Taxes can have a considerable impact on portfolio

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1 Diversificatio i the Presece of Taxes There are substatial risks icurred with cocetrated holdigs. David M. Stei, drew F. Siegel, Premkumar Narasimha, ad Charles E. ppeadu DVID M. STEIN is the chief ivestmet officer ad maagig director at Parametric Portfolio ssociates i Seattle (W 98104). NDREW F. SIEGEL is the Grat I. utterbaugh professor of fiace ad maagemet sciece at the Uiversity of Washigto i Seattle (W 9811). PREMKUMR NRSIMHN is the director of research ad advaced techology at Parametric Portfolio ssociates i Seattle (W 98104). CHRLES E. PPEDU is a assistat professor of fiace at Georgia State Uiversity i tlata (G 30303). IT IS ILLEGL TO REPRODUCE THIS RTICLE IN NY FORMT Taxes ca have a cosiderable impact o portfolio value, ad ivestmet decisios should be made with a clear uderstadig of the tax-adjusted performace of the alteratives uder cosideratio. The impact of taxes has received icreased attetio i recet years with regard to portfolio performace (Stei [1998]); asset allocatio (Jacob [1995]); maager selectio (Jeffrey ad rott [1993]); ad tax efficiecy (Dickso ad Shove [1993]). Oe commo ad key problem that taxable ivestors face is diversifyig a low cost basis sigle-asset or cocetrated portfolio. I the tax-exempt case, moder portfolio theory is very clear o the beefits of diversificatio, ad there have evolved useful idustry stadard methods for addressig this, such as the meavariace approach to optimal portfolio diversificatio (Markowitz [1987]). I the presece of taxes, however, there are o stadard approaches for arrivig at a cosidered choice. Taxes complicate the aalysis because capital gais taxes are icurred at the time of diversificatio. The tax resultig from the sale of a portio of the iitial asset reduces the possibility of future returs, ad may or may ot outweigh ay ucertai future beefit from diversificatio. I proposig a approach to solvig the taxable ivestor s diversificatio dilemma, we cosider a very much simplified problem i which there are just two possible assets: the iitial holdigs ad a diversified bechmark portfolio. Our framework cosiders the ivestmet FLL 000 THE JOURNL OF PORTFOLIO MNGEMENT 61 Istitutioal Ivestor, Ic. ll rights reserved.

2 EXHIIT 1 FTER-TX HORIZON LIQUIDTION VLUE INITIL $1 CONCENTRTED SECURITY WITH VOLTILITY 4 Frequecy decisio with iitial taxes, ad shows how it ca be viewed as a equivalet but much simpler ivestmet decisio without iitial taxes. We do this by creatig a tax-deferred ivestor (with differet ivestmet opportuities) who does ot pay iitial capital gais taxes, but whose fial ivestmet performace is idetical to that of the actual ivestor. 1 Idetifyig the best diversificatio decisio available to the tax-deferred ivestor leads to a decisio for the actual ivestor. The tax-deferred ivestor is i a sese a tax-deferred equivalet of the actual ivestor, facig a equivalet but simpler problem. We preset results to the problem uder specific umerical assumptios, ad ivestigate the sesitivity of the results to the key parameters, amely, the risk of the iitial holdig, the ivestmet horizo, the cost basis, excess expected retur, ad riskless rate. I a example, we cosider diversifyig a portfolio so as to reduce trackig error risk usig a mea-variace optimizer. appedix provides techical details. RISKS ND ENEFITS OF DIVERSIFICTION: N EXMPLE We assume iitially a ivestor with a iitial high-risk holdig who ows $1 millio cocetrated i a risky stock with a zero cost basis. We compare the distributios of ucertai ed-of-horizo future values without ad with diversificatio, ad cotrast these with the much simpler compariso (of yearly expected retur ad risk) that may be used to decide the level of diversificatio for a tax-deferred ivestor % 15% 1 5% Most Likely Value $0.5 (Mode) Expected Fial Value $ Liquidatio Value ($) Expected Fial Value $4.5 Stadard Deviatio $13.0 Max. Likelihood $0.5 Probability of $1 or less 43% Probability of $ or less 6 First cosider the iitial udiversified holdig. Let us assume that the risk is 4, measurig risk as the aual stadard deviatio (volatility) of rate of retur, ad that the ivestmet horizo is 0 years. Let us further assume that the stock returs a expected 1 per year, of which 7% is price appreciatio ad 3% divided yield. Each year, divideds are taxed at 39.6%, ad the after-tax divided proceeds are reivested i the portfolio. We also assume that the ivestor liquidates the holdigs ad icurs capital gais taxes of at the horizo. The ivestor s fial wealth is ucertai, ad Exhibit 1 shows its distributio obtaied from a Mote Carlo simulatio i which the security prices follow a logormal process. While the ivestor ca expect $1 millio to grow o average to $4.5 millio after taxes, the distributio of fial wealth is uattractively broad. The mode (most likely value) of the distributio is oly $0.5 millio; the probability of edig up with less tha the iitial $1 millio is 43%; ad the probability of ot keepig up with iflatio is 6. Next, cosider the cosequeces of a decisio to diversify. Suppose that the ivestor liquidates the risky holdig, payig taxes at the rate, ad ivests the remaiig $800,000 i a diversified portfolio with a aual stadard deviatio of 15%, but with the same expected price ad divided returs as the risky stock. fter 0 years, tax is paid at the rate (reduced by a cost basis of $800,000). Exhibit shows the fial wealth distributio. The ivestor ca expect to have oly $3.8 millio, o average, after 0 years. While this expectatio is lower tha the value i Exhibit 1 because oly 8 of the iitial DIVERSIFICTION IN THE PRESENCE OF TXES FLL 000 It Is Illegal To Reproduce This rticle I y Format. ---> reprits@iijourals.com for reprits or permissio. Istitutioal Ivestor, Ic. ll rights reserved.

