Cash Flow Power

To fairly compare the value of real equity buildup from the mortgage (Σ_n) with monthly cash flows, we need to sum cash flows with weighting factors for the opportunity cost ( $ω$ ) of other investments.

From the price floor model, the gross monthly income of a property minus taxes (q) is expected to grow with inflation (π), while the monthly mortgage payment (p) remains fixed. We choose to adjust for this by discounting the real value of the mortgage payment by the inflation factor.

The difference each month between the gross income less taxes and the mortgage payment could be invested elsewhere (one month later) for a some average monthly return (Ω) with monthly return factor $ω = 1 + Ω$ . We define Q_n as the inflation adjusted sum of all these gains over months 1 to n of a mortgage.

Q_{n} = \sum_{m = 1}^{n} (q - p π^{1 - m}) ω^{m - 1}

Rebasing the index with $k = m - 1$ , defining $γ = ω / π$ and substituting the closed form solution for the geometric series produces the exponential gains one expects from any dividend-like investment.

Q_{n} = \sum_{k = 0}^{n - 1} (q - p π^{- k}) ω^{k}

Q_{n} = \sum_{k = 0}^{n - 1} q ω^{k} - p γ^{k}

Q_{n} = q \frac{ω^{n} - 1}{ω - 1} - p \frac{γ^{n} - 1}{γ - 1}

To simplify the presentation we substitute $Ω = ω - 1$ and $Γ = γ - 1$ . To make the equation mortgage-size agnostic by dividing through by the monthly mortgage payment (p). For the same reason we then replace the ratio of mortgage-free net cash-flow to monthly mortgage payment (q/p) with the personal credit factor to mortgage preference factor ratio (f/C), which were defined in the price floor model.

\frac{Q_{n}}{p} = \frac{R_{N} l}{R_{N}^{*} C} (\frac{ω^{n} - 1}{Ω}) - (\frac{γ^{n} - 1}{Γ}) where \begin{matrix} Ω = ω - 1 \\ γ = ω / π \\ Γ = γ - 1 \end{matrix}