Growth Optimal Investment Strategy Efficacy: An application on long run Australian equity data

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1 8 Investment Management and Fnancal Innovatons, 1/005 Growth Optmal Investment Strategy Effcacy: An applcaton on long run Australan equty data Ben F. Hunt Abstract A number of nvestment strateges desgned to maxmse portfolo growth are tested on a long run Australan equty data set. The applcaton of these growth optmal portfolo technques produces mpressve rates of growth, despte the fact that the assumptons of normalty and stablty that underle the growth optmal model are shown to be nconsstent wth the data. Growth optmal portfolos are constructed by rebalancng the portfolo weghts of 5 Australan lsted companes each month wth the am of maxmsng portfolo growth. These portfolos are shown to produce growth rates that are up to twce those of the benchmark, equally weghted, mnmum varance and 15% drft portfolos. The key to the success of the classc, no short-sales, growth optmal portfolo strategy les n ts ablty to select for portfolo ncluson a small number of Australan stocks durng ther hgh growth perods. The study ntroduces a varant of rdge regresson to form the bass of one of the growth focused nvestment strateges. The rdge growth optmal technque overcomes the problem of numercally unstable portfolo weghts that dogs the formaton of short-sales allowed growth portfolos. For the short sales not allowed growth portfolo, the use of the rdge estmator produces ncreased asset dversfcaton n the growth portfolo, whle at the same tme reducng the amount of portfolo adjustment requred n rebalancng the growth portfolo from perod to perod. Key words: Growth Optmal Portfolos, Australan Equty Returns, Feasble Investment Strategy, Rdge Regresson. Introducton The expected rate of growth of value s consdered by many nvestors to be the preemnent characterstc of an nvestment portfolo. Ways to construct portfolos that maxmse expected growth are well documented 1. Consderng the mportance of expected portfolo growth to both professonal and retal nvestors, t s surprsng that so few examples of studes that focus on the emprcal strateges to maxmse portfolo growth exst. Ths study ams to redress ths defcency by applyng growth optmal technques to long-run Australan equty data. As a frst step n the study, let us set out a stochastc model of asset prce evoluton, upon whch the growth optmsng nvestment strategy wll be based. Suppose that nvestment choce s confned to n assets, each governed by geometrc Brownan moton (generalsed Wener process). That s, the value of the th asset, V, evolves as dv ( t) µ V ( t) dt V ( t) dz, (1) where, s the th asset s rate of drft and z s a Wener process wth zero mean and varance,. The expected rate of growth of the asset, E[G ], over tme t, can, usng Ito s lemma, be derved as The varance of ths growth s V ( t) 1 E[ G ] Eln µ t V. () (0) t. (3) G 1 See Hakansson (1971), Luenberger (1998) and Hunt (00). The dervaton of the portfolo dynamcs follows Luenburger (1998, pp ).

2 Investment Management and Fnancal Innovatons, 1/005 9 Consder the dynamcs of a portfolo constructed usng specfc asset weghts, w. The rate change of portfolo value s thus the weghted sum of the rates of change of the ndvdual assets: dv Vp p n 1 w dv V. (4) Assumng that the n assets are correlated through the Wener process,.e., covarance (dz, dz j),j dt, the value of the portfolo, V p (t), also follows geometrc Brownan moton wth per perod, expected growth, g p, gven by g p Vp ( t) 1 Eln µ t Vp (0) t T 1 T w µ w w, 1 p t 1 p where, w T = (w 1,, w n ) s the vector of portfolo weghts, T = ( 1,, n) s the vector of asset drft parameters and s a matrx contanng n varance and covarance terms,,j. It s evdent from (5) that the rate of growth of a portfolo of assets s governed by the choce of the ndvdual asset weghtngs, w. Naturally, the structure of w may be fashoned to maxmse the expected rate of growth. The portfolo, w*, that maxmses expected portfolo growth s referred to as the growth optmal portfolo. A strategy desgned to maxmse expected growth has an obvous and ntutve appeal. Moreover, maxmsng expected growth has strong theoretcal support. Consder the broad class of power utlty of wealth functons: (5) W ( ). (6) U 1 W It s easly shown that for ndvduals possessng a utlty functon such as (5), the problem of maxmsaton of expected utlty of wealth after n perods, W n, reduces to the myopc strategy of the maxmsaton of wealth over one perod, W 1 1. Further, f s small, the expected value of power utlty E[U(W)] s closely approxmated by E[ U ( W1 )] E[ln( W1 ) (lnw1 ) ] E[ g] ( ( E[ g]) Thus, when s small, t follows that the only two varables of nterest n the quest to maxmse expected utlty of n perod wealth, are expected growth rate and the varance of the growth rate. Investors wth a log utlty functon U ( W n ) ln( Wn ), whch s the lmt case of (7) when 0, wll choose between nvestments based solely on expected portfolo growth. Investment technques based on optmsng expected growth have appeal to both theorsts and practtoners as they: are consstent wth asset dversfcaton, are consstent wth n perod utlty maxmsaton, maxmse expected termnal value of wealth, and mnmse the expected tme requred for accumulated wealth to reach any specfed threshold value. ]. (7) 1 See Luenberger (1998, pp ). Luenberger (1993) provdes a broader ratonale for basng portfolo choce on expected growth usng so called tal preference theory.

