PIECEWISE LINEAR RISK FUNCTION AND PORTFOLIO OPTIMIZATION

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1 Joural of the Operatios Research Society of Japa Vol. 33, No. 2, Jue he Operatios Research Society of Japa PECEWSE LNEAR RSK FUNCON AND PORFOLO OPMZAON Hiroshi Koo okyo stitute of echology (Received April 3, 1989, Revised October 12, 1989) Abstract A ew portfolio optimizatio model usig a piecewise liear risk fuctio is proposed. his model is similar to, but has several advatages over the classical Markowitz's quadratic risk model. First, it is much easier to geerate a optimal portfolio sice the problem to be solved is a liear program istead of a quadratic program. Secod, iteger costraits associated with real trasactio ca be icorporated without makig the problem itractable. hird, it eables us to distiguish two distributios with the same first ad secod momet but with differet third momet. Fourth, we ca geerate the capital-market lie ad derive CAPM type equilibrium relatios. We compared the piecewise liear risk model with the quadratic risk model usig historical data of okyo Stock Market, whose results partly support the claims stated above. 1 troductio Harry Markowitz, i his semial work [7], formulated the portfolio optimizatio problem as a quadratic programmig problem i which variace of retur out of the portfolio is miimized subject to the costrait o the average retur. his formulatio which has log served as the basis of fiacial theory is kow to be valid if a ivestor's utility fuctio is quadratic ad/or the distributio of the rate of retur of stocks is multivariate ormal. t tured out, however that solvig a large scale quadratic programmig problem is ot easy if ot impossible. his computatioal burde was oe of the reasos why this model was ot used i practice to determie a optimal portfolio cosistig of more tha a few hudred stocks. Also this model suggests ivestors to purchase a large umber of differet stocks. his is rather icoveiet sice maagig a portfolio with a large umber of differet stocks is weary ad expesive. Yet eve more computatioal difficulty arises if we take ito accout the costrait o the miimal uit associated with each trasactio. fact, we have to solve a itractable iteger quadratic programmig problem particularly whe the amout of fud is small compared to the umber of stocks. Also, the detailed study [4] of the historical data i the stock market shows that the distributio of the rate of retur of stocks is either ormal or eve symmetric. additio, recet studies o portfolio isurace show that most ivestor do ot purchase "efficiet" portfolio implied by Markowitz's model. hey usually buy a portfolio apart from the efficiet frotier. his meas that quadratic utility model eed ot apply to all ivestors. hese observatios motivated us to itroduce L1 risk fuctio [5] which has the followig 139

2 140 H. Koo advatages over its couterpart. First, the associated optimizatio problem is a easy liear program istead of a "ot so easy" quadratic program. ypically, the model with several hudred stocks may be solved o a real time basis by usig curret state-of-the-art techique. Secod, this model is expected to suggest ivestors to purchase sigificatly fewer umber of differet stocks tha its couterpart. Also it is much easier to icorporate iteger costraits associated with miimum trasactio uits. hird, this model ca icorporate ivestors' subjective perceptio agaist risk ad hece is more operatioal tha the traditioal model. Fially, our model is essetially equivalet to the traditioal model if the rate of retur of stocks are multivariate ormally distributed. hus our model ca be used as a good proxy of Markowitz model. Sectio 2, we itroduce compoud Ll risk model ad derive some of its importat properties. Sectio 3, we derive the "capital-market lie" ad equilibrium coditios for compoud Ll risk model. t will be show that the classical Sharpe-Liter-Mossi type relatio betwee idividual stock ad market portfolio holds for our model as well. Sectio 4 will be devoted to the geeralizatio of the model i which the risk fuctio ca distiguish positive skewess ad egative skewess of the uderlyig distributio of the rate of retur. Fially, some computatioal results usig the historical data of okyo Stock Market will be preseted i Sectio 5. 2 Compoud Ll Risk Fucitos ad Portfolio Optimizatio Let there be stocks deoted by Sj (j = 1,..., ) ad let Rj be the radom variable represetig the rate of retur of Sj. Also let Xj be the amout of moey ivested i Sj out of his total fud Mo. he average rate of retur asscociated with this ivestmet is give by (2.1) r(xl,..., x) = E[L Rj Xj] = L E[Rj] Xj where E[ ] stads for the expected value of a radom variable i the bracket. A ivestor wats to make r( Xl,...,X) as large as possible. At the same time, he wats to miimize his "risk". Markowitz [7] employed the variace of retur (2.2) V(Xl,..., x) = E[{L Rj Xj - E[L Rj Xj]V] as the measure of risk ad formulated the portfolio optimizatio problem as a quadratic programmig problem Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

