This paper provides a new portfolio selection rule. The objective is to minimize the

Transcription

1 Portfolio Optimizatio Uder a Miimax Rule Xiaoiag Cai Kok-Lay Teo Xiaoi Yag Xu Yu Zhou Departmet of Systems Egieerig ad Egieerig Maagemet, The Chiese Uiversity of Hog Kog, Shati, NT, Hog Kog Departmet of Applied Mathematics, The Hog Kog Polytechic Uiversity, Kowloo, Hog Kog Departmet of Applied Mathematics, The Hog Kog Polytechic Uiversity, Kowloo, Hog Kog Departmet of Systems Egieerig ad Egieerig Maagemet, The Chiese Uiversity of Hog Kog, Shati, NT, Hog Kog xcai@secuhkeduhk mateokl@polyueduhk mayagx@polyueduhk xyzhou@secuhkeduhk This paper provides a ew portfolio selectio rule The objective is to miimize the maximum idividual risk ad we use a l fuctio as the risk measure We provide a explicit aalytical solutio for the model ad are thus able to plot the etire efficiet frotier Our selectio rule is very coservative Oe of the features of the solutio is that it does ot explicitly ivolve the covariace of the asset returs (Portfolio Selectio; Risk Averse Measures; Bicriteria Piecewise Liear Program; Efficiet Frotier; Kuh-Tucker Coditios) Itroductio The portfolio selectio problem is of both theoretical ad practical iterest Markowitz (952) laid the foudatios for this lie of research with his mea-variace (M-V) model While Markowitz used the portfolio variace as a risk measure, other risk defiitios have bee proposed Koo (990) ad Koo ad Yamazaki (99) used the mea absolute deviatio as their risk measure The mea absolute deviatio correspods to a l fuctio, whereas the variace correspods to a l 2 fuctio I this paper we propose a more coservative portfolio selectio rule whereby the ivestor miimizes the maximum risk of the idividual assets This risk measure correspods to a l fuctio The classic M-V model ca be solved aalytically for the efficiet frotier (see Merto 972) whe short sellig is permitted It has bee foud that the compositio of the optimal portfolio ca be very sesitive to estimatio errors i the expected returs of the uderlyig assets; see Chopra ad Ziemba /00/4607/0957$ electroic ISSN (998), Hesel ad Turer (998), Chopra et al (992) ad Best ad Grauer (99a, 99b, 992) I the case of large-scale optimizatio problems the relatioships betwee the iputs ad the optimal portfolio teds to be obscured (Best ad Grauer 99a) Our model provides a clear coectio betwee the expected returs of the assets ad their importace i the optimal portfolio Uder our decisio rule there are two steps i the solutio First we rak the idividual assets i terms of their expected returs ad risks Secod, we compute the optimal properties based o the iformatio cotaied i the rakigs The rakig rule cosists of ieualities amog the expected returs This eables us to see more clearly how the compositio of the portfolio varies There are two importat differeces betwee our model ad covetioal models, such as the M-V model I our model we do ot allow for short sellig We impose this restrictio to obtai a simple aalytical solutio It is a weakess of the model I our model correlatios amog the Maagemet Sciece 2000 INFORMS Vol 46, No 7, July 2000 pp

2 Portfolio Optimizatio Uder a Miimax Rule assets are ot take ito accout This is i cotrast to the M-V approach where diversificatio helps to reduce risk However, we argue that total portfolio risk uder the covetioal defiitio will be kept small if our risk measure is kept small I some respects our approach is related to that of Youg (998) I both models there is o short sellig ad the asset correlatios do ot eter explicitly ito the solutios The mai differeces are: (i) Our model miimizes the expected absolute deviatio of the future returs from their mea while Youg s model maximises the miimum portfolio retur over a set of past returs (ii) Youg s solutio ivolves liear programmig whereas we are able to provide a aalytical solutio 2 A New Risk Measure Based o l I this sectio, we itroduce our risk measure ad formulate the correspodig portfolio optimizatio problem with this measure Assume that a ivestor has iitial wealth M 0, which is to be ivested i possible assets S j, j,, Let R j be the retur rate of the asset S j, which is a radom variable Let 0 be the allocatio from M 0 for ivestmet to S j (Note that by assumig 0 we are cocered with the situatio where short sellig is ot allowed) Thus, the feasible regio for the portfolio optimizatio problem is x x,,x : M 0, 0, j j,, (2) Let E(R) deote the mathematical expectatio of a radom variable R Defie r j ER j ad ER j r j Namely, r j ad deote the expected retur rate of the asset S j ad the expected absolute deviatio of R j from its mea, respectively The expected retur of a portfolio x ( x,,x ) is give by E rx,,x j R j x j j ER j r j (22) j The l measure we propose is defied as follows Defiitio 2 The l risk fuctio is defied as w x max ER j r j (23) j Let x The w (x) ma E(R j E(R j )) ma This fuctio is explicitly kow if the distributio of each radom variable R j is give For example, if R j is ormally distributed, the it is easy to verify (see Koo ad Yamazaki 99) that w x max 2 j j, (24) where j is the stadard deviatio of R j Historical data ca also be used to estimate r j ad We assume that ivestors wish to maximize expected retur while miimizig their risk level This is a optimizatio problem with two criteria i coflict Uder the l risk measure as defied above, our portfolio optimizatio problem ca be formulated as a bicriteria piecewise liear program as follows, which is deoted as POL (the Portfolio Optimizatio problem with the l risk measure) Defiitio 22 The bicriteria portfolio optimizatio problem POL uder the l risk measure is formulated as: 2 Miimize max j subject to x,, j r j x j where a feasible portfolio x ( x,,x ) is said See, for example, Başer ad Berhard (995), for more discussio o the l otatio ad other related miimax measures 2 Miimize ( A, B) idicates that A ad B are the two criteria to be miimized 958 Maagemet Sciece/Vol 46, No 7, July 2000

