Mortgage Equity Gain

The prior two articles used the linearity of $\frac{P_{0}}{p} = R_{N}$ to model a relationship between home prices and rent. However to fit this instantaneous home price model to real data, the differences from actual market prices were lumped into a previously unelaborated-upon scalar (c). This scalar suggests that Americans have a relatively consistent preference (i.e. pay more) for mortgages and their associated costs than for renting an equivalent home.

To make use of this scalar (c) we need to account for a major difference between owning with a mortgage and renting—building up equity in a loan.

For fixed rate mortgages, the principal (P) remaining after the m_th monthly payment is defined in terms of the previous month's principal, the monthly interest rate (i) and monthly payment (p).

P_{m} = i P_{m - 1} - p

In the mortgage calculator geometric series we found the closed form solution for the principal remaining after the m^th month of an N-month mortgage.

\frac{P_{m}}{p} = {\begin{matrix} \frac{1 - i^{(m - N) / months}}{i - 1} months, & i \neq 1 \\ N - m, & i = 1 \end{matrix}

The nominal equity gain in the m^th month from the mortgage is to the difference in the balance owed after the payments at the end of months m - 1 and m and due to exposure to inflation (home appreciation) of the remaining loan liability (where π is monthly home appreciation factor).

Δ E_{m} = P_{m - 1} - P_{m} + (π - 1) P_{m - 1}

Δ E_{m} = P_{m - 1} - (i P_{m - 1} - p) + (π - 1) P_{m - 1}

Δ E_{m} = P_{m - 1} (1 - i + π - 1) + p

Δ E_{m} = p + (π - i) P_{m - 1}

Substituting the closed form solution for fixed rate mortgages and defining new variables for nominal monthly interest (I) and home appreciation (Π) rates and real interest rates (R) results in

\frac{Δ E_{m}}{p} = {\begin{matrix} 1 + \frac{π - i}{I} [1 - i^{(m - 1 - N) / months}], & i \neq 1 \\ 1 + π Π (N - m + 1) / months, & i = 1 \end{matrix} where \begin{matrix} I = i - 1 \\ Π = 1 - π^{- 1} \end{matrix}

for $m \in [1 ... N]$ .

It is desirable to discount these nominal equity increases for inflation such that home values at any time m are priced in terms of dollars at the initial purchase/loan. Assume the burden of the monthly mortgage payment and principal remaining is reduced by the inflation factor (π) each month. We will focus on the case i ≠ 1.

\frac{Δ {E'}_{m}}{p} = (1 + \frac{π - i}{I}) π^{- m} + (\frac{π - i}{I i^{N}}) \frac{r^{m - 1}}{π} where r = \frac{i}{π}

\frac{Δ {E'}_{m}}{p} = (\frac{i - 1 + π - i}{I}) π^{- m} - (\frac{R}{I i^{N}}) r^{m - 1} where R = r - 1

\frac{Δ {E'}_{m}}{p} = (\frac{Π}{I}) π^{1 - m} - (\frac{R}{I i^{N}}) r^{m - 1}

Summing the inflation-discounted monthly equity gains from the first to n^th month and once again using a geometric series (and aritmetic series) yields the inflation-discounted equity buildup Σ'_n from the fixed rate mortgage:

\frac{{Σ'}_{n}}{p} = \sum_{m = 1}^{n} \frac{Δ {E'}_{m}}{p}

\frac{{Σ'}_{n}}{p} = (\frac{Π}{I}) \sum_{m = 1}^{n} π^{- m} - (\frac{R}{I i^{N}}) \sum_{m = 1}^{n} r^{m}

\frac{{Σ'}_{n}}{p} = (\frac{Π}{I}) \frac{1 - π^{- n}}{1 - π^{- 1}} - (\frac{R}{I i^{N}}) \frac{r^{n} - 1}{r - 1}

\frac{{Σ'}_{n}}{p} = (\frac{Π}{I}) \frac{1 - π^{- n}}{Π} - (\frac{R}{I i^{N}}) \frac{r^{n} - 1}{R}

Recombining this solution with the other case i = 1,

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

The semantic description of this equation is a little tricky. It describes the real equity change from the mortgage due to paying off and inflation-discounting principal while the underlying portion of the asset owned with debt maintains constant value. Further, this mortgage equity gain is adjusted for inflation into dollars equivalent to those at loan origination.

Phantom Capital Gains

We have calculated equity buildup from paying off the mortgage plus capital gains from inflation of the mortgaged portion of property, but what about capital gains of the owned portion of propery due to inflation and appreciation?

It can be shown that, solving a similar capital gain summation adjusted for inflation, the following equation is true:

{E'}_{Inflation Discounted Equity Dollars Paid/Gained} + {Σ'}_{Inflation Discounted Equity Capital Gains} = E_{"Inflation Adjusted" Equity Purchased}

Consider the capital gains from the portion of property owned by the down payment (E₀), the nominal equity gain after the first n months due to inflation would be

Σ_{ECG(n)} = E_{0} (π^{n} - 1)

Adjusting this gain for inflation to make it equivalent in value to dollars at the beginning of the loan period,

{Σ'}_{ECG(n)} = π^{- n} E_{0} (π^{n} - 1)

{Σ'}_{ECG(n)} = E_{0} (1 - π^{- n})

There remains the nominal base value that you would not get taxed for, but after discounting for inflation its real value has decreased.

{E'}_{EDP(n)} = E_{0} π^{- n}

Summing the discounted value of the initial down payment and capital gains from this portion of the property valuation due to inflation shows that, although the taxman may disagree, there were no real capital gains here.

{E'}_{EDP(n)} + {Σ'}_{ECG(n)} = E_{0} π^{- n} + E_{0} (1 - π^{- n})

{E'}_{EDP(n)} + {Σ'}_{ECG(n)} = E_{0}

In conclusion, if home prices appreciate only with inflation, expressing all dollar values in "inflation back-adjusted" dollars keeps the numeric value of equity built-up from the mortgage and from the down payment steady over time.

What if home prices appreciate faster than inflation? This series breaks home price appreciation into three components:

Underlying rent inflation (π), as determined by market measurement,
A scalar (R_th) predicted by interest rates and other inputs to the price model and
A market forces/human factor term (c), which we call the mortgage preference factor.

This series assumes only underlying inflation (affecting rents and other material inputs to the price model) to calculate capital gains because both interest rates and mortgage preference factors are expected to oscillate and should be accounted for but not counted upon to cause appreciation over the long term.

Put politically, the Federal Reserve may make you feel richer or poorer and scale wealth inequality in nominal terms, but in the long term it is better to account for these actions so you can survive them, take advantage of them, and vote intelligently.