Inspect the Arithmetic — Required Return to Offset Investment and Management Fees

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 answers two related questions under a simple “net = gross − fee” return model:

What additional annual return (alpha) is required to fully offset an annual fee?
If that excess return is not earned, how much extra cash must be contributed each year to end up at the same value you would have had with no fee?

Definitions

Let:

P = starting balance (initial principal, CAD)
c = baseline annual contribution (CAD), applied at the end of each year
c_extra = additional annual contribution required to offset the fee
t = investment horizon in years
r = gross annual return baseline (decimal)
f = annual fee rate (decimal)
α = additional annual return (alpha) required to offset the fee
r_net = net annual return after fee = r − f
FV(P, r, t, c) = future value with annual compounding and end-of-year contributions

Core Equations

1. Future Value with Annual Compounding

The calculator uses the same closed-form future value function as the fee-cost pages:

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. Reference Paths

Two reference paths are defined:

No-fee path:
FV_{no_fee} = FV(P, r, t, c)
With-fee path (no extra return, same contributions):
r_net = r − f
FV_{with_fee} = FV(P, r_net, t, c)

The “ending value lost to fees” shown on the calculator page is simply FV_{no_fee} − FV_{with_fee}.

3. Required Excess Return (Alpha)

Under the “net = gross − fee” model, offsetting the fee on a percentage basis requires that the net-with-fee return match the no-fee return:

r = (r + α − f) ⇒ α = f

The calculator therefore reports:

α = f
Required gross return = r + α = r + f

4. Extra Annual Contribution Required

If excess return α is not achieved, the calculator solves for c_extra such that a higher contribution level exactly compensates for the fee drag:

FV(P, r, t, c) = FV(P, r − f, t, c + c_extra)

For r − f ≠ 0, define:

g = (1 + (r − f))^t
A = \frac{g − 1}{r − f}

Then, expanding both sides and solving algebraically for c_extra gives:

c_extra = \frac{FV(P, r, t, c) − P·g}{A} − c

When r − f = 0, the net-with-fee path is a flat-growth path and the formula reduces to:

c_extra = \frac{FV(P, r, t, c) − P}{t} − c

Implementation Notes

The functions TLM_FeeMath.fvAnnual, TLM_FeeMath.endingValueWithFee(), and TLM_FeeMath.extraAnnualContributionToOffsetFee() in /assets/js/fee-math.js implement the equations above.
The page-specific logic in initRequiredReturnPage() inside /assets/js/fee-ui.js computes:
- α (as a percentage) and the required gross return r + α,
- FV_{no_fee}, FV_{with_fee}, and their difference,
- c_extra and the associated SEO/callout sentence.
All values are computed in continuous decimal form and then formatted as percentages or whole-dollar CAD amounts at the display layer.

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