3 Portfolio Optimizatio Uder a Miimax Rule to be efficiet if there exists o y ( y,,y ) such that max y j max, j j j r j y j r j, j ad at least oe of the ieualities holds strictly Accordigly, the fuctio value (ma, j r j ) is said to be a efficiet poit I words, a efficiet poit is such that there exists o solutio better tha it with respect to both criteria The efficiet frotier is the collectio of all efficiet poits By a simple trasformatio, oe ca show that POL is euivalet to the followig Bicriteria Liear Programmig (BLP) problem Miimize y, j r j subject to y, j,,, x Now we covert the bicriteria liear programmig problem BLP ito a parametric optimizatio problem with a sigle criterio For a fixed, where 0, the Parametric Optimizatio problem of BLP, deoted as PO(), is as follows: Miimize F x, y y j subject to y, j,,, x r j The euivalece relatio betwee BLP ad PO() is give below (cf Yu 974 for proof) Propositio 2 Cosider the problems BLP ad PO() The pair (x, y) is a efficiet solutio of BLP if ad oly if there exists a (0, ) such that (x, y) is a optimal solutio of PO() Oe ca thik of as a ivestor s risk tolerace parameter the larger the, the more risk the ivestor is to tolerate Because of the euivalece betwee POL ad BLP, there exists the same euivalece relatioship betwee POL ad PO() Thus, a optimal solutio for PO() gives, accordigly, a efficiet solutio for POL To obtai the efficiet frotier of POL, oe has to kow the optimal solutios of PO() for all (0, ) I 3 ad 4 below, we will show that a optimal solutio for PO() ca be derived aalytically, ad coseuetly the whole efficiet frotier of POL ca also be determied aalytically 3 A Simple Optimal Ivestmet Strategy Cosider the problem PO() with a give (0, ) Note that the parameters r j E(R j ) ad E(R j E(R j )), j, 2,,, are costats i PO() We assume that r r 2 r (3) Furthermore, to avoid ambiguity, we assume that there do ot exist two assets S i ad S j, i j, such that r i r j ad i (if such two assets do exist i the origial problem, we may treat them as a sigle aggregate asset) 3 All Assets Are Risky I this subsectio we cosider the case where all the assets are risky We preset our mai result ad provide some discussio of its meaig Theorem 3 For ay (0, ), a optimal solutio to PO() is give by M 0 x* j l* l, j *, 0, j*; y* M 0 l* l (32), (33) where *() is the set of assets to be ivested, which is determied by the followig rule: (a) If there exists a iteger k [0, 2] such that r r, (34) Maagemet Sciece/Vol 46, No 7, July

4 Portfolio Optimizatio Uder a Miimax Rule r r k ad r r k the r r 2 r r 2, (35) r r k r r k r k r k k, (36) r k r k k r k r k k, (37) *,,, k (38) (b) Otherwise, if the coditio above is ot satisfied by ay iteger k [0, 2], the *,,, (39) The proof of this theorem is give i Appedix A We ow discuss the meaig of the results by examiig how chages i the portfolio affect its expected retur ad risk Suppose we have a portfolio x 0, i which 0 0if j 0 ad 0 0ifj 0 Namely, 0 () is the set of assets to be ivested Assume there is a asset S h that is ot icluded i 0 () We will ow aalyse how the performace of the portfolio will chage if the allocatio x h to the asset S h is icreased Let us costruct a ew portfolio x as follows: 0 j, if j 0 ; h, if j h; 0, otherwise (30) Further, suppose y 0 is the correspodig risk of the portfolio x 0 (that is, (x 0, y 0 ) costitutes a solutio for the problem PO()) We costruct a ew solutio (x, y) for PO() by creatig x as above ad settig y y 0 y (3) To meet the costraits of the problem PO(), we should have i x 0 j i 0 j y 0 y, j h M 0, j 0 x h h h h y 0 y for j 0, Thus, we ca choose j, h ad y as positive umbers satisfyig h j, (32) j 0 j y, j 0, (33) h y 0 y / h (34) Recall F (x, y) is the objective fuctio of the problem PO() We have F x, y y 0 y r j x 0 j j j 0 r h h F x 0, y 0 y r h j j 0 r j j j 0 F x 0, y 0 F, (35) where F y r j j r h j j 0 j 0 (36) It is easy to see that i the above euatio, { y } idicates the decrease i the risk, {( ) j 0 () r j j } is the decrease i the expected retur due to the decreases i the allocatios to the assets j 0 (), ad {( ) j 0 () r h j } idicates the icrease i the expected retur due to the additio of the asset S h to the set of assets to be ivested From (33), we ca rewrite (36) as F r j r h j 0 j y (37) Defie a { j: j 0 () ad r j r h } ad b { j: 960 Maagemet Sciece/Vol 46, No 7, July 2000

5 Portfolio Optimizatio Uder a Miimax Rule j 0 () ad r j r h } The, (37) ca be rewritte as where ad F F a F b, (38) a F b F r j r h j j a y, (39) r h r j j j b y (320) We ow have the followig few scearios Case : a F 0 From (39), we ca see that a F 0 meas that the decrease i the risk value, { y }, is greater tha the decrease i the et expected retur This together with b F 0 gives us F 0 Thus, it follows from (35) that F (x, y) F (x 0, y 0 ) This meas that the portfolio x 0 ca still be improved if the asset S h is icluded ito the set of assets to be ivested This justifies (36) Case 2: a F 0 ad the set b is empty I this case, F a F 0 This meas that icreasig the allocatio for the asset S h will either result i a decrease i the et expected retur that is greater tha the decrease i the risk value (if a a F 0), or yield o beefit (if F 0) Hece, the asset S h should ot be icluded i the set of assets to be ivested This justifies (37) Case 3: a F 0 ad the set b is ot emptyas b is ot empty, there must exist at least oe elemet m b such that r m r h We ca show, followig a similar idea as above (that is, alterig the portfolio by removig the asset S m ad the icreasig the allocatios to other assets accordigly), that i this case the asset S m should ot be icluded i the set 0 () for ivestmet This, together with the observatio that the asset S m must meet the coditio (37) (this is because r m r h ad a F 0), justifies (37) The above gives a justificatio of the rakig rule of Theorem 3 We ow provide several remarks regardig the optimal portfolio as determied by the theorem Remark 3 It may be a bit surprisig to observe that the ivestmet strategy give by Theorem 3 always suggests icludig those assets with higher retur rates first I other words, a asset with a higher retur rate should always be cosidered before a asset with a lower retur rate is cosidered for selectio The reaso for this apparetly couterituitive result is that the actual amout ivested i a particular asset also depeds o the risk of that asset Hece, it is possible that the actual ivestmet to a asset with a high retur rate is early zero, eve if it may have bee selected accordig to the rules give by Theorem 3 To see this more clearly, cosider a example which we assume satisfies (r r )/ /( ) ad (r r 2 )/ (r r 2 )/ /( ) Thus, it follows from Theorem 3 that a optimal ivestmet strategy is to select assets S ad S oly Further, accordig to Theorem 3, the actual amouts of ivestmet for the assets will be respectively: x* x* M 0 / / M 0 /, ad (32) M 0 / / M 0 / (322) Clearly, if is much greater tha, the it is possible that x* is early zero while x* is early eual to M 0 The optimal strategy as described i Theorem 3 is a two-phase decisio I the first phase, the assets are selected accordig to their retur rates The i the secod phase, the actual amouts allocated to those selected assets are determied based o their risks Whe the trasactio cost for ivestig a asset is take ito cosideratio, a very small allocatio of fudig to the asset may mea that it should i fact be omitted Thus, uder the ivestmet strategy of Theorem 3, a asset may be elimiated i either Phase or Phase 2 I Phase, it may be elimiated if its retur rate is too low, while i Phase 2, it may also be elimiated if its risk is too high Remark 32 The optimal ivestmet strategy give by Theorem 3 has the property that x* j y* for all j *() (ad x* j 0 for all j *()) This meas that, for the assets selected for ivestmet, we Maagemet Sciece/Vol 46, No 7, July

