ndex Mutual Funds andExchange-Traded FundsA comparison of two methods of passive investment.

Leonard Kostovetsky

80

LEONARD KOSTOVETSKY is a

doctoral student in economicsat Princeton Univenicy inPrinceton (NJ 08540).lkostove@princeton.edu

MUTUAL FUNDS AND EXCHANGE-TRADED FUNDS

Exchange-traded funds (ETFs), once a phe-nomenon, have emerged as a viable alternativetor investors seeking to tie their holdings to amajor market index. By the end of 2000, the

market for ETFs totaled over $75 billion, up 82% fix)mthe previous year, this in a climate of less than stellar stockreturns. Just one ETF, the S&P Depository Receipts500, has assets of over $28 billion. While ETFs still rep-resent only a small slice of the $1.5 trillion index fundpie, their growth in popularity among retail and large-scale investors prompts more research on their advantagesand disadvantages.'

One subject not adequately understood is theexplicit and implicit costs incurred by ETFs and how thesecompare to the costs of index mutual fiands. I develop asimple one-period model that is useftil in examining themajor differences between ETFs and index funds,depending on investor trading preferences, tax implica-tions, and other characteristics. Then I expand the one-period model to multiple periods, and also review somequalitative differences between ETFs and index funds thatcannot be incorporated into this model.

PRIOR RESEARCH

Mutual fund performance has certainly not beenignored in the economic literature. Ever since mutualfunds emerged in the early 1960s, the question of theirperformance and fund manager selectivity skills has inter-ested economists. Sharpe [1966], Treynor [1966], and

SUMMER 2003

Jensen [1968] conclude that mutual fund performance, netof expenses and after risk adjustment, is poorer than whatinvestors could achieve using a naive buy-and-hold strat-egy. While authors like Chen and Stockum [1986] and Leeand Rahman [1990] find that a limited number of fundmanagers have the selectivity and market-timing skillsrequired to beat the market, analysis by Malkiel [1995| andBogle [1998] has shown that without prior knowledge ofttiese few superior fund managers, investors would dobest to stay in index funds. As Bogle writes [1998, p. 38].an investor would be "a bit of a fool" not to seriously con-sider limiting fund selection to low-expense funds. Themost recent study by Frino and Gallagher [2001] onceagain concludes that in the past five years, S&P 500 indexmutual funds earned a better risk-adjusted, expense-adjusted return than actively managed funds.

Of course, it would be wrong to say that views onindex fund superiority are unanimous. Minor [2001]notes that, depending on the time horizon of data, it ispossible to find periods when active funds outperformedtheir index fund cousins.

Whichever view one favors, the keys to comparingactive funds and index fiands are the costs of activity:turnover costs, expense ratios, and transaction costs. This1% to 2% per year can often make the difference betweenbeating the market or falling behind it. As a result, researchon transaction costs has been substantial.

Ferris and Chance [ 1987] find that 12b-1 charges (feescharged by mutual funds to pay for sales and advertising)are a deadweight loss borne by the shareholders. Grinblattand Titman [1994] conclude that there is no correlationbetween loads and performance; i.e., there is no addi-tional return premium for buying funds with higher costs.Finally, Dellva and Olson find that "in general, investorsshould not select funds with front-end loads, 12b-l fees,deferred sales charges, and redemption fees, but they shouldnot expect that the avoidance of these fees will coincidewith superior risk-adjusted return" [1998, p. 101]. On bal-ance, the research suggests that avoidance of extra feesremoves deadweight losses, and thus improves returns.

Another area of research deals specifically with thecosts of index funds and exchange-traded funds. Whileall the research cited suggests that active fund managersgenerally do not have superior selectivity skills, but insteadincur extra costs that penalize fund shareholders, analystshave not examined the problems inherent in indexedinvestments. As Frino and Gallagher point out, "Despitethe significant attention to active funds in the perfor-mance evaluation literature, empirical research evaluating

index funds is surprisingly scarce" [2001, p. 45].Frino and Gallagher discuss the main problem of

tracking error. The main factors driving index fundtracking error are transaction costs, fund cash flows, div-idends, benchmark volatility, corporate activity, and indexcomposition changes.- These factors prevent index fundsfrom perfectly replicating the performance of the under-lying index.

One of the most surprising findings in Frino andGallagher [2001 ] is that the extent of tracking errors is sea-sonal in nature. This suggests that some seasonal effectslike December tax-loss selling and quarterly dividend dis-tributions have particularly strong effects on index fundtracking error.

Since the appearance of ETFs in early 1999, muchhas been written about them in the popular businessjournals.-' Barron's, BiisiuessWeek, Money, and Forbes haveall praised ETFs for their efficiency and versatility.Gastineau [2002], one of the developers of ETFs at theAmerican Stock Exchange, outlines their history andmechanics.

The only academic article I am aware of is Dellva[2001], who compares ETFs with index funds, and con-cludes that ETFs are not attractive for small investorsbecause of brokerage commission costs. Because Dellva[2001] does not attempt to quantitatively model the dif-ferences in costs, I focus my attention on that issue.

WORKINGS OF INDEX FUNDS AND ETFs

The goal of index fijnds and ETFs is essentially thesame: to provide investors with a way to own a well-diver-sified indexed portfolio by using economies of scale to buylarge quantities of stock at low cost. They accomplish thisgoal in two very different ways.

Index funds work exactly like other active mutualfunds. They accept cash deposits fiiam outside investors,and in return issue shares of the net asset value (NAV) ofthe fund. Then, they use these deposits to purchase sharesof stock in firms in the index or to pay back investors whoare redeeming shares. Clearly, for most investors, this isfar superior to paying huge transaction costs for buying30 or 500 or even 5,000 different stocks in the index thatthey want to track.

