Buy-Down

[Albert Chan]

(10-27-2023 12:55 PM)SlideRule Wrote:  " Payments by the buyer to the lender will drop from $1264.45 to $1028.62. The $235.83 difference is not simply multiplied by 36 months, or $8490. to arrive at the total buy-down amount. This is because of the present value of money concept; that is, money paid in a lump sum today will be put to productive use.

(10-16-2020 04:02 PM)Albert Chan Wrote:  How to estimate car payments ?

XCas> C := I*N / (1 - (1+I)^-N)       // C = |N*PMT/PV|, "compounding factor"
XCas> series(C,I,polynom)

\(1
+\frac{I(N+1)}{2}
+\frac{I^2 (N^2-1)}{12}
+\frac{I^3 (-N^2+1)}{24}
+\frac{I^4 (-N^4+20N^2-19)}{720}
+\frac{I^5 (N^4-10N^2+9)}{480}\)

For car payments, N is usually not too big, we can use: \(\large C ≈ {(IN + 3)^2 + 3\over 12}\)

This estimate does not require converting annual interest rate to monthly rate.

Using C approximation, we can get very good buy-down estimate. (IN = I*N)

C ≈ ((IN+3)^2 + 3)/12 = 1 + (IN/2) * (1 + (IN/2)/3)

IN = 0.15 * 3 = 0.45
C ≈ 1 + 0.225 * (1 + 0.075) ≈ 1.242

Buy-down = N*PMT/C ≈ $8490. / 1.242 = $6836.

(10-17-2020 11:27 AM)Albert Chan Wrote:  C-1 ≈ IN*(6+IN)/12 + I/2

Monthly rate, I = 15%/12 is very high. We may need +I/2 correction.

C ≈ 1 + 0.225 * (1 + 0.075) + (0.15/12)/2 ≈ 1.248

Compare this with exact calculations:

C = I*N / (1-(1+I)^-N) = 0.45 / (1-(1+.15/12)^-(3*12)) ≈ 1.248

Buy-down = N*PMT/C ≈ $8490. / 1.248 = $6803. (matched my previous post)