3 EXHIIT HORIZON FTER-TX LIQUIDTION VLUE INITIL $0.80 DIVERSIFIED PORTFOLIO WITH VOLTILITY 15% Frequecy 3 5% 15% 1 5% EXHIIT 4 RISK OF SELECTED INVESTMENTS S&P % Exxo 16% GE 1% IM 3 Microsoft 35% Micro Techologies 7% OL 84% mazo.com 14% Risk is measured by aualized stadard deviatio. ased o Most Likely Value $.5 Expected Fial Value $3.8 Stadard Deviatio $.4 Expected Fial Value $3.8 Max. Likelihood $.5 Probability of $1 or less % Probability of $ or less 1% Liquidatio Value ($) EXHIIT 3 TRDE-OFF ETWEEN EXPECTED RETURN ND RISK IN SENCE OF TXES Expected Retur r f Maximum Sharpe ratio sset 1 Total Risk (std. dev.) sset value is available for compoudig, the probability distributio is oetheless more attractive. The mode icreases from $0.5 millio to $.5 millio; the chace of edig up with less tha the iitial $1 millio drops from 43% to just %; ad the chace of ot keepig up with iflatio falls from 6 to 1%. The graphs show that diversificatio boosts performace whe taxes are icurred eve though the asset expected returs are idetical. How ca this be? Diversificatio makes the expected retur more readily achievable. I Exhibit 1, with a high volatility, the expected retur is i the tail of the distributio ad is ot likely to be achieved; i Exhibit, with a lower volatility, the expected retur is more likely to be achieved. 3 Comparig Exhibits 1 ad, we ask whether the iitial tax is justified. How should the ivestor trade off risk (lesseed by diversificatio) agaist aticipated future wealth (also lesseed by iitial taxes)? I particular, how much of the iitial holdig should be diversified, ad by what priciple ca this be justified? Our approach provides a solutio that is both aalytical ad ituitive. The approach is based o the fact that the taxexempt diversificatio problem is simpler tha that faced by a ivestor who must pay iitial taxes i order to diversify. Exhibit 3 shows the well-kow tax-exempt tradeoff betwee risk ad retur. 4 The Sharpe ratio criterio is a commo oe for selectig a diversificatio level; the ivestor chooses the fractio of iitial asset (sset ) to sell (purchasig shares of sset 1) so that the resultig portfolio has the highest ratio of excess retur (above the risk-free rate r f ) to stadard deviatio of excess retur. FLL 000 THE JOURNL OF PORTFOLIO MNGEMENT 63 It Is Illegal To Reproduce This rticle I y Format. ---> reprits@iijourals.com for reprits or permissio. Istitutioal Ivestor, Ic. ll rights reserved.

