# The Arithmetic of Investment Expenses

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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

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

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

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