Portfolio Risk Management Using the Lorenz Curve

Transcription

1 October 5, 203 Portfolo Rsk Management Usng the Lorenz Curve by Ham Shalt Abstract Ths paper presents a methodology for usng the Lorenz curve n fnancal economcs. Most of the recent quanttatve rsk measures that abdes by the rules of second-degree stochastc domnance such as Gn s mean dfference and Condtonal Value-at-Rsk are assocated wth the Lorenz curve. Wth fnancal data, the Lorenz curve s easy to calculate snce t requres only sortng asset returns n ascendng order. Therefore, the fnancal analyst can derve the statstcs necessary to carry out a study of rsk analyss to establsh a set of effcent and most preferred portfolos by all rsk-averse nvestors. JEL Codes G32 D8 G shalt@bgu.ac.l Department of Economcs, Beer-Sheva Unversty of the Negev

2 Portfolo Rsk Management Usng the Lorenz Curve The Lorenz curve was developed n 905 for the purpose of measurng relatve nequalty n ncome dstrbuton. Snce then the curve has been wdely used n welfare economcs to calculate the share of natonal ncome earned by proportons of the populaton as ranked by ther relatve ncome. The applcaton of the Lorenz curve to portfolo rsk management s rooted n ts ablty to measure the dstrbuton and varaton of asset returns. In ths paper I show how some recent quanttatve rsk measures can be derved from the Lorenz curve n order to manage rsk and construct effcent portfolos. In rsk studes, the analyss s based on the absolute Lorenz curve (hereafter referred to as the Lorenz) whch ranks condtonal expected returns wth respect to the cumulatve probabltes of gettng these returns. The Lorenz n rsk analyss orgnated from Shorrocks (983) who used absolute curves to derve second-degree stochastc domnance (SSD) condtons. Most portfolo theory and rsk management results derved from the Lorenz have appeared n the fnance lterature wth respect to Gn s mean dfference (GMD) and the Gn ndex. Fsher and Lore (970 were the frst to apply Gn statstcs usng the standard Lorenz curve to study the varablty of sngle stocks and portfolos. Later, Shalt and Ytzhak (984) used the curve to characterze rsky assets, apply Gn s mean dfference n fnance theory, and derve the mean-gn CAPM. The Lorenz s also very useful for expressng safety-frst rsk quantle measures such as Value-at-Rsk (VaR) and Condtonal Value-at-Rsk (CVaR) that have become very popular n the bankng ndustry. Ths feature s partcularly advantageous when the analyst does not need to specfy a partcular dstrbuton functon. Otherwse VaR and CVaR measures are qute cumbersome to compute. In ths paper, the rsk measures are obtaned from the Lorenz by usng dscrete probabltes whch are commonly avalable n fnancal data. The plan of the paper s as follows: The next secton presents the Lorenz and ts relaton to SSD. Secton 2 dscusses the Lorenz and ts relaton to GMD. In Secton 3, I show the lnk between VaR, CVaR, and the Lorenz. In Secton 4, I present an nvestment example to show how to manage portfolo rsk wth the Lorenz. 2