4 I order to place the 4 stadard deviatio example of Exhibit 1 i cotext, Exhibit 4 shows recet stadard deviatio risks of some well-kow stocks. The 4 volatility of Exhibit 1, while higher tha that of may large-cap securities, is ot particularly high for a techology or small compay. Our base bechmark volatility of 15% is similar to the volatility of the S&P 500 idex over a recet five-year period. N PPROCH TO THE PROLEM 64 We ow specify a aalytical framework for the opportuities ad choices of the actual ivestor ad the tax-deferred ivestor, ad idicate how to match them so as to have very early the same future cash flows. This matchig allows us to view the diversificatio decisio i the presece of taxes (faced by the actual ivestor) as a covetioal risk-retur trade-off without iitial tax complicatios (faced by the tax-deferred ivestor). We use the maximum Sharpe ratio as the decisio criterio for makig the optimal diversificatio choice of the taxdeferred ivestor. If this is the best choice for the taxdeferred ivestor, ad the actual ivestor s future cash flows closely match, it follows that we have also idetified a optimal choice for the actual ivestor. I the simplified problem, the ivestor holds a iitial portfolio,, with market value W 0, which may be sold i whole or i part to purchase shares i a fully diversified bechmark portfolio,. The ivestor s goal is to select the best fractio x (betwee 0 ad 1) of the iitial portfolio to be sold, ad ivest the after-tax proceeds i. The resultig positio is the held for the preset ivestmet horizo of years, durig which time ucertai rates of retur for the iitial asset ad the bechmark are observed ad compouded. t the ed of the ivestmet horizo, the positio is liquidated ad taxes are paid. This defies a probability distributio of fial after-tax values for the actual ivestor to which we will match a taxdeferred ivestor. 5 Imagie, ow, a tax-deferred ivestor who ca diversify without iitially payig capital gais taxes, but who pays taxes o liquidatio ad whose after-tax horizo ivestmet performace matches that of the actual ivestor very closely. The tax-deferred ivestor must face differet ivestmet opportuities; i particular, the taxdeferred assets must pay a lower expected rate of retur to compesate for taxes paid by the actual ivestor. The tax-deferred ivestor holds a iitial portfolio (with market value W 0 ), sells a fractio x, ad (without payig iitial taxes or chagig the cost basis) uses the proceeds to purchase shares of the bechmark,. The resultig portfolio is held for years, ad the positio is the liquidated ad taxes are paid. This defies a probability distributio of fial after-tax values for the tax-deferred ivestor. 6 Iputs to the model are as follows. The expected rates of retur for ad are µ ad µ. Their aual stadard deviatios of retur are σ ad σ, ad the beta of with respect to the bechmark is β. The horizo is fixed at years, after which the ivestmet positio is liquidated. The tax rate o log-term capital gais is τ ; there are o divideds; ad the risk-free rate is r f.. The mathematical formulatio ad its solutio are i the appedix, where we derive the tax-deferred ivestor s rate of retur µ x ad stadard deviatio σ x aalytically i terms of our iputs for each value of the diversificatio fractio x uder the assumptio of joit logormal asset returs. To match the fial cash flow distributios of the actual ad tax-deferred ivestors, we choose x to match diversificatio exposure, ad set the joit performace of ad to compesate for taxes paid by the actual ivestor. THE DIVERSIFICTION SOLUTION: EXMPLE We provide a example to study the actual ivestor s diversificatio decisio, as chose by applyig the maximum Sharpe [1964] ratio criterio to the taxdeferred ivestor. We the use sesitivity aalysis to show that greater diversificatio is associated with: greater iitial asset volatility, loger ivestmet horizo, higher cost basis, lower expected retur of the iitial asset, ad a lower risk-free rate. Less diversificatio is eeded whe the ivestor receives a step-up i basis at the horizo. s our base case, we set umerical values for the iitial asset ad the bechmark as follows: Expected returs: µ µ 1 Volatilities: σ 5%, µ 15% Horizo: 0 years Tax rate: τ o capital gais Risk-free rate: r f 6% Iitial cost basis C 0 0 Exhibit 5 shows the after-tax aual expected retur ad risk trade-off faced by the tax-deferred ivestor i a represetatio aalogous to Exhibit 3. I this case, DIVERSIFICTION IN THE PRESENCE OF TXES FLL 000 It Is Illegal To Reproduce This rticle I y Format. ---> reprits@iijourals.com for reprits or permissio. Istitutioal Ivestor, Ic. ll rights reserved.

