How Efficient is Naive Portfolio Diversification? An Educational Note

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2 How Efficiet is Naive Portfolio Diversificatio? A Educatioal Note by Gordo Y. N. Tag Departmet of Fiace ad Decisio Scieces Hog Kog Baptist Uiversity Kowloo Tog Kowloo HONG KONG Tel: (85) Fax: (85) gytag@hkbu.edu.hk

3 How Efficiet is Naive Portfolio Diversificatio? A Educatioal Note Abstract Stadard textbooks of Ivestmet/Fiacial Maagemet will tell you that although portfolio diversificatio ca help reduce ivestmet risk without sacrificig the expected rate of retur, the beefit of diversificatio is exhausted with a portfolio size of 0 to 5. Sice by the, most of the diversifiable risk is elimiated, leavig oly the portio of systematic risk. How valid is this "commo" kowledge? What is the exact value of "most" i the above statemet? This paper examies the issue o aive (equal weight) diversificatio ad aalytically shows that for a ifiite populatio of stocks, a portfolio size of 0 is required to elimiate 95% of the diversifiable risk. However, a additio of 80 stocks (i.e., a size of 00) is required to elimiate a extra 4% (i.e., 99% total) of diversifiable risk. This result depeds either o the ivestmet horizos, samplig periods or the markets ivolved. Keywords: Naive Diversificatio, Efficiecy, Portfolio, Diversifiable risks

4 . Itroductio Over the past years, portfolio diversificatio is oe of the mai moder ivestmet theories that have bee developed. It may ot be the most importat theory beig developed, however, it is o doubt that holdig portfolios is the most widely accepted ivestmet cocept ad the most beig practised kowledge i real life by geeral ivestors. All uiversity busiess graduates lear that through portfolio diversificatio, ivestmet risk ca be reduced without sacrificig the expected retur. This cocept ca be easily applied without ay complex techiques via aive diversificatio. That is, oe ca hold a diversified portfolio by radomly select a certai umber of stocks ad ivest equal amout of moey i each of them. While this is simple eough, however, stadard textbooks of Ivestmet/Fiacial Maagemet will also tell you that the beefit of diversificatio is mostly exhausted with a portfolio size of 0 to 5. Sice by the, most of the diversifiable risk is elimiated, leavig oly the portio of systematic risk. The existece of systematic risk is the reaso why the beefit of portfolio diversificatio ca be exhausted o matter how large is the portfolio size sice by defiitio, systematic risk caot be elimiated through diversificatio. All stocks would be affected at the same time by some ecoomy-wide factors. Hece, studyig the relatioship betwee the portfolio risk ad the portfolio size is importat as this will dictate the ecessary umber of stocks required i aive diversificatio to obtai the largest beefit. I theory, oe should go for holdig as may stocks as possible as log as the portfolio's variace keeps o decreasig. But i practice, the additioal beefit gaied through further risk reductio may ot be large eough to offset the extra trasactio costs ivolved. Ivestors have to make the trade-off betwee the reduced risks due to more effective diversificatio versus the additioal trasactio costs that mea lower returs from addig more stocks to their portfolios. If idividual ivestors ideed ca obtai most of the beefits of diversificatio by holdig small-size portfolios, say 0-5 stocks as suggested by most Ivestmet/Fiacial Maagemet textbooks, effective diversificatio ca be accessed easily ad directly. The those uit trust ad fud maagers would eed great effort to justify their existece ad the high maagemet fees charged through either superior selectivity or good market timig. So, it becomes iterestig ad

5 importat to aswer these questios: How valid is this "commo" kowledge? What is the exact meaig of "most" i the statemet that most of the diversifiable risk is elimiated? Ca we obtai a exact relatioship betwee the portfolio's variace ad the icreasig portfolio size? Does it matter o the effectiveess of diversificatio with a ifiite populatio or a fiite populatio of stocks? These questios are iterestig because they have ever bee clearly ad directly addressed i the curret textbooks of Ivestmet ad Fiacial Maagemet despite of its direct relevace to geeral ivestors. They are also importat i the sese that they have major implicatios for the busiess of uit trusts ad fud houses ad for the behaviour of geeral ivestors. This paper aims to examie these issues. The rest of this paper is orgaised as follows: Sectio summaries the textbooks' recommedatios; Sectio 3 presets a aalytical relatioship betwee portfolio size ad risk, ad provides a aswer to "most" i the above statemet; the differeces betwee ifiite ad fiite populatios o our results are discussed i Sectio 4; Sectio 5 gives a remark o portfolio diversificatio ad cocludes the paper.. Textbooks' Recommedatios Before reviewig the textbooks' recommedatios, let us first describe the stadard approach i studyig the relatioship betwee risk ad portfolio size through aive diversificatio. For empirical aalysis o a selected market, the populatio of stocks (populatio size) is first defied. For example, the 500 stocks of the S&P 500 Idex i U.S. or it ca be the total umber of atioal stock market idices if iteratioal diversificatio is beig studied. A stock is selected radomly from the populatio ad its risk is measured by the variace (or stadard deviatio) calculated from the series of stock returs. Aother stock is the selected from the populatio to form a portfolio of size with the first stock. The portfolio's variace is calculated by assigig equal weight to these two stocks. Stock 3, stock 4, ad so o are selected radomly from the populatio i sequece without replacemet. At each time, equally weighted portfolios are formed ad the portfolios' variaces are calculated. Hece, a series of portfolios with sizes ragig from to say, 00 are obtaied. The whole process is the repeated for may times, say 00 times. This meas that we

6 have obtaied 00 portfolio's variaces for each idividual portfolio size. For each idividual portfolio size, the average portfolio's variace is calculated. By plottig the average portfolio's variace agaist the portfolio size, the relatioship betwee risk ad the umber of stocks i the portfolio is thus obtaied. The above research approach is metioed i all major Ivestmet/Fiacial Maagemet textbooks. After this, the optimal umber of stocks required i a diversified portfolio is stated out at which the authors claim that most of the beefits of diversificatio will the be obtaied. Table lists the recommedatios o this "magic" umber from te Ivestmet textbooks ad te Fiacial Maagemet textbooks. These textbooks are believed to be the most represetative ad widely adopted by uiversities for ivestmet/fiace courses. Colum shows the authors' ames ad years of publicatio (or years of latest editio). Colum idicates the page umbers respectively for each textbook where the required iformatio ca be foud. Colum 3 shows the recommeded optimal umber of stocks i portfolio. Of the te Ivestmet textbooks, the miimum umber is 8 while the largest umber is aroud 40. Most of them recommed a size of 0 to 5. For the te Fiacial Maagemet textbooks, the miimum ad maximum umbers are 0 ad 40 respectively. The most commo recommedatio is 0 to 0. A closer look at these textbooks' recommedatios reviews that most of their recommedatios are actually based o some academic joural articles. Colum 4 of Table presets the mai sources of referece cited i each textbook. 3. Size vs Risk: Aalytical Relatioship The geeral formula for the variace of a portfolio with size is ormally stated as follows: σ p = wi w jcov(r i,r j ) () i= j= where w i ad w j are the ivestmet proportios o assets i ad j respectively; r i ad r j are the returs of assets i ad j respectively, ad cov(r i, r j ) represets the covariace betwee returs of assets i ad j. I aive diversificatio where a equally weighted portfolio is formed, we have w i = w j = / ad equatio () ca be rewritte as: 3

7 σ ) () p = σi + Cov( ri, r j i= i= j= i j I equatio (), we kow that there are variace terms ad (-) covariace terms. Let the average variace ad average covariace be as follows: σ = σi (3) i= Cov = ( - ) i= i j j= Cov( r i, r j ) (4) The equatio () ca be expressed as - σ p = σ + Cov (5) From equatio (5), it is clear that whe icreases, that is, whe the portfolio size icreases, the first term o the right had side teds to zero while the secod term teds to the average covariace (as (-)/ teds to ). Now suppose N is the populatio size ad equatio (5) ca be used to describe the variace of the portfolio composed of equal ivestmet i each of the N stocks of the populatio. For a portfolio with size < N, the same equatio ca also describe the portfolio variace. However, ulike holdig the whole populatio, we have a total of N C possible portfolios with the same portfolio size. We are iterested i the average portfolio variace for size. For example, for a populatio of 0 stocks, we have 45 ( 0 C ) portfolios of size, 0 ( 0 C 3 ) portfolios of size 3, etc. The variace of each portfolio is first calculated ad the the average variace is obtaied by averagig all variaces with the same portfolio size. By usig all possible combiatios of each portfolio size, the average mea retur is guarateed the same regardless of the portfolio size. I fact, Tag ad Choi (998) employed this methodology to examie empirically the portfolio effect o the stadard deviatio, skewess ad kurtosis of iteratioal stock idex portfolios. However, the limitatio of this methodology is that the populatio size must be restricted to avoid a huge amout of computatioal work i all possible combiatios of stock portfolios. Takig all possible combiatios of portfolios ito cosideratio is the same as takig the 4

