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Calculating the IRR

The IRR can be found either by using trial-and-error techniques or with the aid of a sophisticated financial calculator or a computer. Here we demonstrate the trial-and-error approach. Calculating the IRR for an annuity is considerably easier than calculating it for a mixed stream of operating cash inflows. The steps involved in calculating the IRR in each case are given in Table W9.1. These steps are illustrated by the following example.

Example

The two-step procedure given in Table W9.1 for finding the IRR of an annuity can be demonstrated by using Bennett Company's project A cash flows given in Table 9.1.

Step 1: Dividing the initial investment of $42,000 by the annual cash inflow of $14,000 results in a payback period of 3.000 years ($42,000 ÷ $14,000 = 3.000).

Step 2: According to Table A-4, the PVIFA factors closest to 3.000 for 5 years are 3.058 (for 19%) and 2.991 (for 20%). The value closest to 3.000 is 2.991; therefore, the IRR for project A, to the nearest 1%, is 20%. The actual value, which is between 19 and 20%, could be found by using a calculator or computer or by interpolation; it is 19.86%. (Note: For our purposes, values rounded to the nearest 1% will suffice.) Project A with an IRR of 20% is quite acceptable, because this IRR is above the firm's 10% cost of capital.

The seven-step procedure given in Table W9.1 for finding the internal rate of return of a mixed stream of cash inflows can be illustrated by using Bennett Company's project B cash flows given in Table 9.1.

Step 1: Summing the cash inflows for years 1 through 5 results in total cash inflows of $70,000. That amount, when divided by the number of years in the project's life, results in an average annual cash inflow of $14,000 [($28,000 + $12,000 + $10,000 + $10,000 + $10,000) ÷ 5].

Step 2: Dividing the initial outlay of $45,000 by the average annual cash inflow of $14,000 results in an "average payback period" (or present value of an annuity factor, PVIFA) of 3.214 years.

Step 3: In Table A-4, the factor closest to 3.214 for 5 years is 3.199, the factor for a discount rate of 17%. The starting estimate of the IRR is therefore 17%.

Step 4: Because the actual early-year cash inflows are greater than the average annual cash inflows of $14,000, a subjective increase of 2% is made in the discount rate. This makes the estimated IRR 19%.

Step 5: By using the present value interest factors (PVIF) for 19% and the correct year from Table A-3, we calculate the net present value of the mixed stream as follows:

Year (t) Cash inflows
(1)
PVIF19%,t
(2)
Present value at 19%
[(1) x (2)]
(3)
1 $28,000 .840 $23,520
2 12,000 .706 8,472
3 10,000 .593 5,930
4 10,000 .499 4,990
5 10,000 .419 4,190
  Present value of cash inflows $47,102
  - Initial investment 45,000
  Net present value (NPV) $2,102

Steps 6 and 7: Because the net present value of $2,102, calculated in Step 5, is greater than zero, the discount rate should be subjectively increased. Because the NPV deviates by only about 5% from the $45,000 initial investment, let's try a 2 percentage point increase, to 21%.

Year (t) Cash inflows
(1)
PVIF21%,t
(2)
Present value at 21%
[(1) x (2)]
(3)
1 $28,000 .826 $23,128
2 12,000 .683 8,196
3 10,000 .564 5,640
4 10,000 .467 4,670
5 10,000 .386 3,860
  Present value of cash inflows $45,494
  - Initial investment 45,000
  Net present value (NPV) $494

These calculations indicate that the NPV of $494 for an IRR of 21% is reasonably close to, but still greater than, zero. Thus a higher discount rate should be tried. Because we are so close, let's try a 1 percentage point increase, to 22%. As the following calculations show, the net present value using a discount rate of 22% is - $256.

Year (t) Cash inflows
(1)
PVIF22%,t
(2)
Present value at 22%
[(1) x (2)]
(3)
1 $28,000 .820 $22,960
2 12,000 .672 8,064
3 10,000 .551 5,510
4 10,000 .451 4,510
5 10,000 .370 3,700
  Present value of cash inflows $44,744
  - Initial investment 45,000
  Net present value (NPV) -$256

Because 21 and 22% are consecutive discount rates that give positive and negative net present values, we can stop the trial-and-error process here. The IRR that we are seeking is the discount rate for which the NPV is closest to $0. For project B, 22% causes the NPV to be closer to $0 than 21%, so we will use 22% as the IRR. If we had used a financial calculator or a computer or interpolation, the exact IRR would be 21.65%; as indicated earlier, for our purposes the IRR rounded to the nearest 1% will suffice.

Project B is acceptable, because its IRR of approximately 22% is greater than Bennett Company's 10% cost of capital. This is the same conclusion reached by using the NPV criterion.

It is interesting to note in the example that the IRR suggests that project B, which has an IRR of approximately 22 percent, is preferable to project A, which has an IRR of approximately 20 percent. This conflicts with the rankings of the projects obtained in an earlier example by using NPV. Such conflicts are not unusual. There is no guarantee that these two techniques--NPV and IRR--will rank projects in the same order. However, both methods should reach the same conclusion about the acceptability or nonacceptability of projects.




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