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The Arithmetic of Fees

an interactive working paper · drewbreyer.com

In investing, you get what you don’t pay for.
— John C. Bogle

Abstract

We consider an investor with $10,000 who contributes $500 monthly for 40 years, earning a 6.4% nominal gross return with inflation at 2.5%. Raising the annual fee from 0.03% to 0.66% reduces terminal wealth from $434,479 to $363,095 in today’s dollars — the difference, $71,384, amounts to 20.7% of the lifetime investment gains the market provided. The fee difference compounds geometrically. In this model every simulated scenario applies the same recurring fee drag to the same market path; unlike future returns, that cost is known in advance and remains under the saver’s direct control.

Cost of the fee
$71,384
0.03% → 0.66%, real $
Share of gains consumed
20.7%
of the market's lifetime gains
Extra months to catch up
3 yr 1 mo
for 0.66% to reach the gross balance
Terminal wealth ratio
0.83
0.66% balance ÷ frictionless

I. Assumptions

Saver profile

$
$
%/yr
yr

Market assumptions — the world

%
%
%
%
%
%

Portfolio: 80% equities · 20% bonds

Fee scenarios — the subject

0.03%Total-market index0.07%Peer index0.66%Typical active fund

II. What fees cost

Fig. 1
Fig. 1 — Wealth paths under 3 fee levels, real dollars.
ScenarioTerminal (real)Cost of feeShare of gainsCatch-upWealth ratio
Gross (frictionless)$438,242———1.00
0.03%· Total-market index$434,479−$3,7631.1%2 mo0.991
0.07%· Peer index$429,516−$8,7272.5%4 mo0.980
0.66%· Typical active fund$363,095−$75,14721.8%3 yr 1 mo0.829
The miracle of compounding returns is overwhelmed by the tyranny of compounding costs.
— John C. Bogle, The Little Book of Common Sense Investing (2007)

III. Small numbers, large consequences

Four basis points sounds like nothing. Held for 40 years, the difference between 0.03% and 0.07% on this saver’s path is $13,326 in nominal terms — $4,963 in today’s dollars. Whether that is “nothing” is a question about your hourly wage, not about arithmetic.

0.03%$1,166,604$434,479
0.07%$1,153,277$429,516
Difference$13,326$4,963
FeeNominalReal

IV. Under uncertainty

Fig. 2
Fig. 2 — simulating; 0.03% scenario, real dollars. Dashed line is the deterministic path.
Fig. 3
Fig. 3 — Per-path gap between and under common random numbers, real dollars. The gap is positive on every path: under identical markets the lower fee wins with probability 1; only the size varies.

V. Inflation and purchasing power

Inflation is not an investment fee; it is a loss of purchasing power. At a constant 2.5% annual rate, money loses half its purchasing power in about 28.1 yr. This paper therefore reports real dollars by default.

Fig. 4
Fig. 4 — The 0.03% path in nominal and real dollars; the vertical gap at year 40 is what inflation removed.
Purchasing-power half-life
≈ 28.1 yr
money loses half its value every 28.1 yr at 2.5% inflation

VI. Allocation and the risk premium

Equities have historically paid a premium over bonds for bearing greater risk. An investor cannot choose that premium; the controllable decision is how much exposure to accept. The ladder below shows the modeled wealth effect of that allocation under the stated return assumptions.

Fig. 5
Fig. 5 — Median (filled) and 5th-percentile (open) real terminal wealth by equity weight; the premium rises with equity, and so does the price of a bad outcome. real dollars.
Equity weightDeterministic (real)MC medianMC 5th pct.P(beat inflation)
0% equity$232,261———
20% equity$270,433———
40% equity$315,853———
60% equity$369,958———
80% equitycurrent$434,479———
100% equity$511,489———

VII. What actually matters

Fig. 6
Fig. 6 — Δ real terminal wealth under each one-at-a-time change. Returns and inflation move the number most, but are not yours to choose; the fee is the largest term you control.
Money management has become a loser’s game.
— Charles D. Ellis, “The Loser’s Game,” Financial Analysts Journal (1975)
▸Notes on method

Deterministic engine. Monthly recursion with the contribution at month end, net growth factor ((1+g_p)(1−f))^(1/12), contributions growing annually by g_c. The portfolio return is the blend g_p = w·g_e + (1−w)·g_b (6.40% here); the deterministic engine ignores rebalancing and volatility interactions — the Monte Carlo handles those properly. Real figures deflate by (1+π)^(m/12).

Monte Carlo. A two-asset monthly model with a geometric (median-growth) parameterization: the median compound growth equals g, while the arithmetic mean is higher by the volatility drag σ_p²/2 = 0.84%. Monthly log-returns use a Box–Muller pair per month from a mulberry32 stream.

Common random numbers. One set of monthly draws is shared by every fee scenario — fees do not move markets — so scenario differences are pure fee effects. This is the paper’s methodological signature. Every base path is paired with its sign-flipped antithetic twin for variance reduction.

Seed & paths. Seed 42 — same seed and inputs reproduce bit-identical results. Currently 10,000 effective paths.

Analytic cross-check. For a pure lump sum, the terminal-wealth ratio between two fees is exactly ((1−f₂)/(1−f₁))^T = 0.776566. On this saver’s contributing path the realized ratio is 0.835703 (contributions enter at different times, so the two differ slightly).

Delay cost. The smallest number of extra months the fee-paying saver must keep contributing (at the year-T rate, growth still net of fee) for the balance to reach the frictionless terminal, capped at 1,200 months.

Limitations. No taxes, no stochastic inflation, no withdrawal phase, no glide paths, and a single annual fee drag per scenario. Total contributions here are $250,000; the two extreme loaded fees are 0.03% and 0.66%. Educational model output, not advice.

References

  1. 1.Sharpe, W. F. (1991). “The Arithmetic of Active Management.” Financial Analysts Journal 47(1). source ↗
  2. 2.Bogle, J. C. (2007). The Little Book of Common Sense Investing — the “tyranny of compounding costs.”
  3. 3.French, K. R. (2008). “Presidential Address: The Cost of Active Investing.” Journal of Finance 63(4). source ↗
  4. 4.Investment Company Institute, Fact Book (fund expense-ratio averages). source ↗
  5. 5.Damodaran, A., historical returns and equity risk premium data (pages.stern.nyu.edu/~adamodar). source ↗
  6. 6.S&P Dow Jones Indices, SPIVA U.S. Scorecard (active vs index persistence). source ↗
  7. 7.Ellis, C. D. (1975). “The Loser’s Game.” Financial Analysts Journal 31(4), 19–26. source ↗

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← Previous · No.9The Bearer Asset: Bitcoin, Crypto, and the Price of ControlA field guide to Bitcoin value, sleeve sizing, custody, wrappers, stablecoins, programmable networks, privacy, and failure modes.Next · No.2 →A Wide & Deep PondWhy diversification changes the range of outcomes even when expected return looks similar.

Solved Problems in Personal Finance

№1 The Arithmetic of Fees·№2 A Wide & Deep Pond·№3 The Yield Illusion·№4 The Myth Ledger·№5 Retirement Money Secrets: An Evidence Review·№6 The Employer Stock Transition Guide·№7 The House Percentage: A Note for Michael Batnick·№8 What Money Is For·№9 The Bearer Asset: Bitcoin, Crypto, and the Price of Control

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