5 EXHIIT 5 CTUL INVESTOR S DIVERSIFICTION DECISION Expected Total Retur (tax-adjusted) 9% 8% 7% 6% Maximum Sharpe ratio: sell 86% of iitial portfolio Udiversified iitial portfolio Fully diversified : sell 10 of iitial portfolio 5% 1 15% 5% Total Risk (tax -adjusted) EXHIIT 6 SENSITIVITY TO INITIL STOCK VOLTILITY RISK-RETURN CURVES T DIFFERENT LEVELS OF VOLTILITY Mea Retur % 9. X 9% X 96% X 86% X 65% 8.5% 1 15% Total Risk Iitial volatility Iitial volatility 5% Iitial volatility 3 Iitial volatility 4 EXHIIT 6 SENSITIVITY TO INITIL STOCK VOLTILITY DIVERSIFICTION S FUNCTION OF VOLTILITY 10 Diversificatio Iitial Trackig Error the maximum Sharpe ratio criterio recommeds that 86% of the iitial holdig be sold. Greater diversificatio correspods to a lower expected rate of retur for the tax-deferred ivestor because it requires that higher taxes be paid iitially. The taxdeferred ivestor must ear a lower aual rate of retur i order to experiece the same ed-of-period performace as the actual ivestor. What ow is the sesitivity of this solutio to chages i the base umerical parameters? Of key importace is the risk of the iitial holdig, σ. Exhibit 6 shows the riskretur trade-off ad optimal diversificatio x for a rage of risk levels. The more risky the stock, the more it should be diversified. May securities, such as those with volatility of more tha 3, should be almost completely diversified. There is less eed to diversify low-volatility iitial assets. The horizo is also importat to the diversificatio decisio, as show i Exhibit 7. The loger the horizo, the more importat risk becomes, ad the more it should be diversified. For high-volatility iitial holdigs, the decisio is ot very sesitive to the horizo, ad we recommed diversifyig most of the asset. For low-volatility iitial holdigs, the horizo is more importat. While the cost basis C 0 is importat, too, its effect is straightforward. If the iitial asset has a cost basis higher tha zero, the diversificatio is cheaper. Thus, the diversificatio x icreases with the cost basis, as show i Exhibit 8. The relatioship betwee cost basis ad degree of diversificatio is close to liear. Whe the iitial asset has a higher expected retur, i.e., µ µ + α with α > 0, we would expect that the recommeded diversificatio x decrease with the excess retur α because a higher expected retur makes the iitial asset more valuable as compared to FLL 000 THE JOURNL OF PORTFOLIO MNGEMENT 65 It Is Illegal To Reproduce This rticle I y Format. ---> reprits@iijourals.com for reprits or permissio. Istitutioal Ivestor, Ic. ll rights reserved.

6 EXHIIT 7 SENSITIVITY TO HORIZON DIVERSIFICTION S FUNCTION OF HORIZON 66 Diversificatio Iitial volatility 5% Iitial volatility 3 Iitial volatility Horizo (years) EXHIIT 8 SENSITIVITY TO COST SIS DIVERSIFICTION S FUNCTION OF COST SIS Diversificatio 10 95% 9 85% 8 75% 7 65% Iitial volatility 3 Iitial volatility Iitial volatility 5% Cost asis (as a percetage of iitial value, C 0 ) EXHIIT 9 SENSITIVITY TO EXCESS RETURN DIVERSIFICTION S FUNCTION OF EXCESS RETURN Diversificatio Iitial volatility Iitial volatility 4 Iitial volatility 3 Iitial volatility 5% -6% -4% -% % 4% 6% ual Excess Retur, α, (% per year) the bechmark. Exhibit 9 shows this. For example, if σ 5% ad α > 3.5% per year for 0 years, o diversificatio would be recommeded. If σ 3 ad α % per year for 0 years, the model would recommed diversifyig about 7 of the holdig. The effect of a lower risk-free rate r f is to icrease the level of diversificatio, as is clear from the covexity of the curve i Exhibit 5. We have assumed liquidatio ad the paymet of capital gais taxes at the horizo. If the ivestor receives a stepup i basis at this time, diversificatio is more costly; the ivestor ca avoid payig capital gais taxes by retaiig the sigle stock, but is taxed for diversifyig. It is iterestig to compare the model s recommeded level of diversificatio i the step-up case with the liquidatio case. s expected, for ay give set of parameters our model always suggests less diversificatio i the step-up case. Differeces are most proouced for short horizos, as show i Exhibit 10. The ituitio is simple; if the horizo is short, the risk i holdig the stock is relatively low, while the tax impact of sellig it is relatively high. We fid that for horizos loger tha 0 years, differeces are ot very great, ad the risk of holdig the sigle stock quickly overwhelms the tax beefit of retetio. Thus, over log ivestmet horizos, it is particularly uwise to icur the risk of cocetratio, eve i the step-up case. RELTED PROLEM: REDUCING TRCKING ERROR related problem cocers the holdig of a low cost basis iitial portfolio that is oly partially diversified, ad how it tracks a specified bechmark. The suitable measure of risk i this case is trackig error, rather tha total stadard deviatio. 7 DIVERSIFICTION IN THE PRESENCE OF TXES FLL 000 It Is Illegal To Reproduce This rticle I y Format. ---> reprits@iijourals.com for reprits or permissio. Istitutioal Ivestor, Ic. ll rights reserved.

