(17B/19B) GPM: initial payment

[Albert Chan]

(05-25-2024 08:40 PM)Albert Chan Wrote:  Another way is let r = pv(k) / pmt(k), eliminated the need for s^p

lua> r = -1 / c300 -- = pv6/pmt6
lua> for k=1,5 do r = (d12*r*s - 1) / c12 end
lua> pv / r -- = pmt
-474.82512976281373

We can remove loops, and get direct formula for pv/pmt

Let q = s*d12/c12 = s / (1+i)^12

r6 = -1/c300
r5 = q*r6 - 1/c12
r4 = q^2*r6 - (1 + q) / c12
r3 = q^3*r6 - (1 + q + q^2) / c12
r2 = q^4*r6 - (1 + q + q^2 + q^3) / c12
r1 = q^5*r6 - (1 + q + q^2 + q^3 + q^4) / c12

--> pv/pmt = -q^5 / c300 - (q^5-1)/(q-1) / c12

This matched Gil's formula (sign convention and typo corrected)

PV/PMT = (v^(360-12*5)-1)/i * (s*v^12)^5 + (v^12-1)/i * sum((s*v^12)^j, j=0..(5-1))

where s = 1.075, i = 0.01, v = 1/(1+i)