3 10 Investment Management and Fnancal Innovatons, 1/005 The theoretcal attractveness of maxmal growth portfolos s clear. What s less apparent s whether or not nvestment strateges based on growth portfolos are effcacous. The am of ths paper s to examne the sutablty of growth optmal portfolo technques to the Australan equty nvestment envronment. Data Hakansson (1971) suggested that growth optmum portfolos domnate all other portfolos n the long run. Whle Merton and Samuleson (1974) ponted out the fallacy n ths argument, t remans true that t s easer to dentfy the characterstcs of alternatve nvestment portfolos when observed over a long perod of tme. The desre to test the effcacy of growth-orented nvestment led us to seek out a long run Australan equty data set. The study apples the growth optmal portfolo nvestment strategy technques to 5 years of monthly data, startng n Aprl 1977 and endng n March 00. The data were obtaned from Reuter s Australa s Beacon data servce. The data set comprses prce observatons on 5 Australan lsted companes. These companes selected themselves, beng the only corporatons currently ncluded n the ASX s 150 largest captalsed companes (as of March 00) whose prces from March 197 were recorded n the Beacon data tables 1. The prce data were transformed nto measures of perodc growth (or returns) by usng the formula for contnuous compoundng: g, t ln( P, t / P, t1), (8) where P t s the prce of asset n month t, and g,t s the growth of asset n month t. Table 1 dsplays annualsed statstcs on rates of growth and volatlty of growth for the companes ncluded n the data set. The annual rate of growth of asset, ĝ, was estmated as the sample aggregate growth dvded by the 5 years of the sample. The estmate of the asset drft rate,, was computed as ĝ ˆ / ˆ, ˆ, where s the estmate of the th asset varance. As expected, there s some survvor effect to be evdent n the 5-company data set. Table 1 shows that the All Ordnares Index grew at a rate of 9.68% p.a. whereas the equally weghted portfolo grew at a rate of 11.71% p.a. Australan Equty Performance Summary Statstcs Table 1 Code Name Growth (g, % p.a.) Rank Drft* (, p.a.) Rank Volatlty (, % p.a.) AGL Australan Gas Lght 13.8% % % 8 AMC Amcor 8.6% % 19.80% 3 ANZ ANZ Bank 13.09% % % 19 BHP BHP Bllton 1.61% % % 17 BIL Brambles Industres 14.74% % 7 4.5% 0 CML Colonal Mutual 11.49% % 16.8% CSR Colonal Sugar 3.95% % 7.55% 15 FGL Fosters Brewng 11.10% % % 10 GMF Goodman Felder 3.7% 4 7.8% % 16 Rank 1 Although the growth technques were appled to 300 monthly observatons, begnnng n Aprl 1977, the actual data set employed by the study starts n Aprl 197, to facltate hstorcal parameter estmaton. The expected return over a very short perod of tme s t. However, over a longer perod of tme the expected return s - /. As Hull (000, pp ) notes, the term expected return s ambguous. It can ether refer to or - /. When the term expected return ( or symbol r) s used n ths paper t s n reference to the drft term,, n a generalsed Wener process.

4 Investment Management and Fnancal Innovatons, 1/ Table 1 (contnuous) GPT General Property Trust 4.9% 5.57% % 5 LLC Lend Lease 1.48% % % 14 MAY Mayne Ncholas 9.04% % % 11 MIM Mt. Isa Mnng -.70% % % NAB Natonal Australa Bank 1.67% 9 15.% 15.60% 4 NCP News Corporaton 3.49% 3.86% 43.9% 1 ORI Orca 6.05% % % 1 PDP Pacfc Dunlop 6.05% % 1 8.7% 13 QBE QBE Insurance 18.60% 3 3.1% % 9 RIO Ro Tnto 10.57% % % 7 SRP Southcorp 14.35% % % 18 STO Santos 15.33% % % 5 WBC Westpac Bank 9.54% % % 1 WMC Western Mnng 9.86% % % 4 WPL Woodsde Petroleum 11.68% % % 6 WSF Westfeld Holdngs 38.8% % % 3 ZAORD All Ordnares Index 9.68% 11.61% 19.65% Equal Equally weghted portfolo 11.55% 13.35% 18.95% * The mpled of equaton (1) was computed for each stock by usng equaton (4) as the rate of growth plus the half the varance. Some mpressve ndvdual growth performances are evdenced n Table 1. Westfeld Holdngs and News Corp have grown on average 39% p.a. and 3% p.a. respectvely. At the other end of the performance spectrum les MIM who managed an almost 3% p.a. declne n value over the 5 year perod. Table 1 contans some superfcal evdence of a postve relatonshp between hstorcal share growth and the volatlty of that growth. For example, the two hghest growth stocks are also the two most volatle ones. Further analyss reveals that the correlaton and the rank correlaton between growth and volatlty for the 5 stocks are 0.37 and 0.1 respectvely. Testng the Assumptons of the Growth Model There are a number of assumptons mplct n the model of growth upon whch the nvestment strateges tested n ths paper are based. Most obvously, the Wener process of equaton (1) assumes nvestment returns are normally dstrbuted. Possbly less obvous s the model s relance on the stablty of stochastc process parameters of and. The degree to whch these assumptons are consstent wth the features of the hstorcal data set ought to provde a gude to the lkely success or otherwse of growth optmal nvestment strateges. Normalty The 5 companes and the benchmark equally weghted portfolo and All Ordnares Index seres were tested for normalty of returns wth the results recorded n Table. Three tests of normalty, based on skewness and kurtoss measures, were appled to the data set. The results of these tests reveal that the perod-by-perod returns n the data set were far from normally dstrbuted. out of the 5 stocks dsplayed sgnfcant skewness. In addton, all 5 stocks had returns that were sgnfcantly leptokurtc (at the 1% level). Naturally, the Jacque-Berra statstcs, whch jontly tests for skewness and kurtoss, rejected normalty n all cases, ncludng the two benchmark seres.

