TVM solve for interest rate, revisited

[Albert Chan]

(05-13-2022 10:34 PM)Albert Chan Wrote:  lua> n,pv,pmt,fv = 10,50,-30,100
lua> pmt/fv, -pmt/pv
-0.3      0.6

Lets try sign test, to see which way possible roots are located.

(06-14-2024 03:29 PM)Albert Chan Wrote:  sgn(-(pv+fv)/n) = sgn(pv*(i - pmt/-pv)) = sgn(-fv*(i - pmt/fv))

-(pv+fv)/n = -(50+100)/10 = -15

sgn(i - pmt/-pv) = sgn(-15/50) = -1      → i < pmt/-pv = 0.6
sgn(i - pmt/fv) = sgn(-15/-100) = 1      → i > pmt/fv = -0.3

True rates are 0.582038 and -0.284436 ... all checked out.

Lets try split loan "y" method

(06-13-2024 01:05 PM)Albert Chan Wrote:  

Code:

Loan:  n  pv        pmt     fv
#1     n  pv+y      0       fv-y        --> (1+i)^n = -(fv-y)/(pv+y)
#2     n  -y        pmt     y           --> i = pmt/y

(pv+y) * (fv-y) < 0
(y+pv) * (y-fv) > 0
(y+50) * (y-100) > 0

pmt = -30, i = pmt/y:

(y<-50)      → i < -30/-50 = 0.6
(y>100)      → i > -30/100 = -0.3      -- same edges and direction, without memorized formula!