Real Performance with Mortgage Leverage

This is the sixth part of a series presenting the assumptions and equations behind the mortgage calculator. The fourth article found a function for the buildup of real equity from a mortgage (Σ'_n) by its n^th month, which was adjusted for inflation to be in terms of dollars at the beginning of the loan period.

\frac{Q_{n}}{p} = \frac{R_{N} G_{L}}{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}

The fifth article found a function for the buildup of real wealth from monthly net cash flow (Q_n) after the n^th month, including the opportunity cost of a monthly mortgage payment vs. owning the property mortgage-free.

\frac{{Σ'}_{n}}{p} = {\begin{matrix} \frac{1}{I} (1 - π^{- n} + i^{1 - N} (r^{n} - 1)), & i \neq 1 \\ i = 1 \end{matrix} for \begin{matrix} r = i / π \\ I = i - 1 \end{matrix}

The fractional performance over $n$ months, from purchase to sale, includes the real equity built-up from the mortgage (Σ'_n), real wealth gained from monthly cash flows (Q_n), capital gains from equity appreciation, plus the initial equity owned (E₀) and less a selling transaction fee (F₂), all relative to the down payment and initial purchase costs (E₀ + F₁).

y^{n} = \frac{{Σ'}_{n} + Q_{n} + \sum_{Equity Appreciation Capital Gains} + E_{0} - F_{2}}{E_{0} + F_{1}}

If all of these values are expressed in real terms, such that they are inflation adjusted to be equivalent to dollars at the beginning of the loan period and if, at any time, equity owned is assumed to appreciate at the rate of underlying inflation (π), then there will be no capital gains from equity appreciation when expressed in these real terms.

y^{n} = \frac{{Σ'}_{n} + Q_{n} + E_{0} - F_{2}}{E_{0} + F_{1}}

Using constant dollars allows us make another simplification, by treating your future selling fee as equivalent to those of the person selling us the property now, we are able to think about the cost of both fees, the buyer's fee ( $F_{1}$ ) and seller's fee ( $F_{2}$ ) as single overall transaction cost.

\frac{E_{0} - F_{2}}{E_{0} + F_{1}} = \frac{E_{0} - F_{2} + F_{1} + F_{2} - (F_{1} + F_{2})}{E_{0} + F_{1}} = 1 - \frac{F_{1} + F_{2}}{E_{0} + F_{1}}

y^{n} = 1 + \frac{{Σ'}_{n} + Q_{n} - (F_{1} + F_{2})}{E_{0} + F_{1}}

To normalize the fractional performance such that it is independent of property value and applies to all loan sizes, we define the effective leverage ratio $L = \frac{p \times R_{N}}{E_{0}+F_{1}}$ as the market value relative to the initial investment (down payment plus buyer's fees).

y^{n} = 1 + \frac{L (E_{0} + F_{1})}{p R_{N}} \times \frac{{Σ'}_{n} + Q_{n} - (F_{1} + F_{2})}{E_{0} + F_{1}}

y^{n} = 1 + \frac{L}{R_{N}} (\frac{{Σ'}_{n}}{p} + \frac{Q_{n}}{p} - \frac{F_{1} + F_{2}}{p})

Next, the differential performance after n months is the fractional performance minus one (Y_n = yⁿ - 1).

Y_{n} = \frac{L}{R_{N}} (\frac{{Σ'}_{n}}{p} + \frac{Q_{n}}{p} - f R_{N}) where L = \frac{P_{0}}{E_{0} + F_{1}} and f = \frac{F_{1} + F_{2}}{P_{0}}

Finally we substitute in the real equity buildup from the loan and exponential opportunity gains from positive cash flow:

Y_{n} = \frac{L}{R_{N}} [\frac{1 - π^{- n}}{I} - \frac{r^{n} - 1}{I i^{N}} + \frac{R_{N} l}{R_{N}^{*} C} * \frac{ω^{n} - 1}{Ω} - \frac{γ^{n} - 1}{Γ} - f R_{N}] for i \neq 1, π \neq 1, ω > 1, γ > 1 .

Y_{n} = L * [(\frac{G_{N}^{*} l}{C}) \frac{ω^{n} - 1}{Ω} + \frac{1}{R_{N}} (\frac{1 - π^{- n}}{I} - \frac{r^{n} - 1}{I i^{N}} - \frac{γ^{n} - 1}{Γ}) - f]

This equation describes the differential performance of a mortgage investment after n months. For example Y_n = 0.4 would mean a gain of 40% of the initial investment after n months.

We can see how performance is affected by the following factors:

the leverage ratio (L) due to the down payment percentage,
the ratio (R_N) of initial mortgage pricipal compared to the monthly mortgage payment, which is a function of the nominal interest rate (i) and loan period (N),
inflation (π) induced mortgage equity buildup due to the reduced burden of the remaining loan pricipal relative to home value,
principal payoff in terms of real interest rates (r = i/π) reflecting the larger fraction of the monthly payment going toward principal each month.
yield of other investments (ω) potentially made from postive monthly cash flows, affected strongly by the mortgage preference factor (c),
opportunity cost (γ = ω/π) of the monthly mortgage payment, which is less than the yield of other investments over time due to discounting of the monthly mortgage payment due to inflation (π) and
a recoupment period (T) of the number of monthly mortgage payments equal to the total transaction costs of buying a home and loan origination.