Inspectable Arithmetic for the Pay Off Mortgage vs Invest Calculator (Cash-Flow Neutral)

Version 1.2
Last verified: May 2026

Transparent arithmetic is the operating system of this calculator.

This document publishes the full 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 mortgage paydown versus investing under explicit rules. The user chooses one of two mutually exclusive ways to supply discretionary cash in addition to the regular mortgage payment P:

Monthly extra cash: a fixed extra amount E every month. While a mortgage exists, P + E is split between the mortgage (scheduled payment plus allocated extra paydown) and the portfolio. After payoff, the full monthly budget P + E is invested each month.
Lump sum at start: a one-time amount S applied at the beginning of month 1 only (before that month’s mortgage interest and scheduled payment). There is no recurring E in this mode. The same allocation slider splits S between extra principal and the portfolio. After payoff, only P is invested each month (because there was no recurring extra in this mode).

The calculator reports mortgage balance, investment portfolio value, and net worth over time under constant return assumptions.

For the narrative framing behind the cash-flow-neutral comparison, see Pay Off Your Mortgage Faster or Invest?

Definitions

All accruals below are monthly time steps unless noted.

Inputs

B₀: initial mortgage balance
P: regular mortgage payment per month
Mode (exclusive): either monthly extra cash (inputs E and optional monthly budget tied to P) or lump sum (input S ≥ 0). The engine sets E = 0 in lump mode regardless of other fields.
E: extra cash available per month (monthly mode only; E = 0 in lump mode)
S: lump sum at start of month 1 (lump mode only; S = 0 in monthly mode)
a: allocation slider (0–100), interpreted as % of the relevant discretionary cash allocated to investing (right end of slider). The remainder goes to extra mortgage principal.
Recurring extra to mortgage: E_M = (1 − a/100) E
Recurring extra to invest: E_I = (a/100) E
r: Canadian nominal annual mortgage rate, compounded semi-annually (%)
R: expected investment return before fees (annual, %)
f: investment fees (annual, %)
g: home price growth rate (annual, %)
H₀: current home value
T: time horizon in years
N = 12T: time horizon in months

Derived constants

Mortgage monthly rate:

i = (1 + (r/100) / 2)^1/6 − 1

Canadian mortgage rates are generally quoted as nominal annual rates compounded semi-annually. The calculator converts the entered mortgage rate to an equivalent monthly rate before applying mortgage interest.

Net annual investment return (after fees):

R_net = (R − f) / 100

Monthly geometric investment return:

μ = (1 + R_net)^1/12 − 1

(Clamped internally so that R_net ≥ −0.999.)

Monthly budget used after the mortgage is paid off (same symbol in both modes):

Budget = P + E

In lump mode, E = 0, so Budget = P. In monthly mode, Budget = P + E.

Mode A: Mortgage Payment Computed From Mortgage Inputs

If Use mortgage calculator is enabled (the mode that reuses payment logic from the Canada Mortgage Calculator), the calculator computes the regular mortgage payment P using the standard amortization formula.

Let:

L = H₀ − DownPayment
i = (1 + (r/100) / 2)^1/6 − 1
n = 12 · AmortYears

Then:

P = L/n if r = 0
P = L · i / (1 − (1+i)⁻ⁿ) if r > 0

The simulation then proceeds using B₀ = L and that computed P.

Mode B: Mortgage Payment Entered Manually

If Use mortgage calculator is disabled, the user supplies: B₀ (current mortgage balance), r (Canadian nominal annual mortgage rate), P (monthly payment). The payoff time becomes whatever amortizes under those values.

Fallback: If B₀ ≤ 0 in manual mode, the engine estimates a principal from payment using a rough default (25 years, 5% Canadian nominal annual mortgage rate). This is a non-ideal fallback intended to keep the calculator functioning.

Core Simulation (Monthly Recurrence)

The simulation advances month-by-month for t = 1, 2, …, N. State at the end of month t is written B_t, V_t; month 0 records initial B₀, V₀ = 0.

Lump-sum pre-step (only month t = 1, only if S > 0)

If lump mode is active and S > 0, the engine applies the following before interest due and the scheduled payment for that month. Let B⁻ denote the mortgage balance at the start of the iteration (for t = 1, this is B₀), and V⁻ the portfolio at the start of the iteration (for t = 1, this is V₀ = 0).

Split the lump using the same slider a as for recurring extra:

S_M = (1 − a/100) S (portion intended for mortgage principal)
S_I = (a/100) S (portion intended for investing)

Extra principal from the lump cannot exceed the outstanding balance. Any mortgage-intended amount beyond what is needed to prepay the balance is redirected to the portfolio:

LumpPrincipal = min(max(0, S_M), B⁻)
Overflow = S_M − LumpPrincipal

Update balance and portfolio before the rest of the month:

B′ = max(0, B⁻ − LumpPrincipal)
V′ = V⁻ + S_I + Overflow

For t = 1, the simulation then continues using B′ and V′ as the starting balance and portfolio for that same month’s interest, mortgage payment, contributions, and compounding (below). For t > 1, this pre-step is skipped (S is not applied again).