6 Portfolio Optimizatio Uder a Miimax Rule shall ivest them with the amouts such that they have the same risk y* (ote that E(R j r j ) represets the risk of ivestig a amout i asset S j ; see Defiitio 2) Note that our problem uder the l measure is to miimize the maximum idividual risk, amely, w (x) ma E(R j r j ) Theorem 3 implies that to achieve this objective, a optimal ivestmet strategy should ivest the assets i such a way that their risks are eual Oe ca see that, if this is ot the case, that is, there exists some asset whose risk is less tha that of aother asset, the the allocatio to this asset ca be icreased Such a chage of allocatio will ot icrease the maximum risk, while the overall expected retur will be icreased (for details, see the proof of Lemma 4 i 4 the part o the iefficiecy of the solutio (x*, y*) if the risks of the assets selected uder (x*, y*) are ueual) Remark 33 Aother property of the optimal portfolio give by Theorem 3 is that the amouts of x* j, j *(), do ot deped o the retur rates r j, as log as the set *() is selected This idicates that the expected retur rates determie the set of ivestable assets, but do ot ifluece the magitudes of the allocatios This property does ot seem to exist i portfolio solutios uder other models such as the classic M-V model Why is this a sesible property? The aswer is: (i) the iformatio of the expected retur rates is exploited already i the selectig rules of Theorem 3 whe determiig the set of ivestable assets; ad (ii) after the assets selected for ivestmet have bee determied usig the selectig rules, how to miimize the risk of the ivestmet will become the major cocer Uder our model, the maximum idividual risk is to be miimized As we discussed above, to achieve this goal, a sesible way is to have all the assets ivested carryig the same risk This leads to the allocatios that are idepedet of the expected retur rates Remark 34 It is easy to see from Theorem 3 that a case where we ivest all fud M 0 i a sigle asset is whe r r (325) Nevertheless, it should be oted that there exist other cases where almost all the fud M 0 should be ivested i a sigle asset See Remark 3 above A case where all the fud is ivested i a sigle riskless asset will be further addressed i 32 below Remark 35 Note that the case where we will select all the assets S j, j, 2,,, for ivestmet is whe Coditio (37) is ot satisfied by ay 0 k 2 I this case, *() {,,,2,} ad the proportios of the assets i the efficiet portfolio are give by (32) More specifically, if we assume M 0, from (32) we have x* i / i, i, 2,, (326) / k k There exists a iterestig relatioship 3 betwee this portfolio ad the global miimum variace (GMV) portfolio uder the M-V model (see, eg, Hauge 997) This is aalysed as follows Suppose the variace of asset S i is 2 i, i, 2,,, ad assume that the assets are ucorrelated I this case we ca show that the proportios of the assets, x { i, i the GMV portfolio are give below: x i { k / 2 i / k 2, i, 2,, (327) Moreover, i the situatio whe the asset returs are ormal, we have i (2/) i (see (24)), ad thus (326) ca be rewritte as x* i / i, i, 2,, (328) / k k This has a clear aalogy with (327) Uder our model, the efficiet portfolio uses /, while i the M-V model, the GMV portfolio uses / 2 Remark 36 I recet years, it has bee show (see Chopra ad Ziemba 998, Hesel ad Turer 998, Chopra et al 992, ad Best ad Grauer 99a, 99b, 992) that the compositio of a efficiet M-V portfolio ca be extremely sesitive to errors i problem iputs I particular, it has bee foud that errors i 3 We are very grateful to Professor P P Boyle, the departmet editor for Fiace, for poitig out this relatioship The material here has bee based o a ote kidly provided by him 962 Maagemet Sciece/Vol 46, No 7, July 2000

