zmVault/timestamped/2026-01-29_17-57-00.md

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id	title	tags	daily
2026-01-29T17:57:00-0500	2026-01-29 17:57:??		2026-01-29

2026-01-29 17:57:??

Calculating Utility of Above-Minimum Mortgage Payment

Follow-up to 2026-01-25#Calculating Monthly Principal & Interest Payment:

Homeowners are often advised to make elective mortgage payments to reduce the total interest paid on the loan, but an unrelated investment with a sufficient return could outweigh the reduced loss.

Suppose you have a budget surplus of E dollars and are deciding whether to make an elective payment on your mortgage or to invest in a promising opportunity.

The return on electing to pay E to the mortgage is the ... (i.e. interest that will no longer accrue) at the end of the loan is:

R_{\text{mortgage}} = E(1+i)^{n}

[!info]- Explanation This formula may seem suspiciously straightforward, but suppose you did not contribute E. That portion of the principle would accrue interest every month at rate i. After n months, the interest accrued by that portion is given by:

E(1+i)^{n}

If the same E is invested elsewhere at monthly return j, its future value after n months takes the same form:

\text{FV}_{\text{investment}} = E(1+j)^{n}

Therefore, j must exceed i for the alternative investment to be preferable to elective payment.

Note that i and j are adjusted rates, including respect for taxes and utility.

On second thought, in a utility context, time preference could make j preferable even when slightly lower.

Short-term investments may be favored when liquidity is needed during the term, and tax deferred investments (IRA) are strongly favored over elective payment since interest is deductible (effective interest < nominal).

Calculating Effect of Elective Payment on Term Length

The monthly payment and interest rate are fixed, so the term length must decrease

\begin{align}
A &= P \cdot \frac{i(1+i)^n}{(1+i)^n-1} \\
P &= A \cdot \frac{(1+i)^n-1}{i(1+i)^n} \\
n &= \frac{\ln\left(\frac{A}{A-Pi}\right)}{\ln(1+i)} \\
\end{align}