Home value

Home value is compounded in monthly steps:

H_t = H₀ · (1 + g/100)^t/12

Equity and net worth (end of month)

Equity_t = max(0, H_t − B_t)
NetWorth_t = Equity_t + V_t

where V_t is investment portfolio value.

Case 1: Mortgage Still Outstanding at Start of Month t After Any Lump Pre-step

For t = 1, “B_t−1” in the interest formula below is the balance after the lump pre-step when S > 0 (i.e. B′ above); for t > 1 it is the prior month’s ending balance. The same applies to V when composing contributions.

Step 1 — Mortgage interest due

InterestDue_t = B_t−1 · i

Step 2 — Intended mortgage payment

Regular payment always goes to the mortgage, plus the allocated portion of extra cash:

M_t^intended = P + E_M

But payment cannot exceed what is needed to close the mortgage this month:

M_t^actual = min(M_t^intended, B_t−1 + InterestDue_t)

Step 3 — Principal paid and new balance

PrincipalPaid_t = max(0, M_t^actual − InterestDue_t)
B_t = max(0, B_t−1 − PrincipalPaid_t)

If M_t^actual > 0, interest paid increments by InterestDue_t.

Step 4 — Investment contribution while mortgage exists

The investment contribution has two parts: the allocated investing portion of extra cash E_I, and any remainder from an over-large intended mortgage payment in the final payoff month:

Remainder_t = M_t^intended − M_t^actual
C_t = E_I + Remainder_t

Step 5 — Investment update (contribution first, then compound)

V_t′ = V_t−1 + C_t
InterestEarned_t = V_t′ · μ
V_t = V_t′ · (1 + μ)

Investment value is clamped to never go below zero. For t = 1 with S > 0, V_t−1 is the portfolio already increased by the lump pre-step (S_I + Overflow).

Case 2: Mortgage Paid Off (B_t−1 ≤ 0)

Once the mortgage balance reaches zero (including after the lump pre-step), the engine invests the entire monthly budget:

C_t = Budget = P + E

In lump mode, E = 0, so each post-payoff contribution is C_t = P.

Then applies the same “contribution first, then compound” update:

V_t′ = V_t−1 + C_t
InterestEarned_t = V_t′ · μ
V_t = V_t′ · (1 + μ)

Reported Outputs

At the end of the horizon t = N, the calculator reports:

Mortgage balance: B_N
Investment portfolio value: V_N
Home value: H_N = H₀(1 + g/100)^T
Net worth: max(0, H_N − B_N) + V_N
Total interest paid: Σ InterestDue_t over months where a mortgage payment is made
Total interest earned: Σ InterestEarned_t

Key facts (extreme allocations)

The “Key facts” section computes two additional full simulations using identical inputs: 100% mortgage allocation (a = 0) and 100% investing allocation (a = 100). In monthly mode, that means all of E to mortgage versus all of E to investing. In lump mode, that means all of S to extra principal versus all of S to the portfolio at the start of month 1. The same recurrence (including the lump pre-step when applicable) is used.

Break-even gross return

The calculator may report a gross return R (before fees) at which the two extreme strategies tie at the horizon: a = 0 versus a = 100, holding all other inputs fixed. In monthly mode, both paths use the same E. In lump mode, both paths use the same S and E = 0. If there is no discretionary cash (E = 0 and S = 0), this quantity is not defined.

Assumptions

All calculations are performed in monthly time steps.
The entered mortgage rate is treated as a Canadian nominal annual mortgage rate compounded semi-annually, then converted to the equivalent monthly rate before interest is applied to the outstanding balance.
Investment returns are modeled as constant and deterministic.
Investment fees are modeled as a constant annual % subtracted from expected return, then converted to a monthly geometric return.
Contributions are applied before investment growth each month (beginning-of-period contributions).
In lump-sum mode, S is applied only once, at the beginning of month 1, before that month’s mortgage interest and scheduled payment; it is not repeated in later months.
Home price growth is modeled as constant compounding at rate g (no volatility).
Taxes, transaction costs, refinancing, prepayment penalties, and behavioral changes are not modeled unless explicitly added elsewhere.

Implementation Notes

The slider allocates discretionary cash only: recurring (E) in monthly mode, or the lump (S) in lump mode. The regular payment P always goes to the mortgage until payoff.
Monthly and lump inputs are mutually exclusive in the UI; the engine does not apply both E and S in the same run.
The mortgage recurrence uses i = (1 + (r/100) / 2)^1/6 − 1 for monthly mortgage interest, not simple APR/12 division.
In the payoff month, any portion of the intended mortgage payment not required to close the mortgage is automatically redirected into investing (“remainderFromMortgage”).
After payoff, the engine invests Budget = P + E each month (so P only when E = 0).
Projected payoff month search uses the same lump pre-step on month 1 when S > 0, then the same monthly mortgage recurrence with E_M from recurring E only.

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)

If any discrepancy is identified between this documentation and the calculator output, the arithmetic in the engine governs.