5 1 Investment Management and Fnancal Innovatons, 1/ Frequency % -110% -80% -50% -0% 10% 40% 70% 100% 130% 160% Annualsed contnuously compounded returns Fg. 1. Dstrbuton of Returns of an Equally Weghted, 5 Stock Portfolo Fgure 1 depcts the dstrbuton of returns for the equally weghted portfolo compared to ts equvalent normal dstrbuton. The typcal peaked centre of a leptokurtc dstrbuton s clearly dsplayed. Tests of Normalty 1 and Stablty Table Code Skewness Kurtoss Jacque-Berra 1 ANOVA Kruksal-Walls Varance Rato AGL ** 7.0 ** ** ** AMC ** 4.76 ** ** * ANZ **.05 ** 40.3 ** BHP ** 0.73 ** BIL ** 4.50 ** ** CML ** 7.19 ** ** * CSR **.74 ** 48. ** ** FGL ** ** ** ** GMF ** 4.41 ** **.49 * ** GPT ** 4.1 ** ** LLC -1.6 ** ** ** ** MAY ** 6.7 ** ** MIM ** 3.8 ** ** ** NAB **.76 ** ** * NCP ** 9.84 ** **.67 * ** ORI ** 5.41 ** ** PDP ** 3.34 ** ** 3.06 * 1.59 * 1.87 QBE -.14 ** ** ** ** RIO -1.4 ** ** ** ** 1 The skewness and kurtoss tests are based on the followng. For a normal dstrbuted random varable, x, the skewness 3 3 coeffcent, 1 E [( x ) ]/ estmated from a sample of sze n s dstrbuted as ˆ1 N( 0,6 / n). The 4 4 coeffcent of kurtoss, E[( x ) ]/ s dstrbuted as N(3,4 / n), where E s the expectaton operator, s the mean and s the standard devaton. The Jacque-Berra statstcs, J, 3 where: ˆ ˆ 3 1 J n () 6 4 ˆ

6 Investment Management and Fnancal Innovatons, 1/ Table (contnuous) SRP -0.6 ** 7.3 ** ** STO ** ** 3.16 * 9.50 * 7.01 ** WBC ** 3.64 ** ** * WMC * 4.5 ** ** ** WPL ** 5.7 ** ** WSF 5.9 ** ** ** ** ZAORD -3.7 ** ** ** ** Equal -.80 ** 5.30 ** ** ** * ndcates sgnfcance at the 5% level, ** ndcates sgnfcance at the 1% level. The kurtoss fgure dsplayed was computed usng Excel s KURT() functon and s equal to the tradtonal measure of kurtoss less 3. The results of the analyss of skewness and kurtoss allow us to confdently conclude that the data upon whch we are to test the growth optmal portfolo strateges are non-normal. Exactly how the non-normalty wll mpnge upon the nvestment results s problematcal. For example, t s not clear that excessve kurtoss wll have a deleterous effect on the growth optmal nvestment strateges. Of more concern s the queston as to the stablty of the dstrbutonal statstcs overtme. Seral Stablty The expected growth rate for each stock, the varance of that growth rate and the covarances between each stock s growth rates are essental nputs to the process of determnng growth optmal portfolo weghts. Thus any seral nstablty n these nput parameters wll mperl the success of any nvestment strategy based on an assumpton of parameter constancy. The growth optmal strategy reles on the stablty of the nput parameter estmates. Sub-perod Statstcs Table 3 Perod All Ordnares Index Equally weghted portfolo Growth Volatlty Growth Volatlty % 19.79% 13.5% 18.45% % 19.76% 14.06% 18.04% % 9.38% 4.50% 7.76% % 1.91% 8.59% 1.3% % 1.93% 6.61% 13.4% Table 3 sets out estmates of the average growth and volatlty of growth, for the two benchmark seres, for the fve equal sub-perods that make up the overall data set perod. Casual analyss of the range of sub-perod estmates suggests parameter nstablty. However, the result of applyng formal tests for nstablty of the mean of growth for the ndvdual stocks does not lead to the concluson that these are unstable. Analyss of varances ndcates nstablty n only 4 out of the 5 stocks. The Kruksal-Walls test, whch s the more sutable test gven the non normalty of the data, rejects the hypothess of constancy of growth rates n all fve sub-perods for only out of the 5 stocks. The proposton that varance of growth rates s dentcal n each of the 5 sub-perods, was checked by usng Hartley s test for homogenety of varance. The rato of the largest sub-perod varance to the smallest sub-perod varance, whch s the key statstcs n Hartley s test, s ds-

