Question

Internal Rate of Return (IRR) can be understood as the discount rate that should be applied to a project such that Net Present Value (NPV) = $0. If the discount rate applied in a certain 5 year project is 10% and the resultant NPV is $50K, what is the IRR?

Answer 1

Net Present Value (NPV) = Present Value of Cash Inflow (PVCI) - Present Value of cash Outflow (PVCO)

= Present Value of Net Cash Inflows (PVNCI)

=> NPV = PVCI - PVCO = PVNCI

At IRR, NPV = 0

=> NPV = PVCI - PVCO = 0

=> PVCI - PVCO = 0

Let us assume that there is only initial investment in the project. So, PVCO = Cost of initial investment.

Given that, At Discount Rate of 10%, NPV = 50,000.

Also, project term, n = 5 years.

At i = 10%, NPV = 50000

Future Value of Net Cash Inflows = 50,000 / (1+0.10)⁵ = 80,525

At i = 10%, NPV = 50,000

The difference in FVNCI and NPV (or PVNCI) is 30,525

At IRR, NPV should be zero. So, the rate at which 80,525 reduces to 30,525 is our IRR.

When the Rate is 20%, NPV = 80,525 / (1+0.20)⁵ = 32,361

When the Rate is 21%, NPV_a = 80,525 / (1+0.21)⁵ = 31,045 .....(lower discount rate, r_a )

When the Rate is 22%, NPV_b = 80,525 / (1+0.22)⁵ = 29,794 .....(lower discount rate, r_a )

We realise that the IRR is between 21% and 22%. By using interpolation,

IRR = r_a + [ NPV_a (r_b - r_a) ] / (NPV_a - NPV_b )

=> IRR = 0.21 + [ 31,045 (0.22-0.21) ] / (31,045 - 29,794)

=> IRR = 0.214 = 21.4%

[ Therefore, IRR is 21.4% . At IRR, Present Value of Cash Inflows = Present Value of Net cash Outflows = 80,525/ (1+0.214)⁵  = 30,525 ]