Question

You are considering opening a new plant. The plant will cost $102.5 million upfront and will take one year to build. After​ that, it is expected to produce profits of $29.5 million at the end of every year of production. The cash flows are expected to last forever. Calculate the NPV of this investment opportunity if your cost of capital is 7.8 %. Should you make the​ investment? Calculate the IRR. Does the IRR rule agree with the NPV​ rule?

I know the NPV is 248.34 million but I need to calculate the IRR and I need someone to really dumb it down step by step on how to calculate the IRR. I know the answer is 23.34% but how did they come up with that. Can you please detail this out both with and without excel?

Answer 1

NPV = present value of all cash flows associated with the project

= - present value of cost + present value of all benefits

Present value of costs = $102.5 million (as cost is incurred upfront, the PV is the same)

As the plant takes one year to build, the profit starts coming in from the end of year 2 onwards

Present value of benefits = $29.5 million/1.078² +$29.5 million/1.078³ +...... to infinity (as profit

As this is a perpetuity with 1st term =  $29.5 million/1.078² and common ratio = 1/1.078 which is less than 1

the sum of perpetuity is given by S = first term/ (1- common ratio)

So,   

Present value of benefits = $29.5 million/1.078² / (1- (1/1.078))

= $ 350.84 million

So, NPV = - 102.5 + 350.84 = $ 248.34 million

IRR is the interest rate at which NPV = 0

So, in the NPV equation, if rate of interest was unknown and NPV = 0

we can write

0 = -102.5 + 29.5/(1+r)² / (1- (1/1+r))

=> 29.5 / (r *(1+r)) = 102.5

=> r² + r -0.2878 =0

By the formula of quadratic equation

r = [ -1 + sqrt ( 1 + 4*1*0.2878) ] / 2

= ( -1 + 1.4667) / 2

= 0.23335 or -1.23335

As interest rates can't be negative, the IRR is 0.23335 or 23.34%