As the Vanguard 500 Fund Prospectus points out,however:

An index fiind does not always perform exactly likeits target index. Like aU mutual funds, index funds

SUMMER 20(13 THE JOURNAL OF PORTFCM,IO MANAGEMENT 8 1

have operating expenses and transactions costs.Market indexes do not, and therefore will usuallyhave a slight perfonnance advantage over fiinds thattrack them."'

It is important to enumerate these operating expensesand transaction costs that make index funds imperfectinstruments for tracking indexes.

Index Funds

Bid-ask spreads and other liquidity costs are the pri-mary source of tracking error for index fund managers.For example, when there is a large inflow of funds, man-agers must invest these funds, paying fees {in the form ofbid-ask spreads) to market makers. Likewise, when thereare redemptions that cannot be met with the cash avail-able on hand, fund managers have to sell stocks and onceagain incur costs. Very often, some constituent stocks ofan index are illiquid, forcing managers to suffer high coststo trade in them.

The movement of cash in and out of index funds isa secondary cause of tracking error. An effect known ascash dra^ arises because index fund managers have to keepa certain percentage of assets uninvested to meet redemp-tion needs. Furthermore, because it's impossible to imme-diately invest all incoming funds, there is a short periodwhen inflows remain in cash. Futures are often used toalleviate cash drag, but if futures aren't used or are unavail-able, cash drag could become a significant source of track-ing error.

Critics may argue that this effect is insignificantcompared to the large price movements that occur in thestock market every day. Yet competition in the index-tracking industry is so intense that every basis point indeviation fi-om the target index can be significant.

A third factor causing tracking error Hes in dividendpolicies. Some paper indexes assume an immediate rein-vestment on the ex-dividend date, but because indexflinds must wait a certain time to receive these cash div-idends, there is often a short lag that contributes to track-ing error. This effect has diminished in previous years, asdividend yields have fallen to their lowest levels in manydecades. Yet, for some indexes full of high-dividendstocks, the effect is not negligible, and must be includedas a component of tracking error.

Research has suggested that in-and-out trading canbe a sizable cost drag for long-term mutual fijnd share-holders. Since most mutual fiinds allow trading until 4:00

PM and calculate NAVs as of that time, it is often possi-ble for arbitrageurs to time their trades to take advantageof momentum and stale prices.

Zitzewitz [2002J estimates it is possible for these arbi-trageurs to earn excess returns of 40% to 70% in inter-national funds at the expense of other shareholders. Edelen[ 1999] relates in-and-out trading to liquidity, showing thatthe indirect costs of providing liquidity to investors in anasymmetrically informed market can have a significantnegative impact on mutual fund returns. Although thisproblem isn't as important for domestic index funds, andis not relevant at all for the Vanguard index funds, it canstill be a meaningful influence on index fund trackingerror.

Finally, the last important factor contributing totracking error is rebalancing costs due to index changesor corporate activity. If a company leaves an index becauseit merges with a different firm, for example, timing mis-matches can occur between the time the company leavesthe index and when the index fund is able to seU all itsshares and buy the shares of the company replacing it. Ifcorporate activity such as a spin-off drastically changes themarket value of a firm, the index fund must suffer trans-action costs in rebalancing its portfolio.

Exchange-Traded Funds

An exchange-traded fund works in a completelydifferent way. Unlike an index fiand, an ETF does not needto pay to obtain shares of constituent stocks, operatinginstead through a process known as creation/redemptionin-kind. This means that large investors can purchase a siz-able number of shares of ETFs only by supplying a stockportfolio that matches the target index in weights andthat has the same value as those shares. For example, theSPDR ETF that matches the S&P 500 can be created onlyin 50,000 share chunks {and redemptions work in thesame way).

The advantage for the ETF is that it gets constituentshares without liquidity costs. The advantage for the largeinvestor is that one can obtain a large number of ETFshares without moving its price in the secondary market.

Creations/redemptions in-kind are also importantbecause they provide arbitrage opportunities that preventthe ETF price from diverging too much fiom the net assetvalue of the constituent shares. If there is a substantial devi-ation, arbitrageurs will step in and create or redeem shares,bringing the market back to equilibrium. Most smallinvestors, however, are unable to meet the size require-

8 2 INDEX MUTUAL FUNDS AND EXCHANGE-TRADED FUNDS SUMMER 2003

ments for creations and redemptions in kind, and mustconduct all transactions in the secondary market.

Fund transaction costs are nearly non-existent becauseof creation/redemption in kind, although there is some cashdrag, far smaller than the 2% estimated in index funds.Because the prices of ETFs and constituent stocks changenearly every second, any difference between the value ofthe round nuniber of shares of the ETF {e.g., 50,000) andthe value of the supplied portfolio must be equalized witha cash component. The cash-balancing amount can bepositive or negative, and it is this uninvested componentthat can contribute to the tracking error of ETFs.

The problems arising fi"om ETF dividend policyare similar to those for index funds. They face the samecosts and timing mismatches as index funds when a con-stituent firm is replaced in an index or when corporateactivity such as a secondary public offering changes themarket cap of a stock and increases its weight in theindex. These three sources of tracking error, althoughminor in comparison to market movements, are impos-sible to avoid in whatever form of index tracking aninvestor chooses.

