End-of-term wealth on each path, at every investment return. The crossing is your break-even; left of it cash wins, right of it the mortgage does. Hover to explore.
Both buyers get the same house on day one, so the house cancels out and only the money paths differ. The cash buyer pays the full price, then invests a mortgage-payment-sized amount every month (the money the other buyer sends to the bank). The mortgage buyer puts the down payment in, invests the rest of the cash as a lump, pays the mortgage from income, and — if the deductibility chip is on — invests the monthly tax shield too. At the end of the term both own the house free and clear; the portfolios are compared. The break-even return is found by bisection: the return at which the two paths tie exactly.
Constant rate and constant return — no refinancing, no prepayment, no crash in year two. That last one matters most: sequence risk is invisible to an average. If the invested lump halves early, the mortgage path feels nothing like the smooth line simulated here, while the paid-off house pays its dividend — the cancelled payment — in every market. Investment taxes on the portfolio and PMI below 20% down are not modeled; both nudge the answer toward cash.
payment = loan × j / (1 − (1+j)^−m) j = rate/12, m = months
cash path: wealth = payment invested monthly, m months
mortgage path: wealth = (price − down) grown m months
+ tax shield invested as it arrives
break-even return: found by bisection where the paths tie