7 14 Investment Management and Fnancal Innovatons, 1/005 played n Table 1. Hartley s test ndcates that the presence of seral nstablty of varance of growth rates exsts n many of the sample stocks. The null hypothess of equalty of sub-perod was rejected, at the 5% level at least, for 17out of the 5 companes. The varance for each stock s an nput nto the formaton of growth optmal portfolos. However, t s the full covarance matrx that s the essental nput tem and the varances represent only a small proporton of the larger covarance matrx. However, the precedng evdence of varance nstablty justfed further research to ascertan whether the varance nstablty was also mrrored n covarance nstablty. A test of the hypothess of equalty of sub-perod covarance matrces employs the Box s M statstc, where M n m 1 m ln n ln, (8) s 1 where m s the number of sub-perods, n = m n s s the number of observatons n the full sample, n s s the number of observatons n each sub-perod, s the determnant of the overall, p dmensoned, covarance matrces and s the determnant of the th sub-perod covarance matrx. Pearson (1969) shows that for large p, M s dstrbuted as b F f1,f 3. The sample M/b was computed as Ths s to be compared to the 1% crtcal F of Hence, t must be concluded that the sample data covarance matrx s not statonary. The precedng results do not allow much scope for optmsm as to the successful applcaton of nvestment technques based on growth optmal portfolos, as the technques rely on an assumpton of normalty of perod-by-perod growth rates, and an mplct assumpton of the stablty of the dstrbutonal parameters contaned wthn the expected growth rates and the covarance matrx of growth rates. Contrary to these assumptons, the analyss has shown that long run Australan equty data are leptokurtc and somewhat skewed and are charactersed by a nonstatonary covarance matrx. However, despte the facts of the stuaton, we proceeded to test effcacy of growth optmal portfolo nvestment technques usng the hstorcal Australan data. Applcaton of Growth Optmal Portfolo Investment Technques Ths paper attempts to test a smple, practcal nvestment strategy based on portfolos selected to have maxmum expected growth rate. Testng any proposed nvestment strategy on the hstorcal data nvolved steppng through each of the 300 monthly observatons on the return of 5 Australan companes. At any perod, k, the followng steps are undertaken: 1. The data on the prevous n perods are employed to provde estmates of the expected growth rate for each stock n the sample and to estmate each element of the 5 x 5 growth rate covarance matrx.. The expected growth and covarance matrx estmates are used to produce growth optmal portfolo weghts, w k. 3. The return on ths portfolo n the next,.e. k+1, perod s computed. 4. The tme-frame s moved forward one observaton. Steps 1 to 4 are repeated untl the data set s exhausted. 1 The rato of the hghest to the lowest sub-perod varance s theoretcally dstrbuted as F n1,n, where n1 s the number of sub-samples and n s the number of observatons n each sub-sample, e F 5,60 n our case. For more fulsome explanaton of Hartley s test see Berenson and Levne (199, pp ). A covarance matrx contans n varance terms and n (n-1) covarance terms. n = 5 n ths study. Thus, for ths study the varances represent only 5/600=4% of the terms n the covarance matrx. 3 The calculaton of M/b s rather dauntng wth f1, f and b havng more terms than a Mettalca tour contract. See Pearson (1969, p. 19). An example of the use of M to test equalty of covarance matrces can be found n Morrson (1976, pp. 5-53) (note, however, the error n Morrson s equaton ()).

8 Investment Management and Fnancal Innovatons, 1/ Short-sales Allowed Portfolos Growth optmal portfolos le on a mnmum varance fronter formed when portfolo varance s mnmsed for a range of expected portfolo drft (Fgure ). The short-sales allowed, growth optmal portfolo, w*, vector has the followng structure 1 : w* Aµ b, (9) where: T 1 1 A I, T 1 1 and s the unt vector. b T 1 It s our am to proceed through the hstorcal data set, estmatng and, usng these estmates to calculate the growth optmal weghts, w*, and to use these weghts to produce a set of one-step-ahead returns for each of the 300 observatons n the data set. The success or falure of the growth optmal nvestment technques wll be judged on the nature of the one-step-ahead returns produced by the strategy. The returns on three alternatve nvestment strateges wll provde a base aganst whch to measure the growth optmal technques. These benchmark portfolos are as follows: 1) the equally weghted portfolo, ) the mnmum varance portfolo, and 3) the portfolo wth an expected drft of 15% p.a. The equally weghted portfolo s a smple passve nvestment strategy and represents the absolute mnmum bar aganst whch alternatves ought to be measured. The mnmum varance pont (MVP) strategy ams to mnmse portfolo varance regardless of the expected level of portfolo drft. Weghts for the mnmum varance portfolo (MVP), w MVP are gven by: 1 w MVP b. (10) T 1 The fnal benchmark portfolo s one wth an expected drft rate of 15% pa. The fgure of 15%, whle beng arbtrary, s consstent wth the hstorcal record and s n general accord wth Australan nvestors expectatons of reasonable share market returns. Fg.. Growth Portfolo and the Benchmark Portfolos 1 The structure of short-sales allowed growth optmal portfolos s extensvely explored n Hunt (00).