Non-Tracking Error Differences

Now, let s assume that ETFs and index flinds are ableto perfectly replicate the performance of the market. Aninvestor would still have an important choice to makebecause of three non-tracking error differences betweenETFs and index funds: management fees, shareholdertransaction costs, and taxation costs. While tracking errorsources are nearly impossible to quantify, it is fairly sim-ple to model the effect of these three non-tracking errorsources on investor returns.

Management fees are an inescapable cost of indirectinvestment in the stock market. For active mutual funds,the expense ratio, which measures management fees as apercentage of total managed assets, can be as high as 2%,but for index funds, expense ratios are usually below0.5% per year. Exchange-traded funds have been able tooffer even lower expense ratios than the cheapest of indexfunds.

For example, the SPDR ETF has an expense ratioof 0.12% while the Vanguard 500 Fund has an expenseratio of 0.18%. The Barclays iShares500 ETF, which tracksthe S&P 500, has an even lower expense ratio of only0.09%.

The main reason ETFs are able to offer lowerexpense ratios is that they are not in charge of shareholder

accounting. The task of keeping track of shareholdertransactions and other such paperwork is a large percent-age of the expense ratio; for ETFs, these tasks are per-formed by the brokerage house of the shareholder.Gastineau [2001] suggests that the elimination of share-holder accounting can save ETFs anywhere from 5 to 35basis points in expense costs.

Shareholder transaction cost is another factor that isdifferent for ETFs and index fiinds. No-load index fundsdo not charge commissions on transactions, and sincethe vast majority of index funds are no-load, an investorcan easily fmd an index fund that does not charge a load.

ETFs, on the other hand, have to be purchased onthe secondary-' market (except for large investors who canperform creations/redemptions in-kind) where theinvestor has to pay a commission to the brokerage houseand a fee to the market makers through the mechanismof the bid-ask spread. Brokerage house commissions canbe as high as 2% for full-service brokerages like MerrillLynch, although competition among discount brokersand on-line brokers has cut commissions dramatically. Itis possible to make transactions now for extremely low flatrate commissions.

Bid-ask spreads on ETFs are the other componentof transaction costs for shareholders. As of now, the largestETFs (SPDRs and QQQs) are so liquid that bid-askspreads are estimated to be below 2 cents per share.Smaller ETFs are much less liquid, and experts believe thatin the future they may suffer even worse liquidity {andhigher bid-ask spreads) as volume shifts to the more pop-ular ETFs.

The last factor that distinguishes ETFs and indexflmds is their tax efficiency. When redemptions exceedadditions, the index fund manager is forced to sell stocksand distribute capital gains to shareholders. These capitalgains are immediately taxed and create substantial costs forthe shareholders. An ETF, on the other hand, rarely if everdistributes capital gains.

Because of creation/redemption in-kind, ETFsalways give away the stock with the lowest basis (the onethat appreciated the most and has the most capital gainstaxes to be paid), and keep the stock with the highest basis.When they need to sell stocks in order to rebalance, theycan sell that stock and not incur capital gains because ithas a high basis. Of course. Congress may some timechange the law to close this loophole, but until then taxefficiency favors the ETF, and taxable investors shouldn'tignore this advantage.

SUMMER 2003 THEJOLJRNAL OF PORTFOLEO MANAGEMENT 8 3

E X H I B I T 1Summary of Cost Comparisons

l^pes of Costs

Fund Transaction Costs onPurchases and Sales by the FundCash Inflows and Outflows

Dividend Policy

In-and-Out Arbitrage Trading

Index Fund ChangesCorporate ActivityManagement Fees

Shareholder Transaction Costs

Taxation Costs

Exchange-lVaded FundsFund Costs

None. All creations and redemptionsare In-kindDeviations in value of creations andredemption in-kind are paid in cashLag between ex-dividend date andreceipt of dividendsNone. Arbitrage eliminated bycreation/redemption in-kindETFs must incur costs to rebalanceETFs must incur costs to rebalanceETFs have very low expense ratiosbecause all accounting is done ai theshareholder level

Shareholder Costs

Brokerage transaction fees + bid-askspreads on ETFsCapital gains are distributed veryrarely (almost never)

IVaditional Index Funds

Bid-Ask spreads (as fees to marketmakers, etc.)Cash drag. Small percentage (~2%)of assets is uninvested.Lag between ex-dividend date andreceipt of dividendsCan be important for some domesticindex funds. None at Vanguard.Similar costs to rebalanceSimilar costs to rebalanceIndex funds have slightly higherexpense ratios because shareholderaccounting is done at the fund level

None, except for index funds withloads, which is rareSignificant share of capital gains getsdistributed especially in bull markets

Cost Comparisons

Exhibit 1 provides a summary of ETF and index fundcosts. There are important differences on many levels.

ONE-PERIOD MODEL

Although the differences in the costs of ETFs andindex fiands are small, they are still very important toanalyze. For actively managed funds, itwestors can alwaysmake the argument that they are paying a higher cost fora better manager: for the passively managed funds that weare looking at, though, costs are the only factor in decid-ing which instrument to pick. Thus, we see funds com-peting by reducing their expense ratios only a few basispoints and substantially increasing their market share.

Take an investor who wants to invest an amount /in an index tracking asset for a period of length f. Eitherbecause he wants to use dollar-cost averaging or becausehe receives this sum in installmetits over the entire period(e.g., he is investing a part of his monthly salary over anentire year), he makes N purchases at prices P^, ..., P ..(P. is the price of the fund at each transaction.) He alsopays a flat rate C in cotnmissions for each purchase.

