The Arithmetic of Investment Expenses

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1 Fiacial Aalysts Joural Volume 69 Number CFA Istitute The Arithmetic of Ivestmet Expeses William F. Sharpe Recet regulatory chages have brought a reewed focus o the impact of ivestmet expeses o ivestors fiacial well-beig. The author offers methods for calculatig relative termial wealth levels for those ivestig i fuds with differet expese ratios. Uder plausible coditios, a perso savig for retiremet who chooses low-cost ivestmets could have a stadard of livig throughout retiremet more tha 20% higher tha that of a comparable ivestor i high-cost ivestmets. A recet issue of this publicatio featured a editorial by Charles Ellis (2012), titled Ivestmet Maagemet Fees Are (Much) Higher Tha You Thik, i which Ellis argued that as a percetage of assets, such fees do look low, but calculated correctly, as a percetage of returs, fees o loger look low.... Ivestors should cosider fees charged by active maagers ot as a percetage of total returs but as icremetal fees versus riskadjusted icremetal returs above the market idex. (p. 4) Why? Because extesive, udeiable data show that idetifyig i advace ay oe particular ivestmet maager who will after costs, taxes, ad fees achieve the holy grail of beatig the market is highly improbable. (p. 6) The latter poit is cosistet with decades of research i both academe ad the ivestmet idustry. As the director of fud research at Morigstar, a leadig provider of mutual fud data ad aalysis, oted, If there s aythig i the whole world of mutual fuds that you ca take to the bak, it s that expese ratios help you make a better decisio. I every sigle time period ad data poit tested, low-cost fuds beat high-cost fuds.... Ivestors should make expese ratios a primary test i fud selectio. They are still the most depedable predictor of performace. (Kiel 2010, pp. 2 3) William F. Sharpe is professor emeritus of fiace at Staford Uiversity ad director emeritus at Fiacial Egies, Ic., Suyvale, Califoria. Ivestmet expeses have log bee reported by mutual fuds, although their impact is udoubtedly igored or misuderstood by may ivestors. I the Uited States, those savig for retiremet with 401(k) defied cotributio plas ofte have isufficiet iformatio about such expeses. However, rules recetly issued by the U.S. Departmet of Labor require that such 401(k) ivestors receive reports detailig may of these charges. Give this reewed focus, it is more importat tha ever for ivestors to uderstad the possible impact of ivestmet expeses o their future wealth. I my research, I cocetrated o comparisos of two possible ivestmets oe with high expeses ad the other with low expeses. 1 I cosidered the impact of ivestmet expeses i two settigs: whe a sigle ivestmet is held for a umber of years ad whe recurrig ivestmets are made over may years. For each settig, I looked at cases i which both the low-cost ad the high-cost ivestmets provided the same gross returs (before expeses) ad cases i which such returs might differ. Lump-Sum Ivestmets with Equal Gross Returs The Vaguard Group provides ivestors with computatioal tools to assess the possible impact of mutual fud expeses i cases i which a lump sum is ivested for a umber of years with o withdrawals or cotributios. Figure 1 depicts a example of the Vaguard Total Stock Market Idex Fud Admiral Shares (with a miimum ivestmet of $10,000), desiged to track the broad U.S. stock market. As Figure 1 shows, the Vaguard fud s expese ratio is 0.06% a year, whereas the average expese ratio of similar fuds is assumed to be 1.12% a year. Note that the key outputs are the dollar fees paid over the holdig period. These results deped o the mai assumptios (the expese ratios ad the umber of years the ivestmet is held) as well as the CFA Istitute

