Questions that come up in scenarios related to buying or refinancing a home:
- Is mortgage A better than mortgage B?
- Each mortgage may be amortized over a different duration (e.g., $15$ vs. $30$ years)
- Given that I'm part-way through mortgage C, should I refinance to mortgage D?
Setting aside the emotional aspects, there is almost always a clear answer based on the Net Present Value ($NPV$) calculation of the cashflows related to the principal, interest, and loan fees. For a fixed-rate mortgage with no taxes or unusual features:$$\begin{align*}NPV &= F + P_0 + \sum_{k=1}^N \frac{PMT_k}{(1+d)^k}\\&=F + P_0 + PMT\left[\frac{1-(1+d)^{-N}}{d}\right]\;\text{for a fixed-rate loan}\end{align*}$$where
- $F$ is the amount in fees paid to acquire the loan
- $P_0$ is the initial cash outflow ("downpayment"), if any, outside of fees
- $N$ is the total number of payment periods
- For a $30$ year mortgage with monthly payments, $N = 30 \times 12 = 360$
- $PMT_k$ is the payment due at the end of the $k^\text{th}$ period
- The $PMT$ is a constant for a fixed-rate mortgage
- $d$ is the chosen discount rate per period
The $NPV$ formula, as specified above, assumes that the first payment will be due at the end of the first period immediately following the loan origination.
I've made available a Mortgage Comparison Template.ods file that implements the calculations referred to in this post.