At the end of period (, a share a of his capital gainsis distributed. We assume he earned k in capital gains, ofwhich CCk is distributed and a percentage {cckjXi^, is paid intaxes to the government before reinvesting. Also, at the

end of period t, a part of his initial investment is distributedin dividends d and he must pay the percentage dtj in taxesto the governmetit before reinvesting. Finally, what's leftover is charged an expense ratio e. Note that d and k arenot dollar values but ratios of the total post-commissioninvesttnent ( / - CN), while C is a constant that is inde-pendent of the initial investment /.

Using this information, we can develop a formulafor the final value of the investment. The capital gainsearned are directly related to the price distribution by:

N (1)

The value of the investment at time t before divi-dend and capital gains taxes are paid is:

(2)

The value of the capital gains taxes that have to bepaid is:

a{{I-CN)k}t, . (3)

84 INDEX MUTUAL FUNDS AND EXCHANGE-TRADED FUNDS SUMMER 2003

The value of the dividend taxes that have to bepaid is:

(4)

The value before expenses is simply Expression (2) -Expression (3) - Expression (4):

- {{I ' (5)

We take out the / - CN term and multiply Expres-sion (5) by (1 - e) to get the final value:

Final Value = (1 - e){l - CN) x

This formula works for both index funds and ETFs. Forexample, for index funds, one would usually set C = 0because index funds almost never charge commissions ontransactions. For ETFs. one Vv-ould usually set « = 0because ETFs almost never distribute capital gains.

I make four significant assumptions that must beexplained (see Appendix A for a summary). First, I amassuming that the investor reinvests all after-tax dividendsand distributed capital gains. If the investor sold all his sharesin the asset, he would have to immediately pay capital gainstaxes on the capital gains that weren't distributed, and thiswould eliminate the advantage that the tax-efficient (lowa) funds have over tax-inefficient (high a) funds.

Second, I assume that dividends are paid as a per-centage of the initial investment and not of the finalinvestment. This is irrelevant for my analysis and is cho-sen purely for purposes of simplification.

Third, I am assuming that C is constant and is thusindependent of the value of the transaction. While this isactually untrue for ETFs. which have bid-ask spreads, iassume that we are discussing only liquid ETFs like QQQsor SPDRs, which trade at negligible bid-ask spreadscompared to commissions charged by brokers. Also, Iassume that these charges are flat-rate commissions Ukemost discount and on-line brokerage fees.

Finally, I assume N is not correlated with k, soincreasing N is a deadweight loss (since it increases onlythe total brokerage fees paid). In reality. Equation (1)shows that this is not true, and increasing N can haveunpredictable effects on k, depending on the price dis-tribution. Because my purpose is not to model the effec-tiveness of dollar-cost averaging, and because changes inN have the same effect on k for both index funds andETFs, it is not incorrect to make this assumption forcomparisons.

The nine independent variables in the model aresummarized in Exhibit 2. I call a variable that is thesame across all investors and across all funds tracking thesame target index global. Because tracking error is verydifficult to model, I will assume that ETFs and index ftindscan track the market identically, and thus the returns kand d (the return on capital gains and the dividend yield)are equal for all funds tracking the same index. If onedoesn't wish to assume perfect tracking, it's still possibleto consider k and d as global by including the differencein tracking error as part of the difference in the expenseratio e.

E X H I B I T 2Analysis of Variables

Formula Variablek = return from capital gain.';d = relLim from dividendsC - flat brokerage commissione " expense ratioa = capital gains distribution ratio

/ - initial investment/V - # of purchases•51 = tax rate on capital gains

%i = tax rate on dividends

Distribution of prices P/t - time

Variable TypeGlobalGlobalSemiglobalSenniglobalSemiglobal

LocalLocalLocal

Local

DependentOther

Variable DescriptionEqual to the capital gains return on the tarj et indexEqual lo (he dividend return on the target index$0 for index funds. ETFs: Ranges from $10 to 2%Can include iraL'king error0 for most ETFs. Up to 0.3 for some index funds

Assume less than $500,000Assume N is uncorrelated with k; see below

Usually 20% for most investors

Can range from 0% (for pensions) to nearly 40%

Dependent on kSubscript: Indicates prices at different times.

SUMMER 2003 THE JOURNAL OF PORTFOLIO MANAGEMENT 85

E X H I B I T 3Changes in Initial Investment

S400.00 •

$300.00

$200.00

£• $100.00

S- $0.00

^ ($100.00)

a ($200.00)

(S300.00)

(S400.00)

($500.00)

($600.00)

($700.00)-

Initial Investment (/)

1 call a variable that is the same for all investors butdiffers for assets semiglobal. The flat-rate brokerage fee C,expense ratio e, and capital gains distribution ratio a areall variables that are asset-dependent. Finally, I call a vari-able that differs among investors local. The initial invest-ment /, the number of purchases N, and the tax rates T,and Tj vary for different investors, and are thus localvariables.

The one set of variables not included in this hst isthe distribution of prices P , ..., P . These variables areincluded in k.

An examination of Exhibit 1 indicates that all themajor differences between index funds and ETFs are rep-resented in the models semiglobalv^riahle^. The differencein management fees is in e; the difference in shareholdertransaction fees is in C; and the difference in taxation effi-ciency is in a. If we assume that the difference in track-ing error is a part of the expense ratio e, this model fullyaccounts for all the key quantitative differences betweenindex funds and ETFs.

An investor will choose an index fund over an ETFif the Final Value (index fund) - Final Value (ETF) > 0.The only variables different for the FK (index fund) andFK(ETF) are the semiglobal variahlas. We will indicate allsemigtobal variables for index funds by the subscript (i) andall semiglobal variables for ETFs by the subscript (e). Forexample, the expense ratio of an index fund would be e.,and the expense ratio of an ETF would be e .

