On the Optmal Selecton of Portfolos under Lmted Dversfcaton Jay Sankaran Department of Management Scence and Informaton Systems Unversty of Auckland New Zealand j.sankaran@auckland.ac.nz C. Krshnamurt Department of Fnance and Accountng Natonal Unversty of Sngapore, 10 Kent Rdge Crescent, Sngapore 119260. Ajay A. Patl Computervson Corporaton Bedford, MA 01730, U.S.A. Abstract We address the problem of selectng portfolos that are optmal among all those portfolos that comprse at most a pre-specfed number, k, of securtes. We consder two crtera: maxmzng the rato of the average excess return to the standard devaton; and maxmzng the correlaton wth a specfed market-ndex. Under standard assumptons of the behavour of stock returns, we develop procedures that are of polynomal complexty and that requre mnmal data. 1 Introducton One basc mplcaton of modern portfolo theory s that nvestors hold well-dversfed portfolos. However, there s emprcal evdence that ndvdual nvestors typcally hold only a small number of securtes. Blume, Crockett, and Frend [7] found that only 11% of ndvdual U.S. nvestors held more than ten stocks. Conne, Jensen, and Tamarkn [9, footnote 2, pp. 1004-1005] cte several studes that provde emprcal evdence that the majorty of ndvdual nvestors n the U.S. hold hghly undversfed portfolos. Bark [5] states that one reason for the nadequacy of the Sharpe-Lntner-Mossn captal asset prcng
model n the Korean stock market s that the portfolos of Korean nvestors are also hghly undversfed. The absence of dversfcaton has also been observed wth regard to ndex-trackng portfolos. Rudd [26] provdes evdence whch ndcates that the majorty of Amercan ndex funds do not actually hold all the stocks n the chosen ndex - some of them hold as lttle as 35 stocks to match the 500-stock S&P 500 ndex. Market mperfectons such as fxed transacton costs provde one explanaton for the prevalence of undversfed portfolos. Indrect transacton costs, such as the cost of analyzng securtes, also mltate aganst dversfcaton; a small nvestor who chooses to nvest n only a lmted number of securtes can devote more attenton to the ndvdual behavor of those securtes and ther mean-varance characterstcs. Moreover, there s evdence [16, 18, 19] that dversfcaton beyond 8-10 securtes may not be worthwhle provded these securtes are chosen not randomly but through a systematc, optmum-seekng procedure. Ctng Szegö [30], Sengupta and Sfer [27] observe that t may be superfluous to enlarge the number of securtes n a portfolo beyond a lmt because the varance-covarance matrx of the returns on the securtes n a portfolo that has a large number of securtes tends to conceal sgnfcant sngulartes or near-sngulartes. 2 Problem Defnton We consder two related problems n optmal portfolo selecton. The frst, referred to as P1, s to fnd portfolos that are mean-varance effcent and that comprse at most a pre-specfed number of securtes. The second problem, referred to as P2, s to fnd portfolos that optmally track a specfed market-ndex among all those portfolos that contan at most a prespecfed number of securtes and whose average returns equal the average return on the ndex. We determne the effcency of trackng by measurng the correlaton coeffcent between the returns on the chosen market ndex and the returns of the market-trackng portfolo. 3 Exstng Algorthms for P1 and P2 and Varants Thereof Mao [22] and Jacob [19] formally address P1 but develop ther selecton procedures under somewhat restrctve assumptons and rather hgh degrees of approxmaton. For nstance, to compute the number of securtes that optmally trades-off dversfcaton aganst (fxed) transacton costs, Mao assumes that both the average excess return over the rskless rate and the standard devaton of the return are the same for all the securtes n the portfolo. Further, for selectng the best portfolo among those that comprse a pre-specfed number of securtes, he assumes that for all of these portfolos, the nonsystematc rsk s fully dversfed