3 . Stochastc Domnance and the Lorenz The man advantage of the Lorenz n fnancal analyss les n ts smplcty to rank and evaluate rsky assets accordng to stochastc domnance (SD). Hanoch and Levy (969), Hadar and Russell (969), and Rothschld and Stgltz (970) developed ndependently SD rules that provde portfolo effcency under expected utlty maxmzaton but wthout resortng to specfc utlty functons. SD rules are obtaned by comparng the cumulatve probablty dstrbutons of asset returns. Frst-degree stochastc domnance (FSD) s desgned for nvestors wth ncreasng utltes who are ether rsk-averters or rsk-lovers. Second-degree stochastc domnance (SSD) beng manly for rsk-averse nvestors s the most common model used n portfolo selecton. Usng cumulatve probabltes, SSD rules provde the necessary and suffcent condtons under whch rsky assets are preferred by all rsk-averse expected utlty maxmzers. Nonetheless, the optmal choce of assets s more evdent wth the Lorenz than wth the tradtonal SSD rules as I now explan. A B Consder two rsky assets A and B whose returns x for =,..., N and x for =,..., N are sorted from ther lowest yelds to ther hghest. Returns are dstrbuted wth probabltes q A (x A ) and q B (x B A ) respectvely such that q( x ) 0 and B q( x ) 0 and and N = k B B = k B q( x ) =. The cumulatve probabltes are computed as N A q( x ) =, = k q( x A ) A = p = k q( x ) = p for k =,..., N. Followng Levy (2006), we can state the SSD rules by comparng the areas under the cumulatve probabltes namely: Asset A s preferred to asset B by all rsk averters f and only f k k A = = B p p for all k =,..., N. () Computng the areas under the cumulatve probabltes s not straghtforward snce t must be done for all the probabltes and then comparng between the cumulatve areas. Followng Shorrocks (983) usng the Lorenz to establsh SSD nvolves comparng only between curves as we state. For all rsk-averse nvestors to prefer portfolo A over portfolo B, the Lorenz of A must le above the Lorenz of B. In other words, asset A domnates asset B f and only f: L ( p) L ( p ) for all probabltes 0 p, (2) A B 3

4 where L ( p) and L ( p ) are the Lorenz curves for asset A and B respectvely formulated usng A B Equatons (3) and (4) as follows: In Equaton (3), the Lorenz s obtaned by summng up the returns x tmes ther probabltes q(x ) up to the counter k(p): k ( p) = (3) = L( p) x q( x ) for all p from 0 to where the counter k(p) s obtaned mplctly by: k ( p) = ( ) for all. (4) = p q x x Equaton (4) expresses the cumulatve probablty p that returns are less than a gven value specfed by k(p). Let us now explan the Lorenz that s delneated n Fgure. Cumulatve probabltes are exhbted on the horzontal axs ndcatng that returns are ranked n ncreasng value. On the vertcal axs, we have cumulatve rates of returns weghted by probabltes as expressed by Equaton(3). The Lorenz starts at the orgn of axes (0,0) and accumulates the sorted returns multpled by ther probabltes untl all the returns are used up. Snce the lowest returns can be losses, the Lorenz may result n negatve values. The curve ends at the mean return E(x) on the parallel vertcal axs snce at that pont all returns are used up and multpled by ther probabltes. By usng a contnuous cumulatve dstrbuton functon F(x) and ts nverse, Gastwrth (97) expressed the Lorenz wth the followng sngle equaton : p ( ) = ( ) for 0 L p F t dt p 0 4

5 Fgure : The Lorenz LSA E(x) L(p) L(α) CVaR(α) 0 α p From ts defnton n Equaton (3), the Lorenz captures the condtonal expected return E( x p) gven the probablty p snce L( p) = E( x p) p where 5 E( x p) q( x ) = k ( p) x = p s the mean of returns when ranked returns add up to k(p). When all returns are accounted for,.e., k( p) = N, the Lorenz at p = s the uncondtonal mean return of asset E( x ). Now, we can look at what can be ganed by usng the Lorenz n fnance. The ratonale for usng the Lorenz n SSD s rooted n the manner by whch the Lorenz characterzes rsk and mean return of nvestments for rsk-averse nvestors. Such nvestors have concave utlty functons that express declnng margnal utlty. The horzontal axs n Fgure shows the probabltes of asset returns ranked from those generatng the lowest returns wth the hghest margnal utlty to those generatng the hghest returns wth the lowest margnal utlty. The rankng of asset returns s the only nformaton needed to sort an asset accordng to decreasng margnal utlty. Ths orderng s specfed by the cumulatve returns multpled by

