How Mortgage Leverage Changes with House Price Growth

Summary

The document frames a mortgage from the borrower's perspective, expressing the value of the house less the present value of remaining mortgage payments. The original question asks how down payment, interest rate, and maturity affect that value, and how to calculate the borrower's exposure to house-price returns. It also connects the calculation to a broader comparison between mortgage costs and real estate returns.

The response gives a simplified limiting case with an infinite mortgage term. In that case, the liability is treated as a fixed amount tied to the original purchase price and down-payment share. The resulting return sensitivity is the house's percentage price move multiplied by a leverage factor that depends on current price relative to initial price and the initial equity share. Under positive constant house-price growth, the factor declines toward one; with no growth it remains at its initial level, while negative growth can make the expression unstable. This is an illustrative model, not a treatment of finite maturity, amortization, taxes, or changing rates.

Key ideas

• Borrower equity can be represented as house value minus the present value of remaining mortgage payments.
• The response derives leverage for an infinite-term mortgage as a function of house-price appreciation and initial equity.
• With positive constant growth, the leverage factor approaches one over time.
• With zero growth the factor stays at its initial level, while negative growth can produce unstable behavior.
• The simplified result does not resolve the finite-maturity mortgage case or include other ownership costs.

Tags

# What's the rate of return on a mortgage?


# What's the rate of return on a mortgage?












I'm trying to understand mortgages from first principles, from the perspective of a borrower.

Let $S_t$ be the price of the asset bought with the loan at time $t$ (i.e. house). Let $\alpha$ be the downpayment ratio, $r$ the interest rate, $T$ the maturity of the mortgage and $c$ the continuously paid (for simplicity) coupon rate.

The borrower's value at time $0 \le t \le T$ is given by \begin{align} V_t &= S_t - S_0 c \int_t^T e^{-r(u-t)} \mathrm{d} u \\ &= S_t - S_0 \frac{c}{r} \left[ 1 - e^{-r(T-t)} \right] \end{align}

The par coupon with downpayment $\alpha S_0$ is given by $$ c = r \frac{ 1 - \alpha}{1 - e^{-rT}} $$ so that $V_0 = \alpha S_0$.

I can't seem to answer simple questions like

- Of the 3 parameters $\alpha$, $r$, $T$ can one be eliminated wlog?

- Is there a closed form or approximate way to get the leverage $\frac{\mathrm{d}V_t/V_t}{\mathrm{d}S_t/S_t}$, even for a simple model like $\mathrm{d}S_t/S_t = \text{const}$?

I would like to eventually get to understanding the breakeven between mortgage rates and real estate returns.

Would appreciate any pointers!

## Answer by user357269 (score 2)

https://quant.stackexchange.com/a/79504

If $T = \infty$, $V_t$ simplifies to $$V_t = S_t - S_0 (1 - \alpha) $$ and we have $$ \frac{\mathrm{d} V_t}{V_t} = \frac{\mathrm{d} S_t}{S_t} \frac{1}{1 - \frac{S_0}{S_t}(1-\alpha)}.$$ Letting $S_t = S_0 e^{\mu t}$, the leverage is $$ \frac{1}{1 - \frac{S_0}{S_t}(1-\alpha)} = \frac{1}{1 - e^{-\mu t }(1-\alpha)}.$$ which starts at $1/(1-\alpha)$ and tends to 1 if $\mu > 0$, stays at its initial value if $\mu = 0$, gets weird if $\mu < 0$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.