The investor choice equation is:

-n = [(1 - .,)(/- qN){\ + k(\ - a,r,) + d(\ - X,)]] -

](7)

Although this equation may look complicated, itssimply the difference in value between buying an indexfund and buying an ETF. By adding subscripts to Equa-tion (6), we can quantify the profit of holding indexfunds instead of ETFs (a negative number v^ould indicatethe profit of holding an ETF instead of an index fund).

What happens if we graph the value of Equation (7)with changing /. while keeping all the other variables con-stant? The results are shown in Exhibit 3. (See AppendixB for the actual values of the constants and why the val-ues were chosen.).

First, we can clearly assert that those investing morethan $59,635 will choose to invest in ETFs, while thoseinvesting less will choose index funds. This value is actu-ally not very interesting because the threshold level whereEquation (7) ^ 0 is dependent on the values chosen forthe other variables (see Appendix B).

More important, the first derivative of Equation (7)with respect to / is negative and constant, so this graphwill slope down linearly for all reasonable choices of con-

8 6 INHEX MUTUAL FUNDS AND EXCHANGE-TRAILED FUNDS SUMMER 2003

E X H I B I T 4Boundary Condition Analysis

(1)

Minimum/(P'Deriv,)

a,=0

$217,876(-1,203)

$93,641(-2,799)

Oi = 0.2

$59.635(-4.395)

a, = 0.3

$43,748(-5.991)

Cd = 0.4

$34,546(-7.587)

Oi = 0,5

$28,542(-9.183)

(2)Minimum /(r'Deriv.)

c, = $o$0

(-4.395)

G = $10$29,817(-4.395)

C, = $20$59.635(-4,395)

C, = $35$104,362(-4,395)

C = $50$149,089(-4.395)

C, = $200$596,357(-4.395)

(3)Minimum/(V Deriv.)

e, = 0.05%$118,372(-2.214)

e, = O.I5%$79,313(-3.305)

a = 0.25%$59.635(-4.395)

e, = 0.4%$43,461(-6.031)

a = 0.6%$31,919(-8.211)

e,- = 1 %$20,846(-12.573)

(4)Minimum /d" Deriv.)

$4,969(-4.395)

N = 2$9,939

(-4.395)

A' = 4$19,878(-4.395)

N = t$29,816(-4,395)

^ = 1 2$59.635(-4,395)

A ' = 2 4

$119,271(-4.395)

(5)

Minimum /(I"'Deriv,)

rt = O%

$217,876(-1,203)

r*=10%

$93,641(-2.799)

Xk = 20%

$59.635(-4.395)

Tk = 28%

$46,210/\(-5.67^

/

/

stants. The value ofthe first derivative for the choice ofconstants in Appendix B is 0.00439 so for every extra$10,000 to be invested, ETFs will provide an extra $43.90more in value than index funds.

Finally, Exhibit 3 shows that as initial investment sizegrows, ETFs become far superior to index funds. Forexample, for a person investing $500,000, ETFs will pro-vide $1,935 more than an index fund, suggesting that ETFsshould be an important and useful tool for large investors'portfolios.

The second element ofthe analysis is to adjust sev-eral semiglobal and local variables and see the effect of thison the threshold level. Exhibit 4 provides a summary ofthe minimum investment (/) needed to make ETFs prefer-able to index funds.

The underlined and boldfaced figures are identicalto the minimum / in Exhibit 3 because they are simplyderived from the constants in Appendix B. The numbersin parentheses show the first derivative with respect to /multiplied by 1,000. In other words, this is the change indollars in Equation (7) upon increasing the initial invest-ment by $1,000.

Most of the results seen in Exhibit 4 are as wouldbe expected. Increasing index fund alpha, index fundexpense ratios, and capital gains tax rates has both abso-lute and marginal benefits for ETFs. In all cases, thederivative is negative, because higher initial investmentsshould alvrays benefit ETFs.

SUMMER 2003

Exhibit 4 also highlights some interesting implica-tions that are not as obvious. First, note that increasing Nor increasing C increases the absolute value ofthe mini-mum / needed to switch to ETFs, but has no marginaleffects (since the derivative stays constant). This indicatesthat whatever benefits dollar-cost averaging provides(which I don't look into here) must be weighed againstan initial fixed cost if one decides to invest in ETFs,

Another interesting implication is that the tax rate oncapital gains and the capital gains distribution ratio have anidentical effect on the minimum value of /. Thus, in thismodel, lowering afrom 0.2 to 0.1 is the same as if the gov-ernment had reduced the capital gains tax rate by half, from20% to 10%. This is an important implication that showsjust how critical tax efficiency can be. In multiperiod mod-els, this one-to-one correspondence disappears becausetaxes on the undistributed capital gains will eventually haveto be paid when the investor sells the investment.

The last result that I find surprising is the superior-ity of index funds over ETFs for small investors underalmost any conditions. Assuming rather unusual conditionssuch as an index fund expense ratio o r i% or transactionfees as low as $10 still does not change the preference ofsmall investors (< $20,000) for index funds over ETFs.Tliis su^ests that there is no reason for small investors whowant to invest for a short period of time to choose ETFs.

THEJOURNAL OF PORTFOLIO MANAGEMENT 8 7

MULTIPERIOD MODEL

Does a multiperiod model change the conclusions,or should small investors always prefer index funds toETFs? Take an investor who wants to invest an amount /in an index-tracking asset for a period of n years. Again,she makes N purchases at prices Py,..., P , (P is the priceof the fund at each transaction) all during the first year.Then, in all the following years, she simply reinvests allher distributed dividends and capital gains and does notadd any new money. After ti years, she sells her shares andpays capital gains taxes on the difference between her finalvalue and the cost basis.