6 the probabltes of gettng these returns. Ths s bascally the Lorenz. The prncple of dstrbutng resources accordng to decreasng margnal utlty or decreasng margnal product ensures that fnancal resources are allocated optmally. Usng the Lorenz to manage portfolo rsk guarantees that objectve. Because the curve expresses asset behavor not as a functon of returns over tme but as the occurrence of havng lower and hgher returns, t provdes much more relevant nformaton about rsk and return than perodcal charts. 2. Rsk and Gn s Mean Dfference Orgnally, Gn (92) defned the mean dfference as the expected dstance between observaton pars as follows: GMD = E x x, (5) 2 where x and x 2 are ndependent replcates of the random varable x. When GMD s dvded by the mean t becomes the well-known Gn coeffcent whch s known to evaluate ncome nequalty. GMD s also an attractve measure of rsk because t depends on the spread of the returns among themselves and not on ther devatons from a central value as the mean. The measure has many dfferent representatons and formulatons, most of whch can be found n Ytzhak (998). In fnance and portfolo rsk management t s more convenent to use one half of GMD whch s usually referred to as the Gn Γ. As shown n Equaton (6), ts formulaton uses the covarance between the returns and the cumulatve probabltes of gettng these returns: GMD / 2 Γ= 2cov[ x, p] (6) Beng a statstc the Gn has some advantages snce t allows expressng rsk wth a sngle number and constructng optmal portfolos that are SSD. Furthermore as I wll show, the Gn of an asset can be obtaned drectly from ts Lorenz. Indeed, when nvestors analyze the features of rsky assets, they would lke to decompose the returns nto two components: one that embraces only the rsk of the asset and the other only the safe return. Wth the Lorenz ths task s easly accomplshed. To show ths, let us construct a vrtual asset that has the same mean return as asset x but has no rsk whatsoever as for each probablty the vrtual safe asset always yelds the same mean return E(x). Ths rskless asset s depcted n Fgure by ts Lorenz as a straght lne that orgnates at (0, 0) and ends at the mean 6

7 (E(x) ). Ths lne s called the lne of safe asset (LSA) because t expresses the expected return E(x) multpled by the probablty p. We can now enuncate the rsk of asset x as the dfference between ts LSA that yelds the expected return E(x) and the Lorenz of asset x. It s obvous that for every probablty p nvestng n the rsky asset earns the cumulatve expected return along the Lorenz whle nvestng n the rskless asset earns a hgher cumulatve expected return along the LSA. The rsk of the asset s quantfed by the vertcal dfferences between the LSA and the Lorenz. Ths area s calculated as the dfference of the area below the LSA and the area below the Lorenz. Therefore, the farther the LSA s from the Lorenz, the greater s the rsk assumed by the asset. Snce E( x) p s the LSA equaton and L(p) s the Lorenz equaton, the area between the two lnes s one half the Gn as shown here: N N N (7) E( x) p L( p ) = [ E( x) x ] p = cov[ x, p] = Γ. 2 = = = From Equaton (7), the Gn can be understood as the pure rsk nherent n an asset and therefore can be used together wth the mean to characterze nvestments. Snce the mean and the Gn are statstcs derved from the Lorenz, they can facltate the rankng of rsky assets. Indeed, usng the Lorenz for SSD defnes only a partal orderng of nvestment opportuntes. When Lorenz curves ntersect no clear domnance between rsky assets can be determned and therefore the relaton between all the nvestments cannot be establshed. Sometmes, a complete orderng s requred although these results can only provde the necessary condtons for SSD. Ths s the case when the mean and the Gn are used to establsh necessary condtons for SSD. To clarfy ths argument, consder non-ntersected Lorenz curves and ther relaton to SSD. If we choose a lnear utlty functon to determne the optmal portfolo, a necessary condton for the rsky portfolo to be preferred by all expected utlty maxmzers s that t s preferred by the rsk-neutral nvestor whose margnal utlty s a constant. As such, only the last data pont on the Lorenz, whch s the mean, s the relevant gauge for choosng among assets. Ths explans the frst necessary condton for SSD statng that the mean of the preferred asset s greater than the mean of the domnated asset. The other necessary condton for SSD s that the area below the Lorenz of the preferred asset be greater than the area below the Lorenz of the domnated asset. Ths area s one-half the mean return subtracted by 2Γ= cov[ x, p]. These two requrements explan the necessary 7

