What's the math formula that is used to calculate maximum mortgage amount like in this calculator?

Question

What's the math formula that is used to calculate maximum mortgage amount like in this calculator?

Visual Reference: Take a look at these images to see which tool I'm specifically referring to:

Purchase Budget with $0\%$ Interest

Purchase Budget with $1\%$ Interest

Situation: Right now I can calculate everything correctly while the interest rate is set at $0\%$. I'm having difficulty deriving the formula that takes into account the interest rate and how it impacts the Mortgage Amount/Purchase Budget.

The calculator I linked to has the Debt-to-Income Ratio set to $0.36$, so in my example images, a $\$10,000$ yearly income equates to $\$300,000$ in $30$ years, and $36\%$ of that $\$108,000$, which you can see in the first image with $0\%$ interest set.

All the other fields have a linear correlation to how much they affect the Mortgage Amount/Purchase Budget. For example, a $\$1$ monthly debt would result in a $\$360$ reduction of the budget ($\$1 \times 12$ months $\times 30$-year term).

Attempts:I tried seeing if the correlation was with the Mortgage Amortization, but it doesn't appear so from what I've tried.

Context & Motivation: I'm building my own personal finance/budgeting tool so that I can have every financial calculation in one place rather than utilizing various tools spread across various sites. An additional motivation is that the knowledge and understanding I gain from working on this can help someone else too.

Update: I solved this shortly after posting the question and posted my answer below. I see that someone else also provided a solution that can work too, but the one I posted fit better with my personal project.

Answer 1

I was able to solve this using this resource.

That resource referenced the Foner Books Formulas.

Below is what I used to solve:

$G$ : Gross Annual Income

$g = \frac{G}{12}$ : Gross Monthly Income

$DTI$ : Debt-to-Income Ratio (Typically 36% of $g$)

$d$ : Monthly Debts (Car loan, School loan, Gas, Utilities, etc)

$f$ : Monthly Fees (Property Tax, Home Insurance, PMI, HOA)

$I$ : Interest Rate of Loan

$L$ : Loan Term (Eg. 30 Years)

$i = \frac{I}{12}$ : Monthly Interest Rate

$n = 12 \times L$ : Total Number of Payment Months

$y = (i+1)^n$

Answer:

Affordable Mortgage Principal ($P$) Based on Interest Rate and DTI

$P = \dfrac{DTI - d - f}{\dfrac{(i \times y)}{(y-1)}}$

Answer 2

Your maximum monthly payment is 3% of your annual income.

The net present value of 360 monthly payments discounted at the rate of the loan gives your maximum mortgage balance.

$\text{Ballance} = \text{Payments}\sum_{n=1}^{360} \frac {1}{(1+\frac {y}{12})^n}$

Evaluating the sum of a geometric series.

$B = P\left(\frac {(1+\frac y{12})^{-1}(1- (1+\frac y{12})^{-360})}{1-(1+\frac {y}{12})^{-1}}\right)$

Which simplifies:

$B = P\left(\frac {1- (1+\frac y{12})^{-360}}{\frac {y}{12}}\right)$

Based on a $1\%$ loan and $\$300$ payment, the ballance would be $\$93,272.12$ which is within rounding of the $\$93,300$ in the link.