Question

If you learn a foreign language, you can improve your compensation at work. Given you are not good at learning languages, you estimate today you would need to spend $5,000 to learn a language. You then believe you can earn an extra $3,000 per year for 4 years starting *today* before you move onto a new job. (The last $3,000 occurs three years from today.) However, you will also pay back $2,000 to the company exactly 3 years from today since some of your clients will leave with you. Your discount rate is 6% APR.

1. What is the NPV of learning a new language?

2. What is the IRR?

3. What is the MIRR if you use the discount rate as the reinvestment rate?

Answer 1

1. NPV = PV of all cash flows

NPV = Σ CFⁿ/(1+r)ⁿ , where CF is the cash flow in period n and r is the APR

= -5000/1.06⁰ + 3000/1.06⁰ + 3000/1.06¹ + 3000/1.06² + 3000/1.06³ - 2000/1.06³

= -5000 + 3000 + 2830.19 + 2669.99 + 2518.86 - 1679.24 = $4,339.9

2. IRR is the rate of return when NPV = 0

Let IRR = i

=> -5000/(1+i)⁰ + 3000/(1+i)⁰ + 3000/(1+i)¹ + 3000/(1+i)² + 3000/(1+i)³ - 2000/(1+i)³ = 0

Solve for i to get i = 126.117%

3. Reinvestment Rate = financing rate = 6%

FV_{Positive Cash Flows} = 3000(1.06)³ + 3000(1.06)² + 3000(1.06)1 + 3000(1.06)⁰ = 13123.85

FV_{Negative Cash Flows} = 5000(1.06)³ + 2000(1.06)⁰ = 7955.08

MIRR = (FV_{Positive Cash Flows}/FV_{Negative Cash Flows})^1/n - 1

= (13123.85/7955.08)^1/3 - 1 = 1.1816 - 1 = 0.1816 = 18.16%