Question

You have just landed your dream summer internship, and your boss asks you to analyze a project that has an investment cost of $2,000,000, to be paid today (t = 0), and will generate a cash-flow of $200,000 in the first year (t = 1). The cash-flow will then grow at 10% per year for the next six years (the last time the cash-flow grows at 10% is from t = 6 to t = 7). Afterwards, as competition increases, cash-flow growth is expected to be only 2% per year in perpetuity. (The cash-flow grows at 2% from t = 7 to t = 8 and forever thereafter.) The discount rate is 20%

1. (10 points) What is the NPV of this project?
2. (4 points) Write down the equation for the IRR of this project. (You don’t have to solve for the IRR, just write the equation whose solution is the IRR.)

Answer 1

Solution a) Calculation of the Net Present Value of the Project:

Initial Investment = $2,000,000

Step 1: Calculation of Present Value of Cash Flows from Year 1 to year 6:

Year	Cash Inflows $	Discounting Factor @ 20%	Present Value of Cash Inflows $
1	200000.00	0.8333	1,66,666.67
2	200,000 + 10% = 220,000.00	0.6944	1,52,777.78
3	220000 + 10% = 242,000.00	0.5787	1,40,046.30
4	242,000 + 10% = 266,200.00	0.4823	1,28,375.77
5	266,200 + 10% = 292,820.00	0.4019	1,17,677.79
6	292,820 + 10% = 322,102.00	0.3349	1,07,871.31
	Total Present Value of Cash Inflows $		8,13,415.61

Step 2: Calculation of the value of the Total Cash Inflows (From Year 6 to Infinity) at the end of year 6 when the Cash inflows are growing at a constant rate.

Cash Inflows (At the end of year 7) ₌  322,102 ( 1+0.02) = = $328,5440.04

Using constant growth Model,

Value of Cash Inflows (Perpetuity) at the end of year 6

₌ Cash Inflows (At the end of year 7)/ (Discount Rate– Constant Growth Rate)

= $328,5440.04 / (0.2 – 0.02) = $1,825,244.67

Therefore, the Present Value of $1,825,244.67 today = $1,825,244.67 x 0.3349
                                                                                      = $611,270.75

Total Present Value of the Project

= Total Present Value of Cash Inflows $( From Year 1 to 6) + Present Value of Cash Inflows (Perpetuity) from Year 7

= $813,415.61 + $611,270.75 = $1,424,686.36

Therefore, the Net Present Value of the Project = Total Present Value of the Project – Initial Investment

= $1,424,686.36 - $2,000,000 = ($575,313.64)

Therefore, the Net Present Value of the Project is ($575,313.64), As the Net Present Value of the Project is negative, the company is not recommended to invest into the project.

Solution b) Following is the equation to calculate the Internal Rate of Return:

For calculating IRR, we require two discounting rates in such a way that with one discounting rate we get a positive Net Present Value and with the other discounting rate, we get a negative Net Present Value.

IRR = DF with positive NPV + [(Positive NPV / Difference between Positive and Negative NPV) * (Difference in two rates)]