9 16 Investment Management and Fnancal Innovatons, 1/005 A stylsed representaton of the relatve postons of the growth portfolo and the three benchmark portfolos s depcted n Fgure 1. The nverse of the covarance matrx, -1, s necessary for the determnaton of the weghts of the growth optmal portfolo, the MVP and the 15% drft portfolo. Unfortunately, a problem arses n the computaton of -1 due to the mult-collnear nature of the perodc stock growth rates. The emprcal estmate of the 5-stock covarance, s at tmes close to beng sngular. The near sngularty of results n a loss of numercal precson whch n turn results n estmates of ndvdual stock weghts, w*, that gyrate wldly from observaton to observaton. The replacement of wth an amended covarance matrx, + n the estmaton process provdes a soluton to the mult-collnearty problem, where: d I, (11) where d s a scalar and I s the dentty matrx. The approach emboded n (11) s analogous to the rdge soluton to mult-colnearty n regresson analyss. The use of a non-zero d n (11) produces based estmates of the growth optmal portfolo and the benchmark portfolos. As the pvot d ncreases, the rdge estmate of growth optmal portfolo weghts, w + *, s based away from the classc growth optmal portfolo weghts towards the equally weghted portfolo. That s, n the lmt: lm w * 1, (1) d n where n, the number of assets n the set, s 5 n our case. In other words, the rdge estmator produces weghts that are a combnaton of the classc estmator weghts and those of the equally weghted portfolo. A decson to use a rdge estmator necessarly requres a partcular value for d. A common approach n rdge regresson analyss s to choose a value for d that provdes stablsed estmates of the system parameters. We have taken a smlar approach n choosng a sutable d on the bass of ts nfluence on portfolo length defned by 3 : T length ( n w w). (13) Portfolo length Rdge constant, d. Fg. 3. Portfolo Length Versus the Sze of the Rdge Constant 1 Hunt (00) shows that the growth optmal portfolo les on the mnmum varance fronter drawn n expected drftvarance space. Judge (1985, pp ) provdes an exhaustve revew of rdge estmators. An alternatve form of rdge estmator, + =+d D where d s a dagonal matrx of ndvdual stock varances, was also consdered. The weghts produced by ths rdge estmator are nversely proportonal to the stock varances n the lmt as d. 3 Under ths defnton the mnmum length portfolo,.e. the equally weghted portfolo, has unt length.

10 Investment Management and Fnancal Innovatons, 1/ Fgure 3 plots the growth portfolo length (estmated over the entre sample) aganst d. It was decded, on the bass of ths plot, that settng d equal to 0.35 (35%) represented a reasonable compromse between portfolo length and the portfolo weght bas 1. The results of applyng the MVP, the 15% drft and the growth optmal strategy, wth the rdge constant set to both zero and 0.35, for parameter estmaton perod lengths of 3, 4 and 5 years, are set out n Table 4. These results need to be measured aganst the equally weghted portfolo, whch s shown n Table 1 to have an average growth of 11.55% p.a. (and thus an aggregate growth of 88.7%) wth a volatlty of 18.9% p.a. Estmaton perod = 3 years Short-sales Allowed Portfolo Strategy Returns Rdge constant = 0.00 Rdge constant = 0.35 Table 4 MVP 15% drft Growth MVP 15% drft Growth Aggregate 13.8% 8.0% 1018.% 9.% 84.1% 359.0% Average 8.6% 11.3% 480.7% 11.7% 11.4% 14.4% Volatlty 5.0% 3.7% 549.9% 18.7% 17.7%.4% Estmaton perod = 4 years Aggregate 195.% 5.9% 140.5% 90.1% 81.0% 344.4% Average 7.8% 9.0% 496.8% 11.6% 11.% 13.8% Volatlty 0.4% 0.4% 1057.% 18.7% 18.% 1.8% Estmaton perod = 5 years Aggregate 169.0% 54.9% % 89.1% 91.4% 337.9% Average 6.8% 10.% 45.% 11.6% 11.7% 13.5% Volatlty 19.8% 0.0% 699.9% 18.7% 18.1% 1.5% Equally weghted portfolo MVP (d=0.35) 15% drft(d=0.35) Growth portfolo (d=0.35) Value Mar-77 Mar-8 Mar-87 Mar-9 Mar-97 Mar-0 Fg. 4. Short -sales Allowed Portfolo Growth n Value The frst notable result s the volatlty assocated wth the non-rdge (.e. d=0.00) growth portfolos. Whle the results n Table 4 show that the non-rdge portfolos were numercally attanable, they do not provde evdence that a growth orented, short-sales allowed, nvestment 1 The growth portfolo estmated wth d=0.35 has a length that s less than % of that of the classc, d=0.0, growth portfolo. Wth 5 assets n the portfolo, the maxmum rank of the covarance matrces s n-6, where n s the number of months n the estmaton perod. The portfolo weghts estmated over the 3-year perod are thus partcularly susceptble to the problem of multcollnearty.

