Refinance Tangent

This is the seventh of a twelve-part series for comparing real estate to other investments like stocks. This part focuses on how long a mortgage should be held before refinancing if all of the terms stay the same, interest, inflation, market valuation after inflation, etc.

We previously found the nominal equity buildup ∑_n in mortgages with monthly payment p and the performance G_n of real estate investments after n months, in terms of the mortgage months-to-value ratio (R_N), inflation (π) and an opportunity cost (ω) of other investments.

G_{n} = \frac{L}{R_{N}} (\frac{ω}{c} (\frac{ω^{n} - 1}{ω - 1}) - \frac{ω}{π} (\frac{{(\frac{ω}{π})}^{n} - 1}{(\frac{ω}{π}) - 1}) + \frac{Σ_{n}}{p} π^{- n} - T) for ω > π \geq 1

for $c = \frac{p}{c}$ mortgage preference factor, $T = \frac{F_{1} + F_{2}}{p}$ time underwater and $L = \frac{p \times R_{N}}{E_{0} + F_{1}}$ effective leverage ratio.

For example if $G_{n} = 0.1$ , this would be like saying your IRA is up 10% after $n$ months. Like other lowercase/capital variables, we define $g = 1 + G$ as the fractional performance of a real estate investment over $n$ months. From this, the average fractional yield ( $y$ ) is a geometric average, i.e. the $n$ th root of $g$ .

y = g^{1 / n} = g^{h}

When bought with debt, the yield of real estate investments depends on $n$ , the number of months a mortgage is held before selling or refinancing. To optimize this we take the derivative of the fractional yield using partial derivatives and the chain rule.

\frac{d y}{d n} = \frac{\partial y}{\partial g} \times \frac{d g}{d n} + \frac{\partial y}{\partial h} \times \frac{d h}{d n}

\frac{d y}{d n} = h g^{h - 1} \times \frac{d g}{d n} + ln (g) g^{\frac{1}{n}} \times \frac{- 1}{n^{2}}

\frac{d y}{d n} = \frac{g^{1 / n}}{n} (\frac{1}{g} \frac{d g}{d n} - \frac{ln g}{n})

Setting the derivative to zero and substituting for G gives our optimization equation for n, for $n > 0$ .

n \frac{d G}{d n} = (G + 1) ln (G + 1)

Next we define $Ω = ω - 1$ , $Π = π - 1$ , $ρ = ω / π$ , and $Ρ = ρ - 1$ and make five assumptions to simplify the optimization function.

Monthly opportunity cost yield is positive but significantly less than one and monthly inflation is positive but less than private opportunity cost.
$0 < Ω ≪ 1 and 0 \leq Π < Ω ∴ 0 < R < Ω$
This allows us to ignore the linear edge cases from the geometric series in the performance equation.
Based on assumption (1), we make the algebra and computation simpler by taking Taylor Series expansions for ln(x) about x=1. Thus for ω, π, and ρ in place of x,
$ln x \approx X where X = x - 1$
For $ln (G + 1)$ , we assume it can be approximated by a first order Taylor Series about a performance of zero ( $G \approx 0$ ). Thus the optimization function becomes:
$n \frac{d G}{d n} = (G + 1) G$
The power functions that make up G can be approximated by first order Maclaurin series, and assumption (2) for the resulting logarithm can be reapplied:
$x^{n} \approx 1 + n ln x \approx 1 + X n$
We assume real interest rates are zero, thus $Σ_{n} = pn$ .

Applying the above assumptions simplify the performance function to:

G_{n} = \frac{L}{R_{N}} (A n - T) where A = c ω + 1 - ρ

Next we solve $(n \frac{d G}{d n} = G^{2} + G)$ with a sixth assumption, (6) that the derivative of G can be further approximated by the derivative of the linear approximation of G.

\frac{L}{R_{N}} A n = {(\frac{L}{R_{N}})}^{2} {(A n - T)}^{2} + \frac{L}{R_{N}} (A n - T)

T \frac{R_{N}}{L} = {(A n - T)}^{2}

n = \frac{T + \sqrt{\frac{R_{N} T}{L}}}{c ω + (1 - ρ)} \approx \frac{1}{c} (T + \sqrt{\frac{R_{N} T}{L}})

Qualitatively we see the optimum refinance period (n) grows longer with larger refinance fees (T) and lower nominal interest rates or longer loan periods (R_N) and grows shorter with higher leverage ratios (L) and cash-flow ratio (c).

Since the leverage ratio (L) and cash-flow ratio (c) are inversly related in terms of down payment and the mortgage months-to-value ratio (R_N) and months-of-rent fee (T) are inversly related in terms of loan period, options available to the consumer have little impact on the optimum refinance period, with the exception of points that both lower the interest rate and raise upfront fees. Note this equation does not account for interest rates and inflation changing.

For example for my first two properties, on the one above I chose to maximize points on a 30-year mortgage because I though interest rates would rise. However on the property below I chose no points and a 15-year mortgage in hopes that interest rates lower in the next 5 years.

The examples show this simplified quadratic model is usually ok for a yield estimate, but it has plenty of room for improvement especially at estimating the time. The next article improves upon this quadratic estimate by revisiting assumptions 3-6 to make the cubic estimate also shown on the examples above.

This is useful for improving the calculators but has too much detail to gain any insigts. The next interesting article takes

This is the seventh part of a series Should I Buy or Rent?

As a powerful argument on the nature of fiat currency, as real interest rates trend toward zero, the optimum period for owning a mortgage liability becomes approximatable analytically!

The inflation (i) adjusted yield depends on the leverage ratio (L) and the equity buildup Σ(n)/p, and these equation and all the other terms are defined in those articles. For here, the combined yield (Y) is composed of two parts, the cash flow dividend (X) made each month and an ammortized dividend from equity buildup (y) when you sell or refinance. These two parts are demonstrated in the graph below.

To help determine an approximate time for peak yields, there are three assumptions we will use.

Inflation and interest rates are nearly equal.
Inflation is low enough a first order approximation of the natural log holds.
The leverage ratio is sufficiently large and external yield (X) sufficiently flat that the peak net yield (Y) occurs nearly concurrently with equity yield (y).

Maximizing Equity Yield

The rate of equity buildup, in inflation back-adjusted terms is:

If we assume real interest rates are zero, or that inflation equals the nominal interest rate (i = r), then equity buildup becomes linear, Σn/p = n, and the first intersection point of an "refinance window" can be found quadratically. This optimum "window" represents the tangent point for capital gains from equity buildup (blue), equivalently compared to regular dividends from stocks or rental cash flow (dotted black).

Optimum Point Derivation

To make the algebra easier, we define some helper variables and then take the derivative of the equity yield (y) to find its optimum.

Setting the derivative equal to zero, substituting n for Σn/p, and assuming g is near 1 to approximate ln(g) = g - 1 results in a quadratic equation we can solve.

The long tangent of the last three articles was to reach this point. Now the mortgage calculator can estimate yields from the loan liability for comparisons to stocks or other investments!