# Sensitivity Analysis

In corporate finance, sensitivity analysis refers to an analysis of how sensitive the result of a capital budgeting technique is to a variable, say discount rate, 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.

Sensitivity analysis differs from scenario analysis in that scenario analysis is more complex because it allows us to change more than one variables at once.

## Steps in Conducting Sensitivity Analysis

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 V
_{1}) of the input for which we intend to measure 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 V
_{2}) 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 evaluate sensitivity for another input (say cash flows growth rate) while keeping the rest of inputs constant. We continue 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 produced 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 WACC, calculate net present value at WACC of 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).

To work out the NPV, we need to find the annual net cash flows:

Incremental Cash Flows = 365 × $1M × (1 - 3%) = $354 million

Next, we can work out the NPV at 12.1% discount rate:

NPV_{12.1%} = $354M × | 1 − (1 + 12.1%)^{-100} | − $2,000M = $926M |

12.1% |

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%). This translates to a sensitivity of -2.4.

Sensitivity = | -24.01% | = -2.4 |

10% |

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 value 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.

by Obaidullah Jan, ACA, CFA and last modified on