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 question #1, the Loan Estimate document 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.

I've made available a Mortgage Comparison Template.ods file that implements the calculations referred to in this post.

Discount rate

For the NPV calculation you'll need an appropriate discount rate (aka, opportunity cost). This would be the expected gains on an investment that you'd funnel your cash into if you didn't have to pay the mortgage.

I use a conservative 7.1 %/year (nominal rate which includes inflation).

If your investments are more aggressive, then a nominal rate of 9-10 %/year may be a reasonable choice.

Buying a house

Inputs needed:

• 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:

Present value of the loan =

NPV of cashflows for principal +

NPV of cashflows for interest +

Loan fees

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

One Image Summary of Quantum Mechanics

A 'Spherical Cow' in a Simple Harmonic Oscillator Potential is pretty much Quantum Mechanics in a nutshell.

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Image generated using ChatGPT-5 on 2025-Oct-19.

Random musing about Classical Projections

Assume I have a measuring device to measure the electric field intensity $\vec{E}$ in a specific direction, say, $z$. If $\vec{E}$ is oriented at an angle $\theta$ to the measuring device (or, equivalently to the $z$ direction), then the measuring device will register an intensity of $E\cos\theta$,  $E$ being the magnitude of $\vec{E}$. So far this is in line with classical physics and the parallelogram law of vector addition.

However, no measuring device is instantaneous. Every measuring device takes a finite amount of time to interact with the quantity-under-measurement before settling to a final value. This may not seem terribly relevant as the combined system comprising the measuring device and quantity-under-measurement converges to a steady state in short order. In classical physics, this steady state is real and the measuring device is truly measuring the underlying quantity.

A different perspective

One could posit a different take on the above measurement process: Since the measurement takes a finite time, one could argue that even in the steady state a measuring device is measuring a range of values - in rapid fashion - and is only able to present an average value as (macroscopic) measuring devices simply cannot respond fast enough.

Here's how one might do this. Note that

$$E\cos\theta = E\left[\cos^2(\frac{\theta}{2}) - \sin^2(\frac{\theta}{2})\right] \\ = E\cos^2(\frac{\theta}{2}) + (-E) \sin^2(\frac{\theta}{2})$$

So far this is just mathematical manipulation. However, this new form lets us look at the measurement process from a different perspective.

Perhaps the measuring device only ever measures $+E$ or $-E$ but with respective probabilities $P(+E) = \cos^2(\frac{\theta}{2})$ and $P(-E) = \sin^2(\frac{\theta}{2})$. Note that the individual probabilities add up to $1$, and the average value of $E$

$$ \langle E \rangle = E\cos^2(\frac{\theta}{2}) + (-E) \sin^2(\frac{\theta}{2}) = E\cos\theta$$

is exactly what's presented as the measured value by the measuring device.

Your point being ... ?

It may seem silly to look at it this way but it forms a nice bridge to the discussion of spin-$\frac{1}{2}$ in quantum physics. The only difference being that an individual spin-$\frac{1}{2}$ measurement yields a specific value (either $+\hbar/2$ or $-\hbar/2$) as it's the result of an instantaneous interaction between a Stern-Gerlach apparatus and the spin-$\frac{1}{2}$ particle under measurement. However, over an ensemble, the individual measurements yield the same average (‘expectation value’) as the classical case above.