7 Portfolio Optimizatio Uder a Miimax Rule the asset meas ca be much more damagig tha errors i other parameters Therefore, a similar uestio we may face is how sesitive the solutio of our model could be to chages i the asset meas We ow discuss this uestio Assume that m k is the asset that satisfies the Coditio (37) I other words, *() {,,,m } ad asset S m is the first oe excluded from the set *() (see Theorem 3) Let us aalyse the followig three categories of assets, where i deotes a perturbatio of r i (a) O the assets S j with j m: All these assets are ot selected for ivestmet by Theorem 3 Furthermore, this solutio remais uchaged as log as r j j r m This meas that the optimal portfolio is uchaged as log as the perturbatios are withi the followig rages: j r m r j, for j m (329) (b) O the asset S m : Clearly, if (i) r m m r m ad (ii) r r m m r r m m r m r m m m, the the optimal portfolio is uchaged Let m km (r k r m )/ k /( ) The, to satisfy the above two coditios, it is sufficiet for us to have m mi m, r / m r k m (330) km (c) O the assets S j with j m: Let j /( ) kj (r k r j )/ k, for j m Bear i mid that we wat to determie a iterval for j such that the coditios of Theorem 3 remai satisfied After some developmet, we ca show that a sufficiet coditio is as follows: max j kj max j / k, m, r j r j j mi m, r j r j, if j m ; (33) kj / k, m, r j r j j r j r j, if j m (332) I summary, the aalysis above idicates that if the perturbatio i of a asset mea r i is withi the iterval of (329) (332), the the optimal portfolio uder our model will remai uchaged Note that (329) (332) are sufficiet coditios, ad i may cases these may ot be satisfied at all (but the portfolio may still remai uchaged if they are ot satisfied) Geerally speakig, because the rakig rule for selectig the assets to be ivested uder our model is give by a set of ieualities (34) (37), our model exhibits some robustess agaist errors i the problem iputs O the other had, we should emphasize that our model ca also be uite sesitive i some cases to errors of the problem parameters, such as the meas To illustrate, let us cosider a example i which there are three assets with estimated meas r r 2 r 3 Suppose (r 3 r 2 )/ 3 /( ) ad (r 3 r )/ 3 (r 2 r )/ 2 /( ) The, by Theorem 3, we choose Assets 2 ad 3 ad x* 0, x* 2 3 /( 2 3 ), ad x* 3 2 /( 2 3 ) Further, suppose 2 is much greater tha 3 The, x* 2 0, ad x* 3 M 0 However, there is a error i r ad the actual mea r of Asset satisfies r 2 r r 3 ad (r 3 r )/ 3 /( ) ad (r 3 r 2 )/ 3 (r r 2 )/ /( ) I this case, we should actually choose Assets ad 3 ad the portfolio should be: x* 2 * 0, x* * 3 /( 3 ), ad x* 3 * /( 3 ) Further, suppose 3 is much greater tha The, x* * M 0, ad x* 3 * 0 I this example, a error i the estimatio of r has chaged the portfolio almost completely 32 Iclusio of a Riskless Asset We ow cosider the case where there exists a riskless asset Without loss of geerality, we may assume that this riskless asset has the lowest retur, amely, i (recall our Assumptio (3) ad ote that all risky assets could be elimiated from cosideratio if their returs are ot greater tha that of the riskless asset) Uder the assumptio above, we have 0 To geeralize the result i 3, we first assume that 0, where is a sufficietly small umber We the obtai our result by lettig 3 0 Let us ow cosider the followig two cases Case Accordig to the rule give i Theorem 3 (with 0), we fid that *() I this case, Maagemet Sciece/Vol 46, No 7, July

8 Portfolio Optimizatio Uder a Miimax Rule it is obvious that the optimal solutio for PO() as give i Theorem 3 is uchaged Case 2 Accordig to the rule give i Theorem 3 (with 0), we fid that *() I this case, the optimal solutio for PO() becomes x* j M 0 x* j 0, j*,, j *, lk,l l y* M 0 lk,l l Lettig 3 0, we obtai x* j 0 for all j, x* M 0 ad y* 0 Clearly, the case where the riskless asset S is selected ito the set *() for ivestmet happes oly whe r r r r r 2 r 2 (333) Combiig Case with Case 2 above, we have the followig result Theorem 32 Give ay (0, ) If (333) is ot satisfied, the the set *() of assets to be selected should be determied by (34) (38) ad the optimal solutio should be computed by (32) ad (33) Otherwise, if (333) is satisfied, the optimal ivestmet strategy should be to ivest all fud M 0 i the riskless asset (where y* 0) 4 Tracig Out the Efficiet Frotier We ow discuss how to determie the efficiet frotier of the problem POL Correspodig to the results i 3 ad 32 respectively, let us carry out our aalysis i the followig two subsectios 4 No Riskless Assets Are Ivolved First, defie: k k, k, 2,,, (4) k r r k r r k r k r k k, k, 2,, (42) It is easy to verify that k k k r k r k, k, 2,, 0 0, (43) where k ca be computed by the followig recursive relatio: k k, k, 2,, k (44) 0 0 It is clear that (34) (37) reduce to determiig a iteger k [0, 2] such that:,, k, k (45) Because E(R j E(R j )) 0 for ay j, k 0 for k, 2,, (see (44)) Thus, otig that r j r j for j, 2,,, we kow that 0 2 (46) Therefore, the Coditios (45) reduce to: k Or, euivaletly, Lettig k, k, (47) k k k k (48) k k ad k k k, (49) we see that the Coditios (45) (or (34) (37)) are further euivalet to k, k (40) Bear i mid that we wat to determie the efficiet 964 Maagemet Sciece/Vol 46, No 7, July 2000

