Inspect the Arithmetic — Active vs Passive Break-Even

Version 1.1
Last verified: May 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 an active strategy to a passive strategy under a simple “net = gross − fee” model. It reports:

the break-even excess return (alpha) implied by the fee difference between active and passive,
the break-even active gross return that would exactly match passive net performance, and
ending portfolio values for both strategies under the user’s assumed active portfolio return and at the break-even return.

Definitions

Let:

P = starting balance (initial principal, CAD)
c = contribution amount in CAD per period (weekly, monthly, or annual, depending on user selection)
m = contribution periods per year (52 for weekly, 12 for monthly, 1 for annual)
t = investment horizon in years
r_passive = passive portfolio return before fees (decimal)
r_active = assumed active portfolio return before fees (decimal)
f_passive = passive annual fee rate (decimal)
f_active = active annual fee rate (decimal)
α_break = break-even excess return required by active over passive
r_active,break = break-even active gross return
r_passive,net = r_passive − f_passive
r_active,net = r_active − f_active
r_{active,net,break} = r_active,break − f_active
FV_annual(P, r, t, c) = future value with annual compounding and one end-of-year contribution of c when m = 1
FV_periodic(P, r, t, c, m) = future value when the net annual rate r compounds m times per year and contribution c is added at the end of each of the m·t periods (same convention as the Simple Investment Calculator)

Core Equations

1. Future Value with Contributions

When contributions are annual (m = 1), ending values use one compound period per year and a single end-of-year contribution:

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

When the user selects weekly (m = 52) or monthly (m = 12) contributions, the implementation uses the same effective per-period rate as the Simple Investment Calculator: i = (1 + r)^1/m − 1, with N = m·t periods and contribution c at the end of each period:

FV_periodic = P(1 + i)^N + c · \frac{(1 + i)^N − 1}{i}, if i ≠ 0
FV_periodic = P + c · N, if i = 0

2. Break-Even Alpha from Fee Difference

Break-even is defined as the point where passive and active net returns are equal:

r_passive − f_passive = r_active,break − f_active

Rearranging for the break-even active gross return and identifying the implied excess return:

r_active,break = r_passive + (f_active − f_passive)
α_break = f_active − f_passive

The calculator displays α_break as “Break-even extra return / yr (fee gap)” and r_active,break as “Break-even active return (gross)”.

3. Net Returns Used for Ending Values

The net annual returns used in the future value calculations are:

r_passive,net = r_passive − f_passive
r_active,net = r_active − f_active
r_{active,net,break} = r_active,break − f_active

4. Ending Values and Differences

Using the net returns and contributions, the calculator computes three ending values:

FV_passive = FV(P, r_passive,net, t, c, m)
FV_{active,assumed} = FV(P, r_active,net, t, c, m)
FV_active,break = FV(P, r_{active,net,break}, t, c, m)

Here FV denotes FV_annual when m = 1 and FV_periodic when m = 12 or 52, using the corresponding net return r in the formula for that path.

The difference between active and passive under the assumed active return is:

ΔFV = FV_{active,assumed} − FV_passive

Implementation Notes

All future values are computed via TLM_FeeMath.endingValueWithFee() in /assets/js/fee-math.js, using the specified gross return and fee pair for each path. Optional contribFreq of "monthly" or "weekly" selects periodic compounding; otherwise contributions are treated as annual end-of-year cash flows.
The page-specific logic in initActiveVsPassivePage() inside /assets/js/fee-ui.js computes α_break, r_active,break, and the three ending values, then formats them for display.
The equality of net returns at break-even means that, under this model and horizon, FV_passive and FV_active,break are equal up to floating-point and rounding noise.

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

Sources and References

Canada Revenue Agency (CRA)
Government of Canada
Statistics Canada
Bank of Canada
CMHC (housing and mortgage context, where relevant)
OSFI (financial institution supervision, where relevant)
SPIVA Canada (long-horizon fund performance context, where relevant)
Robert Shiller — historical data (market and long-horizon data context, where relevant)