away. Jacob assumes that the weghts for the securtes n the nvestor s portfolo are all equal to each other. She also lnearzes portfolo rsk n terms of the weghts. For the problem of determnng mean-varance effcent portfolos under the sngle-ndex model [28] of securty returns and an upper lmt on the number of stocks, Faaland [17] develops an algorthm based on nteger programmng, whch s bettered by the mplct enumeraton algorthm of Blog et al. [6]. Cooper and Farhangan [11] develop a dynamc programmng approach for an extenson of ths problem that ncorporates fxed costs of transacton. Assumng that the captal asset prcng model [21, 24, 29] holds, Brennan [8] presents an algorthm for determnng the optmal number of securtes under fxed transacton costs. However, the valdty of that assumpton n the presence of fxed transacton costs has been questoned [25]. Patel and Subrahmanyam [25] develop an effcent algorthm for the problem under the assumpton that the correlaton coeffcent s the same for all pars of securtes [12, pp. 168-169]. Aneja, Chandra, and Gunay [4] show how the average parwse correlaton coeffcent can be effcently estmated usng a portfolo approach. We are not aware of more recent algorthms for P1 or any of ts varants. Nevertheless, smple rankng procedures akn to the well-known EGP algorthms [13, 14, 15] contnue to be developed for selectng mean-varance effcent portfolos n other contexts such as restrcted short-sellng [2, 3] and nsttutonal norms for short-sellng [20]. Procedures for P2 or varants thereof have been propounded by Adcock and Meade [1], Connor and Leland [10], Meade and Salkn [23], and Rudd [26] among others; however, they use optmzaton only to the extent of determnng the portfolo weghts for a pre-selected set of securtes. Thus, for nstance, n one of ther procedures, Meade and Salkn [23] randomly sample a small number of stocks, and then use quadratc programmng to determne the optmal portfolo weghts. 4 The Results To formally present the problems and algorthms, we employ the followng notaton. n: The number of securtes n the unverse. N: The set of securtes n the unverse; thus, N = {1,,n}. k: A pre-specfed upper lmt on the number of securtes n the portfolo (1 k n). x : The weght of securty, = 1,...,n. R: The rate of return on the rskless asset. R : The expected rate of return on securty, = 1,...,n. s : The standard devaton of the rate of return on securty, = 1,...,n.
b : The rato of the average excess return to the standard devaton of securty, = 1,...,n; thus, b = (R - R)/s. p: An estmate of the average correlaton coeffcent of any par of securty returns (we assume that p s non-negatve). The assumpton of non-negatvty for p would appear to be very mld because a negatve value of p would mply that the varance of an equally-weghted portfolo wth, say, m securtes, s negatve for suffcently large values of m (see, for nstance, [12, p. 60]). 4.1 Problem P1 Under the assumpton of constant parwse correlatons, P1 may be formulated as: Maxmze n = 1 s 2 n = 1 x 2 ( R R). x + p. n n = 1 j = 1 j s s x x such that at most k of {x : = 1,...,n} are non-zero, and f short-sellng s dsallowed, then x 0 for = 1,...,n. In the maxmand, the numerator s the excess return on the portfolo whle the denomnator s the standard devaton of the return on the portfolo; by varyng the rskless rate of return, R, we can generate the mean-varance-effcent fronter. We assume that for any gven value of R, the n stocks n the unverse are numbered n descendng order of {b : = 1,,n}. If short-sellng s allowed, then the set of stocks to select s of the form {1,,t, n-k+t+1,..., n}, where 0 t k. Thus, the correct algorthm s to evaluate the k+1 portfolos and choose the best among them. If short-sellng s not allowed, then the correct algorthm s, qute smply, to execute the well-known EGP rankng algorthm untl k securtes are ncluded. We also establsh that the optmal objectve value of the above problem ncreases wth k, but at a decreasng rate; n other words, the margnal beneft from dversfcaton decreases wth the number of stocks n the portfolo. 4.2 Problem P2 The problem, P2, s to fnd a portfolo: () whch uses at most k securtes from the unverse of n stocks; () whose expected return equals that on the gven market-ndex; and j j
