As explained before, in sensitivity analysis, we aim to discover the magnitude of change in one variable (here, output variables) with respect to change in other variables (here, input parameters). Then, we can rank the variables based on their sensitivity. It helps the decisionmaker to understand the parameters that have the biggest impact on the project.
The following example introduces a single variable sensitivity analysis. Please note that here we assume variables are independent and have no effect on each other. For example, it is assumed that the magnitude of initial investment doesn’t affect the operating costs.
Example 61:
For a project, the most expected case includes an initial investment of 150,000 dollars at the present time, an annual income of 40,000 for five years (starting from the first year), and a salvage value of 80,000. Evaluate the sensitivity of the project ROR to 20% and 40% increase and decrease in initial investment, annual income, project life, and salvage value.
Beforetax cash flow of this investment can be shown as:
$150,000

$40,000

$40,000  $40,000  $40,000  $40,000 
$80,000



0

1

2

3

4 
5

The most expected ROR based on the most expected initial investment, annual income, and salvage value can be calculated as:
$150,000=40,000\left(P/{A}_{i,5}\right)+80,000\left(P/{F}_{i,5}\right)$
The most expected ROR will be 20.5%.
A) Sensitivity Analysis of initial investment
$$\begin{array}{l}\text{40\%decrease:initialinvestment}=\left(10.4\right)*150,000=90,000\\ 90,000=40,000\left(P/{A}_{i,5}\right)+80,000\left(P/{F}_{i,5}\right)\\ ROR=43.5\%\end{array}$$ $$\begin{array}{l}\text{20\%decrease:initialinvestment}=\left(10.2\right)*150,000=120,000\\ 120,000=40,000\left(P/{A}_{i,5}\right)+80,000\left(P/{F}_{i,5}\right)\\ ROR=29.6\%\end{array}$$ $$\begin{array}{l}\text{20\%increase:initialinvestment}=\left(1+0.2\right)*150,000=180,000\\ 180,000=40,000\left(P/{A}_{i,5}\right)+80,000\left(P/{F}_{i,5}\right)\\ ROR=13.8\%\end{array}$$ $$\begin{array}{l}\text{40\%increase:initialinvestment}=\left(1+0.4\right)*150,000=210,000\\ 210,000=40,000\left(P/{A}_{i,5}\right)+\text{}80,000\left(P/{F}_{i,5}\right)\\ ROR=8.6\%\end{array}$$Initial investment  Change in prediction  ROR  Percentage change in 20.5% ROR prediction 

90,000  40%  43.5%  112.7% 
120,000  20%  29.6%  44.8% 
150,000  0  20.5%  0% 
180,000  20%  13.8%  32.6% 
210,000  40%  8.6%  57.8% 
As you can see, changes in ROR with respect to changes in initial investment are considerably high. In general, parameters that are close to time zero have a higher impact on the ROR of the project.
B) Sensitivity Analysis of project life
$$\begin{array}{l}\text{40\%decrease:projectlife}=3\text{years}\\ 150,000=40,000\left(P/{A}_{i,3}\right)+80,000\left(P/{F}_{i,3}\right)\\ ROR=12.9\%\end{array}$$ $$\begin{array}{l}\text{20\%decrease:projectlife}=4\text{years}\\ 150,000=40,000\left(P/{A}_{i,4}\right)+80,000\left(P/{F}_{i,4}\right)\\ ROR=17.7\%\end{array}$$ $$\begin{array}{l}20\%\text{}increase:\text{}project\text{}life=6\text{}years\\ 150,000=40,000\left(P/{A}_{i,6}\right)+80,000\left(P/{F}_{i,6}\right)\\ ROR=22.2\%\end{array}$$ $$\begin{array}{l}\text{40\%increase:projectlife}=7\text{}years\\ 150,000=40,000\left(P/{A}_{i,7}\right)+80,000\left(P/{F}_{i,7}\right)\\ ROR=23.4\%\end{array}$$Project life  Change in prediction  ROR  Percentage change in 20.5% ROR Prediction 

