Inspect the Arithmetic — Cost of a 1% Annual Fee

Version 1.0
Last updated: March 2026

Transparent arithmetic is the operating system of this calculator.

This document publishes the formulae, computational structure, and assumptions used to generate the outputs displayed on the calculator page.

No opinions. No hidden assumptions. Just arithmetic.

Purpose

This calculator compares the ending value of an investment with no annual fee versus the ending value with a 1% annual fee, under a simple “net = gross − fee” return model with annual compounding and optional end-of-year contributions. It reports the dollar gap and the percentage reduction in ending value caused by the fee.

Definitions

Let:

P = starting balance (initial principal, CAD)
c = annual contribution (CAD), applied at the end of each year
t = investment horizon in years
r_gross = gross annual return (decimal)
f = annual fee rate (decimal)
r_net = net annual return after fee (decimal)
FV_{no_fee} = ending value with no fee
FV_{with_fee} = ending value with fee
ΔFV = dollar difference = FV_{no_fee} − FV_{with_fee}
PctReduction = ΔFV / FV_{no_fee} (reported as a percentage)

Core Equations

1. Future Value with Annual Compounding

Both paths (with and without fee) use the same closed-form future value formula with annual compounding and contributions at the end of each year:

FV(P, r, t, c) = P(1 + r)^t + c · \frac{(1 + r)^t − 1}{r}, if r ≠ 0
FV(P, 0, t, c) = P + c · t, if r = 0

2. Ending Value with No Fee

With no fee, the calculator uses the gross return r_gross directly:

FV_{no_fee} = FV(P, r_gross, t, c)

3. Ending Value with a 1% Annual Fee

Under the simplified “net = gross − fee” model, a 1% annual fee is treated as a constant subtraction from the annual return:

f = 0.01
r_net = r_gross − f
FV_{with_fee} = FV(P, r_net, t, c)

This simplified model treats the fee as a constant reduction in annual return rather than modeling the fee as a proportional deduction from assets each year.

4. Dollar Difference and Percent Reduction

The calculator reports the fee impact in dollars and as a percentage of the no-fee ending value:

ΔFV = FV_{no_fee} − FV_{with_fee}
PctReduction = \frac{ΔFV}{FV_{no_fee}}

Implementation Notes

The core arithmetic is implemented in TLM_FeeMath.fvAnnual and TLM_FeeMath.endingValueWithFee() inside /assets/js/fee-math.js, and is wired to the page by initFeeCostPage() in /assets/js/fee-ui.js.
Inputs are parsed from the fields P, years, rGrossPct, feePct, and contrib. The fee cost pages fix feePct at 1.0% or 0.5% by default but allow user edits; the underlying math is identical.
All calculations use JavaScript’s standard floating-point arithmetic. Display formatting (rounding to whole dollars and formatted percentages) occurs only in the UI layer.

If any discrepancy is identified between this documentation and the live calculator engine, the engine’s arithmetic (TLM_FeeMath + initFeeCostPage()) is the source of truth. This page will be updated to match the engine.