3 Piecewise Liear Risk Portfolio Model 141 (2.3) miimize V(X1'.,x ) subject to r(x1,"" x) ~ p Mo LXj = Mo Xj ~ 0, j = 1,..., where p is a parameter represetig the miimal rate of retur specified by the ivestor. his model has several ice properties from the theoretical poit of view, but it has ot bee used extesively i practice by practitioers. Amog the more importat reasos are (i) solvig a large scale quadratic programmig problem o a real time basis was difficult, at least utill very recetly, (ii) may ivestors were ot coviced of the validity of the quadratic risk fuctio [6], (iii) the solutio of (2.3) may cotai too may ozero variables to be practical particularly whe Mo is relatively small. (2.4) Figure 2.1 hese observatios led us [5] to itroduce a alterative measure of risk W,,(X1,"" x) = E[ L R j Xj - E[L R j xj]l-l -a E[ L Rj Xj - E[L Rj xjll+l where a is a positive parameter represetig the degree of risk aversio of a ivestor ad (2.5) ~+ = { ~, if ~ ~ 0 0, if ~ < 0 (2.6) ~- = { ~~, if ~ ~ 0 if ~ < 0 heorem 2.1 f (R l,..., R) are multivariate O'mally distributed with mea (ll,"" Jl) ad variace-covariace matrix L == (O"ij) the Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

4 142 H. Koo Proof. t is well kow [8] that the radom variable Y = E.i=l Rj Xj is ormally distributed with mea J1. == E'J=1 J1.jXj ad variace u 2 == Ei=1 E'J=1 UijXiXj. hus W,,(Xl,"" x) = E[lY - E[Y]_] - a E[Y - E[Y]+] 1 /0 u 2 1 LOO u 2 = -- u exp--du - a-- u exp--du..j2;u -00 2u 2 V21ru 0 2u 2 u = J'<C(1 - a) v21r o his theorem implies that miimizig W,,(X,..., x) is equivalet to miimizig t'(xl,..., x,,) if a < 1 ad (R'... ' R) are multivariate ormally distributed. Let us ext proceed to the represetatio of the fuctio W,,(X,..., x) usig historical data or projected data. Let jt be the realizatio of radom variable R j durig period t (t = 1,..., ), which we assume to be available from historical data or from some future projectio. We also assume that the expected value of the radom variable ca be approximated by the average derived from these data. particular, let (2.7) j == E[Rj] = L jt/ t=1 he (2.8) (Xl'... ' x) = E[L Rj Xj] = L j Xj Let = E[ L Rj Xj - E[L Rj xj]_] - a E[ L Rj Xj - E[L Rj Xj]l+] = L{ L(jt - j)xjl_ - a L(jt - j)xjl+}/ t=1 (2.10) et = L(jt - j)xj, t = 1,..., ad let he (2.12) W,,(Xl,..., x) = L{letl- - a 161+}/ = L g,,(6)/ t=1 t=1 Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

5 Piecewise Liear Risk Portfolio Model 143 t ca be see from Fig. 2.2 that a = -1 correspods to absolu'te value (Lt} risk. Also a = 0 is associated with a ivestor who oly cares about "below the average" retur. Moreover, a > represets a ivestor whose risk associated with "below the average" retur is compesated to some extet by "above the average" retur. fact, a ivestor whose a is greater tha oe may be viewed as a risk proe ivestor. (b) a = 0 (c) a = 1/2 Figure 2.2 (d) a = 2 Replacig quadratic risk fuctio of (2.2) by compoud L1 risk fuctio (2.4), we obtai a alterative class of portfolio optimizatio problems pea): (2.13) miimize :E{ :E ajt xjl_ - al :E ajt xjl+} t=l subject to :E rj Xj 2: p Mo :EXj = Mo Xj 2: 0, j = 1,..., where ajt=rjt-rj,,...,; t=l,..., heorem 2.2 he class of optimizatio problems P(a) have the same optimal solutio JO' all a E (0,1). Also they have the same optimal solutio fol' all a gl'eater tha oe. Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

