A Comparison of Risk Return Relationship in the Portfolio Selection Models

Transcription

1 Proceedngs 59th ISI World Statstcs Congress, 5-30 August 03, Hong Kong (Sesson CPS00) p.3495 A Comparson of Rsk Return Relatonshp n the Portfolo Selecton Models C. W. Yang, Ken Hung,Yfan Zhao Claron Unversty of Pennsylvana, Claron, Pennsylvana 64 Pekng Unversty HSBC Busness School, Shenzhen, Chna Claron Unversty of Pennsylvana, Claron, Pennsylvana 64 Correspondng author: Ken Hung, e-mal: hungkuen@gmal.com Abstract In ths paper, we calculate four dfferent knds of means- AM, GM, HM, and GDM- to nvestgate the rsk-return contour usng Markowtz rsk mnmzaton and Sharpe s angle maxmzaton models. or a gven k value (target portfolo return), the rank order of rsk or varance-covarance (υ) can change. In the vertcal segment of an effcent fronter curve, we observed v(gdm)> v(hm)> v(gm)> v(am). At hgher k values, the rank changes to v(gdm)> v(hm)> v(am)> v(gm). That s to say, rankng a portfolo usng dfferent knds of means may well gve dfferent rankngs dependng on what k value one s evaluatng. It s also shown the harmonc mean should not be used n the case of a small negatve growth rate n stock prces. Keywords: arthmetc mean, geometrc mean, golden mean, harmonc mean, Markowtz rsk mnmzaton, Sharpe s angle maxmzaton, I. Introducton The foundaton of modern nvestment theory s lad upon the quadratc program portfolo selecton model developed more than half century ago by Harry Markowtz (95, 956 and 959). The optmzaton (rsk-mnmzaton) process over mean-varance-covarance space can trace out the effcent fronter curve, whch provdes the soluton space for nvestors. However, an exact soluton cannot be found wthout the knowledge of a rsk free rate on a government bond and an nvestor s atttude toward rsk. To ths end, Sharpe (964) formulated and solved the angle-maxmzaton model n whch the rsk (standard devaton) adjusted portfolo return (net of rsk free rate) s maxmzed. The Sharpe model provdes a convex combnaton of rsk free government bonds and a portfolo of stocks selected based on the crteron of rsk mnmzaton. Atttude toward rsk such as 0% on bond and 80% on stocks wll gve nvestor an exact soluton wthout the knowledge of the ndfference (soutlty) curve. Yang et al. (00) proved that Markowtz rsk mnmzaton and Sharpe angle-maxmzaton models are mathematcally equvalent gven some requred portfolo returns and rsk free bond rate. Does the choce of mean returns of stocks n the portfolo matter n the selecton process? If so, how dfferent are the optmum soluton sets? In ths note, we frst apply the wellknown means: arthmetc, geometrc and harmonc means to fve companes stocks. In addton, we add a golden mean to the smulaton for comparson. The organzaton of the paper s as follows. Next secton ntroduces the Markowtz rsk mnmzaton and Sharpe angle maxmzaton models. Secton III descrbes data and four dfferent means. Secton IV performs computer smulatons va LINGO to trace out correspondng effcent fronter curves. Secton V gves a concluson. II. Portfolo selecton models wth dfferent means