11 18 Investment Management and Fnancal Innovatons, 1/005 strategy was realstc or feasble. The short-sales allowed, zero rdge constant, growth portfolos had gearng ratos that any nvestor would fnd mpractcally hgh. For example, the average length of the 4-year estmaton perod, zero rdge constant, growth portfolo exceeded a value of 300. The strategy routnely requred an asset to be short-sold more than 1000%. The hghly geared portfolos were naturally charactersed by hgh volatlty of nvestment returns. The short-sales allowed growth portfolos, whose weghts were estmated wth a rdge factor of 0.35, were much better behaved than ther zero rdge factor counterparts. The growth strategy portfolos outperformed the other bench marks by more or less than 3% p.a. dependng on the length of the estmaton perod. It s worth notng smlarty n performance of each of the three benchmark portfolos for d=0.35. The equally weghted portfolo, the MVP portfolo and the 15% drft portfolo each produced a rate of growth of a lttle under 1% p.a., wth an assocated volatlty of about 18% regardless of the length of the estmaton perod 1. In fact the performance of all four strateges, ncludng the growth strategy, appears to be relatvely ndependent of the length of the estmaton perod for both the classc, and the rdge non-rdge portfolos Fgure 4 shows the dollar extent of the superor performance by the short-sales allowed, rdge constant=0.35, 4-year estmaton perod, portfolo over the 5 years of the data set. As prevously stated, the presence of short-sold shares n the portfolos of ether professonal or retal nvestors s not typcal. An analyss of the results of growth portfolos where short-sellng s not allowed wll provde a more practcal test of the growth nvestment strategy. Short-sellng not allowed growth portfolos Whle the short-sellng of stock n most equty markets, ncludng Australan, s allowed, t s not typcal. Trallng growth optmal portfolos where a no short-sales restrcton s mposed on portfolo weghts, s a more realstc test of the strategy. The results from testng no short-sales growth portfolos are set out n Table 5. The no short-sales growth portfolo performances are mpressve. The statstcs recorded n Table 5 shows that the classc no short-sales, non-rdge (.e. d=0), estmator produces portfolo growth rates n excess of 0% p.a. and up to 31% p.a., dependng on the length of the nput estmaton perod. No Short-sales Allowed Portfolo Strategy Returns Table 5 Rdge constant = 0.00 Rdge constant = 0.35 MVP 15% drft Growth MVP 15% drft Growth Estmaton perod = 3 years Aggregate 60.6% 89.% 79.5% 9.% 95.7% 358.1% Average 10.4% 11.6% 31.7% 11.7% 11.8% 14.3% Volatlty 16.% 15.4% 45.7% 18.7% 17.8%.% Estmaton perod = 4 years Aggregate 58.7% 78.3% 563.0% 90.1% 84.8% 345.% Average 10.3% 11.1%.5% 11.6% 11.4% 13.8% Volatlty 16.5% 16.3% 37.7% 18.7% 18.5% 1.8% Estmaton perod = 5 years Aggregate 6.8% 76.8% 569.0% 89.1% 64.6% 337.8% Average 10.5% 11.1%.8% 11.6% 10.6% 13.5% Volatlty 16.7% 18.1% 36.% 18.7% 19.0% 1.5% 1 The equally weghted portfolo performance s of course ndependent of the estmaton perod.

12 Investment Management and Fnancal Innovatons, 1/ The aggregate growth for the -year, 3-year and 4-year estmaton perod growth portfolos s depcted n Fgure 5 1. The extent to whch the growth portfolos outpaced the benchmark portfolos s clearly evdent. The growth portfolos performance s even more mpressve when stated n dollar terms. One dollar nvested n the no short-sales, 3-year, 3-year and 5-year estmaton perod, growth optmal portfolo strategy n March 1977 would have returned $,764, $78 and $95 respectvely at the end of March 00. These fgures grossly exceed the return on the All Ordnares Index and the equally weghted portfolo, of $11.4 and $18.61 respectvely. 1000% 900% 800% All ordnares ndex Equally weghted portfolo Growth (est.per=3yrs, d=0.00) Growth (est.per=4yrs, d=0.00) Growth (est.per=5yrs, d=0.00) 700% 600% 500% 400% 300% 00% 100% 0% Mar % Mar-8 Mar-87 Mar-9 Mar-97 Mar-0 Fg. 5. Short-sales Not allowed Aggregate Growth Rates The mpressve performance of the no short-sales growth strategy begs further analyss. A couple of ponts about the performance of the growth portfolos can be made wth reference to Fgure 5. Frst the growth strategy accumulaton of value s slower n the second half of the perod than t s n the frst half. Moreover, the performance n the last couple of years has been notably negatve. Table 6 sets out growth and volatlty statstcs for the classc and rdge short-sales not permtted portfolos and for each of the benchmarks, for the 4-year estmaton perods. It s clear from Table 6 that the growth orented strateges, whle havng consderably hgher growth rates than the benchmark strateges, also have much hgher volatlty than the benchmark MVP and 15% growth and equally weghted strateges. The drect relatonshp between growth and volatlty s also evdent n the 3-year and 5-year estmaton perod as Table 5 shows. The evdence shows that no sngle nvestment strategy clearly domnates any other strategy. Indeed, the results of ths study provde strong support for what Luenberger (1998) calls the log mean-varance model 3. Low growth portfolos are assocated wth low volatlty and hgh growth portfolos are assocated wth hgh volatlty. The pont s, however, that regardless of the cost n terms of volatlty, the portfolos desgned for maxmal growth dd produce qute remarkable rates of growth. It s worth nvestgatng the source of ths growth. It s nsghtful to examne the average portfolo length and average number of ncluded assets n the no short-sales growth optmal portfolos. The length of a no short-sales allowed portfolo s nversely ndcatve of ts dversty 4. The length of a no short-sales, 5-stock portfolo, can take values between one and fve. The maxmum portfolo length of 5 s acheved, for any 5- stock portfolo, when 100% of portfolo value s held n a sngle stock. At the other end of the spectrum s the maxmally dversfed, equally weghted portfolo wth unt length. 1 From here on, the analyss of results s restrcted, for the sake of brevty, to the -year and the 3-year estmaton perod. The choce of these two estmaton perods s justfed as they yeld both the hghest and lowest growth rates respectvely. Perhaps the poor recent performance of the growth portfolos may be summarsed n the old adage, f one lves by the sword (n ths case an nstrument fnely crafted by Messes Murdoch and Lowey) one des by the sword. 3 Luenburger (1998, pp ). 4 The use of vector length to measure dversty s related to Fernholz s measure of dversty D p =(w p ) 1/p. See Fernholz, Garvy and Hannon (1998).

