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+# 2026-01-29 17:57:??
+
+#topic/finance
+
+### Calculating Utility of Above-Minimum Mortgage Payment
+
+See [[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
+
+$$
+\begin{align}
+A &= P \cdot \frac{i(1+i)^n}{(1+i)^n-1} \\
+P &= A \cdot \frac{(1+i)^n-1}{i(1+i)^n} \\
+n &= \frac{\ln\left(\frac{A}{A-Pi}\right)}{\ln(1+i)} \\
+\end{align}
+$$