6 144 H. Koo Proof. Let us ote the idetity he 1 lel+ = 2{lel + e} E- = ~{E- E} -a l+a W",(X1,...,x) = {-- L L ajtxjl---llajtxj}/ 2 t=1 2 t=1 sice L L ajt Xj = L L(rjt - rj)xj = 0 t=1 t=1 hus for fixed, miimizig W",(X1,..., x) is equivalet to miimizig L L ajt xjl for t=1 all a less tha oe. Also for fixed it is equivalet to maximizig L L ajt x j for all a t=1 greater tha oe. 0 We thus eed to solve the followig two problems: (2.14) (2.15) miimize LLajtxjl t=1 subject to L X > pmo J J- LXj = Mo XJ 2: 0, j = 1,..., ma.xlmlze LLajtxjl t=1 subject to L X J J- > pmo LXj = Mo X j 2: 0, j = 1,..., t is well kow [2] that (2.14) is equivalet to the followig liear program: Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

7 Piecewise Liear Risk Portfolio Model 145 (2.16) miimize L Yt t=1 subject to Yt + L ajt Xj ~ 0, t = 1,..., Yt - L ajt Xj ~ 0, t = 1,..., Lrjxj ~ pmo LXj = Mo Xj ~ 0, j = 1,..., Remark. he dual of the problem (2.16) is: (2.17) maximize p Mo Z1 + Mo Z2 subject to 2 L ajt et + rj Z1 + Z2 ~ 0, j = 1,..., t=1 ~ et ~ 1, t = 1,..., Z1 ~ hus, we had better solve this problem istead of (2.16) if is less tha. 0 Problem (2.15)' o the other had caot be coverted ito a liear program. heorem 2.3 here exists a optimal solutio xj (j = 1,..., ) of {2.15} for which at most two idices j satisfy xj > 0. Proof. he objective fuctio of (2.15) is covex. hus there exists a optimal solutio amog extreme poits of the feasible regio [2], which has the stated property. 0 his theorem implies that the optimal portfolio ca be obtaied by eumeratig all extreme poits i O( 2 ) steps. 3 Capital Market Lie ad Equilibrium Model for L1 Risk Fuctio We showed i Sectio 2 that compoud L1 risk miimizatio is eqivalet to the L1 risk miimizatio problem (2.14) for all a less tha oe. this sectio, we will derive capital market lie ad the equilibrium relatio betwee idividual stock ad market portfolio [9] for L1 risk model. For this purpose, let us first assume without loss of geerality that Mo = 1. Also let us remove the oegativity costrait o each variable i problem (2.14). his meas that a ivestor is allowed to purchase each stock as much as he wats. Also, he is allowed to sell each stock short (i.e., regardless of whether he ows it or ot). Uder these assumptios, L1 risk miimizatio problem (2.14) is substat.ially simplified as follows: Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

8 146 H. Koo L: L: ajt xjl miimize t=1 (3.1) subject to L:rjxj 2: p L: Xj = 1 Let Xj(p) (j = 1,..., ) be a optimal solutio of this problem. Also let (3.2) f(p) = L: L: ajt xj{p)1 t=1 heorem 3.1 f(p) is a o-decreasig piecewise liear covex fuctio. Proof. f(p) is obviously o-decreasig. hat f(p) is piecewise liear covex fuctio follows from the stadard result of liear programmig [2] by otig that (3.1) is equivalet to the followig liear programmig problem: (3.3) miimize L: Yt t=1 subject to Yt + L: ajt Xj 2: 0, Yt - L: ajt Xj 2': 0, L: rj x j 2': P Ej'=l Xj = 1 0 t = 1,..., t = 1,..., Fig. 3.1 shows the piecewise liear covex fuctio f(p), which we call the efficiet frotier. l(p) -+ -L~ -L p Figure 3.1 Next, we itroduce a special asset So whose rate of retur is costat throughout the etire period, i.e., Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

