Time Value of Money, Discounted Cash Flow Analysis (NPV) & Internal Rate of Return

1. Time Value of Money,Discounted Cash Flow Analysis (NPV) & Internal Rate of Return

2. John has $100 that he can invest at 10% per annum. In one year this amount will grow to $110 = ($100 x 10%) + $100 (NOTE: 10% is used as it is easy for calculation – interest rates are currently much lower!) • In two years it will grow to $121 = ($110 x 10%) + $110 • And in three years, to $133.10 = ($121 x 10%) + $121 • Every year, the amount of interest gets larger ($10, $11, $12.10) because of compound interest (interest on interest)

3. Notice that • 1. $110 = $100 x (1 + .10) • 2. $121 = $110 x (1 + .10) = $100 x 1.1 x 1.1 = $100 x 1.12 • 3. $133.10 = $121 x (1 + .10) = $100 x 1.1 x 1.1 x 1.1 = = $100 x 1.13 • In general, the future value, FVt, of $1 invested today at i% for t periods is FVt= $1 x (1 + i)t • The expression (1 + i)t is the future value interest factor.

4. Conversely, if you were offered $100 today, $110 to be paid in one year, $121 to be paid in two year or $133.10 to be paid in three years, you should be indifferent as to which you would choose as the $100 invested at 10% would grow to $110 in 1 year, to $121 in 2 years and $133.10 in three years. • Extending this further, if you were offered $100 today versus $100 in three years, you should select $100 today as the $100 today will grow to $133.10 in 3 years • This is known as the time value of money • Comparing money received in different time periods is like comparing apples and oranges - they have different values because of the differing time periods • So when we analyze projects with cash flows over several years, we need to adjust for this

5. Extending the analogy, at a 10% potential rate of investment:: • $110 in 1 year is worth $100 today • $121 in 2 years is worth $100 today • $133.10 in 3 years is worth $100 today • Notice that • 1. $100 = $110 x 1/(1 + .10) • 2. $100 = $121 x 1/(1 + .10)2 • 3. $100 = $133.10 x 1/(1 + .10)3 • In general, the present value, PVt, of $1 received in t periods when the potential investment rate is i% is PVt = $1 x 1/(1 + i)t

6. The expression 1/(1 + i)t is called the present value interest factor (also commonly called the “discount factor”) • This is the same as the discount factor that is referred to on page 91 of your text book • When we analyze projects that have cash flows in several years, we need to convert all the dollar amounts into today’s dollars • We do this by using these discount factors to convert future dollars to their value in today’s dollars so that you are comparing apples and apples – NOT apples and oranges

7. Internal Rate of Return • definition - the discount rate that makes net present value equal to zero • If you invest $100 today and receive $110 in one year, what is the rate of return? PV = FVt/(1 + i)t • 100 = 110(1 + i)1 • Solving this equation, you find that i is 10%

8. Another more difficult example • Suppose you deposit $5000 today in an account paying r percent per year. If you will get $10,000 in 10 years, what rate of return are you being offered? • Set this up as present value equation: FV = $10,000 PV = $ 5,000 t = 10 years PV = FVt/(1 + i)t $5000 = $10,000/(1 + i)10 • Now solve for i: (1 + i)10 = $10,000/$5,000 = 2.00 i = (2.00)1/10 - 1 = .0718 = 7.18 percent An easier way! Use the Excel IRR function!

9. Your turn ….. • An e-commerce project requires a cash outlay of $350,000 today but achieves net benefits (revenues less expenses) in the next five years of $20000, 50000, 100000, 150000, 200000 • Calculate the internal rate of return of this project using Excel