2 Proceedngs 59th ISI World Statstcs Congress, 5-30 August 03, Hong Kong (Sesson CPS00) p.3496 Gven a set of n selectable stocks, the purpose of the Markowtz portfolo model s to mnmze the weghted rsk n terms of varance and covarance of n stock returns, or Mnmze x () Subject to Where () (3) (4) x x R j x R x 0 k j x x j j = the weght or proporton of nvestment n stock = varance of returns n stock = covarance of return between stock and j = expected or average rate of return of stock k = target portfolo rate of return Pt Pt Note that rate of return s frequently calculated as on whch mean, varance and P covarance are calculated. Optmum weghts ( x t * * *, x,..., xn ) for x 0 are the framework under whch weghted rsk υ can be calculated. Along wth a set of k values, we have geometrc means calculated accordng to the growth rate formula (next secton) and a set of rsk-return values on whch an effcent fronter curve can be traced. However, the exact locaton cannot be determned wth a rsk-free bond rate R. To expand to rsk mnmzaton model, Sharpe (964) proposed an angle-maxmzng model wth a hghest straght lne from R that s tangent to the effcent fronter derved from the Markowtz model. x ( R R ) Maxmze tan (5) / ( x x x ) j Subject to x (6) j x 0 (7) As s shown by Yang et al. (00), a gven R corresponds to a k value n equaton () of the Markowtz model. urthermore, the denomnaton of equaton (5) s the square root of the objectve functon of equaton (). Thus, the Markowtz and Sharpe models exhbt recprocal relatons for a gven set of R, and k value. III. Descrpton of data and characterstcs of dfferent mean returns R j *

3 Proceedngs 59th ISI World Statstcs Congress, 5-30 August 03, Hong Kong (Sesson CPS00) p.3497 Monthly stock prce of 5 companes from September of 007 to August of 008 are calculated to obtan 55 (*5) rates of return. The arthmetc means (AM) and assocated varance and covarance of stock returns are reported n Table. Table. Means (%) and Varances (%) of ve Stock Returns MA IBM JNJ MCD WMT AM % % 0.940%.7303% 3.083% GM 4.635% % %.60%.956% HM % % 3.584% 4.577% 0.983% GDM.563%.45%.934%.55% 5.867% REG 4.57%.300% 0.60% 0.935% 3.0% ܯܣ ߪ.7700% % 0.36% 0.573% 0.56% ܯܩ ߪ.7768% % 0.37% 0.575% 0.56% ܯܪ ߪ 3.38% 8.750% 0.808% 0.3% 0.36% ܯܦܩ ߪ.333% % 0.45% 0.636% 0.07% ଶ.7790% % 0.38% 0.64% 0.56% ߪ AM=arthmetc mean GM=geometrc mean HM=harmonc mean GDM=golden mean REG= regresson-based mean locaton. ଶ = varance of stock returns usng AM as the central ߪ MA= Mastercard Incorporated IBM=Internatonal Busness Machnes Corp. JNJ= Johnson & Johnson MCD= McDonald's Corp. WMT= Wal-Mart Stores Inc. An examnaton on fve arthmetc means ndcates the hghest return s by Masters Charge (MA), followed by Wal-Mart (WMT), McDonalds (MCD), Johnson & Johnson (JNJ) and Internatonal Busness Machne (IBM). As an alternatve to arthmetc mean s P P3 P4 Pn P n n n the often used geometrc mean: ( ) ( ). Note that when P P P P P P P 3 exceeds (falls short of ) one t mples a poston (negatve) growth rate as s P P P measured by. Vewed n ths lght, an arthmetc mean (AM) s P P derved addtvely whereas ts geometrc mean (GM) s calculated multplcatvely usng P the same measure of rate of return,.well-known n statstcs, AM s more P senstve to outlers than s GM and as such GM may be preferred n such cases. Durng the sample perod, the fve stock prces underwent substantal changes. or example, the greatest monthly prce change was 8.4% whle the largest drop regstered 3.97% for Masters Charge. rom the perspectve of rsk averseness, AM mght not be preferred. n 3