8 condtons for SSD usng the mean and the Gn. As establshed by Ytzhak (982), these necessary condtons are expressed as: E( x ) E( x ) A E( x ) Γ E( x ) Γ A A B B mplyng that f portfolo A s SSD preferred to portfolo B, then the mean and the rsk-adjusted mean return of A cannot be less than the mean and the rsk-adjusted mean return of B when rsk s measured by the Gn of the portfolo. 2 B (8) 3. The Lorenz and CVaR As a popular measure of rsk VaR quantfes exposure to rsk as the amount of cash to be held n a safe asset to overcome the ncdence of total loss for a portfolo. It s a safety-frst rsk measure defned as the quantle of a gven probablty p, formulated mplctly as the return VaR(p) such that k ( p) p= q( x ) x VaR( p) (9) = As seen from Equatons (3) and (4), VaR(p) s only one sngle element of the Lorenz that can be obtaned drectly from the cumulatve probabltes p. It s surprsng that VaR s so prevalent n fnance beng that t lacks the followng basc propertes of a rsk measure ρ(x) for t to be coherent ( see Artzner et al. (999)):. Translaton nvarance, ρ( X + R ) = ρ( X ) R, where R F s a safe return F. Subaddtvty, ρ( A+ B) ρ( A) + ρ( B). Postvely homogenety, ρ( λ X ) = λρ( X ) v. Monotoncty X Y ρ( Y ) ρ( X ) Indeed, unless the returns dstrbuton s normal, VaR lacks coherence because t fals to satsfy the subaddtvty axom that would prevent rsk reducton n portfolo dversfcaton. To crcumvent VaR s lack of coherence fnance researchers developed Condtonal Valueat-Rsk (CVaR)) (Rockafellar and Uryasev 2000, Acerb and Tasche, 200). The basc dea of F 2 Ytzhak (982) also showed that the mean-gn condtons for SSD are suffcent whenever cumulatve probablty dstrbutons functons ntersect at most once. 8

9 ths measure s to calculate CVaR(p) as the mean of all the quantles below the orgnal VaR n the lower tal of the cumulatve probablty dstrbuton from 0 to p. In formal terms: k ( p) CVaR( p) = VaR( q ) q( x ) for all q( x ) p (0) p = 0 By comparng Equaton (3) and Equaton (0), we see that CVaR s easly obtaned from the Lorenz as follows: L( p) CVaR( p) =. () p Let us consder a specfc probablty α between 0 and. Fgure shows how CVaR(α) for the probablty α s expressed by the slope of the straght lne connectng the orgn (0,0) to the pont (α, L(α)). Ths slope can also be measured on the vertcal axs at p= by the segment from the horzontal axs up to the pont labeled CVaR(α). As such, t s easer to calculate CVaR for a gven asset snce the technque s not restrcted to specfc probablty dstrbutons. Under these provsons, CVaR s obtaned from a specfc value of the Lorenz whch, for a gven data set, s estmated by sortng and summng up the returns. 4. Managng Rsky Assets: An Investment Example To show how the varous rsk measures are obtaned, I calculate the Lorenz for varous traded stocks. Ths s an easy task for a sample of dscrete observatons snce t nvolves only rankng returns n ascendng order and then, for each gven return, summng all the lower returns up to that observaton. To llustrate the relevance of the Lorenz n rankng securtes wth respect to rsk and return we use the 250 daly returns of the 30 stocks of the Dow Jones Industrals Average from January 3, 202 to December 3, 202. The Lorenz curves for these 30 stocks are calculated as descrbed n Equaton (3) above where the probablty of occurrence s /250 for each return. For sake of clarty n Fgure 2 only 0 Lorenz curves are plotted. Now we can solate the set of SSD domnated stocks whch are the ones wth the Lorenz curves lyng on top of the chart. The set of effcent stocks nclude JNJ, KO, IBM, and PFE whose Lorenz curves form the non-domnated set. The worst stocks accordng to SSD are the ones lyng on the bottom of the chart and nclude AA, HPQ, CAT, and CSCO. Bear n mnd that t s not only the poston of the Lorenz curves that s relevant but also whether or not they ntersect. 9