8 expectatio of equatio (5). I fact, this is the result metioed by Elto ad Gruber (977) as equatio B of Appedix B ad therefore, equatio (5) ca be regarded as the correct formula for the average variace of a portfolio with stocks regardless of what the populatio size is. The oly modificatio is that ow the average variace ad average covariace will be calculated from all stocks i the populatio. To make it more specific, the relatioship ca be re-stated as follows: where - σ = σ + Cov (6) = the umber of stocks i the portfolio, =,,..., N N = the umber of stocks i the populatio ó = the average portfolio variace with portfolio size ó = the average variace of all stocks i the populatio Cov = the average covariace of all stocks i the populatio Oe implicatio from our results is that for a ifiite populatio of stocks (i.e., N teds to ifiite), whe the portfolio size, icreases from, the first term i the right had side of equatio (6) will become smaller ad smaller ad teds to zero, while the secod term will become larger ad larger ad teds to Cov. Hece, the first term is the diversifiable (o-systematic) part while the secod term is the o-diversifiable (systematic) part. I order to illustrate the above poit more clearly ad to idicate the efficiecy of aive portfolio diversificatio, we compute the relative average variaces of portfolios with differet portfolio sizes (see Tag ad Choi, 998). This is accomplished by dividig all average portfolio variaces by the average variace of portfolio with size equals to oe. From equatio (6), whe equals, we have σ = σ. The result is obvious. Hece, dividig equatio (6) by σ, we have RV = +( - ) X (7) where RV = relative average portfolio variace, σ σ X = the ratio of average covariace to average variace of all stocks i the populatio, 5

9 Cov σ Rearragig terms i equatio (7), we obtai Now it is clear that X is the relative systematic risk that caot be elimiated through diversificatio while ( - X) is the relative o-systematic risk that ca be elimiated completely through aive diversificatio. Whe teds to ifiite, ( - X) is completely goe, leavig oly X, the systematic part. Several implicatios are draw from the above result. First, the power of aive diversificatio o risk reductio is iversely proportioal to the portfolio size. With oly a portfolio size of two, half of the diversifiable risk is elimiated o average. With a size of 0, 90% of diversifiable risk is elimiated ad 95% of such risk ca be elimiated with a portfolio size of 0 o average. Hece, the value "most" i the statemet which appears i may fiacial/ivestmet maagemet textbooks, that most of the diversifiable risk is elimiated with 0-5 stocks i the portfolio ca ow be aswered specifically ad directly. Secod, the effectiveess of aive diversificatio o reducig diversifiable risks is idepedet either of the samplig periods, ivestmet horizos or of the markets ivolved. It does ot matter whether the stocks are hold for oe moth, two moths, or oe year or whether we are ivestig i the U.S., U.K. or the Japaese stock markets or eve are ivestig i iteratioal stock markets. The oly relevat factor is the portfolio size. Third, the part o o-systematic risk caot be completely elimiated uless we have a ifiite stock populatio. However, the margial beefit of larger portfolio size due to further risk reductio is a decreasig fuctio of. I fact, for a portfolio size of, a additio stock to the portfolio will further elimiate /( + ) (or / - /(+)) of the diversifiable risk, that is, ( - X) i equatio (8). Fourth, that previous empirical results which foud that the impact of diversificatio o portfolio risk varies across differet stock samples ad differet periods is because of the variatios i the relative systematic risk, that is, X i equatio (8), the ratio of average covariace to the average variace of all stocks i the populatio. The power (or effectiveess) of aive diversificatio o reducig diversifiable risks has ot chaged. RV = X +(- X) (8) 6