7 EXHIIT 10 SENSITIVITY TO HORIZON WITH ND WITHOUT COST SIS STEP-UP DIVERSIFICTION S FUNCTION OF HORIZON INITIL VOLTILITY σ 5% Diversificatio Without basis step-up With basis step-up Horizo (years) EXHIIT 11 EMPIRICL TRCKING ERROR VERSUS TX COST Tax Cost 8% 6% 4% % 1% % 3% 4% 5% 6% 7% 8% -% Trackig Error The ivestor reduces trackig error by sellig first the tax lots that realize few taxes but that also provide the best opportuity for diversificatio. I geeral, the tax cost icreases as the portfolio is squeezed dow to track the bechmark. Mea-variace optimizatio ca be used to miimize the tax cost ad to provide a tradeoff betwee tax cost ad trackig error. Exhibit 11 shows a empirical plot of the tax cost versus trackig error for a portfolio with a iitial trackig error of 6.8%. few iitial tax lots have urealized losses, ad these allow the trackig error of the portfolio to be reduced to 4.% before ay et taxes are realized. Thereafter, the tax cost icreases as trackig error decreases. I this example, which poit o the curve is best? Oce agai, by defiig a tax-deferred equivalet ivestor, we ca develop a solutio. I this case, the similar taxdeferred problem is that faced by a ivestor who is seekig a active portfolio maager, tradig off trackig error σ for excess retur α. Such a ivestor typically seeks either a iformatio ratio that is high eough (where the iformatio ratio is measured by the slope α/σ) or maximizatio of utility α λσ for some give λ, the ivestor s risk preferece. Our aalogous approach works for the tax maagemet decisio described here. The choice of λ is somewhat differet from that described, for example, by Griold ad Kah [1995] because whe we compare α values at differet diversificatio levels, the differeces are ot due to ucertai estimated portfolio performace, but istead come from kow iitial taxes paid. y reviewig the choices made by large umbers of ivestors, it would be possible to idetify values of λ that are implicitly revealed by their prefereces. SUMMRY ND CONCLUSIONS We have itroduced a framework for tradig off risk ad retur whe diversifyig low-basis taxable holdigs. I the case of a risky sigle asset, we aim to reduce the total risk (stadard deviatio) of the asset. I the case of a iitial portfolio that seeks to track a specified bechmark, we aim to reduce trackig error to a optimal level. I each case, we weigh the risk improvemet agaist its tax cost. Whe the iitial asset has substatially more risk tha the bechmark, our results recommed ear-complete diversificatio, despite a high iitial tax cost. If the iitial asset s total risk is ot much higher tha that of the bechmark, the approach recommeds less diversificatio, because the beefits do ot cover the margial tax cost. Sesitivity aalysis reveals that greater diversificatio is eeded: with greater iitial asset volatility, with loger ivestmet horizo, with a lower expected retur of the iitial asset, with a higher cost basis, ad with a lower risk-free rate. Less diversificatio is eeded whe the ivestor receives a step-up i basis at the horizo. Our approach has bee to formulate a particularly simple decisio problem. We have cosidered a sigle fixed-horizo ivestmet, with oly two possible extreme choices for portfolio formatio. The formula- FLL 000 THE JOURNL OF PORTFOLIO MNGEMENT 67 It Is Illegal To Reproduce This rticle I y Format. ---> reprits@iijourals.com for reprits or permissio. Istitutioal Ivestor, Ic. ll rights reserved.