2 The Arithmetic of Ivestmet Expeses Figure 1. Vaguard Estimates of the Effects of Differet Expese Ratios Fees o $10,000 Ivested over 10 Years ($) 3,000 $2,720 2,000 You save $2,568 1,000 0 Category Average a $153 This Fud b a Average expese ratio of similar fuds = 1.12%. b Expese ratio = 0.06%. Source: Vaguard Group (2012). iitial amout to be ivested ad a assumed costat gross retur o the ivestmet to be eared by each fud. I propose a differet measure for the effects of expeses oe that is both simple ad likely to be more meaigful for may ivestors. For reasos that will become clear, I call it the termial wealth ratio (TWR). For cases such as those cosidered by Vaguard i which the low-cost ad high-cost fuds are assumed to have equal cumulative gross returs, o assumptios eed be made about the levels or patters of such returs over time. Let us cosider a ivestmet that returs r i i year i. Returs are measured i proportios; thus, for a retur of 8%, r i = The tilde over the variable idicates that the value may ot be kow i advace. A dollar ivested at the begiig of year i will grow to 1+ r i by the ed of the year. Assume that expeses are the paid usig a expese ratio of x (e.g., if the expese ratio is 1.12%, x = ). As a result, a dollar ivested at the begiig of the year will grow (et of expeses) to ( 1 x) 1+ r i. Now let us cosider a ivestmet held for years. The termial value per dollar of iitial ivestmet is ( + ) ( + ) ( + ) 1 x 1 r 1 1 x 1 r 2 1 x 1 r. After rearragig, the termial value is 1 x ( 1 r 1) ( 1+ r 2) 1+ r. + The expressio i brackets is the termial value per dollar ivested that would have bee obtaied had there bee o expeses that is, the compouded gross retur, or G. The iitial term i paretheses (1 x) is the proportio of the value retaied each period (if x = , the proportio retaied is ). Thus, the termial value per dollar of ivestmet is. ( 1 x) G Now let us cosider a compariso betwee two fuds with potetially differet compouded gross returs ad differet expese ratios x 1 ad x 2. The fial values are ad 1 x 1 G1 1 x 2 G 2. The ratio of the two fial values is the TWR: TWR = ( 1 x1) G1. ( 1 x2) G2 Rearragig slightly gives ( 1) 1 x TWR = 1 x 2 1. (1) 2 G G I its calculatios, Vaguard assumes that the two ivestmets have equal costat gross returs i each year of the holdig period likely March/April

3 Fiacial Aalysts Joural to be a reasoable assumptio for expected values, although ot ecessarily for actual values, a possibility that we will examie later. I ay evet, whe the compouded gross returs are the same, the secod ratio equals 1. I such a case, the ratio of the termial values does ot deped i ay way o the actual returs durig the holdig period. Oly the umber of years the ivestmet is held ad the two expese ratios matter. Equivaletly, the key igrediets are the retetio ratio (the bracketed expressio i Equatio 1) ad the umber of years the ivestmet is held (the expoet ). More geerally, whe the two ivestmets have the same compouded gross returs, the TWR equals the compouded retetio ratio: ( 1 x1 ) TWR = ( 1 x2 ) if G 1 = G 2. (2) I the Vaguard example show i Figure 1, x 1 = , x 2 = , ad = 10. I such a case, 10 ( ) TWR = = ( ) Breakig the computatio ito its compoets, we see that the retetio ratio is /0.9888, or approximately (a advatage of more tha 1% a year for the lower-cost fud). This ratio, compouded over 10 years, is approximately the TWR for the two ivestmets. Thus, a ivestor who obtais a give termial gross retur over 10 years with a expese ratio of 0.06% a year eds up with 11.25% more wealth tha oe who obtais the same termial gross retur but must pay 1.12% a year i expeses. To paraphrase Ellis (2012), the differece betwee a expese ratio of 0.06% a year ad a expese ratio of 1.12% a year may seem small, but the cumulative result of the lower expese ratio after 10 years is a icrease i wealth of more tha 11%. Ad this outcome is the case o matter what the actual ivestmet returs are from year to year so log as the ivestmets with the differig expese ratios have the same compouded gross returs. Figure 2 shows the TWRs of lump-sum ivestmets held for 10, 20, ad 30 years with retetio ratios of Not surprisigly, the effects of a give retetio ratio are greater if assets are ivested for loger periods. For two fuds with the expeses i our example, the TWR for a holdig period of 30 years is slightly less tha A icrease i termial wealth of almost 38% is huge by ay stadard. Real Returs Most ivestors are iterested i the purchasig power of their termial wealth, ot its omial value. Thus, returs should be measured i real (iflatio-adjusted), ot omial, terms. Fortuately, so log as such returs are computed i the correct maer, Equatios 1 ad 2 cotiue to apply. If we let r i be the omial retur o a ivestmet i period i ad c i be the proportioal chage i a measure of the cost of livig durig the period, the the real retur is computed as follows: ( 1+ r i )= 1+ r i 1+ c i. Dividig each value of 1+ r i i each of the equatios used to derive Equatio 1 by 1+ c i ad simplifyig as before agai give Equatio 1, with the compouded gross returs measured i real terms. Thus, the previously discussed argumets suggest that Equatio 1 holds whether the two ivestmets have the same gross compouded real returs or the same gross compouded omial returs as is also true for Equatio 2. More simply, if oe alterative is worth more tha aother by a give percetage i omial terms, it will exceed the latter by the same percetage i real terms. Recurrig Ivestmets with Equal Costat Aual Returs Let us ow tur to cases i which recurrig ivestmets are made over N years. Ufortuately, the simple formula i Equatio 1 is ot directly applicable, ad so calculatios must be made o the basis of assumptios about ivestmet returs. We ca follow the Vaguard example by assumig the same costat rate of retur from year to year for both the low-cost fud ad the high-cost fud. Let us cocetrate o cases i which each year for N years, a ivestor devotes a particular portio of icome to ivestmet i a retiremet pla, with the goal of usig the total value of the fud N years hece to fiace spedig durig the retiremet years. To simplify, let us assume that the aual cotributios are equal i real terms, although the calculatios ca be easily adjusted to icorporate chages i the real values saved each year, mothly ivestmets, ad other possible aspects of a retiremet savigs pla. I this case, assumig that equal real ivestmets are made for N years, TWR = TWR = N N 1 ( 1 x1 ) + ( 1 1) ( 1 1) 1 GN x GN x G1 N N 1 ( 1 x2 ) GN + ( 1 x2 ) G N ( 1 x2 ) 1. G CFA Istitute