9 Portfolio Optimizatio Uder a Miimax Rule frotier, amely, all the efficiet poits, correspodig to all possible (0, ) Recall Theorem 3 Give a iteger k [0, 2], it is clear that for all values of i a iterval ( k, k ], the set *() of assets selected for ivestmet remais uchaged, ad thus the optimal solutio of PO() as give by (32) ad (33) remais uchaged Accordig to the discussios i 2, such a solutio correspods to a efficiet poit of POL Now the uestio is whether there exist other efficiet poits for ( k, k ] This is euivalet to askig whether there exist other optimal solutios for PO() whe ( k, k ] Our mai idea to aalyse the efficiet frotier cosists of the followig argumets: () For all ( k, k ), the solutio give by (32) ad (33) is the uiue optimum for PO() Thus, there is oly oe efficiet poit for POL (2) For k, there are multiple efficiet poits for POL, which, however, ca be determied aalytically The followig two lemmas establish these argumets respectively For otatioal coveiece, we let below Lemma 4 For ay k 0,,,, if ( k, k ), the the solutio give by (32) ad (33) is the uiue optimal solutio for PO() The proof of this lemma is give i Appedix B By substitutig (32) ad (33) ito the objective fuctios of POL (Defiitio 22), oe ca see that, correspodig to the solutio (x*, y*) for ( k, k ), the efficiet poit of the problem POL is eual to where y* is give by (33), while P* k y*, z*, (4) z* M 0 r j l* j l* Usig the otatio above, we have l (42) Lemma 42 For ay k 0,,, 2, if k, the (y* y, z* y k /( k )), where 0 y y* j* k /, (43) is a efficiet poit of the problem POL The proof of this lemma is give i Appedix C Remark 4 From Lemma 42 ad its proof we ca see that ay solutios which select the assets i *( k ) as well as the asset l k are optimal to PO() The iclusio of a asset which is ot i *( k ) decreases the total retur, which, however, also reduces the risk If k for 0 k 2or ( 2, ), the such a solutio will be domiated by the solutio give by (32) ad (33) (Lemma 4 implies this fact) However, if k for 0 k 2, the reductio i risk (with a weight eual to k ) is just balaced by the reductio i total retur (with a weight eual to k ), ad thus geerates a udomiated (efficiet) poit I summary, we have the followig result o the efficiet frotier Theorem 4 The efficiet frotier of the problem POL ca be determied by cosiderig itervals ( k, k ), k 0,,,, as well as edpoits k, k 0,,, 2 Specifically, the efficiet frotier cosists of () the efficiet poit ( y*, z*) correspodig to each ( k, k ) with k 0,,,, where y* ad z* are give by (33) ad (42) respectively; ad (2) the multiple efficiet poits ( y* y, z* y k /( k )) correspodig to each k with k 0,,, 2, where y is govered by (43) 42 Riskless Assets Are Ivolved Similar to 32, we assume, without loss of geerality, that there is oly oe riskless asset S i0 ; amely, 0 for j i 0 ad i0 0 Accordig to Theorem 32, the optimal solutio for PO() is to allocate all fud M 0 to the riskless asset S i0 whe (322) is satisfied, ie, i0 /( ) This is euivalet to ( i0, ) I this case, it is easy to see that ay other solutio with a x k 0 (ad thus x i0 M 0 ), where k i 0, will be worse tha the solutio with x k 0 ad x* i0 M 0 (because r k r i0 but k i0 0, ay reallocatio of the fud M 0 from the asset S i0 to the asset S k will icrease the objective fuctio value of PO()) This meas that the optimal solutio for PO() is uiue whe ( i0, ) Whe ( i0, ), the asset S i0 is ot selected by Maagemet Sciece/Vol 46, No 7, July

10 Portfolio Optimizatio Uder a Miimax Rule the rule of Theorem 3 ad the relevat aalysis i 42 above is still valid Therefore, we have Theorem 42 The efficiet frotier of the problem POL ca be determied by cosiderig i 0 itervals ( k, k ), k 0,,, i 0, ad ( i0, ), as well as i 0 edpoits k, k 0,,, i 0 Specifically, the efficiet frotier cosists of () the efficiet poit ( y*, z*) correspodig to each ( k, k ) with k 0,,, i 0 or ( k,)with k i 0, where y* ad z* are give by (33) ad (42) respectively; ad (2) the multiple efficiet poits ( y* y, z* ( k /( k )) y ) correspodig to each k with k 0,,, i 0, where y is govered by (43) 5 Total Portfolio Risk ad Covariaces At first sight, it seems that either our ew risk measure w (x), or the optimal solutio derived, depeds o the covariaces betwee assets Also, it seems that oly the risks of the idividual assets, rather tha the risk of the etire portfolio, are take care of This feature of our approach is i marked cotrast to the covetioal approach which explicitly takes accout of the covariaces betwee the assets We should poit out here that the total portfolio risk is i fact cotaied i our model, albeit i a implicit way To be more specific, we will show i the followig that the total portfolio risk is bouded above by our risk criterio w (x) It is well kow that the total portfolio risk is usually modeled as a kid of deviatio of the actual total retur from the expected total retur For example, i Markowitz s M-V model, the total portfolio risk is defied to be the variace as follows: E w 2 x j R j r j 2, (5) j which is the expected suared deviatio of the actual total retur j R j from the expected total retur j r j Now, istead of cosiderig the expectatio of the deviatio, let us cosider the probability that this deviatio is greater tha a prespecified level, amely, P( j r j j R j ), where is a give positive umber Clearly, to make this probability as small as possible is also a way to esure the deviatio of the actual total retur from the expected total retur as small as possible This is a alterative measure of the total portfolio risk By usig the Markov ieuality (cf eg, Leo-Garcia 994, p 37), oe ca obtai P j r j R j x j j E j R j j r j x j ER j r j w x (52) j The above ieuality idicates that the total portfolio risk is bouded by w (x) multiplied by a costat / which is idepedet of the choice of the portfolios The total portfolio risk will be small if w (x) is kept small (evertheless, it may ot be true the other way aroud) The aalysis above shows that the total portfolio risk is compressed by the risk w (x), ad what we propose to do is to miimize w (x) Note that the covariaces amog assets are ivolved i the total portfolio risk Nevertheless, we do ot have to deal with the covariaces directly The advatage of doig so is obvious eve from the implemetatio poit of view: The optimal ivestmet strategy uder our model is much easier to compute ad implemet, ad the whole efficiet frotier is also much easier to costruct A iterestig uestio is how the compositios of the portfolios uder the l model ad the M-V model would chage, ad compare to each other, whe the assets to be ivested have differet degree (or tightess) of correlatios To examie this uestio, we will ow cosider a simple example which cotais two assets Let 2 E[(R r ) 2 ], 2 2 E[(R 2 r 2 ) 2 ], ad COV(R, R 2 ) E[(R r )(R 2 r 2 )] Further, let be the correlatio coefficiet; that is, defie 966 Maagemet Sciece/Vol 46, No 7, July 2000

