The internal rate of return (IRR) is an appraisal technique that utilizes discounted cash flows – taking into account the timing and magnitude of cash flows. It is a rate that the present value of the expected future cash flows with the cost of the investment. In other words, it is the discounting rate that equates NPV to zero. The internal rate of return is also described as the yield on investment, marginal efficiency of capital, time-adjusted rate of internal return, or the rate of return over cost.

**How to Calculate IRR**

The IRR concept is easy to understand if it deals with one project in one period. As an example, consider an investment whose initial cost is $10,000. Assume that the investment will be worth $10,800 after 1 year. The true rate of return for this one-period investment will be given as:

Rate of Return = (10,800-10,000)/10,000 = 0.08 or 8%.

This means that the return on your investment is $10,800 – $10,000 = $800. The rate of return is 800/10,000*100% = *8%.*

So, at the end of 1 year you get your initial investment of $10,000 plus the return of $800. This could be an investment on bonds, shares, property, land, or even fixed deposit in a bank. If your rate of return is negative, then the investment is not worth investing. For instance, if your investment of $10,000 comes to $9,800 after 1 year, the return will be $9,800-10,000 = -200 and the rate of return will be -200/10,000 = *-2%.*

The rate of return of 8% and the rate of return of -2% make the discounted (present) value of the future cash inflows to be equal to the initial investment of $10,000. Thus, 8% makes the future cash inflows of $10,800 to be equal to the initial cost of $10,000. This is the basic idea of internal rate of return.

The formula for the internal rate of return (r) on an investment C_{0} that generates a single cash flow after period (C_{1}) is given as follows:

r = (C_{1}-C_{0})/C_{0} = (C_{1}/C_{0}) – 1………………………………………………………………… (1)

This equation can be rewritten as:

C_{0}/C_{1} = 1 + r

This is also the same as:

C_{0} = C_{1}/(1+r)………………………………………………………………………………… (2)

Equation 2 shows that the rate of return (r) depends solely on the cash flows of the project, and not any other factor. This is why it is known as the internal rate of return. The internal rate of return (IRR) is the rate that equates the investment outlay with the present value of cash inflow received after one period. This also implies that the rate of return is the discount rate which makes NPV = 0. There is no satisfactory way of defining the true rate of return of a long-term asset. IRR is the best available concept.

Based on the equation involving r and C above, the actual formula for IRR is given as:

…………………………………………. (3)

It can be noticed that the ERR equation is the same as the one used for the NPV method. In the NPV method, the required rate of return, k, is known and the net present value is found, while in the IRR method the value of r has to be determined at which the net present value becomes zero.

The acceptable IRR can be calculated using trial and error and interpolation.

*Example: Calculating IRR Using Trial and Error*

A project costs $16,000 and is expected to generate cash inflows of $8,000, $7,000 and $6,000 at the end of each year for next 3 years. Calculate its IRR using the trial and error method.

*Solution:*

We know that IRR is the rate at which project will have a zero NPV. As a first step, we try (arbitrarily) a 20 per cent discount rate. The project’s NPV at 20 per cent is:

NPV = 8,000(PVF_{1, 020}) + 7,000(PVF_{2, 020}) + 6,000 (PVF_{3, 0.20}) – 16,000

= (8,000 x 0.833) + (7,000 x 0.694) + (6,000 x 0.579) – 16,000

= 14,996 – 16,000

= -1,004

The NPV is a negative value at a discount rate of 20%. To get the right rate of return that equates NPV to zero, we try a lower rate, say 16%.

NPV = 8,000(PVF_{1, 0.16}) + 7,000(PVF_{2, 0.16}) + 6,000 (PVF_{3, 0.16}) – 16,000

= (8,000 x 0.862) + (7,000 x 0.743) + (6,000 x 0.641) – 16,000

= 15,943 – 16,000

= -57

The answer is still negative, so we still try a lower rate of return, e.g. 15%

NPV = 8,000(PVF_{1, 0.15}) + 7,000(PVF_{2, 0.15}) + 6,000 (PVF_{3, 0.15}) – 16,000

= (8,000 x 0.870) + (7,000 x 0.756) + (6,000 x 0.658) – 16,000

= 16,200 – 16,000

= 200

This is a positive NPV, which shows that the rate of return that equates NPV to zero is between 15% and 16%. At this point we use linear interpolation to determine the most accurate rate of return as follows:

Where:

L = Lower Discount Rate

H = Higher Discount Rate

N_{L} = NPV at lower discount rate

N_{H} = NPV at higher discount rate

**Decision Criteria**

The accept-or-reject rule, using the IRR method, is to accept the project if its internal rate of return is higher than the opportunity cost of capital (r > k). Note that k is also known as the required rate of return, or the cut-off. The project shall be rejected if its internal rate of return is lower than the opportunity cost of capital (r < k). The decision maker may remain indifferent if the internal rate of return is equal to the opportunity cost of capital. Thus the IRR acceptance rules are:

- Accept the project when r > k.
- Reject the project when r < k.
- The investor is indifferent when r = k.

In case of independent projects, IRR and NPV rules will give the same results if the firm has no shortage of funds.