8 tio ca be geeralized i may pragmatically useful directios. Oe could: Iclude divided yields, which affects the aalysis because of the high rate of divided taxatio. Cosider how a ucertai horizo affects decisiomakig. Ivestors with large low-basis cocetrated holdigs are ofte reluctat to embrace our model s high diversificatio recommedatios. For such ivestors, other pragmatic extesios are iterestig. They might: Seek to compromise by stagig the diversificatio over time; exploit tax-maaged methods as i Stei ad Narasimha [1999] i maagig the diversified slice to reduce the tax burde. Istead of ivestig i a diversified bechmark idex, ivest the liquidated asset i a portfolio that will complete the remaiig udiversified holdigs. That is, seek a portfolio that will have low (or ideally egative) correlatio with the iitial holdigs. I practice, ivestors may also be able to obtai additioal flexibility with derivative securities, exchage fuds, or other ivestmet vehicles. While we have focused o a particular ad simplified aalytical problem, our solutio method ca be quite geerally applied to other portfolio decisios i the presece of taxes. I essece, our method traslates a taxable problem ito a tax-deferred equivalet problem based o aual mea ad stadard deviatio. Istead of maximizig the Sharpe ratio or trackig error utility, oe could choose the portfolio with maximum tax-deferred growth rate (please see edote 3 for a discussio o growth rate), or use ay other criterio for portfolio choice based o yearly mea ad stadard deviatio. Oe ca exted the cocept of a matched tax-deferred ivestor to provide aalytic ituitive solutios to a wide rage of more complex situatios. 68 PPENDIX TECHNICL DETILS We outlie here the assumptios we use i defiig the taxdeferred ivestor, ad we the idetify the future cash flows for both the taxable ad the tax-deferred ivestor. Usig these, we derive closed-form expressios for the tax-adjusted yearly expected rate of retur ad stadard deviatio of retur. Fially, we show how these expressios must be modified for the case i which the ivestor receives a step-up i basis at maturity. We assume that the actual ivestor iitially holds a portfolio with iitial market value W 0 ad iitial cost basis C 0 W 0 so that C 0 represets the iitial cost basis as a fractio betwee 0 ad 1. This portfolio is assumed to grow at a radom realized rate of retur i 1 i year i, so that the total horizo before-tax rate of retur over years is Π i1 i 1. The ivestor is cosiderig sellig a fractio x (betwee 0 ad 1) of the iitial portfolio ad payig taxes of τxw 0 (1 C x ) at rate τ o the proceeds xw 0 less the cost basis xc x W 0 o shares sold. (Note that this formulatio allows high cost basis shares to be chose for sale.) The after-tax proceeds xw 0 [1 τ(1 C x )] are used to purchase shares of a bechmark portfolio with radom realized rate of retur i 1 i year i. The resultig partially diversified portfolio ow has cost basis (C 0 xc x )W 0 i the iitial assets ad full cost basis xw 0 [1 τ(1 C x )] i the ewly purchased bechmark - portfolio. Whe the portfolio is liquidated after years, the ivestor receives the compouded amout (1 x)w0 i 1i + xw0[ 1 τ( 1 Cx) ] i 1i ad pays tax of [ ] + ( ) (-1) τw 0 (1 x) i 1i C0 xcx τxw0 1 τ 1 Cx i 1i 1 (-) resultig i a total after-tax compouded horizo rate of retur equal to r x (1 t)(1 x) i 1i + x(1 t) [ 1 t( 1 Cx )] i 1 + tc + xt(1 t)(1 C ) 1 i 0 x [ ]( ) (-3) I order to replicate the ivestmet performace of this actual ivestor, we seek to costruct a tax-deferred ivestor (who does ot pay tax iitially, but whose after-tax ed-of-horizo ivestmet performace is idetical to that of the actual ivestor) for each choice of x. Such a tax-deferred ivestor iitially holds a portfolio with the same iitial market value W 0 ad iitial cost basis C 0 W 0 as the actual ivestor, but that returs i 1 i year i. DIVERSIFICTION IN THE PRESENCE OF TXES FLL 000 It Is Illegal To Reproduce This rticle I y Format. ---> reprits@iijourals.com for reprits or permissio. Istitutioal Ivestor, Ic. ll rights reserved.