4 The Arithmetic of Ivestmet Expeses Figure 2. Termial Wealth Ratios for Lump-Sum Ivestmets: Alterative Retetio Ratios for Ivestmets over 10, 20, ad 30 Years Termial Wealth Ratio Retetio Ratio 30 Years 20 Years 10 Years The relative wealth obtaied with oe expese ratio vis-à-vis that obtaied with a differet expese ratio depeds o the returs provided by the uderlyig ivestmets. To assess the outcomes, assumptios must be made about likely future ivestmet returs. Figure 3 shows termial real wealth ratios for cases i which equal real amouts are ivested each year for N years while the uderlyig gross ivestmet real retur remais costat. The horizotal axis shows the alterative aual real rates of retur; the vertical axis shows the termial real wealth ratios. Curves are show for 10-, 20-, ad 30-year ivestmet periods. Not surprisigly, the TWRs are higher the loger the period over which ivestmets are made because the effects of higher expese ratios compoud from year to year. Moreover, for a give umber of years, the higher the real rate of retur o the uderlyig assets, the higher the TWR. Why? Because the higher the retur, the greater the termial values of the earlier cotributios relative to those of the later cotributios ad the former provide higher TWRs tha the latter. Recall that for a lump-sum ivestmet held for 30 years, the effect of ivestig with a expese ratio of 0.06% rather tha 1.12% is very large, with a TWR of approximately 1.38%. Whe moey is ivested i equal aual real amouts over 30 years, the effects are smaller because ot all ivestmets experiece the fees for the full period. As Figure 3 shows, for 30-year ivestmet periods, TWRs rage from approximately to 1.26, depedig o the gross retur o the uderlyig assets. For plausible assumptios about the retur o a diversified ivestmet portfolio, the TWR is sigificatly greater tha Almost certaily, most ivestors would cosider it extremely desirable to be able to look forward to havig the fuds saved for their retiremet provide 20% more purchasig power. Mote Carlo Aalysis Let us ow tur to more realistic cases, i which future gross returs ca differ betwee the two fuds, leadig to ucertaity about the termial March/April