9 Piecewise Liear Risk Portfolio Model 147 (3.4) Ot = O, t = 1,..., So is called a risk-free asset. By virtue of heorem 3.1, we ca draw a taget lie l(p) to the efficiet frotier f(p) passig through poit (O, 0). Let us assume i this sectio that M = (M' WM) is the uique poit of taget ad let PM be the portfolio correspodig to this poit. PM is called the market portfolio associated with Ll risk fuctio, which ca be obtaied by solvig the followig fractioal program: (3.5) miimize Er=1 E ajt x j E j Xj - subject to L Xj = 1 his problem is equivalet to the liear fractioal program: O (3.6) miimize subject to Er=l Yt E j Xj - O Yt + Lajtxj ~ 0, Yt - Lajtxj ~ 0, t = 1,..., t = 1,..., so that it ca be solved by a variat of simplex method [1]. heorem 3.2 All Ll risk miimizig ivestors purchase the combiatio of market portfolio ad risk-free asset ad othig else. Proof. Let ox be the amout of fud ivested ito risk free asset So ad let (1 - ox) be the amout offud ivested ito the market portfolio PM. Simple arithmetic shows that Ll risk of this portfolio is give by (1 - ox)wm' his meas that the Ll risk of this portfolio lies o the lie l(p) which lies everywhere below f(p). hus we coclude that the combiatio of risk-free asset ad market portfolio is better tha ay combiatio ofrisky assets 51,"" 5. o We will ext proceed to the derivatio of the equilibrium relat.ios betwee idividual stocks ad the market portfolio. For this purpose, let us defie a fuctio (3.7) h(xl,''''x ) = LLajtxjl t=1 Note that this fuctio is differetiable at (x{"f,...,x~) if (3.8) L ajt xf # 0, t = 1,..., Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

10 148 H. KOrlrlo heorem 3.3 Let h(xl,..., x) be differetiable at (x~,..., x~). he there exists a costat Oi such that (3.9) rj = rm + 8,(rM - ro), i = 1,..., Proof. Market portfolio xf (j = 1,..., ) is by defiitio a optimal solutio of the problem (3.5). hus there exists a costat.a such that ~ h(xl,..., x) -.A = 0 ox, ' '1=1 rj Xj - ro "'j="'f ' from which we obtai i = 1,..., (3.10) Ji - ri WM /( rm - ro) =.x, i = 1,..., M - O where o Ji = -;:;;-h(x1,..., x)",=",m UXi J J MUltiplyig xfl o both sides of (3.10) ad summig for all i, we obtai (3.11),\ = ' ~1 Ji xfl - M WJ,!(M - ro) Equatig both sides of (3.10) ad (3.11), we obtai the relatio (3.12) hus we obtai (3.9) by defiig (3.13) 8 i = [Ji - 2: J xf1"]/wm 0 1=1 ()i will be called "heta" of the stock Sj. Corollary 3.2 Let P = (Sl Xl,..., S x) be a arbitrary portfolio where 2: Xi = 1. he the theta of the portfolio is give by L 8. Xi. i=l Proof. he average retur of P is give by p=lixi=m+(2:8ixi)(rm-ro) 0 i=l.=1 Let us ote the remarkable similarity of (3.9) with the Sharpe-Liter-Mossi relatio (3.14) ri = O + f3i(rm - O) for the quadratic risk formulatio [3] where f3i is the so called "Beta" of Si. Extesio of our result to the o-differetiable situatio will eed more complicated a.alysis ad will be postpoed to the subsequet pa.per. i=l Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

11 Piecewise Liear Risk Portfolio Model Piecewise Liear Risk Fuctio with hree Liear Pieces this sectio, we itroduce a more geeral risk fuctio with three liear compoets depicted i Fig. 4.]. his risk fuctio ca be expressed as follows: p q -p + q et p - f3 q = -~ - et pi + -~ - f3 pi + ( )~ + ( + l)p where p, q, et, f3 are some positive costats. -(1 + p)e + 6 / -e ~--~--~-~ ~. e a p p ~ P Figure 4.1 Example. Let us cosider the special case i which (p, q, a, (3) = (1,1/2,1/2,3/2) Correspodig risk fuctio is give by g(~) = -~ - -pi + -~ - -pl- -~ + -p Let us cosider two discrete radom variables Xl ad X 2 whose desity fuctios are give by: 0.2, e=o 0.1, e=1 Prob{Xl = e} = 0.4, ~=2 0.3, ~=7 0, otherwise Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

