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

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

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

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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

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