10 Figure plots the relative average variace agaist the portfolio size with three differet assumptios o the value of X, the relative systematic risk. We let X equal to 0.75, 0.5 ad 0.5 ad check the impact o the dowward slopig curves. It is clear that the shape is similar to those preseted i textbooks (e.g., Figure 9. (p.9), Fracis, 99; Figure 5.4 (p.3), Piches, 996). However, i our case, we clearly show the exact relatioship betwee portfolio risks ad sizes graphically give the value of relative systematic risk. Whe the relative systematic risk is 0.75, the curve levels off at a portfolio size of 0. However, whe the correspodig value is 0.5 (0.5), Figure shows that the curve levels off at a portfolio size of 5 (0). Hece, this explais why recommedatios from textbooks say that empirically for a portfolio size of 0 to 5, most of the diversificatio beefit is exhausted. 4. Fiite Populatio of Stocks Sectio 3 presets the aalytical relatioship betwee the average portfolio variace ad the portfolio size. Equatio (8) also implies that the part o o-systematic risk caot be completely elimiated uless we have a ifiite stock populatio. However, uder a ormal ivestmet eviromet, ivestors ca oly select stocks withi a populatio of limited size. Furthermore, ivestors may eve wat to restrict their potetial pools of stocks to a smaller size tha the whole populatio for various reasos. For example, fud maagers may have iterest oly o those blue-chip stocks i each market. Hece, a relevat questio is what will be the impact of differet populatio sizes o the umber of stocks required to elimiate a certai percetage of the o-systematic risk. A logical predictio is that a smaller umber of stocks are required to achieve the same level of risk reductio for a populatio with smaller size. Is that true? This sectio aims to give a quatitative aswer. Accordig to equatio (8), eve whe you hold the whole populatio of stocks i your portfolio, you still caot completely elimiate all diversifiable risk. The o-systematic part of relative risk that is still remaied equals ( - X)/N. I other words, all you ca do best is to elimiate [(N - )/N] of ( - X), the maximum diversifiable risk of the whole populatio of size N that ca be diversified away. Similarly, for a portfolio with a size, [( - )/] of ( - X) is reduced through 7

11 diversificatio. Hece, we ca see that the proportio of the maximum relative diversifiable risk elimiated with a portfolio size is equal to [( - )/]/[(N - )/N]. Here, ( - )/ is the part of diversifiable risk of a portfolio with size where (N - )/N is the total (maximum) diversifiable risk of the whole populatio of size N. Lettig this proportio, say a, to vary for differet percetages, we ca solve the value for give a particular umber of N. That is, the umber of stocks required to achieve a certai level of risk reductio for differet populatio sizes ca be foud precisely. Table presets the umber of stocks required i a portfolio to elimiate a certai percetage of diversifiable risk give differet populatio sizes. Our results cofirm that the smaller the populatio size, the smaller is the required umber of stocks. Table shows that if oe wats to elimiate oly 50% of the diversifiable risk, the populatio size really does ot matter sice you still eed stocks (for a populatio of 00 stocks, you eed.98 stocks o average). However, if oe wats to elimiate 95% of the diversifiable risk, the umber of stocks required varies greatly across differet populatio sizes. For a populatio of,000 stocks, you eed 9.6 stocks o average but you oly eed 6.8 stocks o average for a populatio size of 00. If you further restrict your populatio size to 40, what you eed is just 3.6 stocks to achieve the same target. 5. Remarks ad Coclusios Naive diversificatio is a simple but powerful way to reduce your portfolio's risk effectively without sacrificig the expected rate of retur. Busiess graduates kow this result well. However, how true is this fact ad what is the impact of portfolio sizes o the efficiecy of aive diversificatio? This paper shows aalytically that for a ifiite populatio of stocks, a portfolio size of 0 is required to elimiate 95% of the diversifiable risk. However, a additio of 80 stocks (i.e., a size of 00 stocks) is required to elimiate a extra 4% (i.e., 99% total) of diversifiable risk. This result depeds either o the samplig periods, ivestmet horizos or the markets ivolved. For a fiite populatio of stocks, the correspodig portfolio size required is smaller, the smaller the populatio size. Our fidigs have seldom bee metioed or discussed i may Fiace/Ivestmet textbooks. Although results preseted i this paper are importat ad highly relevat to all ivestors, there are some remarks o diversificatio beefits that we should aware. First, our fidigs are 8

