Comparing mortgages for buying or refinancing a house

Questions that come up in scenarios related to buying or refinancing a home:

1. Is mortgage A better than mortgage B?
• Each mortgage may be amortized over a different duration (e.g., $15$ vs. $30$ years)
2. 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.

Sidebar

For question $1$, the Loan Estimate would include an APR value that can be the basis for a comparison if the loan durations being compared are identical. It's not immediately obvious how to use the APR to compare loans of different durations. Also, the APR cannot be used to answer question $2$ related to a refinance.

Sometimes the total (undiscounted) cashflow of interest payments over the span of the loan is used for the comparison, but as you'll see below that may give you the wrong answer.

Discount rate

For the $NPV$ calculation you'll need an appropriate discount rate (aka, opportunity cost). This would be the expected risk-adjusted after-tax return on an investment that you'd redirect your cash into if you didn't have to pay the mortgage. On this everything hinges.

One option is to use the "risk-free" rate of return (yield on T-bills/T-notes of the appropriate duration). For most people this is likely the right choice as a fixed-rate mortgage is effectively as a "risk-free" liability.

I use a conservative after-tax expected portfolio return of $7.1$ %/year (nominal rate which includes inflation). Note that the risk characteristics of a portfolio that includes stocks is completely unlike a fixed-rate mortgage which is a risk-free liability. Ergo, the choice of $7.1$ %/year for the discount rate is a personal one; it is not a universal discount rate.

Also, see discussion on the break-even discount rate below which highlights how the choice of the discount rate can materially change the outcome.

Caveat

While the discount rate is specified in terms of %/year, most mortgages require monthly payments. If so, payments must be discounted on a monthly basis using $1/12^\text{th}$ the assumed discount rate. For the $7.1$ %/year choice above, it would be $7.1/12$ $=$ $0.5917$ %/month.

Buying a house

Inputs needed:

• Initial downpayment
• Starting principal
• Interest rate (%/year)
• Amoritization duration (years)
• Amortization period (usually monthly; to wit, $12$ periods/year)
• Loan fees (origination fees, title/insurance/deed fees, recording fees, etc.)
• Do not include any escrow amounts for home insurance and/or property taxes as they are not pertinent to the cost-benefit analysis

Set-up your amortization table for each mortgage and then calculate:

Net Present Value of the loan = 

Downpayment +

$PV$ of cashflows for principal +

$PV$ of cashflows for interest +

Loan fees

Choose the loan that has the lowest present value. That's it.

What if two loans have the same present value?

If I knew I would never refinance again then I could argue that either loan is equally viable; it's a true coin toss. 

However, practically speaking, there is always a higher chance that I'll refinance again if it's an environment of falling interest rates. Ergo, personally, I would go with the option that has the lowest initial cash outflows as that meaningfully lowers the opportunity cost.

To get more confidence, you can run the $NPV$ calculation for $1$-year, $3$-year, $5$-year etc. durations to determine which loan is the best choice for the intended duration. Remember to include the unpaid principal at the end of the intended duration as a liability and discount it to the present day.

Example $1$ ($30$ year loans).

Consider the following two loans for a starting principal of $\$332,000.00$ which I was offered in Jan-$2026$. Both loans were for a $30$ year duration, with monthly payments.

• Loan #$1$: $5.875$ %/year; loan fees of $\$4,717$
• Loan #$2$: $5.990$ %/year; loan fees of $\$0$
• The bank is willing to absorb the loan fees in order to charge me a higher interest

After setting up the amortization tables, the total interest payments (undiscounted) were:

• Loan #$1$: $\$375,002.93$
• Loan #$2$: $\$383,817.06$

So, I'm paying more in interest with Loan #$2$ (which would be obvious looking just at the interest rate). Even if I include the $\$4,717$ of loan fees in Loan #$1$, it still looks better than Loan #$2$.

What about looking at discounted cashflows for the interest? The $NPV$ calculation (using a discount rate of $7.1$ %/year) for interest alone is:

• Loan #$1$: $\$190,714.10$
• Loan #$2$: $\$194,946.08$

Now, if I include the loan fees ($\$4,717$), Loan #$1$ is starting to look less attractive than Loan #$2$. But, to be honest, it's almost even. The difference is only $\$485.02$.