10 The goal now s to calculate the stocks statstcs from the Lorenz. The last value on the curve s the stock mean return. The Gn s obtaned as the area under a vrtual LSA and the Lorenz for each stock. Alternatvely for the Gn, one can use Equaton (7) and substtute for the cumulatve probablty p the value /N. The statstcs for the 30 DJIA stocks are reported n Table. The rsk-adjusted mean return for each stock s shown as the mean mnus the Gn. Two CVaRs are also exhbted: one for 5% and the other for 0%. The CVaRs are calculated by usng Equaton (0) or extrapolated from the Lorenz usng Equaton (). Note that the CVaR at 5% s greater than the CVaR at 0%. We can also apply the mean-gn condtons for SSD as expressed by the condtons of Equaton(8). Ths s done on Table 2 where the stocks are ranked frst accordng to the mean and then accordng to the mean less the Gn. The lst shows the most desrable stocks ranked accordng to the necessary condtons for SSD. Hence, the top stocks on the lst have hgher means and hgher rsk-adjusted means. As such, the lst provdes a complete orderng of stock choces by weghng rsk and mean return for all rsk-averse nvestors. Furthermore, we compare these results wth the outcomes obtaned n Table 2 by rankng CVaR from the lowest ones (safest stocks) to the hgher ones (rsker stocks). As seen n the table, there s some correspondence between the Lorenz and the mean-gn condtons. However, the comparson s not complete snce CVaR consders only low-returns rsks at a gven probablty whereas Lorenz statstcs consder rsk for the entre dstrbuton of returns and therefore provdes much more nformaton about rsk and mean return. Concludng Remarks Ths paper has shown how the Lorenz can serve as a basc tool to measure rsk and return of ndvdual assets and portfolos. Not only does the Lorenz comply wth SSD, t also facltates the computaton of the mean-gn condtons for SSD when Lorenz curves ntersect. Furthermore, the Lorenz allows for calculatng the CVaRs for all probabltes of occurrence. Hence, stocks and portfolos can be ranked n terms of rsk and return by usng only the Lorenz curves wthout estmatng probablty functons. 0

11 References Artzner, P., Delbaen, F., Eber, J-M., and Heath, D., Coherent Measures of Rsk. Mathematcal Fnance 9, (999), pp Fsher, L. and Lore, J. H., Some Studes of Varablty of Returns on Investments n Common Stocks. Journal of Busness 43, (970), pp Hadar, J. and Russell, W. R., Rules for Orderng Uncertan Prospects. Amercan Economc Revew 59, (969), pp Hanoch, G. and Levy, H., The Effcency Analyss of Choce Involvng Rsk. Revew of Economc Studes 36, (969), pp Gastwrth, J. L., A General Defnton of the Lorenz Curve. Econometrca 39, (97), pp Gn, C. Varabltà e Mutabltà. Stud Economco-Gurdc, Unverstà d Caglar, tpografa d P. Cuppn, Bologna. (92). Levy, H., Stochastc Domnance: Investment Decson Makng under Uncertanty, 2 nd edton. New York: Sprnger Verlag, Lorenz, M. O., Methods of Measurng the Concentraton of Wealth. Publcatons of the Amercan Statstcal Assocaton, 9, (905), pp Ogryczak, W. and Ruszczynsk, A., Dual Stochastc Domnance and Quantle Rsk Measures. Internatonal Transactons n Operatonal Research 9, (2002a), pp Rockafellar, R. T. and Uryasev, S., Optmzaton of Condtonal Value-at-Rsk. Journal of Rsk 2, (2000), pp Rothschld, M. and Stgltz, J. E., Increasng Rsk I: A Defnton. Journal of Economc Theory 2, (970), pp Shalt, H. and Ytzhak, S., Mean-Gn, Portfolo Theory, and the Prcng of Rsky Assets. Journal of Fnance 39, (984), pp Shorrocks, A. F., Rankng Income Dstrbutons. Economca 50, (983), pp Ytzhak, S., Stochastc Domnance, Mean-Varance, and Gn's Mean Dfference. Amercan Economc Revew 72, (982), pp Ytzhak, S., More than a Dozen Alternatve Ways of Spellng Gn. Research on Economc Inequalty 8, (998), pp

