IRR, Internal rate of return

Assessing investments: Economic approach

IRR, Internal rate of return

Internal rate of return

The internal rate of return is a way of expressing the profitability of an investment that uses the concept of net present value. The term is abbreviated as IRR. The internal rate of return calculates the economic profit in the form of the discount rate at which the investment project has a net present value equal to #0#.

If all cash flows are #C_1,\ldots,C_n# positive and the net return is negative, then this assessment makes no sense, because then there is no positive discount rate at which the investment project has a net present value equal to #0#. In general, we assume that cash flows and net return are positive.

The method to calculate the internal rate of return is to solve the equation #NPV = 0# with the discount rate #r# as the unknown:

Internal rate of return For a given investment with duration #n# and cash flows #C_0, C_1, \ldots, C_n# the internal rate of return is defined as the percentage associated with the discount rate #r#, for which #NPV = 0#, that is, the net present value of the investment #NPV# is exactly equal to #0#. Using the net present value formula, this equation can be written as

\[ \sum_{i=0}^{n} \dfrac{C_i}{(1+r)^i} = 0\]

At this discount rate, the cost of the investment #|C_0|# is exactly equal to the sum of the discounted cash flows #\sum_{i=1}^{n} \dfrac{C_i}{(1+r)^i}#.

The equation with the unknown #r#, which we are solving, can often only be solved numerically (i.e. not exactly).

The major advantage of internal rate of return as an assessment method compared to net present value is that this method better enables investors to compare multiple investments. This can best be explained with the help of an example.

Comparing investments

Suppose we have to choose between two investments, investment #1# and investment #2#, both of which have a net present value of #\euro\,1000#. At first glance, the two investments appear equally profitable and an investor should value both options equally.

However, the cost of investment #1# appears to total #\euro\,10\,000# while the cost of investment #2# totals #\euro\,50\,000#. If we take this into account, it becomes clear that investment #1# is a much better option than investment #2#, as the investor has to risk a significantly smaller amount to ultimately make the same amount of profit. In other words, the risk of investment #1# is a lot lower than the risk of investment #2#.

Net present value, a method that expresses the profitability of an investment in an absolute number, is often unsuitable for comparing multiple investments. Unlike the net present value, the internal rate of return expresses the profitability of an investment as a percentage that takes into account the ratio between the cost and returns of an investment. In the above scenario, investment #1# will therefore have a higher internal rate of return than investment #2#.

The calculations below assume a discount rate of #10\%#.

\[\begin{array}{rcl}
NPV_1 &=& -10000 + \dfrac{4423.26}{1.1} + \dfrac{4423.26}{1.1^2} + \dfrac{4423.26}{1.1^3} = 1000\\
&&\\
NPV_2 &=& -50000 + \dfrac{20507.85}{1.1} + \dfrac{20507.85}{1.1^2} + \dfrac{20507.85}{1.1^3} = 1000
\end{array}\]

Thus, on a net present value basis, an investor should be indifferent between the two investments. However, when we look at the values for the discount rate #r# that make the net present values of the investments equal to #0#, there is a clear difference between the two options.

\[\begin{array}{rcl}
NPV_1 &=& -10000 + \dfrac{4423.26}{1.15598} + \dfrac{4423.26}{1.15598^2} + \dfrac{4423.26}{1.15598^3} \approx 0\\
&&\\
NPV_2 &=& -50000 + \dfrac{20507.85}{1.11133} + \dfrac{20507.85}{1.11133^2} + \dfrac{20507.85}{1.11133^3} \approx 0
\end{array}\]

This shows that the internal rate of return of investment #1# is equal to #\pm15,60\%# and that of investment #2# is equal to #\pm11.13\%#. Based on the internal rate of return, an investor will therefore choose investment #1#.

The cash flows of an investment are shown in the table below.
\[\begin{array}{l|c} &\text{Cash flows}\\ \hline\ C_0 & -320\\\ C_1 & 90 \\ \ C_2 & 90 \\ \ C_3 & 90 \\ \ C_4 & 90 \\ \ C_5 & 90 \\ \end{array}\]
Determine the internal rate of return of the investment.

The internal rate of return is between 12% and 13%

The internal rate of return is defined as the value for the discount rate #r# at which the net present value of the investment is equal to zero. Thus, in order to calculate the internal rate of return, the following equation must be solved for #r#:
\[NPV = -320 + {{90}\over{1+r}}+{{90}\over{\left(1+r\right)^2}}+{{90}\over{\left(1+r\right)^3}}+{{90}\over{\left(1+r\right)^4}}+{{90}\over{\left(1+r\right)^5}}=0\]

Substituting #r = 0.12\,# gives #\,NPV = # #4#.
Substituting #r = 0.13\,# gives #\,NPV = # #-3#.

The value of #r# for which #NPV = 0#, is therefore between the 12% and 13%.

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