3  40%  12.9%  36.6% 
4  20%  17.7%  13.5% 
5  0  20.5%  0% 
6  20%  22.2%  8.7% 
7  40%  23.4%  14.5% 
Note that changes in the project ROR become smaller as the project life gets longer.
C) Sensitivity Analysis of annual income
$$\begin{array}{l}\text{40\%decrease:annualincome}=\left(10.4\right)*40,000=24,000\\ 150,000=24,000\left(P/{A}_{i,5}\right)+80,000\left(P/{F}_{i,5}\right)\\ ROR=8.1\%\end{array}$$ $$\begin{array}{l}\text{20\%decrease:annualincome}=\left(10.2\right)*40,000=32,000\\ 150,000=32,000\left(P/{A}_{i,5}\right)+80,000\left(P/{F}_{i,5}\right)\\ ROR=14.3\%\end{array}$$ $$\begin{array}{l}\text{20\%increase:annualincome}=\left(1+0.2\right)*40,000=48,000\\ 150,000=48,000\left(P/{A}_{i,5}\right)+80,000\left(P/{F}_{i,5}\right)\\ ROR=26.5\%\end{array}$$ $$\begin{array}{l}\text{40\%increase:annualincome}=\left(1+0.4\right)*40,000=56,000\\ 150,000=56,000\left(P/{A}_{i,5}\right)+80,000\left(P/{F}_{i,5}\right)\\ ROR=32.4\%\end{array}$$Annual income  Change in prediction  ROR  Percentage change in 20.5% ROR prediction 

24,000  40%  8.1%  60.6% 
32,000  20%  14.3%  30.0% 
40,000  0  20.5%  0% 
48,000  20%  26.5%  29.5% 
56,000  40%  32.4%  58.5% 
Changes in annual income also have a significant effect on ROR because these changes start happening close to present time.
D) Sensitivity Analysis of salvage value
$$\begin{array}{l}\text{40\%decrease:annualincome}=\left(10.4\right)*80,000=48,000\\ 150,000=40,000\left(P/{A}_{i,5}\right)+48,000\left(P/{F}_{i,5}\right)\\ ROR=17.0\%\end{array}$$ $$\begin{array}{l}\text{20\%decrease:annualincome}=\left(10.2\right)*80,000=64,000\\ 150,000=40,000\left(P/{A}_{i,5}\right)+64,000\left(P/{F}_{i,5}\right)\\ ROR=18.8\%\end{array}$$ $$\begin{array}{l}\text{20\%increase:annualincome}=\left(1+0.2\right)*80,000=96,000\\ 150,000=40,000\left(P/{A}_{i,5}\right)+96,000\left(P/{F}_{i,5}\right)\\ ROR=22.0\%\end{array}$$ $$\begin{array}{l}\text{40\%increase:annualincome}=\left(1+0.4\right)*80,000=112,000\\ 150,000=40,000\left(P/{A}_{i,5}\right)+112,000\left(P/{F}_{i,5}\right)\\ ROR=23.5\%\end{array}$$salvage value  Change in prediction  ROR  Percentage change in 20.5% ROR prediction 

48,000  40%  17.0%  17.0% 
64,000  20%  18.8%  8.2% 
80,000  0  20.5%  0% 
96,000  20%  22.0%  17.7% 
112,000  40%  23.5%  14.8% 
We can conclude that salvage value has the least effect on the ROR of the project because salvage value is the last amount in the future and its present value is relatively small compared to other amounts.
The following figure displays a tornado chart that is a very useful method to graphically summarize the results of sensitivity analysis. The right and left hand side of each bar indicate the maximum and the minimum ROR that each parameter generates when changed from 40% to +40%.
Type  Rate of Return Range 

Initial investment  8.6%  43.5% 
Project life  13%  23.4% 
Annual payment  8.1%  32.4% 
Salvage  17%  23.5% 
Please watch the following video (18:02): Sensitivity Analysis.
If you are interested, the following video (10:48) explains how to draw a tornado chart in Microsoft Excel (please watch from 6:10 to 9:00).