# Invest or Pay Extra on Mortgage?

In writing this post, I discovered that this question is very common1,2,3,4, and that my treatment will be relatively simplistic (I almost didn't post it, but this blog has been dark for a while...). There are many issues to consider beyond total worth, including the liquidity of savings versus home equity, tax and tax-sheltered savings, variability in interest rates, etc.

Consider the problem where a certain amount of monthly income $E$ is available to either invest in savings (i.e., savings or money market account, CD, mutual funds, stocks, other unwieldy financial instruments5), or to make an additional payment towards a home mortgage. The relevant quantities are denoted

• $B_l$, $B_s$ - Loan or Savings Balance
• $P$ - Initial Loan (Principal)
• $R_l$, $R_s$ - Monthly Interest Rates
• $A$ - Minimum Monthly Mortgage Payment
• $N$ - Number of Payments / Investments
• $E$ - Unallocated Earnings
• $S$ - Monthly Savings ($S \leq E$)

Hence, $S$ is the portion of $E$ that is invested, where the remainder of $E$ is used to reduce the mortgage balance. Our problem is to select $S$.

The balance formulae for savings and loans are, respectively,

Derivation of these formulae6 relies on the sum of geometric series7. For simplicity, we assume that no savings have been accumulated thus far (i.e., $P_s = 0$). In this scenario, the total accumulated value is the sum of home equity and savings

By factoring $S$, we find that $W = C + S\{[(1+R_s)^N-1]/R_s - [(1+R_l)^N-1]/R_l\}$, where $C$ is constant with respect to $S$. Hence, it's optimal to save all of $E$ (i.e.,$S=E$) when $[(1+R_s)^N-1]/R_s$ is greater than $[(1+R_l)^N-1]/R_l$, but to save none of $E$ (i.e., to apply $E$ towards mortgage principle) when the opposite is true. What's interesting here, and not immediately intuitive, is that optimality depends only on the related interest rates, but not the mortgage balance!