration curve ODA. 0 The 45 line portrays the risk-free asset (if it exists) while OFA represents a stock which is negatively correlated with the portfolio. By construction and definition, the relative Gini coefficient is 1-2 (area under j is the Lorenz curve) and the relative concentration ratio for security area under the concentration curve for security Hence, - 16 whenever concentration curves do not intersect:, the classification o securities by the Gini coefficient and the concentration ratio truly represents their relative riskiness. That is, if cov[ F (y )] > P P cov[y,F (y )1, . security i is said to be aggressive, and it .1S P P P said to be defensive if the inequality is reversed. However, when concentration curves intersect, the relative riskiness of security depends on the investor aversion towards risk. Thus, different investors may disagree about the relative riskiness of securities. 1.7 IV. THE PRICING OF RISKY ASSETS In this sectionwe develop the security valuation theorem for investors 'holding mean-Gini efficient portfoiios. The CAPM market-equilibrium relationship has been formulated for MV efficient portfolios by Treynor •(1961), Sharpe (1960., Lintner (1965), and Mossin (1966). The theorem states that for any security, the higher its nondiversifiable risk, the higher will be its expected return. Nondiversifiable (or systematic) risk is that part of the security's total risk that cannot be reduced by diversifying a portfolio without reducing its expected rate of return. The theorem is stated in the context of competitive financial markets without taxes and without restrictions on short selling and borrowing. In these markets investors trade risky assets whose quantities are known and fixed, to build efficient portfolios that answer their preferences. By doing so, they actin.the securities •• market, building forces that influence and determine the value of these securities. Assuming that investors build their portfolios according to a ,MV utility, ,the familiar CAPM expected return and risk is expressed as [cov(x., y )]/0 m relationship between 18! - where = E(x p is the expected rate of return on security =is the rate of,return on a risk-free security. = E(y a 2 is the expected rate of return on the market portfolio = variance of the rate o return on_theimarket portfolio. Equation (11) was derived under several assumptions of which the most important are (a) single-period analysis, (b) the existence of a risk-free asset, and ) perfect competition in the securities market. Since then, most efforts in modern financial theory have been directed o adapting the model, to different. contexts and testing it empirically. Much less attention has been devoted to whether its theoretical foundations were sound, with the notable exception of stochastic dominance theory (SD). Unfortunately, the SD approach was found to be computationally cumbersome and cannot provide answers to investors' questions on security valuation in terms,of portfolio diversification. The valuation theorem proposed here retains the main assumptions of the classical CAPM. However, instead of, holding MV efficient portfolios, investors construct market portfolios which are SD efficient by the mean-Gini approach. This is done - 19 by choosing a securities Each investor determines his optimum portfolio of the portfolio given its mix that minimizes the Gini's mean difference ' expected rate of return. j's portfolio is given by The rate of return of investor riskless rate, (12) Investors are permitted to borrow and lend at the Eai <1 ix • • •-1-. . Y. =a 11 • where x, is the rate of return on security . is the share of investor 1, wealth invested in security investors' problem is to minimize . + (1 . a3 11 i=1 = , , 007 i. Hence the subject to N (13) •,E Iaj i=1 lio can be written as Recall that the Gini coefficient of the portfo = 2cov[y. J F(y)] = 2 E a cov x , F.(37.)] J i= simply: Thus the necessary conditions for a minimum are N :(15) 2 COV and equation investor a k=1 k cov[xk,Fj(y.j)] 1 is the Lagrange multiplier ,associated with - As we saw earlier, between security 20 - cov[x., F (y )] j 1 is the concentration ratio8 It represents. and .the.portfolio'of individual i the degree by which the risk of security i cannot be 'diversified by including it in j's portfolio. By property 3 of the Gini coefficient, we know that homogeneous of degree one in a4 . is Therefore r. N E a3. 