Solved Problems

№1 in the series

The Arithmetic of Fees

an interactive working paper · drewbreyer.com

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. We show the cost of fees compounds geometrically, survives every simulated market path, and is the largest determinant of terminal wealth 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

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, the fee everyone pays

Inflation is the one fee no provider will waive. At 2.5% annually, money loses half its purchasing power every 28.1 yr — which is why this paper 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. The price of safety

Equities have historically paid a premium over bonds for bearing their volatility. The ladder below prices that premium under our assumptions — and prices the safety of leaving it unclaimed.

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.
▸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).
  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).
  4. 4.Investment Company Institute, Fact Book (fund expense-ratio averages).
  5. 5.Damodaran, A., historical returns and equity risk premium data (pages.stern.nyu.edu/~adamodar).
  6. 6.S&P Dow Jones Indices, SPIVA U.S. Scorecard (active vs index persistence).

Continue the series →

№2 A Wide & Deep Pond

Why diversification is the solved core of retirement investing — and an honest price on gambling anyway.

Solved Problems in Personal Finance

№1 The Arithmetic of Fees·№2 A Wide & Deep Pond·№3 The Yield Illusion

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