# Sensitivity Analysis

Sensitivity analysis is an analysis that finds out how sensitive an output is to any change in an input while keeping other inputs constant.

In corporate finance, it refers to an analysis of how each of the input variables in a capital budgeting decision (such as discount rate, cash flows growth rate, tax rate, etc.) affects the net present value, IRR or any other output while keeping other variables constant.

Sensitivity analysis is useful because it tells the model user how dependent the output value is on each input. It gives him an idea of how much room he has for each variable to go adverse. It helps in assessing risk.

## Explanation: Measurement of Sensitivity

We conduct sensitivity analysis by an approach outlined below:

- Find the base case output (for example the net present value) at the base case value (say V1) of the input for which we intend to measure the sensitivity (such as discount rate). We keep all other inputs in the model (such as cash flow growth rate, tax rate, depreciation, etc.) constant.
- Find the value of output at a new value of the input (say V2) while keeping other inputs constant.
- Find the percentage change in the output and the percentage change in the input.
- Find sensitivity by dividing the percentage change in output by the percentage change in input.

In second round, we test sensitivity for another input (say cash flows growth rate) while keeping the rest of inputs constant. We carry on this process till we get the sensitivity figure for each of the inputs. The higher the sensitivity figure, the more sensitive the output is to any change in that input and vice versa.

## Example

Great Wall Beatle is a company that operates in the mountainous country of Zhongua and constructs tunnels for the country's major road developers. The company is in the process of submitting its bid for construction of the country's longest tunnel on the interstate expressway. The tunnel would be 20 kilometer long and the company bids to receive $1 from each vehicle that crosses the tunnel for 100 years. The company's chief engineer came up with an NPV of $1,218 million for the project assuming cash flows are received at the year end. His estimates include: weighted average cost of capital of 11%, daily traffic of 1,000,000 vehicles, daily operating expenses as 3% of total revenue and initial cost of $2 billion. Find how sensitive the net present value is to each input.

__Solution__

To find sensitivity of net present value to the estimate of WACC, calculate net present value assuming WACC is 12.1% instead of 11% (while keeping daily traffic at 1,000,000, daily operating expenses at 3% and initial costs at $2,000 million). This gives us a net present value of $926 million. Percentage change in output is -24.01% (($926 million × $1,218 million) ÷ $1,218 million) while the corresponding change in input is 10% ((11.1% − 11%) ÷ 11%). Percentage change in output per 1% change in input is -2.4

Similarly, we find that sensitivity estimates for daily traffic, daily operating expenses and initial costs are 2.64, -0.08 and -1.64.

The calculations not only show the relationship between output and input but it also tells how sensitive output is to each input. A negative sensitivity means that the output (net present value) decreases with an increase in that input (such as discount rate).

We conclude that the net present is most sensitive to the estimate of daily traffic and least sensitive to the estimate of daily operating expenses. Knowing the importance of the daily traffic figure in the output, the company should try to estimate the daily traffic with as much accuracy as possible.