Questions:
What is the relationship between Net Present Value (NPV) and Internal Rate of Return (IRR) in capital budgeting?

Let's say I have an investment opportunity that will cost $100, but will generate $20 at the end of the first year and will grow 10% per year in perpetuity onwards. What is the IRR of this opportunity?

Reference answers:
Capital budgeting also known as investment appraisal is a process that businesses to evaluate major projects. The process involves analyzing company's cash inflows and outflows to check whether expected return meets the benchmark. Two major techniques used for this are Net Present Value (NPV) and Internal Rate of Return (IRR). 

NPV is used to calculate current value of expected future cash flows of project. It calculates difference between net cash inflows and net cash outflows over a time period. A positive NPV indicates that a project or investment is profitable when discounting the cash flows by a certain discount rate, whereas, a negative NPV indicates that a project or investment is unprofitable. NPV is calculated as:
NPV = [ Cash Flow / (1+r)^t ] - initial investment (for 1 cash flow from project)
where 
r = discount rate
t = time period 

IRR is a discount rate that makes value of NPV of all cash flows equal to zero. It is in general an annual rate of growth that an investment is expected to generate. Once the IRR is determined it is compared with company's cost of capital and if it is greater than that the project is considered to be a lucrative investment. IRR can be calculated in multiple ways. 

The most common method is to find the discount rate that sets NPV to 0. This is done by taking the formula for an opportunities present value, setting it equal to 0 and then solving for the discount rate.

Another method is interpolation:
IRR = R1 + { [NPV1 x (R2-R1)] / [(NPV1 - NPV2)] }
where
R1 & R2 = randomly selected discount rates 
NPV1 = higher NPV
NPV2 = lower NPV


Both of the methods can be calculated with the single formula, when we have a predefined discount rate we calculate NPV and when NPV is set to 0 we can get IRR.

To find the IRR, we need to find an interest rate r such that the cost of this opportunity equals the present value of its future cash flows. We can treat this opportunity as an annuity with constant growth. The present value of such an annuity is defined as:

PV = C / (r - g)

where:
PV = present value
C = initial return
g = growth rate
r = interest rate

Filling these variables in and setting it equal to the initial cost gives us:

100 = 20 / (r - 0.10)

Solving for r gives us r = 0.3 or r = 30%. Therefore, the IRR of this opportunity is 30%.