# Table W9.1

Steps for Calculating the Internal Rates of Return (IRRs) of Annuities and Mixed Streams

FOR AN ANNUITY

Step 1: Calculate the payback period for the project. a

Step 2: Find, for the life of the project, the present value interest factor closest to the payback value. (Use Table A-4, which shows the present value interest factors for a \$1 annuity, PVIFA.) The discount rate associated with that factor is the internal rate of return (IRR) to the nearest 1%.

FOR A MIXED STREAM b

Step 1: Find the average annual cash inflow by dividing the sum of the annual cash inflows by the number of years in the project's life.

Step 2: Divide the average annual cash inflow into the initial investment to get an "average payback period" (or present value interest factor for a \$1 annuity, PVIFA). The average payback is needed to estimate the IRR for the average annual cash inflow.

Step 3: Find the discount rate associated with the present value interest factor in Table A-4 (PVIFA) for the life of the project that is closest to the average payback period (as described in Step 2 for finding the IRR of an annuity). The result will be a very rough approximation of the IRR based on the assumption that the mixed stream of cash inflows is an annuity.

Step 4:c Adjust subjectively the IRR obtained in Step 3 by comparing the pattern of average annual cash inflows (calculated in Step 1) to the actual mixed stream of cash inflows. If the actual cash flow stream seems to have higher inflows in the earlier years than the average stream, adjust the IRR up. If the actual cash inflows in the earlier years are below the average, adjust the IRR down. The amount of adjustment up or down typically ranges from 1 to 3 percentage points, depending upon how much the actual cash inflow stream's pattern deviates from the average annual cash inflows. For small deviations, an adjustment of around 1 percentage point may be best, whereas for large deviations, adjustments of around 3 percentage points are generally appropriate. If the average cash inflows seem fairly close to the actual pattern, make no adjustment in the IRR.

Step 5: Calculate the net present value of the mixed stream project using the IRR from Step 4. Be sure to use Table A-3 (the present value interest factors for \$1, PVIF), treating the estimated IRR as the discount rate.

Step 6: If the resulting NPV is greater than zero, subjectively raise the discount rate; if the resulting NPV is less than zero, subjectively lower the discount rate. The greater the deviation of the resulting NPV from zero, the larger the subjective adjustment. Typically, adjustments of 1 to 3 percentage points are used for relatively small deviations, whereas larger adjustments are required for relatively large deviations.

Step 7: Calculate the NPV using the new discount rate. Repeat Step 6. Stop as soon as two consecutive discount rates that cause the NPV to be positive and negative, respectively, have been found. d Whichever of these rates causes the NPV to be closer to zero is the IRR to the nearest 1%.

a The payback period calculated actually represents the interest factor for the present value of an annuity (PVIFA) for the given life discounted at an unknown rate, which, once determined, represents the IRR for the project.

b Note that the subjective estimates are suggested in Steps 4 and 6. After working a number of these problems, a "feel" for the appropriate subjective adjustment, or "educated guess" may result.

c The purpose of this step is to provide a more accurate estimate of the IRR. This step can be skipped.

d A shortcut method is to find a discount rate that results in a positive NPV and another that results in a negative NPV. Using only these two values, one can interpolate between the two discount rates to find the IRR. This approach, which may be nearly as accurate as that described before, can guarantee an answer after only two NPV calculations. Of course, because interpolation involves a straight-line approximation to an exponential function, the wider the interpolation interval, the less accurate the estimate.