Points: prepaid interest as a rate trade

A point is prepaid interest, one percent of the principal, paid on day one in exchange for a lower r on the installment. The trade is a packaging of Track 1 lesson 13's spread: instead of accruing a wider spread every month, a slice of that spread is paid up front and the remaining monthly rate is narrower. This lesson describes the identity. It does not choose a side of the trade.

Marrow Street Mortgage writes $265,000 for 360 months. Without points, illustrative 6.75 percent, payment $1,718.78. With 2.00 points, cash of $265,000 × 0.02 = $5,300 on day one, illustrative 6.25 percent, payment $1,631.65. Monthly difference is $1,718.78 − $1,631.65 = $87.13. Break-even, prepaid cost divided by monthly difference, is $5,300 ÷ $87.13 = 60.83 months.

Sixty-one months of $87.13 is $5,314.93, which first exceeds the $5,300. Until that month the prepaid cash is larger than the cumulative difference in installments. After that month the cumulative difference is larger. The identity does not say which loan is held. It says how long the up-front spread takes to equal the monthly spread it replaced.

The $5,300 is not principal. It does not reduce the $265,000. Both substitutions in the installment formula still use P = $265,000. What changed is r, and what paid for that change is a day-one cash figure sitting outside the loan. A spread paid up front and a spread accrued month by month are two wrappers on the same economic gap between two rates.

Break-even is an identity: prepaid cash divided by the monthly difference. It is not a verdict on whether the points were paid.