5 Fiacial Aalysts Joural Figure 3. Termial Wealth Ratios with Costat Real Rates of Retur: Equal Aual Real Ivestmets over Alterative Ivestmet Periods for Fuds with Expese Ratios of 0.06% ad 1.12% Termial Wealth Ratio Costat Real Rate of Retur (% per year) 30 Years 20 Years 10 Years wealth ratio. I this case, I used Mote Carlo aalysis to geerate a millio possible 30-year scearios, computed the TWR for each oe, ad the calculated the rage of the ratios across scearios. Although it is coveiet to assume that fuds with differet expese ratios but comparable ivestmet styles (ad/or bechmarks) will have the same compouded gross returs, this sceario is ulikely to be the case i practice. Istead, the fud with higher expeses is likely to egage i a more active maagemet style tha the fud with lower expeses. A useful way to represet this differece is to assume that i each year, the higher-cost fud (h) will provide a gross retur equal to that of the lower-cost fud (l) plus a trackig error, which will vary from year to year aroud a zero value. I such a situatio, the TWR of the two fuds will be ucertai beforehad, with the actual result depedet o the realized differeces betwee their gross returs. We ca represet the relatioship as follows: r h = r l +ε. (3) Equatio 3 differs from the commo formulatio, i which a alpha term o the right-had side represets the expected additioal retur from active maagemet. I effect, we are assumig that there is o such added retur ad thus omit the alpha term. I the Mote Carlo simulatios, for each year ad sceario, a retur o the low-cost fud ( r l ) is draw radomly from a ormal distributio. This distributio is the same for every year ad sceario; thus, ex ate, these returs are ucorrelated from year to year. I based this retur distributio o the historical performace of a idex of global real stock returs (Dimso, Marsh, Stauto, McGiie, ad Wilmot 2012, p. 14, Table 3). From 1900 through 2011, these stocks provided a aual average real retur of 6.9%, with a stadard deviatio of 17.7% a year. After obtaiig each retur for the low-cost fud, I the drew a value for the trackig error (ε) from a ormal distributio with a expected value of zero ad a specified stadard deviatio. These trackig errors are also ucorrelated ex ate from CFA Istitute

6 The Arithmetic of Ivestmet Expeses year to year. The retur o the high-cost fud is the sum of the retur o the low-cost fud ad the trackig error, as i Equatio 3. Lump-Sum Ivestmets with Ucertai Returs The first set of simulatios assumes that a lump sum is ivested i each of the two alterative fuds for 30 years. As before, let us assume that the low-cost fud has a expese ratio of 0.06% a year ad the high-cost fud has a expese ratio of 1.12% a year. Figure 4 shows the results for three possible levels of active risk, with trackig error stadard deviatios of 0%, 2.5%, ad 5.0% a year. The results for zero trackig error are all the same because the coditios required for Equatio 2 are met, givig a TWR of approximately 1.38% i every sceario. However, each of the two cases with trackig error risk has a rage of possible outcomes. The probability that the TWR will exceed 1.0 is more tha 99% for the case with moderate active risk (0.025) adw more tha 90% for the case with the greatest active risk (0.050). I each case, there is a 50% chace that the TWR will be roughly 1.38% or greater. 2 Note that with moderate amouts of active risk (trackig error stadard deviatios of less tha 2.5% a year), a ivestor i the low-cost fud will almost certaily be richer tha a ivestor i the high-cost fud. Moreover, the disparity is likely to be very large. But there is a small chace that after 30 years, the two will have relatively similar amouts of wealth. Ad whe the active risk is relatively large (trackig error stadard deviatios of at least 5.0%), there is a chace (albeit a small oe) that eve after 30 years, a ivestor i the low-cost fud will be poorer tha a ivestor who chose the high-cost alterative. Although bettig o a relatively active maager with o ability to add value, o average, is a poor choice, the simulatios show why a Darwiia process does ot weed out such maagers with great rapidity. I this case, the odds are eve that a ivestor i the low-cost fud will be Figure 4. Probabilities of Alterative Termial Wealth Ratios for Trackig Errors with Aual Stadard Deviatios of 0, 0.025, ad 0.050: Lump-Sum Ivestmets for 30 Years with Expese Ratios of 0.06% ad 1.12% Probability That Termial Wealth Ratio Exceeds X X StdTE = 0 StdTE = StdTE = March/April

7 Fiacial Aalysts Joural well over a third richer tha a ivestor i the highcost fud after 30 years. But there is a small chace that a ivestor i the low-cost fud will regret ot havig selected the high-cost fud. For those who choose fuds with high expese ratios, hope may sprig eteral. Recurrig Ivestmets with Ucertai Returs Fially, we ca aalyze the rages of TWRs obtaied with equal real ivestmets i each year for a give umber of years. Figure 5 shows the results for ivestmets made over 30 years. It differs from Figure 4 i two respects. First, there is ucertaity about the TWR eve whe the highcost fud has zero trackig error. This result is due to the variatio i the relative weights of the TWRs of the aual ivestmets, which arises from the variatio i the returs of the two fuds from year to year. Secod, the rages of TWRs are arrower tha i the earlier case because the holdig periods for all but oe of the amouts ivested are less tha 30 years. Despite these differeces, there is agai a small but sigificat chace (10%) that after 30 years, a low-cost fud will provide a ivestor with less wealth tha a high-cost fud with a similar ivestmet style but substatial trackig error (5%), eve though both fuds have the same ex ate expected gross returs. That said, the odds are eve that the frugal ivestor will have over 20% more moey to sped durig retiremet. Coclusio May of the umeric results i this article deped o the particular expese ratios used (0.06% a year for the low-cost fud ad 1.12% a year for the high-cost fud). But the formulas ad Mote Carlo procedures ca be applied i other cases. My mai goals here were to advocate the use of the termial wealth ratio a simple yet meaigful measure of the relative outcomes provided by fuds with Figure 5. Probabilities of Alterative Termial Wealth Ratios for Trackig Errors with Aual Stadard Deviatios of 0, 0.025, ad 0.050: Equal Aual Real Ivestmets for 30 Years with Expese Ratios of 0.06% ad 1.12% Probability That Termial Wealth Ratio Exceeds X X StdTE = 0 StdTE = StdTE = CFA Istitute

8 The Arithmetic of Ivestmet Expeses differet expese ratios ad to show how to calculate possible rages of this measure for various types of ivestmets over time. That said, the results I obtaied for the expese ratios cosidered are dramatic. Whether oe is ivestig a lump-sum amout or a series of periodic amouts, the arithmetic of ivestmet expeses is compellig. Although a log-term ivestor may be able to fid oe or more high-cost maagers who ca beat a appropriate bechmark by a amout sufficiet to more tha offset the added costs, the reality is that compared with the readily available passive alterative, fees for active maagemet are astoishigly high (Ellis 2012, p. 4). Maagers with extraordiary skills may exist, but as I argued i this publicatio may years ago (Sharpe 1991), aother exercise i arithmetic idicates that such maagers are i the miority. Ad as Ellis has remided us, they are very hard ideed to idetify i advace. Caveat emptor. I thak Robert Youg, Joh Watso, ad Jaso Scott of Fiacial Egies ad Steve Greadier of Staford Uiversity for helpful commets. This article qualifies for 1 CE credit. Notes 1. The formulas ad simulatio procedures also work if oe of the ivestmets has zero expeses. 2. Give our (traditioal) assumptios about trackig error, the media compouded gross retur for the high-cost Refereces Dimso, E., P. Marsh, M. Stauto, P. McGiie, ad J. Wilmot Credit Suisse Global Ivestmet Returs Yearbook Zurich: Credit Suisse. Ellis, Charles D Ivestmet Maagemet Fees Are (Much) Higher Tha You Thik. Fiacial Aalysts Joural, vol. 68, o. 3 (May/Jue):4 6. Kiel, Russel Morigstar FudIvestor, vol. 18, o. 12 (August):1 3. active strategy is smaller tha that for the low-cost passive strategy eve though the expected returs for both strategies are the same. Sharpe, William F The Arithmetic of Active Maagemet. Fiacial Aalysts Joural, vol. 47, o. 1 (Jauary/February):7 9. Vaguard Group Vaguard Total Stock Market Idex Fud Admiral Shares (https://persoal.vaguard.com/us/ fuds/sapshot?fudid=0585&fuditext=int#hist=tab %3A3; accessed 5 July 2012). March/April

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