12 based o the average portfolio variaces for differet portfolio sizes. There is o guaratee that oe particular portfolio's risk is the same as the average portfolio risk with the same size. Hece, there are additioal sample risks i that your portfolio may ot be the same as the populatio average. Because of this additioal risk, Newbould ad Poo (993) argued that ivestors eed substatially more tha 0 stocks i a portfolio to elimiate diversifiable risk. Secod, eve though the power of aive diversificatio o reducig the percetage of diversifiable risk is idepedet either of the markets ivolved, samplig periods or ivestmet horizos, the actual amout elimiated does vary depedig o the ratio of the average covariace to the average stocks variace i differet markets. Obviously, trasactio costs also matter. The cotributio of this paper is i statig out the efficiecy of aive diversificatio, which is almost left out i may uiversity fiace/ivestmet textbooks, from a educatioal poit of view. 9

13 Refereces Amlig, Frederick, 989, Ivestmets: A Itroductio to Aalysis & Maagemet, 6th editio (Pretice Hall, Ic., Eglewood Cliffs, New Jersey). Arold, Gle, 998, Corporate Fiacial Maagemet (Pitma Publishig, Great Britai). Bodie, Zvi, Alex Kae, ad Ala J. Marcus, 999, Ivestmets, 4th editio (McGraw-Hill Compaies, Ic.) Brealey, Richard A., ad Stewart C. Myers, 996, Priciples of Corporate Fiace, 5th editio (The McGraw-Hill Compaies, Ic., New York). Elto, Edwi J., ad Marti J. Gruber, 977, Risk reductio ad portfolio size: A aalytic solutio, Joural of Busiess 50, Emery, Douglas R., ad Joh D. Fierty, 997, Corporate Fiacial Maagemet (Pretice Hall, Upper Saddle River, New Jersey). Emery, Gary W., 998, Corporate Fiace: Priciples ad Practice (Addiso Wesley Logma, Ic., New York). Evas, Joh L., ad Stephe H. Archer, 968, Diversificatio ad the reductio of dispersio: A empirical aalysis, Joural of Fiace 3, (5), Fabozzi, Frak J., 995, Ivestmet Maagemet (Pretice Hall, Ic., Eglewood Cliffs, New Jersey). Fracis, Jack C., 99, Ivestmets: Aalysis ad Maagemet, 5th editio (McGraw-Hill, Ic., New York). Gitma, Lawrece J., 000, Priciples of Maagerial Fiace, 9th editio (Addiso Wesley Logma, Ic.). Gitma, Lawrece J., ad Michael D. Joehk, 996, Fudametals of Ivestig, 6th editio (Harper Collis College Publishers, New York). Joes, Charles P., 996, Ivestmets: Aalysis ad Maagemet, 5th editio (Joh Wiley & Sos, Ic., Sigapore). Lee, Cheg F., Joseph E. Fierty, ad Edgar A. Norto, 997, Foudatios of Fiacial Maagemet (West Publishig Compay, Mieapolis, St. Paul). Lee, Cheg F., Joseph E. Fierty, ad Doald H. Wort, 990, Security Aalysis ad Portfolio Maagemet (Scott, Foresma ad Compay, Lodo, Eglad). Levy, Haim, 996, Itroductio to Ivestmets (South-Wester College Publishig, Ciciati, Ohio). Mayo, Herbert B., 993, Ivestmet: A Itroductio, 4th editio (Harcourt Brace College 0

