# IRR Calculation

Internal rate of return (IRR) is the annual compound interest rate at which an investment’s net present value is zero. Projects whose IRR is higher than the company’s cost of capital are good candidates for investment and projects with highest IRR must be selected. There are multiple ways in which we can calculate IRR: using Microsoft Excel IRR or XIRR functions, using any financial calculator or through a manual hit-and-trial method.

In manual calculation of internal rate of return, we must use a hit-and-trial method because there is no algebraic formula in which we plug the cash flows and initial investment and get exact IRR. In the hit-and-trial method, we start by arbitrarily selecting a discount rate and working out the net present value of the project by discounting the cash flows using that rate. If the net present value is positive, we call it NPVa. Since there is an inverse relationship between present value and discount rate, select a higher discount rate and rework the NPV. Keep increasing the rate until we get close to zero. If the initial net present value is negative, let’s label it NPVb, decrease the discount rate and recalculate NPV to get close to zero. The discount rate at which NPV = 0 is the IRR.

Instead of going through all these iterations, we can use the two NPVs, one higher and one lower than zero, to approximate IRR using the following formula:

 IRR = ra + NPVa × (rb − ra) NPVa − NPVb

Where ra is the lower interest rate, rb is the higher interest rate and NPVa and NPVb are the net present values calculated using the interest rate ra and rb respectively.

## Example

Your company is interested in a project that will generate cash inflows of \$300,000, \$350,000, \$370,000, \$330,000 at the end of each year for 4 years. Cash outflows before taxes amount to 50% of the cash inflows. The project will require purchase of fixed assets of \$550,000 which is to be depreciated using straight-line method with a salvage value of \$150,000. After the initiation of project, existing equipment with book value of \$80,000 can be sold at \$100,000. The hurdle rate applicable to the project is 12% and the company’s tax rate is 40%. Working capital of \$130,000 is required at time 0 which shall be released at the end of the project.

Let’s calculate the IRR of the project.

The first thing we need is to get a complete picture of the project’s cash flows. The initial investment is calculated below:

 Equipment A 550,000 Working capital B 130,000 Sales proceeds of equipment disposed off C 100,000 Carrying value of equipment disposed off D 80,000 Taxable gain E=C-D 20,000 Tax on gain @ 40% F=E×40% 8,000 After-tax sales proceeds G=C-F 92,000 Total initial investment outlay H=A+B-G 588,000

We also need to find the net project cash flows which are worked out as follows:

Year 0 1 2 3 4
Cash inflows 300,000 350,000 370,000 330,000
Less: operating costs @ 50% 150,000 175,000 185,000 165,000
Operating cash flows before tax 150,000 175,000 185,000 165,000
Less: depreciation (straight-line) 100,000 100,000 100,000 100,000
Income before tax 50,000 75,000 85,000 65,000
Less: taxes @ 40% 20,000 30,000 34,000 26,000
Net income 30,000 45,000 51,000 39,000
Add: depreciation 100,000 100,000 100,000 100,000
Net operating cash flows 130,000 145,000 151,000 139,000
Terminal cash flow 280,000
Initial investment outlay -588,000
Net cash flows -588,000 130,000 145,000 151,000 419,000

The terminal cash flow above equals the sum of salvage value (\$150,000) and recoupment of working capital (\$130,000).

Let’s calculate NPV at 10% and 15%:

Period Cash Flow PV Factor
@ 10%
PV @ 10% PV Factor
@15%
PV @ 15%
0 (588,000) 1.0000 (588,000) 1.0000 (588,000)
1 130,000 0.9091 118,182 0.8696 113,043
2 145,000 0.8264 119,835 0.7561 109,641
3 151,000 0.7513 113,449 0.6575 99,285
4 419,000 0.6830 286,183 0.5718 239,565
NPV 49,648 (26,466)

NPV at 10% is positive and NPV at 15% is negative so we know IRR must be between 10% and 15%. IRR can be estimated using the following formula:

 IRR = 10% + \$49,648 × (15% − 10%) \$49,648 − (−\$26,466)
 IRR = 10% + \$49,648 × (15% − 10%) \$49,648 + \$26,466)

IRR = 10% + 3.26% = 13.26%

If we work out IRR using Excel IRR function, we get a value of 13.15% which is pretty close to our approximation.