11 Portfolio Optimizatio Uder a Miimax Rule COV(R, R 2 )/( 2 ) The, the M-V model ca be formulated as follows: Miimize w 2 x, r x r 2 x 2 (53) subject to x x 2 M 0, (54) x 0, x 2 0, (55) where w 2 (x) is the variace of the total portfolio (see (5) above) I this two-asset problem, oe ca see that w 2 (x) 2 x x x x 2 Itroducig a parameter, where 0, we ca covert the above bicriteria problem to a parametric optimizatio problem as follows: Figure 5 Efficiet Frotiers of l 2 ad l Models i M-V Space (r 05, r 2 025) Miimize 2 x x x x 2 r x r 2 x 2 (56) subject to x x 2 M 0, (57) x 0, x 2 0 (58) Deote the optimal solutio for the above problem as xˆ ( xˆ, xˆ 2) (xˆ correspods to a efficiet poit for the multicriteria model give by (53) (55); cf Yu 974) Applyig Kuh-Tucker coditios, after some simplificatio we ca obtai the followig results: (a) Let A ( ) M 0 (( )/2)(r r 2 ) ad A 2 ( 2 2 ) M 0 (( )/2)(r 2 r ) If A 0 ad A 2 0, the xˆ xˆ 2 A , (59) A ; (50) (b) If A 0, the xˆ 0 ad xˆ 2 M 0 ; (c) If A 2 0, the xˆ M 0 ad xˆ 2 0 Now, let us cosider the followig two cases Case 5 0, amely, assume that the two assets are ot correlated Moreover, assume that the parameters i this case have bee so chose that the two assets are all selected uder both the l ad the M-V models It follows from (59) ad (50) that xˆ M 0 2, (5) 2 r r xˆ M r 2 r (52) O the other had, it is easy to see from Theorem 3 that x* 2 2 M 0, (53) x* 2 2 M 0 (54) Note that the role of i is similar to that of 2 i Therefore, comparig (5) ad (52) with (53) ad (54), we ca see that the relevat solutio xˆ i uder the M-V model has a additioal term (ie, the term (( )/2)((r r 2 )/( )) for x ad (( 2 )/2)((r 2 r )/( 2 2 )) for x 2 ) This ca be regarded as a compesative term, which makes use of the iformatio give by the retur rates r ad r 2 to fie-tue the portfolio ( xˆ, xˆ 2) The effect of the compesatio reduces whe the differece betwee r ad r 2 decreases This argumet is illustrated by Figures 5 ad 52, where we show the efficiet frotiers of the l ad M-V models, 4 i the 4 I each of the figures, we assumed the two assets follow idepedet ormal distributios, with 2 08, , ad 0 The values of ad 2 were computed by the relatio (2/) j ; see (24) We altered oly the values of r ad r 2 i the two figures Maagemet Sciece/Vol 46, No 7, July

12 Portfolio Optimizatio Uder a Miimax Rule Figure 52 Efficiet Frotiers of l 2 ad l Models i M-V Space (r 03, r 2 025) M-V space The two figures cosider respectively the two situatios: the differece betwee r ad r 2 is large (Figure 5), ad small (Figure 52) Note that the l solutio is always domiated by the M-V solutio i the M-V space, ad therefore the efficiet frotier of the l model is always below that of the M-V model Nevertheless, from the two figures we see that the l frotier teds to approach the M-V frotier whe the differece betwee r ad r 2 decreases Case , amely, assume the variaces of the two assets are close to each other I this case, from (59) ad (50) we have xˆ 2 M r r M 0 2 r r 2 4, (55) xˆ 2 2 M r 2 r M 0 2 r 2 r 4 (56) Thus, if, amely, the two assets are highly correlated, the portfolio xˆ uder the M-V model ca be very sesitive to the parameters It is possible that a small differece i some parameters will cause xˆ to allocate all the fud M 0 to a asset (eg, Asset ) ad othig to the other asset The allocatio uder the l model remais as (53) ad (54) (More specifically, x* x* 2 2 M 0,if 2 ) Remark 5 I summary, we have the followig observatios: (i) I situatios i which the assets have low or o correlatios, the allocatio to each asset uder the M-V model may be further tued by the iformatio o the retur rates As compared to the portfolio uder the l model, the portfolio uder the M-V model may yield a higher retur (ii) I situatios i which the assets are highly correlated ad the variaces of the assets are close to each other, the geeratio of a portfolio uder the M-V model ca be highly sesitive to the parameters A small error i the estimatio of the parameters may result i a totally differet portfolio Nevertheless, i these situatios a diversificatio i the portfolio of the l model is still maitaied this should be a desirable feature to avoid the risk of geeratig a wrog portfolio due to some small differece i the parameters (iii) A special case of (ii) above is whe For the M-V model, this correspods to the global miimum variace portfolio (see also Remark 35) I this case, (55) ad (56) reduce to xˆ xˆ 2 2 M 0, ad ow the portfolio uder our l model ad that uder the M-V model become early idetical Sice the global miimum variace portfolio is the most coservative solutio uder the M-V model, this also teds to affirm the argumet that our model is a very coservative oe 6 Cocludig Remarks This article addresses the problem of portfolio selectio for cautious ivestors A portfolio optimizatio model with a ew l risk measure has bee proposed A simple scheme has bee derived, which geerates the efficiet portfolio uder the l model aalytically We have also show how the whole efficiet frotier of the l model ca be traced out aalytically A simple example is discussed to show the portfolio compositios of the l model as compared to the M-V model The aalysis idicates that the l model would be more stable whe assets to be ivested are highly correlated Nevertheless, the M-V model may geerate 968 Maagemet Sciece/Vol 46, No 7, July 2000