13 0 Investment Management and Fnancal Innovatons, 1/005 No Short-sales, 4-year Estmaton Perod, Portfolo Propertes Table 6 Input estmaton perod Classc no short-sales growth portfolos (d=0.00) Rdge no short-sales growth portfolos (d=0.35) Portfolo type Average growth rate Volatlty of growth rate Average portfolo length Average no. of ncluded assets Average turnover of assets Equal weghts 11.55% 18.94% % MVP 10.35% 16.49% % 15% drft 11.13% 16.3% % Growth.5% 37.65% % MVP 11.61% 18.70% % 15% drft 11.39% 18.53% % Growth 13.81% 1.75% % Table 6 shows that the classc no short-sales allowed, growth optmal portfolo wth an average length of 4.36, s at the lower end of the dversty spectrum. Moreover, Table 6 shows that ths portfolo contans on average only 1.71 assets n each perod. Further, Fgure 7, whch shows the dstrbuton of the number of assets held n each perod, reveals that for the majorty of the 300 monthly perods, the no short-sales growth optmal portfolo conssted of a sngle asset. Growth Portfolo Included Companes* Table 7 Company No. of appearances Frst appears Last appears Growth whle ncluded (% p.a.) Overall growth (% p.a.) AGL 4 Jul-9 Dec % AMC 4 Jul-9 Aug-9 8.6% ANZ 1 Nov-9 BHP 10 Dec-91 Mar % BIL 18 Mar-90 Dec % 14.74% CSR 1 Mar % FGL 11 Sep-8 Sep % MAY 4 Jun-78 Jul % MIM 1 Mar-0 NAB 1 Jun-99 NCP 118 Apr-77 Feb % 3.49% ORI 3 Dec-94 Jan-95 QBE 39 Jan-83 Sep % 18.60% RIO 9 Apr-90 Sep % SRP Feb-84 Aug % STO 74 May-77 Jun % 15.33% WMC 3 Dec-88 Jan % 9.86% WPL 6 Feb-89 Sep % 11.68% WSF 155 Feb-79 Mar % 38.8% The statstcs n the table s for stocks ncluded n a no short-sales, classc growth optmal portfolo strategy, employng a 3 year estmaton perod. The low number of assets held n each perod results n an overall low rate of ncluson of ndvdual companes over the 5 years. Table 7 sets out statstcs relatng to ncluded stocks n a growth optmal strategy over 5 years. Only seven out of the 5 stocks are ncluded n the strategy for 1 months or more. Table 7 shows that the 5-year strategy s domnated by three stocks: STO,

14 Investment Management and Fnancal Innovatons, 1/005 1 NCP and WSF. NCP and WSF are Australan share market stellar performers producng farly steady growth over 5 years of 3.49% p.a. and 38.8% respectvely. In contrast, STO was a patchy performer. However, STO was well-performed stock over the perod of ts ncluson n the growth portfolo n the frst fve years of the 5-year tral perod. STO contrbuted to the strategy 55% p.a. whle t was ncluded, compared to a more modest 15% p.a. over the whole perod. The success of the no short-sales allowed, growth optmal strategy appears to le n ts ablty to dentfy companes durng ther perods of hgh growth for portfolo ncluson. 01 Frequency Growth portfolo (d=0.0) Rdge estmator, growth portfolo (d=0.35) Number of ncluded assets Fg. 7. Dstrbuton of the Number of Assets Held n Growth Portfolos Increased transacton costs are a practcal consderaton for any strategy that results n the placement of portfolo value n a few assets. Transacton costs wll be sgnfcant f a strategy requres the flp-floppng of large asset weghts from one asset to another. One can see that ths s the case to a degree wth the no short-sales allowed growth portfolos. Table 6 shows that the mantenance of ths growth optmal portfolo strategy would have requred on average a turnover of stock of about 1% per month. The transacton costs assocated wth any strategy that turns over 1% of a portfolo per month are consderable, and wll sgnfcantly lower the effectve rate of portfolo growth 1. The rdge estmator was employed to produce short-sales allowed portfolo weghts that were numercally stable and produced acceptable gearng levels. The justfcaton for ths use of a rdge estmator does not have the same force for short-sales not allowed portfolos. The no short-sales restrcton consderably reduces the dmenson of the covarance matrx that s nverted to produce portfolo weghts. Classc short-sales not allowed portfolo weghts are numercally stable and are by defnton not geared. There s, however, an argument for the use of a rdge growth estmator n the short-sales not allowed context, on the grounds that t produces more dversfed, less rsky portfolos. The rdge growth portfolo estmator s a combnaton of the classc growth optmal portfolo estmator and the maxmally dversfed, equally weghted portfolo. The sze of the rdge constant, d, determnes the extent to whch a rdge growth estmator s based away from the classc estmator towards the equally weghted estmator. We have computed no short-sales, rdge, growth optmal portfolos usng a rdge constant of 0.35 to facltate comparson wth the short-sales allowed results. Predctably, Table 6 reveals that the no short-sales allowed, rdge, growth portfolo performance les somewhere between the performance of the classc growth optmal portfolo and the equally weghted portfolo. The no short-sales allowed, rdge, growth portfolo contans more assets (see Fgure 6), s less rsky and has a lower growth rate than the classc growth portfolo. Concluson Growth optmal portfolo nvestment strateges were appled to a 5-year data set of 5 Australan companes. Intal statstcal nvestgaton of data provded no reason to be optmstc about the successful applcaton of the growth technques. The growth optmal technque assumptons of nor- 1 For example, the cost of portfolo adjustment would be.9% p.a. f a round transacton cost was % of traded value.

