Mortgage Calculator

Most mortgage calculators help determine is this property affordable? This calculator helps evaluate if it is a good deal.

I.	(yrs) Loan Period	payment coefficient	30 × 12 = 360
	(%) APR	marginal dividend
II.	($) Loan Estimate	Monthly Payment	use loan estimate
	($) Purchase Price	Down Payment %	use % down
	($) Closing Costs	Total Upfront Cost
III.	(yrs) since purchase	internal dividend	÷ equity buildup (-)
IV.	(%) inflation	short dividend	× leverage ratio
V.	($) Monthly Income	external dividend	estimate income
	Year of Income	VI. total dividend

Unlike inferior calculators, this one is organized into six clean dependencies. Results in lower sections depend on values from sections above them but higher sections are independent of inputs lower down.

the marginal dividend is independent of loan size.
for convenience, you can use the loan estimate provided by your lender or estimate total upfront costs yourself with a percent down payment.
the internal dividend is your rate of equity buildup (principal repayment) in the loan and is independent of the housing market after date of purchase.
the short dividend is based on your "risky" exposure to inflation after accepting a loan.
the external dividend can change after date of purchase, depending on home improvements, local markets and management (sublets, rentals, etc.), not just on inflation.
unlike its component dividends, the total dividend is weighted by the mortgage leverage for comparison other investments like stocks. (It may seem large but it shrinks quickly each year as your equity builds up.)

The independence of the payment coefficient/marginal dividend from loan size is proven below; it is important for developing a model to estimate property value vs possible rental income if used as a rental property instead of your home. This relationship between the marginal dividend and external dividend (i.e. rental income less mortgage payment) is explored further in the next article on home prices. This is the first article of a series Should I Buy or Rent?

About the Calculation

The calculation assumes monthly interest compounding, based on an annualized percentage yield, and a fixed monthly payment that covers the entire monthly interest with the remainder reducing principal owed. This is the starting point in the proof below, where Pm is principal remaining in the m'th month, r the monthly compounding rate, and p the monthly payment. Lower case terms r, p, and n number of months (loan period) are constant for the duration of a mortgage.

From the proof, there is a linear relationship between initial loan principal and monthly mortgage payment. I call this (the parenthetical value in the last line of the proof) the payment coefficient, or its inverse the limiting annual dividend. The maximum possible value for the payment coefficient is n, the number of payments made over the lifetime of the loan.

The internal dividend and short dividend are how the owner's equity and resale value grow from the initial down payment, useful for comparing real estate to stock captital gains. The external dividend is based on thinking of yourself as landlord and tenant rather than as a homeowner. If there is a difference between what could be made from a tenant in rent income, compared to the monthly mortgage payment, this difference is like a stock dividend in your pocket!

The equity buildup and its inverse, leverage ratio, are modified slightly from the proof below by the parenthetical fraction reported by the calculator, such that build up is to slightly ≥1 over the loan period due to closing costs but is always >0 at the start of the loan due to closing costs and down payment.

Proof

P_{0} = p R_{N} where R_{N} = {\begin{matrix} \frac{1 - i^{- N / months}}{i - 1} months, & i \neq 1 \\ N, & i = 1 \end{matrix}

For a loan of N months, the principal owed at the Nth month, P(N), is zero. Thus, the monthly payment p is linearly related to intial loan size P(0) by the parenthetical payment coefficient, P(0) = p×R(N).

Finally, the finite geometric series applys for all values of i except i = 1, for which it is just n. In summary: