Time Value of Money, Discounted Cash Flow Analysis (NPV) & Internal Rate of Return
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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 $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)
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.
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
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
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 = FVt/(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 = $10,000 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!
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