Buy-Down

[Albert Chan]

Note: based on post#8, interest is compounded monthly. Redo calculations.

Without builder's help: n = 30*12, i = 15%/12, pv = 100e3, fv = 0      --> pmt = -1264.44

With builder's help effective rate ≈ (12*3 + 15*27)/30 % = 14.7%
But, builder lowered rate is applied to *beginning* of loan, let's say rate = 14%

i = 14%/12      --> pmt = -1184.87

To get exact pmt, we break-up 30 years = 3 + 27, and solve for f = 0:

NFV(n,i,pv,pmt,fv) = pv*k + pmt*(k-1)/i + fv, where k = (1+i)^n

f = NFV(27*12, 0.15/12, NFV(3*12, 0.12/12, 100e3, pmt, 0), pmt, 0)

f(pmt = -1200) = −162343.41
f(pmt = -1100) = +518573.43

Interpolate for f=0, we have pmt = -1176.16

With builder's help, monthly payment $1264.44 → $1176.16, for 30 years

Or, equivalently, interest 15.000 % → 13.900 %
Or, equivalently, loan $100,000.00 → $93,017.96

---

f = 0, with NFV formula applied twice.

(pv*k1 + pmt*(k1-1)/i1 + 0) * k2 + pmt*(k2-1)/i2 + 0 = 0

Solve for pmt, we have:

lua> i1, n1, i2, n2 = 0.12/12, 3*12, 0.15/12, 27*12
lua> k1, k2 = (1+i1)^n1, (1+i2)^n2
lua> pv = 100e3
lua> fv = k1*k2*(-pv) -- if pmt=0
lua> pmt = fv / (k2*(k1-1)/i1+(k2-1)/i2)
lua> pmt
-1176.1581151520215