9 The tax-deferred ivestor will sell a fractio x of the iitial portfolio give by (-4) where x is chose so that the risk-retur exposures of the actual ad tax-deferred ivestors are equal (that is, x is set equal to the ratio, for the actual ivestor, of the after-tax dollar amout i divided by total postdiversificatio portfolio value). The choice of x adjusts for the fact that the tax-deferred ivestor ca ivest the etire proceeds, x W 0, i the tax-deferred bechmark, which returs i 1 i year i. Note that the probability distributio of (, ) may deped o x. The tax-deferred ivestor retais the iitial cost basis of C 0 W 0. Whe the tax-deferred portfolio is liquidated after years, the tax-deferred ivestor receives the compouded amout ad pays tax at the ed of the time horizo i the amout of (-5) (-6) resultig i a total after-tax compouded horizo rate of retur equal to x i 1 i i 1 i r (1 τ)(1 x ) + (1 τ)x + τc 1 (-7) I the fully diversified case (x x 1), we have total rates of retur r ad r for the actual ad tax-deferred ivestors: r (1 τ) [ 1 τ( 1 C0) ] i 1 i + τ(1 τ) + τ C 1 0 [ ( x) ] ( x) x x1 τ 1 C / 1 τx1 C [ ] 0 i 1 i i 1 i W (1 x ) + x 0 [ i 1 i i 1 i 0 ] τ W (1 x ) Α + x Β C i 1 i r (1 τ) + τc 1 0 [ ] (-8) Ucertaity is specified as follows. The distributio of ( i, i ) is joit logormal, idepedet for differet years, with meas (µ, µ ), stadard deviatios (σ, σ ), ad istataeous beta β. Similarly, the distributio of ( i, i ) is joit logormal, idepedet for differet years, with meas (µ, µ ), stadard deviatios (σ, σ ), ad istataeous beta β (which may differ from β due to iitial taxes). To fid the joit distributio (specified by µ, µ, σ, σ, ad β ) for the tax-deferred ivestor, momet coditios are imposed i order to make the joit distributio of after-tax compouded horizo rates of retur (of the partially diversified portfolio ad the bechmark) early idetical for the actual ad the 0 tax-deferred ivestors at this particular value for x. That is, the joit probability distributio of (r x, r ) is closely matched to that of (r x, r ) usig the five momet coditios: E(r x ) E(r x ) E(r ) E(r ) σ rx σ r x σ r σ r Cov(r x, r ) Cov(r x, r ) (-9) (-10) (-11) The yearly expected rates of retur for the tax-deferred ivestor ca the be show to be give by µ (-1) (-13) Defie ν 1 + µ, θ σ + ν, ad similarly for,, ad to reduce the complexity of the equatios to follow, ad also defie δ ν ν b (θ /ν )β ad similarly δ ν ν b (θ /ν )β. The yearly stadard deviatios ad systematic risk β for the taxdeferred ivestor are give by θ σ θ 1 + µ δ {[ ] + } ( 0) + ( 0 ) µ 1 τ 1 C 1 µ τ 1 C 1 τ θ ττ ν τ [ 1 ( 1 C0) ] τ τc0 τ ν τc0 1 1 τ { ( + ( ) ( + ) + ( x) τ τ τx τ τ x x τ + ( ) 1 C 0 C 1 C + 0 τ C τ C + τ + 0 x ν x τ x ν τ { ( 1 ) + [ 1 ( 1 C )] } 1 + x 1 τ 1 C 1 τ 1 C θ ( 1 x) 1 τ 1 C δ [ ] [ ] + ( 1 x) 1+ µ x 1 τ 1 Cx 1 µ x 1 µ τx 1 C 1 x τc0 τc 1 x ν 1 x ν x θ 1 τ 1 τ [ ] x [ ( 0) ] + ( 0) [ + ( + ) ] [ ] 0 FLL 000 THE JOURNL OF PORTFOLIO MNGEMENT 69 )} ν 1 (-14) (-15) (-16) It Is Illegal To Reproduce This rticle I y Format. ---> reprits@iijourals.com for reprits or permissio. Istitutioal Ivestor, Ic. ll rights reserved.

10 β θ δ l ( 1+ µ ) 1+ µ σ µ l 1 + µ { ( 1 τc0 τx 1 C x 1 τ ( 1 x ) + ( ) τc0 τx 1 Cx 1 x ν x 1 τ 1 Cx ν 1 τ + ( ) + + ( ) + ( 1 x) θ + x 1 τ 1 C θ + x( 1 x) 1 τ 1 C δ σ θ 1 + µ (-17) (-18) (-19) The tax-adjusted yearly expected rate of retur ad stadard deviatio may ow be computed usig the values derived above: [ 1 + 1] µ + x µ µ x E 1 x x 1 x ( ) [ ] x + [ 1 + 1] + + ( ) ( ) + σx Var 1 x x 1 x θ x θ x 1 x δ 1 x ν x ν (-0) (-1) If the ivestor receives a step-up i basis at maturity, the the partially diversified ivestor keeps the compouded amout from Equatio (-1) without payig the tax of Equatio (-), resultig i a total after-tax compouded horizo rate of retur equal to rx ( 1 x) i 1i + x[ 1 τ( 1 Cx) ] i 1i 1 (-) i place of Equatio (-3). We keep the defiitio of x from Equatio (-4) uchaged. For the stepped-up tax-deferred ivestor, i place of Equatio (-7) we fid a total after-tax compouded horizo rate of retur equal to rx ( 1 x ) i 1 i + x i 1 i 1 70 [ ] [ ( x) ] 1 τc0 τc0 1 ν ν θ 1 δ 1 τ 1 τ + x x ( x ) x x [ ] )}/ (-3) I the fully diversified case (x x 1), we have total rates of retur r ad r for the stepped-up actual ad tax-deferred ivestors [i place of Equatio (-8)]: (-4) We use the same forms for the joit distributios of ( i, i ) ad ( i, i ) as before, ad use the same five momet coditios [Equatios (-9) through (-11)]. The yearly expected rates of retur for the stepped-up tax-deferred ivestor ca the be show to be give [i place of Equatios (-1) ad (-13)] by µ 1+ µ 1 τ 1 C 1 µ (-5) (-6) Usig defiitios of ν, θ, ad δ as before, the yearly stadard deviatios for the stepped-up tax-deferred ivestor are the give [i place of Equatios (-14) through (-19)] by θ θ 1 τ 1 C0 σ θ 1 + µ β r [ τ( C0) ] i 1 i r i 1 i 1 ( ) + ( ) + ( 1 x) 1+ µ x 1 τ 1 Cx 1 µ x 1 µ 1 x [ ] [ ] ( ) δ l ( 1+ µ ) 1+ µ σ µ l 1 + µ [ ( 0) ] [ ] + [ ( )] ( ) [ ] + ( ) (-7) (-8) x 1 τ 1 C τ θ θ τ δ x 1 1 C0 x ( 1 x) 1 1 C 0 δ 1 x (-9) 1 (-30) DIVERSIFICTION IN THE PRESENCE OF TXES FLL 000 It Is Illegal To Reproduce This rticle I y Format. ---> reprits@iijourals.com for reprits or permissio. Istitutioal Ivestor, Ic. ll rights reserved.