Adding in the discounted cashflows for the principal in addition to the interest and loan fees gives the full picture:

• Loan #$1$: $\$296,951.61$
• Loan #$2$: $\$295,874.32$

IOW, Loan #$1$ is going to cost me $\$1,077.29$ extra in today's dollars over the course of the mortgage. Thus, Loan #$2$ wins, under the assumed discount rate, even though it has a higher interest rate.

The point of this real-life example is that a simplistic view of looking only at (undiscounted) interest payments does not always give the right result.

Example $2$ ($15$ vs. $30$ year loans)

The general idea doesn't change if the two mortgages being compared are over different durations. You would still do the same $NPV$ calculation for principal, interest, and loan fees and compare the two products to determine which is better. This is where the oft-repeated notion that a shorter duration mortgage is preferable over one with a longer duration can fail depending on the choice of the discount rate.

Consider the following loan terms for a principal of $\$332,000.00$ which I was offered in Jan-$2026$. Both loans had the same loan fees of $\$4,717$ and were subject to monthly payments.

• Loan #$1$: $15$ year, $5.500$ %/year
• Loan #$2$: $30$ year, $5.875$ %/year

In terms of (undiscounted) interest payments only:

• Loan #$1$: $\$156,288.79$
• Loan #$2$: $\$375,002.93$

I would be paying almost twice the amount of interest with Loan #$2$, suggesting that I should go for Loan #$1$. This is usually the argument for the shorter duration loans. 

However, comparing the $NPV$ of cashflows for principal, interest, and loan fees over the duration of each loan (using a discount rate of $7.1$ %/year):

• Loan #$1$: $\$304,654.55$
• Loan #$2$: $\$296,951.61$

IOW, Loan #$1$ will cost me $\$7,702.94$ extra in today's dollars. Thus, Loan #$2$ wins under the assumed discount rate.

Break-even discount rate

When the loans under comparison have different durations, there exists a discount rate at which the $NPV$ of both the loans is the same. This is the break-even discount rate.

For the example above, this happens at a discount rate of approximately $6.4453$ %/year. If the discount rate being used for the $NPV$ calculation is much lower than this (e.g., a risk-free discount rate of, say, $4.7$ %/year), then Loan #$1$ would be the winner.

Further, if one uses the "risk-free" rate then it should ideally be matched to the duration of the loan; i.e., the $15$ year loan should use a different discount rate than the $30$ year loan.

This example demonstrates the sensitivity of the final result on the choice of the discount rate. In scenarios like this, there is no objectively correct answer.

Refinancing a mortgage

The idea is the same as before, except here you want to compare the $NPV$ of the remaining cashflows on your current mortage to those of the new (refinanced) mortgage.

When calculating the remaining cashflows on the current mortgage you must ignore past cashflows that have already happened. So, the amortization table should be set-up only for all cashflows pending on the remaining portion of the loan. This should then be compared with the full duration of the refinance option.

Example $3$

My current loan is on an initial principal of $\$400,000.00$ at an interest of $6.490$ %/year that I've been paying down for the last $10$ years. The outstanding balance is now $\$339,019.74$ with $20$ years left on the mortgage. 

I want to compare this to a refinance option on the outstanding balance at $5.875$ %/year with refinance fees of $\$4,717$. (This is the same as Loan #$1$ from Example $1$ above.) A refinance will reset my payment schedule; to wit, I'll carry the mortgage for another $30$ years.

From the amortization tables, total interest payments (undiscounted) were:

• Current loan (pending payments only): $\$267,136.17$
• Refinance option: $\$382,934.74$

Even without counting the refinance fees, it might look like I should continue with my current loan. However, looking at the $NPV$ (using a discount rate of $7.1$ %/year) for principal, interest, and loan fees:

• Current loan (pending interest + principal): $\$323,256.49$
• $\$0$ in fees to continue my current loan
• Refinance option (interest + principal + loan fees): $\$303,129.89$

Now it's obvious that my current loan will cost me $\$20,126.60$ more in today's dollars. Thus, refinance is the better option here under the assumed discount rate. Once again, using a different discount rate could result in the opposite outcome.

If I have the opportunity to refinance again in the future, I would run the exact same calculation. It totally doesn't matter what loan fees were paid in the past as we only look at future cashflows - and that's all that matters.