If we assume all the variables are the same as in theone-period model, the value of the investment at theend ofthe first year is given by Equation (6). In all theremaining years, she receives capital gains k^ and dividendsd^, which vary from year to year. The final value after timet (where I can be any number from 1 to n) is thus:

Value, = (1 -e)'{l-CN) x

+ k,{l - ar,) + d,(\ - T,)] .gj

We get this by compounding the returns for each year(using ri), and multiplying each time by the expenseratio deduction (1 - e).

The cost basis is the part ofthe final value that is nottaxable because it was either already taxed or because itis part ofthe initial investment:

Cost Basis, = {(1 - e)Va\ue^} +

(9)

The final value after redemption (i.e., where ( ~ ri)is the value from Equation (8) minus the liquidation taxesthat have to be paid on the undistributed capital gainsminus the transaction fee for conducting that last trans-action of selling one's shares:

Final Value^ = Value^ -

^- Cost Basis,,) - C (10)

What are some interesting improvements of themultiperiod model over the one-period model? First, wecan look at how different time horizons affect investorchoices by seeing if a higher value of n favors ETFs orindex funds. Second, we can look at how difFerent statis-tical distributions of capital gains {kj, ..., k ) affect investorchoices. For example, if the capital gains returns of a tar-get index have a high variance (e.g., the Nasdaq), we canexamine whether this favors ETFs or index funds.

Finally, we can eliminate the assumption of non-redemption included in the one-period model (Assump-tion 1, Appendix A). This allows our model to betterapproximate non-theoretical real-world conditions.Assumptions 2-(S in Appendix A are still necessary.

We start the analysis by exploring investor choice overdifferent values of n. We can do this by simply graphingthe difference between index funds and ETFs {FV.- FV)using the final values from Equation (10) against the timehorizon.

Exhibit 5 shows these relationships for difFerentchoices ofthe initial investment /. Once again, all othervariables have the values shown in Appendix B. All j^ areequal to 8%, and all d^ are equal to 2%.

The graph in Exhibit 5 highlights some intriguingproperties of investor choice in the multiperiod model.We find that changing the time horizon has a significanteffect on whether index funds or ETFs are preferred,but the effect is not linear like the effect of/ on investorchoice (see Exhibit 3), but rather quadratic. As ti rises ini-tially, index funds become better off since the initial fixed-cost advantage is multiplied, but after an extended periodof time, the superior tax-efficiency and lower expenseratios of ETFs cause a dramatic drop in the graph.

Although it looks as if the graph for / = $500 keepsrising, one can calculate that ETFs will evenuially becomebetter than index funds after 171 periods. Only if/< C Nwill index funds be superior for all time horizons, becausein that case, ETF investors will have to pay out theirentire initial investment in brokerage fees.

Another interesting conclusion is just how much themultiperiod model favors ETFs more than the one-periodmodel. Keeping all other variables constant, one needs atleast $59,635 in investments to prefer ETFs to indexfunds for a holding period of one year. If an investor wantsto stay in the tracking instrument for ten years, however,one needs only $13,019. As most small investors fall inbetween these two numbers, this has an important ram-ification for our comparisons.

Even for investors who want to stay in the tracking

88 INDEX MUTUAL FUNDS AND EXCHANGE-TRADED FUNDS SUMMER 2003

E X H I B I T 5Changes in Time Horizon

$2,000

$1,000

$0

($1,000)

9i

($2,000)

($3,000)

($4,000) J

1 2 3 4 5 6 7

# of Time Periods (n)

instrument for only five years, $33,787 is the minimuminvestment, which is still well below the one-year thresh-old level. As with the one-period model, large investorsstrongly favor ETFs over index funds.

The last interesting point to note about Exhibit 5is derived from the property that the relationship betweenthe FV. — FV and n is quadratic- As a result, the slopeincreases quickly in magnitude (becoming more negative),making marginal decisions over the long term veryimportant.

For example, if we look at the I = $20,000 part ofExhibit 5, lengthening the holding period by one yearfrom 19 years to 20 years reduces FV. - FV^ by more than$600. If we look at the more extreme case of / = $1,000,lengthening the holding period by one year firom 76 yearsto 77 years (unrealistic numbers, I do agree) would changean investor from preferring index funds by more than$1,400 to preferring ETFs by more than $600. This againreminds us how important marginal decisions can be overthe long term.

The second part ofthe analysis compares the effecton investor choice of changing the variance ot capital gainsreturns. We assign M the value of 10, and all other vari-ables have the values given in Appendix B. Then, insteadof fe =8% for all (, k^ alternates between a high value anda low value (with the average remaining at 8%). Forexample, to obtain a standard deviation of 2%, we havek = 9.897% and fe ,, = 6.103% for ti years (ten

cii'it years oaa years '

in this case).Exhibit 6 displays the effect of differing standard

deviation of fe on the minimum initial investment /neededto make ETFs preferable to index funds. From this graph,we can see that the variance (or standard deviation) of cap-ital gains can be very important for investors decidingwhether index funds or ETFs should be preferred. Formore stable indexes like the S&P 500, there is little effectof additional instability, but for volatile indexes like thosein developing markets or the Nasdaq Composite Index,index funds have a dramatic advantage.

It is difficult to say whether the results in Exhibit 6

SUMMER 2003 THEJOURNAI, OF PORTFOLIO MANAGEMENT 89

are simply the consequence of choosing certain values forvariables, or if they hold for any reasonable variable val-ues for ETFs and index funds. I can try to suggest somebasis behind the index fund advantage. Because an ETFdistributes less of its capital gains, the tax burden isimpounded in the value ofthe shares until redemption.More volatile returns thus make the tax burden of ETFsmore volatile, which diminishes their value compared tothe more stable index fund tax burden. Of course, it'simpossible to guess the validity of this hypothesis with-out more analysis of taxations effect on Exhibit 6, a sub-ject beyond my objectives.

In the one-period model, because non-distributedcapital gains were never taxed, changing the capital gainsdistribution ratio of index funds Of. had the same effect aschanging the capital gains tax rate T ,. Clearly, this shouldnot occur in the multiperiod model, as undistributedcapital gains are later taxed at redemption.

Exhibit 7 (similar to Exhibit 4) compares the mini-mum value of/needed to make ETFs preferable to indexfunds for the multiperiod model with n - 10. Derivativesare not given because they are no longer constant.

Not surprisingly, the one-to-one correspondencebetween a. and T^^ disappears in the multiperiod model.Instead, we see that raising the capital gains tax rate hasless of an effect on investor choice than increasing the cap-ital gains distribution of index funds. This is because rais-ing the capital gains tax rate depresses the value of bothETFs and index funds (unlike the one-period case wherecapital gains taxes were charged only on the index fund).

while increasing a. increases the tax burden of only theindex fund.

QUALITATIVE COMPARISON

As I indicated earlier, the models I propose areunable to perfectly replicate real-world conditions. First,there are simplifying assumptions that must be made inorder to be able to conduct mathematical comparisons,and second, there are factors investors consider that justcannot be expressed in terms of numbers. Often, these fac-tors have as much of a part to play in decision-making asthe explicit cost comparison.

One important qualitative advantage of ETFs isconvenience. One can sell or buy at any time of dayinstead of waiting until the end ofthe day for an indexfund. Carty [2001] suggests an example. Imagine aninvestor who put in an order to sell his S&P index mutualfund on the morning of October 19, 1987, expecting toget a price around that ofthe previous day. What did hethink when, the next day, he discovered that his proceedswere 22.4% less than he expected? Certainly, the 1987crash isn't a daily occurrence, but the ease of getting outof an ETF can be a very important advantage, especiallyfor more active larger investors.

A second advantage of ETFs is the ability to buy onmargin and sell short. For many investors, this ability canbe vital, and because ETFs are exempt from the sell-shortuptick rule, they can also be used to great advantage inhedging strategies.

E X H I B I T 6Changes in Standard Deviation

SIOO.OOO

I$80,000

$60,000

$40,000

$20,000

so

7

8 §

Standard Deviation

S.D.{).iMVi

4.()()9;.6.00':;-

10,00'7f

12.0{W<

14.00';;I6.(X)%

20-()()'.i

n.oin24,(KJ';i26,00'^28.00%30.00%

Mini$I3.()I'J$13,047$13,179.$13,422$13.7S8$14,297$14,979$15,878$17,064$18,644

$20,795$23,830$28,345$35.651$49,272$83,018

90 INDEX MUTUAL FUNI>S AND EXCHANGE-TRADED FUNDS SUMMER 2003

E X H I B I T 7Multivariable Boundary Condition Analysis

L(1)

MinimuiTi /

a, = 0$22,738

ov = 0.1

$16,539

01 = 0.2

$13,019

at = 0.3$10,750

Of = 0.4

$9,167

QV = 0.5

$7,998

(2)

Minimum /]i = 0%$22,653

T*= 10%

$16,309

T* = 20%$13,019

T* = 30%

$11,063

r* = 40%

$9,829

Tk = 50%

$9,050

Finally, the fact that ETFs are traded allows investorsto place stop, stop-loss, and limit orders on them. Theseare frequently used tools, and using them on ETFs issomething many investors can take advantage of to pre-vent market makers from providing them bad prices.

Of course, index flinds also have useful features thatare attractive to investors. The most important feature issimplicity. For ETFs, one needs to open a brokerageaccount, deposit cash in the account, set a market orlimit order, and make sure it's executed. For averageinvestors, it is far simpler to send a check to an index fund,and wait for reports in the mail to see how one's invest-ment is doing.

It is often surprising to Wall Street experts howmuch small investors value simplicity and are willing tosacrifice a lower expense ratio or superior tax-efficiencyto achieve it,

CONCLUSION

As Gastineau [2002] points out, ETFs are still evolv-ing. In the near future, fixed-income ETFs and activelymanaged ETFs may once again change the world offinance as equity index ETFs have in the last decade.

My research shows that the key areas of differencebetween the two instruments are management fees, share-holder transaction fees, taxation efficiency, and otherqualitative differences. Tracking error is difficult to modelbecause there isn't a true benchmark for comparison.Comparison with the paper indexes is fallacious becausethey assume efficient paper transactions.

Instead, my one-period mode! specifically looks at theother three quantitative factors and examines their effectson investor choices. A multiperiod model can deal with var-ious problems in the one-period model. Finally, I considerthat qualitative differences between ETFs and index fundscan be extremely significant for decision-making.

Some additional analysis would be interesting, suchas the effects of changing tax rates on investor choices.Other questions are why increasing variances helps indexfunds over ETFs in the multiperiod model, or different

SUMMER 2003

values of the dividend rate and a different mean capitalgains rate affect investor choices, or whether it is possi-ble to relax some ofthe simplifying assumptions.

APPENDIX A

Simplifying Assumptionsfor the One-Period Model

1. The final value is left invested in the asset and not soldto get cash. This is important because it creates anadvantage for funds with a lower OC.