11 θ θ τ θ C θ τ C δ δ ( 1 x) + x 1 ( 1 ) x x x( x) x x x (-31) 1 x σ θ 1 + µ (-3) With these modificatios, the tax-adjusted yearly expected rate of retur ad stadard deviatio i the case of stepped-up basis may ow be computed as before, usig Equatios (-0) ad (-1). ENDNOTES [ ] + t the time of this writig, Charles ppeadu was a portfolio maager at Parametric Portfolio ssociates. 1 We use the term ivestmet performace to refer to the aftertax ed-of-horizo cash flow probability distributio. t a iflatio rate of about 3.5% for 0 years, the iitial $1 millio value doubles to $ millio i 0 years. 3 This pheomeo is due to the fact that the log-term growth rate (see, e.g., Ferholz ad Shay [198]) is less tha the yearly expected rate of retur due to a risk pealty (equal to half the variace i the case of a logormal distributio). Some ituitio ito this paradox is provided by the simple example of gaiig or losig with probability oe-i-two. The expected rate of retur is zero, eve over the log ru. You would have to be very lucky ot to lose moey over the log ru, however, because the particular sequece gai, the lose reduces wealth by 4% [computed as (1 + 0.)(1 0.) 1] or about % each time the example is played (whe there are exactly equal umbers of ups ad dows). This % reductio is ideed half the variace sice 0. / %. Curiously, while the compouded expected rate of retur is equal to the expected compouded rate of retur, over the log ru this rate becomes early impossible to attai due to the risk pealty. 4 ruel [1998] emphasizes that taxable ivestors should use cautio whe usig traditioal efficiet frotier tools directly. 5 For simplicity, we assume o trasactio costs. This assumptio is reasoable whe trasactio costs are small compared to tax costs. 6 The aalysis may also be geeralized to the case of a ivestor who does ot liquidate, but who receives a step-up i cost basis at death. 7 Trackig error is the stadard deviatio of the aual differece betwee the retur of the portfolio ad that of the bechmark. [ ] REFERENCES ruel, J.L.P. Why Taxable Ivestors Should e Cautious Whe Usig Traditioal Frotier Tools. The Joural of Private Portfolio Maagemet, Witer 1998, pp Dickso, J.M., ad J.. Shove. Rakig Mutual Fuds o fter-tax asis. Ceter for Ecoomic Policy Research, Publicatio No. 344, Staford Uiversity, Ferholz, R., ad. Shay. Stochastic Portfolio Theory ad Stock Market Equilibrium. The Joural of Fiace, May 198, pp Griold, R.C., ad R.N. Kah. ctive Portfolio Maagemet Quatitative Theory ad pplicatios. Homewood, IL: Irwi, Jacob, N.L. Taxes, Ivestmet Strategy, ad Diversifyig Low asis Stock. Trusts ad Estates, May Jeffrey, R.H., ad R. rott. Is Your lpha ig Eough to Cover its Taxes? The Joural of Portfolio Maagemet, Sprig Markowitz, H.M. Mea Variace alysis i Portfolio Choice ad Capital Markets. Oxford: asil lackwell, Sharpe, W.F. Capital sset Prices: Theory of Market Equilibrium uder Coditios of Risk. The Joural of Fiace, September 1964, pp Stei, D.M. Measurig ad Evaluatig Portfolio Performace fter Taxes. The Joural of Portfolio Maagemet, Witer Stei, D.M., ad P. Narasimha. Of Passive ad ctive Equity Portfolios i the Presece of Taxes. The Joural of Private Portfolio Maagemet, Fall 1999, pp FLL 000 THE JOURNL OF PORTFOLIO MNGEMENT 71 It Is Illegal To Reproduce This rticle I y Format. ---> reprits@iijourals.com for reprits or permissio. Istitutioal Ivestor, Ic. ll rights reserved.