# Principles of Finance/Section 1/Chapter 6/Corp/Computation

### IRR , computation , Newton's Method

editProject valuation may require finding the Internal Rate of Return, or the discount rate at which a project's present value of future cash flows equals to zero. This is the maximum rate of return achievable, and should be greater than the inflation rate. a simple converging guessing method has been illustrated under the Project Valuation chapter. The use of Newton's method was left as an exercise, and a possible solution is shown below. Newton's method finds the x-axis intersection of the gradient line found at an approximate IRR, which will give a better rate to try for IRR, and iterations will find an acceptable IRR where the NPV is small enough to be equal to zero.

background for derivative function of npv:

Axiom: for bx^{a}, d ( bx^{a}) / dx = ab x^{a-1}

Hence, if, npv = CF0 / R^{0}+ CF1 / R^{1}+ .. + CFn / R^{n}, or npv = CF0 + CF1 x R^{-1}+ CF2 x R^{-2}+ .. + CFn x R^{-n}, then , d(npv)/dR = 0 x CF0 x R^{-1}+ - CF1x R^{-2}+ - 2 CF2x R^{-3}... + -n x CFn x R^{-n - 1}.

```
#copyright SJT, GNU 2014.
# Newton's method
# graph is y-axis NPV and x-axis is rate, so find where NPV = 0, or the project rate function crosses the x-axis.
# Do this by finding where the slope line of the function curve at any given R crosses the x-axis, which gives a
# better R' which is closer to where the function curve crosses the x-axis, and this is done iteratively until
# a close enough R' is found where the npv is practically 0.
def irr(cfs):
def npv(cfs, r):
R = 1. + r/100.
return reduce( lambda x,(i,y): x + y * R**-i , enumerate(cfs), 0)
def deriv_npv( cfs, r):
R = 1. + r/ 100.
return reduce ( lambda x, (i,y): x - i * y * R **(-i-1) , enumerate( cfs), 0)
r = 10.
lim = 0.1 # ten cents is practically zero
while True:
np_v = npv( cfs, r)
print "np_v", np_v
if abs(np_v ) < lim:
break
dnpv = deriv_npv( cfs, r)
r = r - np_v / dnpv
return r
print irr( [ -30000, 3000, 4000, 16000, 15000, 5000] )
```