There are a few ways to look at this question.
- What is the return on equity? At any given time? Or averaged over the course of 30 years?
- What is the return on paying down the mortgage?
- What is the internal rate of return if the owner never sells?
Per the original post's assumptions, this answer:
- Ignores income taxes, property taxes, insurance, property management costs, depreciation, maintenance, changes in rental value, and property appreciation.
- Ignores pre-payment penalties, and opportunities for refinancing.
- Assumes that after taxes and maintenance, the net revenue is enough to cover the initial principal and interest payment, and remains at that level regardless of any financial engineering.
- Assumes a 20% initial equity, with no other out-of-pocket initial costs.
- Assumes a fixed-rate 4.5% APR level-payment mortgage amortized over 360 months.
- 4.5% APR corresponds to 4.594 % APY.
- The first month's $ 810.70 principal and interest payment consists of $ 600 of interest, and $ 210.70 of principal.
In other words, if the owner paid the mortgage on its original schedule, the deal could boil down to a $ 40,000 up-front payment, in exchange for $ 200,000 of equity after 30 years. Or the deal could boil down to a $ 40,000 up-front payment, in exchange for a $ 810.70 monthly payment starting in 30 years.
While the owner is paying down the mortgage, the return on equity is the principal payment divided by the equity. The principal payment is the net rent minus non-financing costs and interest, so it is actually a profit.
The initial return on equity is 6.321 % APR, or 6.507 % APY. This is calculated by dividing the $ 210.70 monthly principal payment by the initial $ 40,000 equity, and converting from monthly return to annual return.
After 30 years, the return on equity is 4.864 % APR, or 4.974 % APY. This is calculated by dividing the $ 810.70 monthly cash flow (which is no longer reduced by mortgage payments) by the $ 200,000 equity after 30 years, and converting from monthly return to annual return.
The cap rate is the same as the return on equity in the absence of debt. In this example, 4.864 % APR, or 4.974 % APY.
The return on equity declines from 6.507 % APY initially to 4.974 % APY after 30 years. This is because the cap rate exceeds the note rate (4.974 % APY vs. 4.594 % APY), and the leverage decreases from 5x to 1x.
The weighted average compound annual growth rate of the equity during the 30 years is 5.511 % APY. Per the original poster's answer, this is computed by taking the 30th root of the 5-fold increase in equity. Because the owner made no extra principal payments (besides those already discussed), the relevant amounts are the initial $ 40,000 owner payment and the final $ 200,000 owner equity. 5.511 % APY corresponds to a 5.377 % APR.
The internal rate of return if the owner never sells can be computed by treating the deal as a $ 40,000 up-front payment, in exchange for an $ 810.70 monthly payment starting in 30 years. The internal rate of return (IRR) is not a very useful number, because it assumes that you can somehow reinvest the eventual dividends at the same rate. In this example, the IRR is 5.172 % APR, or 5.296 % APY. In this example, the IRR is calculated by (iteratively) finding an interest rate for which (initial investment) * (1 + IRR) ^ (number periods before dividends start) = (periodic dividend) / (IRR - growth rate of dividend). For example:
$ 40,000 * (1.004309687)^360 = $ 810.70 / (0.004309687 - 0) = $ 188,111
I then converted the 0.431 % monthly IRR to an annual IRR.
The deal can be thought of as a return on equity, plus a return on paying down the mortgage.
When computing the return from paying down the mortgage, the initial equity is irrelevant. It does not matter whether you start with a $ 160,000 mortgage on a $ 160,000 property, a $ 160,000 mortgage on a $ 200,000 property, or a $ 160,000 mortgage on a $ 1,000,000 property. All that matters is the note rate on the mortgage, which is the applicable compound interest rate.
The return on paying down the mortgage equals the note rate of the mortgage. For a 4.5% note rate, this works out to a 4.594% annual percentage yield (APY).
You can confirm this by looking at your amortization schedule. Suppose you have a $ 160,000 mortgage with a fixed 4.5% APR note rate for 360 months. Your monthly payment is $ 810.70. In the first month, $ 600 goes toward interest, and $ 210.70 reduces the principal. In other words, the $ 210.70 principal payment eliminated the need for a $ 810.70 payment 30 years later. Notice that:
. $ 210.70 * (1 + 0.045 / 12)^360
= $ 210.70 * (1.00375)^360
= $ 210.70 * 3.8477
= $ 810.71
which is within rounding error of $ 810.70.
The interest rate is 3/8 % per month, which is an APR of 4.5%, and an APY of 4.594 %.