13 Portfolio Optimizatio Uder a Miimax Rule higher returs whe the assets have o or low correlatios The ew l model ad the related techiues are easy to maipulate ad implemet i practice For example, our selectio of the efficiet portfolio is based o a simple rakig rule, which evaluates oe asset at a time, to determie whether or ot it should be icluded ito the portfolio This ot oly allows a portfolio maager to evaluate which of the curret assets are ivestable, but also eables him to assess the impact of the itroductio of ay ew asset o the efficiet portfolio Also, from the rakig rule, a portfolio maager ca see the desirable characteristics of those good assets Besides, our selectio of the optimal portfolio does ot ivolve the correlatios amog stocks This releases a portfolio maager from the reuiremet to relate his portfolio selectio to those complicated covariace matrices The whole efficiet frotier of our model ca be costructed aalytically This is useful, as it makes it easy to examie the various possible trade-offs betwee retur ad risk As revealed by recet research i the portfolio selectio literature, a commo problem with portfolio selectio models is their sesitivity to errors i problem parameters, particularly i the meas We have show that, due to the feature that the rakig rule uder our model cosists of a set of ieualities, our model exhibits some robustess to the errors i the problem iputs Some coditios uder which perturbatios i the meas would ot chage the efficiet portfolio have bee give i Remark 36 However, the sesitivity aalysis i Remark 36 is still very brief ad prelimiary Moreover, as we have show i Remark 36, i certai situatios our model ca also be very sesitive to errors i the meas As our model relies heavily o the meas, ad there is widespread evidece that the meas are very difficult to estimate (see the refereces cited i Remark 36), a more thorough sesitivity aalysis o the model is a importat topic for further research Both theoretical aalysis ad computatioal evaluatio should be helpful Our model has the restrictio that short sellig is ot allowed, ad all of our results have bee obtaied with this assumptio While it is well kow that the removal of this restrictio makes the derivatio of a optimal portfolio for the M-V model much easier, it is ot clear whether this is also true i our model As short sellig also represets a importat class of market situatios, it is a iterestig topic for further research to ivestigate how the ivestmet strategy of our model would chage if short sellig were allowed Furthermore, it would be more iterestig to geeralize the model to iclude the costraits that some x i are subject to upper bouds U i, some x i are subject to lower bouds L i, ad some x i are subject to o restrictio 5 5 We wish to express our sicere gratitude to Professor Phelim Boyle, the departmet editor for Fiace, who provided may valuable suggestios, icludig a detailed aalysis revealig a relatioship of our portfolio to the global miimum variace portfolio uder the M-V model, ad drew our attetio to related work i this area This has led to sigificat improvemets i the paper The helpful commets ad suggestios from the aoymous referees ad Professor Robert Heikel, former departmet editor, are also much appreciated Fially, we gratefully ackowledge the support of a RGC Direct Grat (X Cai); two research grats from the PolyU Research Committee (K-L Teo ad X Q Yag), ad a CUHK Mailie Research Grat (X Y Zhou) Appedix A Proof of Theorem 3 We apply the Kuh-Tucker (K-T) coditios to PO() First, let us itroduce the Lagragia of PO(): Lx, y,, 0, y j y 0j r j j M 0 j j j The, the K-T coditios (see, for example, Zeley 98) that a optimal solutio (x, y) must satisfy ca be writte as follows: L y j 0, j (A) L r j j 0 j 0, j,,, (A2) j M 0, (A3) y j 0, j,,, (A4) Maagemet Sciece/Vol 46, No 7, July

14 j 0, j,,, (A5) j 0, j,,, (A6) j 0, j,, (A7) Defie *() { j: j 0} We let 0, for j *() (This is a cojecture, but we shall show i the followig that this is i fact correct i terms of satisfyig the K-T coditios) The, from (A4) we have y/,ifj *() Thus, from (A3) we obtai Therefore, M 0 y M 0 l* l* l l (A8), j *, 0, j* (A9) From (A5), it follows that if 0, the j 0 Thus j *() For j *(), it is clear from (A2) that CAI, TEO, YANG, AND ZHOU Portfolio Optimizatio Uder a Miimax Rule j l* j l* l l l* r j r l l r l r j l lj j r j r l l lk 0, by 36 O the other had, for j, 2,, k, from (A3) ad (A0), we have j r j l* l r l l l* l* 0 l* l l by 37 l* l* r l r j l r l r k l j r j 0 j r j 0 (A0) This together with (A) gives us l*() (/ l )[( )r l 0 ] Therefore, 0 l* l r l l (A) l* Thus, from (A0), j r j ad for j *(), l* r l l* l l, j *, (A2) j r j j 0 r j 0 (A3) Clearly, if oe ca correctly determie the set *() which esures that j ad j as expressed by (A2) ad (A3) are all oegative, the y ad as give by (A8) ad (A9), respectively, will be a solutio satisfyig all the K-T Coditios (A) (A7) Our argumet is, if there exists a iteger k [0, 2] such that (34) (37) hold, the *() as give by (38) is the set that esures j 0 ad j 0 The followig aalysis proves this argumet By (A2), it follows that, for ay j *() {,,, k}, The above shows that the K-T Coditios (A6) ad (A7) are satisfied This together with the fact that the solutio give by (A8) ad (A9) with the set *() of (38) also satisfies (A) (A5) implies that all the K-T coditios are satisfied I the case where there does ot exist ay iteger k [0, 2] such that (34) (37) hold, we ca show that the solutio give by (A8) ad (A9) with *() {,,, 2, }, will satisfy all K-T coditios To do this, we may itroduce a dummy asset S 0 with r 0 L ad 0 L, where L is a sufficietly large positive umber (K-T coditios ca be applied eve if some parameters, such as a r j, are egative) Followig a similar aalysis to the oe above, oe ca show that all the K-T coditios are satisfied I summary, because PO() is a covex programmig problem, the K-T coditios become ecessary ad sufficiet for optimality ad therefore the solutio give by (A8) ad (A9), or euivaletly, (32) ad (33), which has bee show to satisfy all the coditios, is optimal This completes the proof Appedix B Proof of Lemma 4 We first cosider k 0,,, 2 Suppose that (x 0, y 0 ), where x 0 ( x 0,, x 0 ), is a optimal solutio for PO() Let 0 () bea set such that 0 0ifj 0 () ad 0 0ifj 0 () We shall first show that, if 0 () *() (*() is the set determied by Theorem 3), the we ca fid a solutio better tha (x 0, y 0 ) which leads to a cotradictio Note that it will be sufficiet for us to cosider the followig two cases Case 0 () *() ad there exists at least oe h such that h *() but h 0 (), amely, x h 0 0 I this case, by costructig a solutio (x, y) for PO() as (30) ad (3), we ca show, after some developmet, that 970 Maagemet Sciece/Vol 46, No 7, July 2000