2. The dividend d is paid on the total after-brokerage feeinvestment / - CN.

3. C is a flat dollar rate and unrelated to the transactionvalue.

4. k and d are the same for all ETFs and index funds thattrack the same index. Thus, tracking error is not inclu-ded, or it is included as part ofthe expense ratio e.

5. k is unrelated to the value of N.6. All distribution reinvestments are made without trans-

action costs.

APPENDIX B

Choice of Constants

/ = $5,000 (this can be amended for different investors).N = 12 (assumes monthly investing for one year).e = 0.25% (this incorporates expense ratio and tracking

error).e = 0.14% (this incorporates expense ratio and tracking

error).a = 0.2 (from historical data of index fiind distributions),a = 0 (ETFs almost never distribute capital gains).Tjj = 20% (usual capital gains tax rate).T, = 32% (usual income tax rate).C = $0 (index funds almost never charge brokerage fees).C = $20 (usual discount brokerage fee for buying stocks).fej = 8% (from historical data of index capital gains).d^ = 2% (fi-om recent historical data of index dividends).

Some of these values are assigned by looking at current(and historical) data on averages, and others are assigned at ran-

THE JOURNAL OF PORTFOLIO MANAtiEMENT 9 1

dom. Thus, values are liable to change for different indexes,fimds. and investors, but my analysis tries to focus on generalcharacteristics rather than on actual values.

ENDNOTES

The author thanks Gary Gastineau, Alex Kostovetsky, andMarciano Siniscalchi for their excellent advice, contributions,and criticism.

'Statistics come, variously, from Williams [2001];www.indexfunds.com; and Frino and Gallagher [2001].

•These factors are enumerated in Chiang [1998].'The SPUR S&P 500 ETF has been around since 1993

although there was no real demand for it until the mid- or late1990s.

•"Vanguard 500 Fund Prospectus, at www.vanguard.com,^Vanguard tries to protect shareholden by not allowing

transactions in the last hours ofthe trading day,'The SPUR has not distributed capital gains in the last four

yean, and the QQQ has not had a capital gains distribution sinceits inception two yean ago. Vanguard, however, has passedalong 4% of its value to its shareholders in the past three years.

REFERENCES

Bogle, J. "The Implications of Style Analysis for Mutual FundPerformance Evaluation." The Journal of Portfolio Management,Summer 1998, pp. 34-42.

Carty. C. Michael. "ETFs From A to Z." Financial Planning,May I, 2001. pp. 102-106,

Chen, C , and S, Stockuni. "Selectivity, Market Timing, andRandom Beta Behavior of Mutual Funds: A GeneralizedModel."yoMm<j/ of Finandal Research, 9 (1986), pp. 87-96.

Chiang, W. "Optimizing Performance." In A. Neubert, ed.,Indexing for Maximizing Investment Results. Chicago: GPCo.Publishers, 1998.

Dellva, W. "Exchange-Traded Funds Not For Everyone."Journal of Finandal Planning, April 2001, pp, 110-124.

Dellva. W,, and G. Olson. "The Relationship Between MutualFund Fees and Expenses and Their Effects on Performance."77if Finandal Review, 33 (1998), pp, 85-103.

Edelen, R,M. "Investor Flows and the Assessed Perfonnanceof Open-End Mutual Funds..'' Journal of Finandal Economics,September 1999, pp, 439-466,

Ferris, S., and D. Chance. "The Effects of ]2b-l Plans onMutual Fund Expense Ratios: A Note."_/<»«mi3/ of Finance, 42(1987). pp. 1077-1082.

Frino. A., and D.R. Gallagher. "Tracking S&P 500 IndexFunds." TheJournal of Portfolio Management, Fall 2001, pp. 44-54.

Gastineau, G.L. "'Exchange-Traded Funds: An Introduction."The journal of Portfolio Management, Spring 2001, pp. 88-96.

. 77if Exchange Traded Funds Manual. New York: JohnWiley & Sons, 2002.

Grinblatt, M,, and S, Titman. "A Study of Monthly MutualFund Perfonnance Returns and Performance Evaluation Tech-niques." Jowmc/ of Finandal and Quantitative Analysis, 29 (1994),pp. 419-444.

Jensen, M. "The Performance of Mutual Funds in the Period\9AS-\96A." Journal of Finance, 42 (1968), pp. 389-416.

Lee, C , and S. Rahman. "Market Timing, Selectivity, andMutual Fund Performance: An Empirical Investigation."_/' ""'-nal of Business, 63 (1990), pp. 261-278.

Malkiel, B, "Returns from Investing in Equity Mutual Funds1971-1991.">Mm<7/ of Finance, 50 (1995), pp. 549-572.

Minor, D.B. "Beware of Index Fund Fundamentalists." TlieJournal of Portfolio Management, Suimiier 2001, pp, 45-50.

Sharpe, W. "Mutual Fund Vcdoimzncc." Journal of Business, 39(1966), pp. 119-138.

Treynor,J. "How to Rate Management of Investment Funds."Harvard Business Review, 44 (1966). pp. 131-136.

Williams, F. "ETFs: Market up 82% to Nearly $76 Billion."Petisions & Investments, March 5, 2001, pp, 25-30.

Zitzewitz, E. "Who Cares about Shareholders? Arbitrage-Proofing Mutual Funds." Working paper, Stanford UnivenityGraduate School of Business, March 2002.

To order reprints of this article, please contact Ajani Malik atamalik@iijournals.com or 212-224-3205.

9 2 INDEX MUTUAL FUNDS AND EXCHANGE-TKADED FUNIIS SUMMER 2003