12 150 H. Koo 0.3, e= , e=4 Prob{X 2 = e} = 0.1, e=5 0.2, e=6 0, otherwise Simple arithmetic shows that E[Xd = E[X2J = 3 ad V[XJ = V[X2J = 7.4. Hece two radom variables Xl ad X 2 are idetical from the viewpoit of Markowitz's model. t turs out, however that E{X1 - E[Xd}3 = 6.2 here is a good reaso to believe that a radom variable with larger third momet is preferred sice it has a loger tail to the right of the mea. Let us compute the risk of Xl ad X 2 usig risk fuctio g(e): gl = 2"EX1-2"P + 4"EX1-2"P- 4"E[Xtl + 8P g2 = 2"EX2-2"P + 4EX2-2"P- 4E[X2J + 8P able 4.1: P gl g2 gl - g able 4.1 shows the value of gl ad g2 for various value of p. t turs out that the risk of Xl is smaller tha that of X 2 for all values of p. hus Xl is preferred to X 2 i our model. his is compatible with our statemet about the third momet of these variables. 0 (4.2) he optimizatio problem usig the risk fuctio (4.1) ca be writte as follows: p ffillllffilze - :E :E jt Xj - a P Mol 2 t=l q ap-f3q +2" :E :E jt Xj - f3 pmol + ( 2 + l)p Mo t=l subject to :E j Xj 2: p Mo :EXj = Mo Xj 2: 0, j = 1,..., Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

13 Piecewise Liear Risk Portfolio Model 151 his is a covex miimizatio problem whose optimal solutio ca be obtaied by solvig the followig liear program: ( 4.3) miimize p q ap-f3q - LYt + - L Zt + ( + l)p Mo 2 t=l 2 t=l 2 subject to Yt - LjtXj ~ -apmo, Yt + LjtXj ~ apmo, Zt- LjtXj ~ -f3pmo, Zt + L jt X j ~ {3 p Mo, LjXj ~ pmo LXj = Mo Xj ~ 0, j = 1,..., he dual of this problem is maximize -2a p L ~t - 2{3 P L -lt + P Zl + Z2 t=l t=l ( 4.4) p+q subject to 2 L jt ~t + 2 L jt -lt - j Zl - Z2 ~ -2- L jt t=l t=l t=l o S ~t S~, 0 S fit S~, Zl~:: 0 t = 1,..., j = 1,..., Note that this problem has exactly the same umber of costraits as its couterpart (2.17), so that it ca be solved i much the same time as (2.17) by usig upper boudig simplex method. 5 Compariso of Models Usig Historical Data We compared the models proposed i this paper with the Markowitz's quadratic risk model usig historical data of okyo Stock Market. Mothly data of fifty stocks were collected for five years from November 1983 through October hese stocks are the oes icorporated i the future idex called Osaka 50. Fig. 5.1 shows the efficiet frotier of Markowi1.z's Lrisk model. Also Fig. 5.2 shows the efficiet frotiers of Lt-risk model (2.16) ad piecewise liear risk model (4.2), whose parameters are chose as follows: (p,q,a,{3) = (1, 2' 2' 2) All the problems were solved o SUN V system. Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

14 152 H. Koo 0"(%) L-O~.~28~----1~ L2----~~-3L lL----- p(%) Figure 5.1. Efficiet Frotier of L2 Risk Model l! Figure 5.2. Efficiet Frotier of L1 Risk Model. p(%) Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

15 Piecewise Liear Risk Portfolio Model 153 u(%) r ~ ". >< ~. X.@,,, X L j( L ~,, ~ " ~ X LrRisk ~ 'x L X X 0 X 'x - - ~ ~----'--~i ~~ ~~ l p(%) Figure 5.3. Efficiet Frotier of L2 Risk ad L1 Risk ModeL t should be oted that L 1 -risk efficiet frotier ad piecewise liear risk efficiet frotier were geerated withi 20 secods by applyig parametric liear programmig algorithm to the dual of the problems (2.16) ad (4.2). L 2 -risk frotier, o the other had was geerated by solvig 25 quadratic programs associated with 25 differet p values, amely p = 0.017, 0.018, "., Each case took a little less tha oe miute computig time. t could be solved i a couple of miutes if a efficiet parametric programmig code is available. Market portfolios for both of these models are idicated by poit M o each graph. Expected rate of retur of L 2, L1 ad piecewise liear risk models are 2.69%, 2.60%, 2.60% per moth, respectively. Fig. 5.3 shows the efficiet frotiers of L2 ad L1 models i terms of average retulstadard deviatio space. L1 frotier aturally lies above L2 frotier, ad its differece Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