15 Portfolio Optimizatio Uder a Miimax Rule F x, y F x 0, y 0 y jm r j r m jm r m r j (B) Because jm ((r j r m )/ ) /( ) if m *() (see (34) (36)) ad r m r j if j m, (B) gives us F (x, y ) F (x 0, y 0 ), implyig that a feasible solutio (x, y ) exists which is better tha the solutio (x 0, y 0 ) This cotradicts the fact that the solutio (x 0, y 0 ) is optimal This meas that we must have 0 () *() We ow cosider the followig case, the result of which will elimiate the possibility that 0 () *() Case () *(), amely, there is a m with x m 0, m 0 () but m *() I this case, we ca costruct a solutio (x, y) for PO() as follows: x j 0 j, j *, x 0 m m, j m, x 0 j, otherwise y y 0 y, where y, m, ad j are selected to be positive umbers such that j y, j *, (B2) m j* j (B3) It is ot hard to show that the Relatios (B2) ad (B3) together with the feasibility of (x 0, y 0 ) esure that (x 2, y 2 ) is a feasible solutio Similar to (B), oe ca derive F x 0 x, y 0 y F x 0, y 0 y j* r j r m (B4) Because j*() ((r j r m )/ ) /( ) (bearig i mid that ( m, m )), we have F (x 2, y 2 ) F (x 0, y 0 ), which is agai a cotradictio Combiig the results of Cases ad 2, we see that 0 () *(), amely, ay optimal solutio (x 0, y 0 ) must have the same set of assets selected for ivestmet as that i the solutio (x*, y*) After the set 0 () has bee fixed, all assets S j, j 0 (), should be 0 ivested such that their risks are eual, amely, should be determied such that x 0 j y 0 for all j 0 () If this is ot true, that is, there exists a asset S m i 0 () such that y 0 x 0 m m 0, the we ca icrease the allocatio to asset S m by a positive amout m to costruct a ew solutio (x, y) as follows: where j y, m y m y y 0 y, j 0 m, j 0 m j, j 0 ad It is easy to see that such a solutio (x, y) is feasible Similar to the aalysis i Case above, we ca show that that F (x, y) F (x 0, y 0 ) Clearly, if 0 () *(), ad x 0 j y 0 for all j 0 (), the the solutio (x 0, y 0 ) is exactly the same as (x*, y*) This proves the uiueess of (x*, y*) What remais to be cosidered is k Accordig to (39), the set *() becomes {,,, 2, } whe (, ) Thus, Case 2 above is impossible ow As for Case, the proof remais the same I summary, the solutio give by (32) ad (33) must be the oly optimum for PO(), if ( k, k ), k 0,,, This completes the proof Appedix C Proof of Lemma 42 Case i the proof for Lemma 4 cotiues to be impossible eve if k, sice 0 ad thus (B) still implies F (x, y ) F (x 0, y 0 ) Nevertheless, it is possible to have Case 2, amely, to have a asset S l, where l *(), to be selected i a optimal solutio This is show below Clearly (x*, y*) remais a optimal solutio for PO() whe k Now, we ca costruct a ew solutio (x 0, y 0 ) from (x*, y*), by icreasig the value of x l, where l k, to l ad, accordigly, decreasig the values of all by j for j *() ad y by y such that x* j j y* y, j *, (C) l l y* y, x* j j l M 0 j* where l, j, ad y are all positive, satisfyig: (C2) (C3) j y, j *, (C4) x 0 j j, j 0 ad j m, x 0 j m, j m, 0, otherwise, y y* l l, l j* j (C5) (C6) Maagemet Sciece/Vol 46, No 7, July

16 Portfolio Optimizatio Uder a Miimax Rule Similar to (B), we ca show that F x 0 x, y 0 y F x 0, y 0 y j* r j r l (C7) Because j*() ((r j r l )/ ) /( ) whe k, we have F (x 0 x, y 0 y ) F (x 0, y 0 ) This meas that all poits ( y* y, z* ( k /( k )) y ) are efficiet poits for the bicriteria problem POL, as log as y satisfies (C4) (C6) Note that ( y* y, z* ( k /( k )) y ) is the value of the objective fuctio uder the solutio (x 0, y 0 ), as y 0 y* y ad z 0 z* z* j* j* j* z* k y r j x* j j r l l r j j j* r j r l y r l j z* k y cf 42 ad 49 k It ca be see that y withi the rage of (43) satisfies (C4) (C6) This completes the proof Refereces Başer, T, P Berhard 995 H -Optimal Cotrol ad Related Miimax Desig Problems A Dyamic Game Approach Birkhauser, Bosto, MA Best, M J, R R Grauer 99a Sesitivity aalysis for meavariace portfolio problems Maagemet Sci , 99b O the sesitivity of mea-variace-efficiet portfolios to chages i asset meas: Some aalytical ad computatioal results Rev Fiacial Stud , 992 Positively weighted miimum-variace portfolios ad the structure of asset expected returs J Fiacial Quat Aal Chopra, V K, C R Hesel, A L Turer 992 Massagig meavariace iputs: Returs from alterative global ivestmet strategies i the 980s Maagemet Sci , W T Ziembia 998 The effect of errors i meas, variaces, ad covariaces o optimal portfolio choices W T Ziembia ad J M Mulvey, eds Worldwide Asset ad Liability Modelig Cambridge Uiversity Press, Cambridge 53 6 Hauge, R A 997 Moder Ivestmet Theory, 4th ed Pretice Hall, Upper Saddle River, NJ Hesel, C R, A L Turer 998 Makig superior asset allocatio decisios: A practitioer s guide W T Ziembia ad J M Mulvey, eds Worldwide Asset ad Liability Modelig Cambridge Uiversity Press, Cambridge Koo, H 990 Piecewise liear risk fuctio ad portfolio optimizatio J Oper Res Soc Japa , H Yamazaki 99 Mea-absolute deviatio portfolio optimizatio models ad its applicatios to Tokyo stock market Maagemet Sci Leo-Garcia, Alberto 994 Probability ad Radom Processes for Electrical Egieerig, 2d ed Addiso-Wesley, Readig, MA Markowitz, H M 952 Portfolio selectio J Fiace Merto, R C 972 A aalytic derivatio of the efficiet portfolio frotier J Fiacial Quat Aal Youg, M R 998 A miimax portfolio selectio rule with liear programmig solutio Maagemet Sci Yu, P L 974 Coe covexity, coe extreme poits, ad odomiated solutios i decisio problems with multiobjectives J Optim Theory ad Appl Zeley, M 98 Multiple Criteria Decisio Makig McGraw-Hill, New York Accepted by Phelim P Boyle; received March 996 This paper was with the authors 6 moths for 6 revisios 972 Maagemet Sciece/Vol 46, No 7, July 2000