Assessing investments: Accounting approach
ARR, Average accounting rate of return
Average accounting rate of return
The average accounting rate of return is a method of assessing the profitability of an investment and is abbreviated as #ARR#. With this method, the average net return on an investment is expressed as a percentage of the average invested capital:
\[\text{ARR} = \frac{\text{average net return}}{\text{average invested capital}}\cdot 100\%\]
The averages are taken over the entire duration (the number of periods) of the activity.
To calculate the average net return, we divide the total cash flows of an investment by the duration of the project: \[\text{average net return} = \frac{\text{net return}}{\text{duration}} = \frac{\displaystyle \sum_{j=0}^{n} C_j}{n} \] where #n# is the duration of the investment and #C_j# is the cash flow in period #j#.
The average invested capital is given by \[\text{average invested capital} = \frac{|C_0| +RV}{2}\] where #RV# is the residual value and #|C_0|# is the absolute value of #C_0#, i.e. the cost of the investment.
To calculate the average invested capital, we add the residual value of the investment to the initial investment and divide this sum by two.
In this calculation of the average invested capital, we assume that the invested capital #|C_0|# decreases linearly over time to the level of the residual value. The value in the formula is the #y#-coordinate of the midpoint of the line segment of the graph of this linear function between the points #\rv{0,|C_0|}# and #\rv{n,RV}#.
The figure below illustrates this. Here the invested capital is #|C_0|=500#, and the residual value is #RV = 50#. The average invested capital is then #\frac{500+50}{2}= 275#.
Unless otherwise stated, it can be assumed that the residual value of the investment is zero.
\[\begin{array}{c|c} &\text{Cash flows}\\ \hline\ C_0 &-475\\\ C_1 & 155 \\ \ C_2 & 155 \\ \ C_3 & 155 \\ \ C_4 & 155 \\ \ C_5 & 155 \\ \end{array}\]
Calculate the average accounting rate of return for this investment. Give your answer with two decimal places.
To calculate the average accounting rate of return, we use the following formula:
\[ARR = \dfrac{\text{average net return}}{\text{average invested capital}}\cdot 100\%\]
where
\[\begin{array}{rcl}
\text{average net return} &=& \dfrac{\displaystyle \sum_{i=0}^{n} C_i}{n}\\
&&\phantom{xxxx}\color{blue}{\text{formula for average net return}}\\
&=& \dfrac{C_0 + C_1 + C_2 + C_3 + C_4 + C_5}{n}\\
&&\phantom{xxxx}\color{blue}{\text{the summation notation expanded }}\\
&=& \dfrac{-475 + 155+ 155+ 155+ 155+ 155}{5}\\
&&\phantom{xxxx}\color{blue}{\text{substituted the values for }C_0, C_1, C_2, C_3, C_4, C_5 \text{ and }n }\\
&=& 60\\
&&\phantom{xxxx}\color{blue}{\text{calculated and rounded}}\\
\end{array}\]
and
\[\begin{array}{rcl}
\text{average invested capital} &=& \dfrac{|C_0| + {RV}}{2}\\
&&\phantom{xxxx}\color{blue}{\text{formula for average invested capital}}\\
&=& \dfrac{475 + 0}{2}\\
&&\phantom{xxxx}\color{blue}{\text{substituted the value for }C_0 \text{ and } RV=0}\\
&=& 237.50\\
&&\phantom{xxxx}\color{blue}{\text{calculated}}\\
\end{array}\]
This results in the following calculation for the average accounting return:
\[\begin{array}{rcl}
ARR &=& \dfrac{\text{average net return}}{\text{average invested capital}}\cdot 100\%\\
&&\phantom{xxxx}\color{blue}{\text{formula for average accounting rate of return}}\\
&=& \dfrac{60}{237.50}\cdot 100\%\\
&&\phantom{xxxx}\color{blue}{\text{values entered}}\\
&\approx& 25.26\%\\
&&\phantom{xxxx}\color{blue}{\text{calculated and rounded to two decimal places}}\\
\end{array}\]
Or visit omptest.org if jou are taking an OMPT exam.