16 154 H. Koo decreases as the required rate of retur p icreases. Average retur ad stadard deviatio of 50 stocks are plotted o the same graph. Stocks with circles are 'the oes cotaied i the Lt-risk market portfolio. hese stocks together with the stocks with cross marks are the oes cotaied i L 2 -risk market portfolio. N umber of Stocks 35-, o (p) Figure 5.4. Number of Stocks with Positive Weight. Fig. 5.4 shows the umbers of stocks with positive weight i the portfolio. Amog the remarkable observatios are (i) Stocks with positive weight i Ll-risk market portfolio are cotaied i the set of stocks with positive weight i Lrisk market portfolio for all values of p. (ii) Number of stocks icluded i Ll market portfolio is about oe half of the umber of stocks icluded i L 2 -risk portfolio for all values of p. able.5.1: Characteristics of the Market Portfolio Markowitz Lt Risk # of stocks i the portfolio average retur 2.69%/mo. 2.60%/mo. stadard deviatio 3.9% 4.5% stock with maximum share 14.6% 28% stock with miimum positive share 0.4% 2.1% sum of the share of top 5 stocks 47.0% 66.0% sum of the share of top 10 stocks 66.0% 92.2% able 5.1 shows some ofthe importat features of L 2 -risk ad Ll-risk market portfolio. t ca be see from this that Ll-risk model suggests oe to ivest substatial proportio of the Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

17 Piecewise Liear Risk Portfolio Model 155 available fud ito smaller umber of stocks. Moreover, this property holds for all values of p. Massive ivestmet ito smaller umber of stocks is very coveiet whe we cosider t.he itegral costrait imposed o real trasactio. fact, each stock has to be purchased at a iteger multiple of certai miimal uit, usually 1,000 stocks. hus to obtai a itegral solutio, we must either roud the variables to the earest iteger multiple of miimal uit or solve a smaller iteger programmig problem obtaied by elimiatig zero variable i the optimal solutio. either case, Lt-risk model is superior to L 2 -risk model sice the problem to be solved cotais fewer variables. We observe from this prelimiary experimet that Lt-risk model ad piecewise liear risk model ca alleviate some of the difficulties of Lrisk model referred to i the troductio. Numerical experimet usig 1,100 stocks of okyo Stock Market is ow uder way, whose result will be reported subsequetly. Ackowledgemets. he author ackowledges may costructive commets by aoymous referees. his research was supported i part by Grat-i-aid for Scietific Research of the Miistry of Educatio, Sciece ad Culture, Grat No Refereces [1] Chares, A. ad W. W. Cooper, "Programmig with Liear Fractioal Fuctios", Naval Research Logistics Quarterly 9, (1962). [2J Datzig, G.B., Liear Programmig ad Extesios, Priceto Uiversity Press, [3] Elt o, E.J. ad M.J. Gruber, Moder Portfolio heory ad vestmet Aalysis (3rd Editio), Joh Wiley & Sos, [4] Kariya, et. al, "Distributio of Stock Prices the Stock Market of Japa", oyo Keizai Publishig Co., 1989 (i Japaese). [5] Koo, H., "Portfolio Optimizatio Usig Lt Risk" HSS Report 88-9, st. of Huma ad Social Scieces, okyo stitute of echology, September, [6J Kroll, Y., H. Levy ad H. Markowitz, "Mea-variace Versus Direct Utility Maximizatio", he Joural of Fiace 39, (1984). [7] Markowitz, H., Portfolio Selectio: Efficiet Diversificatio of vestmets, Joh Wiley & Sos, [8] Rao, C.R., Liear Statistical ferece ad ts Applicatios (2d ed.), Joh Wiley & Sos, Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

18 156 H. Koo [9] Sharpe, W.F., "Capital Asset Prices: A heory of Market Equilibrium uder Coditios of Risk", he J otlrai of Fiace 19, (1964) [10] Sharpe, W. F., "Mea-Absolute-Deviatio Characteristic Lies for Securities ad Portfolios" Maagemet Sciece, B-1"-' B-13, Hiroshi KONNO: stitute of Huma ad Social Scieces, okyo stitute of echology, Oh-okayama, Meguro-ku, okyo, 152, Japa. Copyright by ORSJ. Uauthorized reproductio of this article is prohibited.

CAPITAL ASSET PRICING MODEL

CAPITAL ASSET PRICING MODEL RETURN. Retur i respect of a observatio is give by the followig formula R = (P P 0 ) + D P 0 Where R = Retur from the ivestmet durig this period P 0 = Curret market price P