1 i=1 * jt (16) • r.. J aa. i or N , 3 F(y)] E a. cov[x' i=1 (16a) implying N N E sot 2 E i=1 k=1 . a3 k .coV xk, F. kt. by equation (14) that the double sum on the right-hand-side vanishes. By multiplying each of the conditions i summing over all the securities (17) = A.E r. j a. N . (15) by its share et4 and we obtain for every investor or (18) F. 3 1=1 1 1 and we have obtained the relation between risk (expressed by the Gini) f the portfolio and its expected return as (19) r. = A. E( .) - rf 3 3 3 - 21 Equation (19) represents the highest feasible straight line • in the (P, r) ions of space given an efficiency frontier constructed by portfolio combinat that space risky assets. ,Since the efficiency frontier is concave within a minimum are due to properties - 3 and 5, the second-order conditions for satisfied. In that context 1/A, is .the investor subjective price of risk portfolio to its since it relates the expected rate of return of the chosen • mix along that line that maximizes o portfoli a chose will investor The risk. •-• his utility. have Now assume a market of similar investors who are risk averse of. identical investment opportunities, and minimize the Gini coefficients their portfolios subject to their expected rates of return. In that case, equation (19) will be identical for all the investors in that market. For a given risk-free rate of return, the unit price of risk will be equal and determined by the slope of the market line in the , r)space (see Figure.3) For an investor which does not borrow nor lend, all his wealth will be invested in risky secdrities whose portfolio is the market portfolio in Figure 3. where p coefficient. investor pictured Thusfbr all investors, the price of risk will be the expected return on the market portfolio and r is its Gini In that case, the Gini coefficient of the portfolio held by is equal to 'since the optimum ranking, remains unchanged whether or not • F (yi) investor risk..-free'security is added and thus for all investors. 'Thus, from (15) is equal to F(y) m m )] F 3 cov[x N ' m m k E a3 (21) Da: k k=1 and the equilibrium condition for every security and thvestor becomes - market portfolio 23 (22) p.A r • 2cov x,, F m 1 M y M / F This is essentially the CAPM valuation relationship for a market o investors using the mean-Gini approach.. •To understand the dependence between systematic risk and expected return, let us rewrite R F im i 2cov[x , F(y)] m m as where - F(y )] , coy m m R. = cov[x. F. x.)] im 1. a sort of rank correlation coefficient and Thus (23) , Therefore, security (24) beta is simply = R. (r./r ) im 1 m As is well known, of return of security represents the degree of responsiveness of the rate to changes in the market, and Rim is the proportion of total risk .(expressed by the Gini coefficients of security I,r that cannot be/eliminated by the market without reducing the expected rate o return. 24 At this point, it is important to draw the analogy between the in (24) and the f3. derived from the MV-CAPM. normally distributed securities, and rn . = /ill I. It can'be shown that for r'=- and aff. where are the standard deviations of i and Therefore, R does converge to. the Pearson correlation coefficient im between security i and the market This assertion is intuitively deduce since in the case of normally distributed prospects coincide for the same set of observations. the MV and MG 'betas' However this is not always true for prospects that are not normally distributed. In general, MV and MG betas will be different; with the MG betas corresponding to SD efficient securities markets. Furthermore when investors use the Tean-extended-Gini (MEG) approach to evaluate risk and construct efficient portfolios, the computation and ",. estimation of betas will depend on their attitude towards risk, as will now be shown, when we" extend the analysis with the MEG. — 25 THE EXTENDED GINI COEFFICIENT In this section we develop the extended Gini coefficient and apply it to portfolio analysis and the CAPM. Gini'a mean difference may be extended into a family of coefficients of variability differing from each other in the decision-maker's degree of risk aversion, which is reflected by the Rewrite equation (2) as parameter JU F(y)] (25.) where 1< v < is a parameter chosen by the user of the method. We define (25) as the extended Gini coefficient whose properties are (Yitzhaki, 1980) The extended Gini coefficient is non-negative and bounded from above 1. v . for all That i 0 S r(v) < 1.1 - a It is anon-decreasing function The proof of this property is immediate if we remember that 1 2. F(x) >'O..for , For simply x < b . , 4 r(v) , x)] X'c)b min(xi) Thus, integers, the extended Gini coefficient is E[min (x.)] .v. i=1,2 1.1 (x.) < yl = i=1,2..v rb 1 [1-F y)] •o) y for since 6b 1 Y -F 7)1 , integer The extended Gini coefficient is sensitive to mean-preserving,'spreads. . If = hen 26 - - • Proof: Since a F for a'> 0 