14 Publishers, Orlado). Moyer, R. Charles, James R. McGuiga, ad William J. Kretlow, 998, Cotemporary Fiace Maagemet (South-Wester College Publishig, Ciciati, Ohio). Newbould, Gerald D., ad Percy S. Poo, 993, The miimum umber of stocks eeded for diversificatio, Fiacial Practice ad Educatio 3, (), Piches, George E., 996, Essetials of Fiacial Maagemet, 5th editio (Harper Collis College Publishers, New York, N. Y.). Rao, Ramesh K. S., 995, Fiacial Maagemet: Cocepts ad Applicatios, 3rd editio (South-Wester College Publishig, Ciciati, Ohio). Ross, Stephe A., Radolph W. Westerfield, ad Bradford D. Jorda, 000, Fudametals of Corporate Fiace, 5th editio (Irwi/McGraw-Hill, Bosto, Massachusetts). Sharpe, William F., Gordo J. Alexader, ad Jeffrey V. Bailey, 995, Ivestmets, 5th editio (Pretice Hall, Ic., Eglewood Cliffs, New Jersey). Statma, Meir, 987, How may stocks make a diversified portfolio?, Joural of Fiacial ad Quatitative Aalysis, (3), Tag, Gordo Y. N., ad Daiel F. S. Choi, 998, Impact of diversificatio o the distributio of stock returs: iteratioal evidece, Joural of Ecoomics ad Fiace, (-3), 9-7. Wager, W. H., ad S. C. Lau, 97, The effect of diversificatio o risk, Fiacial Aalysts Joural 7, (6),

15 Table Recommedatios from Textbooks o the Number of Stocks that ca Elimiate Most of the Portfolio's Diversifiable Risk Author(s) Page # of stocks Mai sources cited A. 0 Ivestmet Maagemet Textbooks. Amlig (989) p Evas & Archer. Bodie, Kae p.0 0 Elto & Gruber & Marcus (999) 3. Fabozzi (995) p.89 ~0 Wager & Lau 4. Fracis (99) p Evas & Archer 5. Gitma & Joehk p , ~40 - (996) 6. Joes (996) p.8 0-5, >30 Evas & Archer; Statma 7. Lee, Fierty & p.7 5 Evas & Archer Wort (990) 8. Levy (996) p Mayo (993) p Evas & Archer 0. Sharpe, Alexader p.5 ~30 - & Bailey (995) B. 0 Corporate Fiace Textbooks. Arold (998) p Brealey & Myers p (996) 3. Emery (998) p Statma 4. Emery & Fierty p (997) 5. Gitma (000) p Wager & Lau; Evas & Archer 6. Lee, Fierty p Evas & Archer & Norto (997) 7. Moyer, McGuiga & p Wager & Lau Kretlow (998) 8. Piches (996) p Rao (995) p Statma 0. Ross, Westerfield p Elto & Gruber; Statma & Jorda (000)

16 Table The Number of Stocks Required i a Portfolio to Elimiate a Certai Percetage of Diversifiable Risk give Differet Populatio Sizes Percetage of diversifiable risk to be elimiated, a 50% 75% 90% 93% 95% 97% 98% 99% Populatio Number of stocks required i the portfolio size, N give the above percetage, Note: The figures i the table are calculated based o the followig formula: [( - )/]/[(N - )/N] = a. Here, ( - )/ is the part of diversifiable risk of a portfolio with size where (N - )/N is the total diversifiable risk of the whole populatio of size N. The table shows that if oe wats to elimiate oly 50% of the diversifiable risk, populatio size does ot matter as you still eed stocks. However, if oe wats to elimiate more tha 95% of the diversifiable risk, the umber of stocks required varies greatly for differet populatio sizes. 3

17 Figure Diversificatio Beefit: Relative Risk vs Portfolio Size. Relative average variace Number of stocks i portfolio X = 0.75 X = 0.5 X = 0.5 This graph plots the relative risk (defied as the ratio of average portfolio variace divided by the average variace of all stocks i the populatio) agaist the portfolio size, give differet values (0.75, 0.5, ad 0.5) of the relative systematic risk, X (defied as the average covariace divided by the average variace of all stocks i the populatio). I all three cases, the curves are decreasig fuctios of, the portfolio size. The graph shows that whe X = 0.75, the curve levels off at a portfolio size of 0 while whe X = 0.5 (0.5), the curve levels off at a size of 5 (0). 4