Npv vs. Irr
NPV Versus IRR W.L. Silber
I.
Our favorite project A has the following cash flows:
-1000 0
0 1
0 2
+300 3
+600 4
+900 5
We know that if the cost of capital is 18 percent we reject the project because the net present value is negative:
- 1000 + 300 600 900 + + = NPV 3 4 (1.18) (1.18) (1.18)5
- 1000 + 182.59 + 309.47 + 393.40 = -114.54
We also know that at a cost of capital of 8% we accept the project because the net present value is positive:
- 1000 +
300 600 900 + + = NPV 3 4 (1.08 ) (1.08 ) (1.08 )5
- 1000 + 238.15 + 441 .02 + 612 .52 = 291.69
II.
Thus, somewhere between 8% and 18% we change our evaluation of project A
from rejecting it (when NPV is negative) to accepting it (when NPV is positive). We can calculate the point at which NPV shifts from negative to positive by searching for the value of r, called the internal rate of return (IRR) in the following equation, which makes the NPV=0.
- 1000 +
300 600 900 + + =0 3 4 (1 + r ) (1 + r ) (1 + r )5
More generally, if CFi is the cash flow in period i, the IRR is that rate, r, such that:
CF0 +
CFt CF1 CF2 + +L+ =0 2 (1 + r ) (1 + r ) (1 + r )t
In our case, CF0 = -1000, CF3 = 300, CF4 = 600 and CF5 = 900. All the other CFi = 0.
III.
The IRR can, in general, only be derived by trial and error. Putting our values for
the CFi into a calculator (very carefully) we find the IRR= 14.668%. We can check this result as follows:
- 1000 + 300 600 900 + + = 3 4 (1.14668) (1.14668) (1.14668) 5
- 1000 + 198.97 + 347 .04 + 453 .97 = -.02
The sum is not exactly zero because of rounding.
IV.
We can now formulate an alternative rule to accepting the project if NPV > 0 and
rejecting it if NPV < 0. In particular, we can recommend rejecting a project if the cost of capital is greater than the IRR (14.668% in this case) and we can recommend accepting a project if the cost of capital is less than the IRR. These two rules are equally acceptable in
I.
Our favorite project A has the following cash flows:
-1000 0
0 1
0 2
+300 3
+600 4
+900 5
We know that if the cost of capital is 18 percent we reject the project because the net present value is negative:
- 1000 + 300 600 900 + + = NPV 3 4 (1.18) (1.18) (1.18)5
- 1000 + 182.59 + 309.47 + 393.40 = -114.54
We also know that at a cost of capital of 8% we accept the project because the net present value is positive:
- 1000 +
300 600 900 + + = NPV 3 4 (1.08 ) (1.08 ) (1.08 )5
- 1000 + 238.15 + 441 .02 + 612 .52 = 291.69
II.
Thus, somewhere between 8% and 18% we change our evaluation of project A
from rejecting it (when NPV is negative) to accepting it (when NPV is positive). We can calculate the point at which NPV shifts from negative to positive by searching for the value of r, called the internal rate of return (IRR) in the following equation, which makes the NPV=0.
- 1000 +
300 600 900 + + =0 3 4 (1 + r ) (1 + r ) (1 + r )5
More generally, if CFi is the cash flow in period i, the IRR is that rate, r, such that:
CF0 +
CFt CF1 CF2 + +L+ =0 2 (1 + r ) (1 + r ) (1 + r )t
In our case, CF0 = -1000, CF3 = 300, CF4 = 600 and CF5 = 900. All the other CFi = 0.
III.
The IRR can, in general, only be derived by trial and error. Putting our values for
the CFi into a calculator (very carefully) we find the IRR= 14.668%. We can check this result as follows:
- 1000 + 300 600 900 + + = 3 4 (1.14668) (1.14668) (1.14668) 5
- 1000 + 198.97 + 347 .04 + 453 .97 = -.02
The sum is not exactly zero because of rounding.
IV.
We can now formulate an alternative rule to accepting the project if NPV > 0 and
rejecting it if NPV < 0. In particular, we can recommend rejecting a project if the cost of capital is greater than the IRR (14.668% in this case) and we can recommend accepting a project if the cost of capital is less than the IRR. These two rules are equally acceptable in