12 Table : Statstcs of Daly Returns for the 30 DJIA Stocks for 202 Symbol Mean Gn Mean-Gn CVaR(5%) CVaR(0%) AA 0.022% 0.990% % 3.682% 3.052% AXP 0.093% 0.72% % 2.799% 2.268% BA 0.027% 0.626% % 2.592%.977% BAC 0.327%.372% -.045% 4.528% 3.693% CAT 0.020% 0.922% % 3.603% 2.952% CSCO 0.057% 0.807% % 3.620% 2.633% CVX 0.026% 0.65% % 2.720% 2.083% DD 0.05% 0.655% % 2.778% 2.75% DIS 0.26% 0.67% -0.49% 2.438%.869% GE 0.084% 0.65% % 2.556% 2.00% HD 0.70% 0.647% % 2.595% 2.003% HPQ -0.99%.57% -.357% 6.3% 4.457% IBM 0.028% 0.53% % 2.373%.759% INTC % 0.743% % 3.94% 2.555% JNJ 0.043% 0.330% %.236%.025% JPM 0.40% 0.958% -0.89% 4.07% 3.04% KO 0.029% 0.455% %.682%.383% MCD % 0.469% % 2.460%.8% MMM 0.066% 0.492% % 2.038%.528% MRK 0.054% 0.528% % 2.9%.678% MSFT 0.03% 0.70% % 2.663% 2.6% PFE 0.078% 0.456% %.567%.270% PG 0.024% 0.427% %.808%.373% T 0.068% 0.473% %.982%.498% TRV 0.094% 0.550% % 2.35%.634% UNH 0.042% 0.732% % 3.23% 2.429% UTX 0.064% 0.683% -0.69% 2.33%.983% VZ 0.054% 0.494% -0.44%.858%.500% WMT 0.068% 0.532% % 2.662%.848% XOM 0.023% 0.54% -0.49% 2.8%.649% 2

13 Table 2: Stocks Ranked w.r.t. mean-gn and w.r.t. CVaR Ranked Ranked Ranked Mean Mean- Gn CVaR(5%) CVaR(0%) BAC 0.327% -.045% JNJ.236% JNJ.025% HD 0.70% % PFE.567% PFE.270% JPM 0.40% -0.89% KO.682% PG.373% DIS 0.26% -0.49% PG.808% KO.383% TRV 0.094% % VZ.858% T.498% AXP 0.093% % T.982% VZ.500% GE 0.084% % MMM 2.038% MMM.528% PFE 0.078% % XOM 2.8% TRV.634% T 0.068% % MRK 2.9% XOM.649% WMT 0.068% % TRV 2.35% MRK.678% MMM 0.066% % UTX 2.33% IBM.759% UTX 0.064% -0.69% IBM 2.373% MCD.8% CSCO 0.057% % DIS 2.438% WMT.848% MRK 0.054% % MCD 2.460% DIS.869% VZ 0.054% -0.44% GE 2.556% BA.977% JNJ 0.043% % BA 2.592% UTX.983% UNH 0.042% % HD 2.595% GE 2.00% MSFT 0.03% % WMT 2.662% HD 2.003% KO 0.029% % MSFT 2.663% CVX 2.083% IBM 0.028% % CVX 2.720% MSFT 2.6% BA 0.027% % DD 2.778% DD 2.75% CVX 0.026% % AXP 2.799% AXP 2.268% PG 0.024% % UNH 3.23% UNH 2.429% XOM 0.023% -0.49% INTC 3.94% INTC 2.555% AA 0.022% % CAT 3.603% CSCO 2.633% CAT 0.020% % CSCO 3.620% CAT 2.952% DD 0.05% % AA 3.682% AA 3.052% MCD % % JPM 4.07% JPM 3.04% INTC % % BAC 4.528% BAC 3.693% HPQ -0.99% -.357% HPQ 6.3% HPQ 4.457% 3

14 Fgure : Lorenz Curves of 0 Select DJIA Stocks usng Daly Returns for AA AXP CAT CSCO HPQ IBM JNJ KO PFE UNH 4