15 Investment Management and Fnancal Innovatons, 1/005 malty and stablty were volated by the nature of the Australan data. Returns on the 5 stocks were found to be skewed and leptokurtc and to have tme varyng varances and covarances. However, the growth optmal technques perform well, despte the assumptons not beng met. The growth optmal portfolos, both short-sales allowed and short-sales not allowed, produced rates of growth that exceeded those of the benchmark portfolos. The classc no short-sales allowed, growth optmal portfolos produced mpressve rates of growth that were more than double those of the benchmark portfolos. Analyss of the structure of these portfolos showed that, at any pont n tme, they conssted of a very small number of ncluded stocks. The secret of the success of these portfolos appears to le n ther ablty to select a few stocks durng ther hgh growth perods. Ths study detals the successful ncluson of a varant of rdge regresson as the bass of a growth optmal strategy. The rdge growth optmal technque facltated producton of numercally stable weghts for short-sales allowed portfolos. When short-sales were not allowed, the use of the rdge estmator produced more dversfed growth portfolos. There are two possble answers to the queston of why the growth optmal technques performed well n the face of non-normalty and nstablty n the data. The frst reason, whch cannot be dsmssed, s that the technques work well on ths partcular data set by pure chance alone. The second explanaton s that the assumptons of normalty and stablty are not necessary to the success of the technque. Whle the model used n ths paper assumes normalty n the Ito process, t may be that growth nvestment strategy s equally effcacous under alternatve stochastc processes that allow kurtoss. Why does the nvestment strategy cope wth dstrbutonal nstablty? Perhaps the use of a movng wndow estmaton process may counter the problems arsng from the nstablty of mean growth rates and growth rate covarances. The study detals the successful applcaton of growth optmal technques. There s, however, no evdence of the general superorty of growth optmal technques. Growth portfolo strateges are also hgh volatlty strateges. Whle the results of an emprcal study such as ths are necessarly lmted to the specfc market and to the specfc tme-frame of the study, the pont that ths study makes s, however, that regardless of ther other propertes and potental drawbacks, the portfolos desgned for maxmal growth dd n fact produce qute remarkable rates of growth. References 1. Berenson, M.L. and Levne, M.L. (1993), Basc Busness Statstcs Concepts and Applcatons, Prentce Hall.. Breman, L. (1961), Optmal Gamblng Systems for Favourable Games, Proceedngs of Fourth Berkeley Symposum on Mathematcal Statstcs and Probablty, Unversty of Calforna Press, Berkeley. 3. Fernholz, R., Garvy R. and Hannon, J., (1998) Dversty-Weghted Indexng, Journal of Portfolo Management, Wnter, pp Golberger, A.S. (1964), Econometrc Theory, Wley. 5. Green, W.H. (1993), Econometrc Analyss ( nd Ed.) McMllan. 6. Hakansson, N.H., (1971), Multperod Mean-varance Analyss: Towards a General Theory of Portfolo Choce, Journal of Fnance, 6, Hull, J.C. (000), Optons Futures and Other Dervatves (4 th Edton), Prentce Hall. 8. Hunt, B.F. (00) Growth Optmal Portfolos: ther structure and nature, UTS mmeo. 9. Kelly, J.L.,(1956), A New Interpretaton of the Informaton Rate, Bell Systems Techncal Journal, 35, Levy, M., Levy, H. and Solomon S. (000), Mcroscopc Smulaton of Fnancal Markets, Academc Press. 11. Luenberger, D.G. (1998), Investment Scence, Oxford Unversty Press, New York, New York. 1. Merton, R.C. and Samuelson P.A. (1974), Fallacy of the Log-normal Approxmaton to Optmal Portfolo Decson-makng Over Many Perods, Journal of Fnancal Economcs 1, pp Pearson, E.S. (1969) Some Comments on the accuracy of Box s approxmaton to the dstrbuton on M, Bometrca, 56, pp Rachev, R. and Mttnk, S. (000), Stable Paretan Models n Fnance, Wley.