This calculator grows two numbers month by month and reports when one catches the other. Your savings grow as an ordinary annuity: each contribution lands at the end of its month and the balance earns the monthly share of your annual return. Starting from what you have saved today, the balance after a given number of months is the starting amount compounded forward plus the future value of the stream of contributions. When the return is zero, that formula would divide by zero, so the calculator switches to plain addition: your starting balance plus your contribution times the number of months.
The target is the part other tools skip. The down payment you need is a share of the home price, and if the price is rising, the dollar amount you need rises with it. So the target starts at the price times your down payment percent, and grows each month at the monthly share of the appreciation rate you enter. Set appreciation to zero and the target holds still, which is the right reading for a flat market. Set it above your savings growth and the target can pull away.
Because the target moves, there is no clean formula for the month the two lines meet, so the calculator steps forward one month at a time and returns the first month your savings reach or pass the target. It stops after 50 years. If your savings have not caught the target by then, it says the goal is not reached rather than printing a month that does not exist, and it tells you which lever changes that: a larger contribution, a higher return, a smaller down payment share, or a market that rises more slowly.
When you do reach the goal, the calculator splits the balance into the money you put in and the money the return added, so you can see how much of the work was yours and how much was the market's.
What this leaves out: taxes on your savings' returns (enter an after-tax return if you want an after-tax answer), closing costs and reserves and earnest money (this is the down payment alone, not the full cash to close), and private mortgage insurance. Putting down less than 20% is a common and valid choice; the down payment calculator does not model the PMI a smaller down payment carries, but the PMI calculator does.
Related reading: First-Time Homebuyer Guide