4 Proceedngs 59th ISI World Statstcs Congress, 5-30 August 03, Hong Kong (Sesson CPS00) p.3498 Calculated geometrc mean returns follow the same rank order as that of arthmetc mean returns (Table ). A harmonc mean may be approprate for a varable that measures rates of change (e.g., velocty). In busness applcatons, number of shares of stocks of a natonal fund, a unt of prce (e.g., $,000,000) can purchase could ft nto ths category. The harmonc mean (HM) n x s calculated as where x represents rate of return for stock. or n= and HM n x >0, the geometrc mean s the square root of arthmetc and harmonc means or GM AM HM. or a set of clustered numbers, the three means produce very smlar values. However, HM can be very based toward a negatve value n the presence of a small negatve return,.e., x =-0.0 mples =-50 whch wll domnate other postve regular returns. Ths s the case we encounter n calculatng HM for IBM and MA.(e.g., x 0.66% for MA and x4 0.9% for IBM). or comparson, we report the HM return n Table as well. The golden mean, also known as golden secton and golden rato from Leonardo da Vnc s a specal case of the geometrc mean on lne segment AC =: a locaton B between AC AB 5 ponts A and C such as whch leads to the soluton AB BC In another word, t s the range ( max x x x ) that determnes the value of golden mean mn mn ( xmax xmn (GDM) snce t s hnged calculated as x ). or nstance, the GDM return for MA s 3.97% (8.47% (3.97%)).56%. When the range plays a key role n determnng a representatve value, GDM may be a vable canddate and are reported n Table. The correspondng varances and covarances of stock returns usng the AM, GM, HM, and GDM are shown n Table. Table. Varance and Covarance Usng ve Means IV. A comparson of smulaton results An effcent fronter generally conssts of two parts: a vertcal secton and a concave part. The vertcal part ndcates equaton () holds wth strct nequalty (>k). That s the portfolo return at optmalty exceeds the mnmum requred rate k. or k = % (annualzed), whch s obvously too low. The optmum portfolo return far exceeds k =% n the case of AM. If we arbtrarly mpose equaton () wth an equalty sgn ( k =%), there wll exst no feasble soluton, for the lowest annual average rate of return s 7.37% (IBM), and as such the vertcal sectons of the effcent fronter curve starts to bend at k 0%, k =9%, k =8%, and k =3% n the cases of AM, GM, HM and GDM respectvely (Table 3). A perusal of Table 3 ndcate that for range of k <8%, the portfolo rsks n the Markowtz model manfest the followng rank order 0.5 (AM)< 0.6(GM)< 0.4(HM)< 0.48(GDM) for each gven k value. 4

5 Proceedngs 59th ISI World Statstcs Congress, 5-30 August 03, Hong Kong (Sesson CPS00) p.3499 As far as mnmum rsk level where effcent fronter s vertcal s concerned, the rsk levels of GM and HM are bounded by that AM curve and GDM. Gven that GDM are hghest n all four means, ts varance-covarance are greatest at lower level of k values. The varances of HM are even greater because of negatve mean returns. However, fve of ten covarances n HM are negatve and thus the portfolo rsk could be reduced from proper dversfcaton. The hgh values of the GDM returns translate nto hgher varances and covarances when compared to that of AM and GM. It s lttle wonder that the effcent fronter has the largest rsk n the vertcal segment or v(gdm)> v(hm)> v(gm)> v(am). As k value ncreases to 30% the rsk become the greatest for HM that has 5 largest varances of all the four means. The rank of the portfolo rsk become v(hm)> v(gm) > v(am)> v(gdm). Note that v(gdm) s stll at ts vertcal segment at k=0.30% (see Table 3). rom k and on the rsk n the case of GM become the greatest because the choce set has dwndled: only WMT (35.436%) and MA (55.477%) have the mean return greater than 34% and as such most of the weghts go to WMT and MA regardless of ther rsk levels. On the contrary rsk of GDM assumes ts smallest value snce most of the fve stocks have hgher returns: 47%,.4%, 30.3%, 3.08% and 7.4% for MA, WMT, MCD, JNJ and IBM respectvely. HM has shown smallest rsk due to negatve covarance. However, t must be ponted out that small negatve returns are based and msleadng n the sense that the double recprocty formula assgns too much weght to small negatve return (slght prce decrease n stock). The rsk assocated wth AM for k 34% s slghtly less than that of GM snce GM s less senstve to outlers than s AM, and as such AM has a slghtly larger choce set (smaller rsk) at hgh k value. The four effcent fronters are shown n gure. or a gven rsk free rate R, the greatest tangent angle attanable between the ray from R and the effcent fronter curve can be calculated from equaton (5). By verfyng R we fnd the optmal tangent angle va LINGO and report the portfolo returns portfolo rsk (υ) and tangent angles (tan θ) n Table4. An nspecton of Table 4 reveals that Sharpe s solutons wth GDM domnates that wth HM for R =3%, 4%, 5% and 6% because return under GDM are greater but rsks smaller than that under HM. or R =7% and 0%, returns and rsks are both greater under GDM n comparson to HM. Hence we calculate Sharpe s ndex or ts tan and used as the selecton ctaton. A comparson between AM and GM ndcate that rsk-return combnatons usng AM domnate that usng GM: hgher return wth less rsk from R =3% through 7%. At R 0%, both the portfolo return and rsk are greater usng AM: 0.36> and 0.84>0.04. As a consequence, we compare tan θ and fnd that under AM s greater > Table 3. Rsk-Return Combnaton Usng ve Means Table 4. Angle Maxmzaton Solutons Usng ve Means ത = Portfolo return ݔ = = Varance and covarance of 5 stocks returns ݒ = Portfolo premum net of rsk free rate per rsk ݒ /( ( ത ݔ = ߠ ݐ 5

6 Proceedngs 59th ISI World Statstcs Congress, 5-30 August 03, Hong Kong (Sesson CPS00) p.3500 V. Concludng remarks In ths paper, we calculate four dfferent knds of means AM, GM, HM, and GDM to nvestgate the rsk-return contour usng Markowtz rsk mnmzaton and Sharpe s anglemaxmzaton models. or a gven k value (target portfolo return), the rank order of rsk or varance-covarance (υ) can change. In the vertcal segment of an effcent fronter curve, we observed v(gdm)> v(hm)> v(gm)> v(am). At hgher k values, the rank changes to v(gdm)> v(hm)> v(am)> v(gm). That s to say, rankng a portfolo usng dfferent knds of means may well gve dfferent rankngs dependng on what k value one s evaluatng. When a rsk free-rate s added to the Markowtz model, we arrve at the anglemaxmzaton soluton. It seems that rsk-return combnatons under GDM domnate those under HM for the former has hgh returns and less rsk. The same can be sad of AM and GM. The combnaton usng AM seems to have greater portfolo returns but less rsk. When these exst a trade-off,.e., hgher rsk coupled wth hgher return, Sharpe s ndex (tan θ) favors both GDM and AM over HM and GM respectvely for greater angle translates nto hgh portfolo return(net of R ) per rsk. Care must be exercsed though, the results from ths paper are lmted to the stocks we take n the sample. However, comparatve evaluatons are needed for a comprehensve analyss on a portfolo performance. In sum, GM s less senstve to outlers and hence s more sutable for conservatve strategy. GDM s determned by the sze of ts range and tends to offer an optmstc forecast on mean return especally when there exst a few unusually large postve returns. nally, HM s not approprate f some of the returns (%) are small and negatve. In that case, HM ought to be removed from the analyss. References gure. Effcent ronter Curves Usng ve Dfferent Means Elton, E. J., Gruber, M. J., Brown, S. J. and Goetzmann. Modern Portfolo Theory and Investment Analyss. 7 th ed. (Hoboken, NJ: John Wley and Sons, Inc.007) LINGO 8.0 Lnear and Nonlnear Optmzer (Chcago, IL, LINGO system Inc. 003) Markowtz, H. (95). Portfolo Selecton. Journal of nance, 7, Markowtz, H. (95). The Optmzaton of a Quadratc uncton Subject to Lnear Constrants. Naval Research Logstcs quarterly, 3,-33. Markowtz, H. (959). Portfolo Selecton: Effcent Dversfcaton of Investment. New York: John Wley & Sons, Inc... Sharpe, W.. (964) Captal Asset Prces: A Theory of Market Equlbrum under Condton of Rsk. Journal of nance, (September), Yang, C. W., Hung K and.a Yang. A Note on the Markowtz Rsk Mnmzaton and the Sharpe Angle Maxmzaton Models. Vol.9 Advances n Investment Analyss and Portfolo Management Ed. by C.. Lee (Amsterdam, Elsever Scence, 00) f 6