() whose return has the greatest correlaton wth that on the ndex, among all portfolos that satsfy condtons () and (). For formalzng the problem and presentng the results, we ntroduce some addtonal notaton: X : The nvestment (not the weght) n securty. π : The correlaton of the return on securty wth that on the market-ndex. s p : The standard devaton of portfolo return. Assumng the presence of a rskless asset, P2 can be transformed to: Mn (s p ) 2 such that at most k from among {X : = 1,...,n} are non-zero, and f short-sellng s dsallowed, all of {X : = 1,...,n} are non-negatve. We assume that stocks are numbered n descendng order of correlaton wth the ndex. Under the sngle-ndex model of stock returns, the correct algorthm s to pck stocks n descendng order of correlaton (f short-sellng s allowed, we use the absolute values of the correlatons) untl the lmt, k, s reached. We can readly establsh that regardless of whether or not short-sellng s allowed, the beneft from dversfcaton decreases wth the pre-specfed upper lmt on the number of stocks n the portfolo. Under the assumpton of constant parwse correlatons, the followng results apply. (a) If short-sellng s allowed, then the set of stocks to select s of the form {1,,t, n- k+t+1,..., n}, where 0 t k. Thus, the correct algorthm s to evaluate the k+1 portfolos and choose the best among them. (b) If short-sellng s dsallowed, the correct procedure s to execute the EGP algorthm (on {π : = 1,,n} rather than {b : = 1,,n}) untl k securtes are ncluded. In ths case, we can establsh that dversfcaton yelds dmnshng returns (no pun ntended). A potental problem n applyng the above procedures s that under constant parwse correlatons, the estmated (n+1)x(n+1) varance-covarance matrx (nvolvng the n securtes and the market-ndex) may not be postve defnte. 5 Summary of Fndngs We have presented algorthms for selectng `small portfolos (namely, those that contan at most a pre-specfed number of securtes) under two dfferent objectve crtera. The frst s to maxmze the rato of excess return to standard devaton (and thereby, generate the meanvarance-effcent fronter), whle the second s to maxmze the correlaton wth a gven market-ndex the correspondng problems are referred to as P1 and P2 respectvely.
Under the assumpton of constant parwse correlatons, exact and effcent (polynomal) procedures exst for both P1 and P2 regardless of whether or not short-sellng s allowed. Further, P2 can also be effcently solved under the sngle-ndex model of stock returns for both the cases (.e., short-sellng allowed and short-sellng dsallowed). It s mportant to emphasze that the requrements n terms of both data and computng power are mnmal for the algorthms presented heren. Moreover, emprcal results on data from the Korean stock exchange suggest that the optmal portfolo weghts are nether too large nor too small. Future research could explore problems P1 and P2 under more general correlaton structures. Under the sngle-ndex assumpton, P1 s NP-hard; thus, the sngle-ndex case mght need enumeraton. At least n the case when short-sellng s dsallowed, both lower and upper bounds on the optmal objectve value of any node n the enumeraton tree can be obtaned through the results of the paper. References [1] C.J. Adcock, N. Meade, A Smple Algorthm to Incorporate Transactons Costs n Quadratc Optmsaton, European Journal of Operatonal Research, 79 (1994), pp 85-- 94. [2] G.J. Alexander, Short Sellng and Effcent Sets, Journal of Fnance, 48 (1993), pp 1497--1506. [3] G.J. Alexander, Effcent Sets, Short-Sellng, and Estmaton Rsk, Journal of Portfolo Management, 21 (1995), pp 64--73. [4] Y.P. Aneja, R. Chandra, E.A. Gunay, A Portfolo Approach to Estmatng the Average Correlaton Coeffcent for the Constant Correlaton Model, Journal of Fnance, 44 (1989), pp 1435--1438. [5] H.K. Bark, Rsk, Return, and Equlbrum n the Emergng Markets: Evdence from the Korean Stock Market, Journal of Economcs and Busness, 43 (1991), pp 353--362. [6] B. Blog, G. van der Hoek, A.H.G. Rnnooy Kan, G.T. Tmmer, The Optmal Selecton of Small Portfolos, Management Scence, 29 (1983), pp 792--798. [7] M.E. Blume, J. Crockett, I. Frend, Stock Ownershp n the Unted States: Characterstcs and Trends, Survey of Current Busness, 54 (1974), pp 16--40. [8] M.J. Brennan, The Optmal Number of Securtes n a Rsky Asset Portfolo when there are Fxed Costs of Transactng: Theory and some Emprcal Results, Journal of Fnancal and Quanttatve Analyss, 10 (1975), pp 483--496. [9] T.E. Conne, Jr., O.W. Jensen, M. Tamarkn, On Optmal Producton and the Market to Book Rato gven Lmted Shareholder Dversfcaton, Management Scence, 35 (1989), pp 1004--1013.
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