Time Value of Money, Discounted Cash Flow Analysis (NPV) & 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!) In two years it will grow to $121 = ($110 x 10%) + $110 And in three years, to $ = ($121 x 10%) + $121 Every year, the amount of interest gets larger ($10, $11, $12.10) because of compound interest (interest on interest)

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

Conversely, if you were offered $100 today, $110 to be paid in one year, $121 to be paid in two year or $ 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 $ 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 $ 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

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  $ 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= $ x 1/(1 +.10) 3 In general, the present value, PV t, of $1 received in t periods when the potential investment rate is i% is PV t = $1 x 1/(1 + i) t

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

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 = FV t /(1 + i) t 100 = 110(1 + i) 1 Solving this equation, you find that i is 10%

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 = $10,000PV = $ 5,000t = 10 years PV = FV t /(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/ =.0718 = 7.18 percent An easier way! Use the Excel IRR function!

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, , , Calculate the internal rate of return of this project using Excel