(17B/19B) GPM: initial payment

[Albert Chan]

(05-25-2024 11:52 PM)Gil Wrote:  [PMT] is plain and easy. Trickier is to find i.

Inspired by Werner's post of finding growth rate, we have the formula for i too.

x = ((s-1)-APY) / (1+APY)

tvm(N, x, USPV(i,300), USPV(i,12), nil) --> PV/PMT

Now, do this in reverse, with nil replaced by PV/PMT, solve for rate x

x = s / (1+APY) - 1
(1+x) * (1+APY) = s            // formula for x and APY have same form!

--> APY = ((s-1)-x) / (1+x)
--> (s-1) = x*(1+APY) + APY

To confirm math, we redo Werner's previous post, solve for growth rate (s-1)

lua> K = fn'i,n: expm1(log1p(i)*n)'
lua> USPV = fn'i,n: -K(i,-n)/i'
lua> i = 0.01
lua> APY = K(i,12)

lua> tvm(5, nil, USPV(i,300), USPV(i,12), -1/i) * (1+APY) + APY
0.008053452004141717
lua> tvm(30, nil, 0, USPV(i,12), -1/i) * (1+APY) + APY
0.003973548652941375