we get ab j {[ a ] - [1 Let Y 2 = Y1 Let y1, y for F ( )]v) =.T"1(v) then be two prospects; then[p-r , 2, 3, • 1 ]- is the necessary condition for first and second degree stochastic dominance (Yitzhaki 1982). • 7. Let yl, y2 be two prospects with cumulative distributions that intersect most once. Then [111 - r i( )] is a sufficient condition for (Yitzhaki y1 - [P2 - r2(v)] v = 1, 2,:3 >,0 for all to stochastically dominate 1982). The interpretation of can best be seen by looking at v Its value for different values of p 'r(v). are v= 1 11(v) = v= 2 V and it is a non-increasing function of v . Thus one can view F(v) as the risk premium that should be substracted from the expected value of the distribution. while v 00 The case V = 1 represents the risk-neutral investor represents the investor who is interested in the minimum value of the distribution, and maximizes this minimum.9 27 portfolio can be decomposed -into • The extended Gini coefficient an equation similar to (6). "That i (26) r v) = -v Proof: E a cov x , 1=1 Let y = E ,a x i=1 P 1 F v and v = )] F (y) u } hen r' (v) "=P Defining • dy . as 1 - F(y)] F(y) (y) • U= y and applying integration by parts, we get )]vy - F (y)]v- yf (y)dy P P [1- F(y - 28 t y) F (y) y dy =-[ thus 1[1 - F 30] 1 dy Yfp Y yfp(y)dy F(y) F = -VCOV 1_ 1 v F (y) v-1 Therefore (27) r (v) = 1 - F cov x., i -1} since y• = 1 Q.E.D. prospect, is its covariance with That is, •the nondiversified risk the portfolio rank the power . The higher the greater the weight given to the performance of the prospect when the yield of the portfolio is low. Note that if v = 2 , then we have the regular Gini coefficient. We derive the CAPM, using the extended Gini of degree v. , the extended-Gini of his portfolio is given by F (v) = -v E a4covf[ 1 i=1 F.(y.)]Vl, x.} 1 J 3 For investor - 29 that chooses Investor F.( ) subject minimize to the expected rate of return on the portfolio given by (12). v . reflects the degree of risk aversion. As •shown, Hence, it is .possible to model a securities market that will exhibit different market portfolios because Of different degrees of risk aversion. For the present, we require that . investors are similar and that they display identical all j . Therefore the necessary conditions for a minimum are -vcovf 1 - J J (28) the market portfolio's extended Gini coefficient o where r() is degree v , and the market-equilibrium relation becomes v-1 (29) If. the !rank ,correlation .ratio degree and the _market is covU1 (v) = ••• cov{0. m m W- -1 , x.} x} v between security. 30 the market—equilibrium valuation for every and any is given by . r(v) - (30) (v) t (11)R. f im r m(v) Thus, (31) = R. im Ti . rm(v) Thus, even if investors have the same attitude towards risk as expressed by v, different efficient market portfolios and different systematic risks for each security will be obtained. estimating betas. This feature must be borne in mind when It must be added that if securities are normally distributed, the MEG 6's will b .identical to the MV betas, independently of But our concern for the existence for different efficient market portfolios is -principally directed towards distributions other than normal such as the log normal and the uniform distribution. For these 'distributions, the MV approach is not consistent with expected utility maximization and stochastic dominance, whereas the mean—Gini i 10 VT. CONCLUSION We have presented a new approach to analyze risky prospects and construct optimal portfolios. This method owns the simplicity'of a two-parameter model with the efficiency f Stochastic dominance. To that extent, we claim that Gini's mean difference is superior to the variance for evaluating the variability of a security. Furthermore, the concentration ratio based on the Gini coefficient permits us to classify different securities with respect to their relative riskiness. Finally we have applied the MG approach to capital markets and the security valuation theorem was derived on a general relationship between average return and risk. By extending we have explicitly the analysis with the mean-extended-Gini method, introduced the degree of risk aversion as a parameter that can determine the specific composition of market portfolios. The main implications of the model reside as whether investors, in general, behave more in a MG or MEG framework rather than follow the MV approach. These implications can be empirically tested by estimating the performance of CAPM for different degrees of these with the results obtained from MV-CAPM. models were tested using regression techniques. v and comparing Two-parameter portfolio This was notably exemplified by Fama and MacBeth (1973) who supported the hypothesis that risk-averse investors hold efficient portfolio in terms of mean and standard -deviation of the returns. Recently, certains doubts were - 32 raised as to whether the ,different regression procedures were to test the CAPM (see Ross, 1980) By proposing the mean7Gini and . mean-extended-Gini models as means of evaluatingcapital market data, we add a new dimension to modern, finance theory, suggesting that we should return to the drawing •board. .