Assessing investments: Economic approach
DPP, Discounted Payback Period
Discounted payback period
The discounted payback period, like the accounting payback period, is a method of calculating how long it takes to recoup the costs of an investment and is abbreviated as #DPP#.
The discounted payback period is therefore the period in which the sum of the discounted cash flows is greater than or equal to #0# for the first time.
When applying this method, projects with a discounted payback period that is shorter than the payback period set by the company's management are acceptable.
Discounted payback period
Given an investment with duration #n#, cash flows #C_0#, #C_1 ,\ldots, C_n# and discount rate #r# the discounted payback period is the period #j# in which for the first time the sum of the discounted cash flows up to and including period #j# is non-negative:
\[\sum_{i=0}^{k}\dfrac{C_i}{(1+r)^i} \lt 0\quad \text{for all }k\lt j\quad \text{ and }\quad \sum_{i=0}^{j}\dfrac{C_i}{(1+r)^i} \geq 0\]
We therefore add up one by one the discounted cash flows of the investment until they are together greater than or equal to #0#. The period in which this requirement is met for the first time is the discounted payback period.
When comparing two or more projects with the same discounted payback period, preference is given to the investment with the highest net return in the period in which the DPP is achieved.
\[\begin{array}{l|c} &\text{Cash flows}\\ \hline\ C_0 & -250\\\ C_1 & 95 \\ \ C_2 & 95 \\ \ C_3 & 95 \\ \ C_4 & 95 \\ \ C_5 & 95 \\ \end{array}\]
Assume a discount rate of #6#%.
Determine the discounted payback period of the investment.
To determine the discounted payback period #DPP#, we first set up the table with discounted cash flows with #r=\frac{6}{100}=0.06#.
\[\begin{array}{l|ccc} i&\dfrac{C_i}{(1+r)^i}\\ \hline\ 0 & -250\\\ 1 & 89.62 \\ \ 2 & 84.55 \\ \ 3 & 79.76 \\ \ 4 & 75.25 \\ \ 5 & 70.99 \\ \end{array}\]
Then set up the table with cumulative discounted cash flows.
\[\begin{array}{l|ccc} j&\displaystyle \sum_{i=0}^{j} \dfrac{C_i}{(1+r)^i}\\ \hline\ 0 & -250\\\ 1 & -160.38 \\ \ 2 & -75.83 \\ \ 3 & 3.94 \\ \ 4 & 79.19 \\ \ 5 & 150.17 \\ \end{array}\]
The discounted payback period is the period in which the sum of the discounted cash flows is greater than or equal to #0# for the first time. Since
- #\sum_{i=0}^{2}\dfrac{C_i}{(1+r)^i} =-75.83 \lt 0# and
- #\sum_{i=0}^{3} \dfrac{C_i}{(1+r)^i} =3.94 \geq 0#,
Or visit omptest.org if jou are taking an OMPT exam.