• • 33 Appendix . (Yitzhaki, 1982). Proposition The condition p i - a necessary condition for y is to dominate y first and second stochastic rules for, v Proof: according to 0. For y l. o dominate y2 according to F§D and SSD rules it is necessary that for all y (A.1) and ry for all y. t dt t)dt 5_ implies The condition (A.3) If af -F v 1-F 37) dy 2 dy for V > 1. v = 1, the proposition is proved directly since (A.3) holds whenever (A.1) and (A.2) exist. For v > 1 v z strictly convex for positive '(A4 [1-F 1 O]v > we know that since the function -F(y)] + z.v v[i-F2 v-1 (z-z ). V z y)] y) is Thus (y)] and fb 5), y > v [ 1 -F a jb 1[1-F (y)]v- (A.5) 1-F a ]dt since a 0 for ally by A.2) afy and [1-F v-1 is non negative and non increasing in y, y)]dy 34 3. 7)]V71. (A.6) (y) - F (y)]dy 2 a 0 for v > 1. by the following lemma. Thus from (A.5) we have . (A.7) .1 whenever Lemma: Proof: dy y) stochastically dominates Let h(z) be a non-negative and non-increasing function of z, and let . g(z) be a :function with the property that ixg(z)dz a). Then I z)-g(z)dz 0 for all x. a 0 Assume at g(x) changes signs X) . ' 2' . n n times between Then g(x)r < x <:x , and so on. ).11(z)dz h jg a we have z)-h z dz 0 and b 0 -for a < < x and g(x) < 0 for x 2 . Thus f g(z)-h(z d ?. h(x )1x2g(z)dz ?.. 0. 1 j a ' ince this argumOnt can be repeated for aT a for all x. for all x. )dz . when i = 0 , - 35 FOOTNOTES For simplicity of presentation, continuous and bounded random variables wiT1 be used throughout the paper, keeping in mind that most of the results can be applied to discontinuous and unbounded distributions. The index I will be omitted wheneverit is not necessary to disting- uish between two variables. mean difference. Note that we use half of Gini's original Note too that we here use the absolute forms of Gini's mean difference and the concentration ratio; that is, we do not follow the more usual practice of dividing by the mean (see Kendall and Stuart, 1977)., 2For derivation of equation (2) for continuous, discrete and unbounded distribution see Dorfman (1979). 3For-the definitions of the stochastic dominance rules see Hanoch and . Levy (1969) and Rothschild and Stiglitz (1970). 'For derivation of formula (3) see Kendall and Stuart (1977) for the continuous case and Pyatt, Chen, and Fei (1980), in the case of discrete distributions. 5For a discussion of the properties of.the concentration curve see Kakwani (1977) and Pyatt et al 6The restrictions g.( ) (1980) 0 and yp > 0 should be interpreted as shift of the origin. 7This proposition and the two corollaries were proved by Kakwani (1977). 8Note that to obtain the concentration ratio .as defined by Pyatt'et al; (1980), covix., F(yi)] 9 should be divided by Ei(xj) Values of 0 < v < 1 represent the case of the risk lover, the extreme case v = 0 representing the investor interested in the maximum value of the distribution, b (the max-max investor) In this paper, we restrict ourselves to risk-averse investors. 10See Yitzhaki (1982) for a consistency analysis of the mean-Gini approach for different distributions. 3 REFERENCES • Atkinson, A.B. "On the Measurement f Inequality.". —Econ. 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"On an Extension of the Gini Inequality Index "Discussion Paper 8013, Falk Institute (June 1980). . PREVIOUS WORKING PAPERS 6901 414- 6)02 Yoav Kislev and Hanna Lifson - An Economic Analysis of Drainage Projects. Yair Mundlak and Ran Mosenson - Two-Sector Model with Generalized Demand. 6903 Yoav Kislev. 'The Econothics of the Agricultural Extension Service. in Hebrew). 7001 D n Yaron and Gideon Fishelson - A Survey of Water Mobility on Moshav . Villages. (Hebrew). Yakir Plessner - Comnpting Eqi:Iilibrium Solutions for Various Market Structures, 7602 7003 7004 Yoav Kislev and Yeshayahu Nun - Economic Analysis of Flood Control Pro ects in.the Hula Valley, Stage One - Final Report. (le rew). Yoav Kislev and Hanna Lifson - Caiital Ad ustment with U-Sha ed Avera e Cost of Investment. Empyi cal Production Functions with a Variable Firm Effect. 7005 .Yair Mundlak 7006 On Some Im Yair Mundlak Functions. 7101 7102 Ila_n1/1.21.2R2. 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