## Definition

Sensitivity analysis is used to evaluate how sensitive the output variable is to the change in one of the variables while other input variables remain unchanged. Sensitivity analysis is widely used in capital budgeting decisions to assess how the change in such inputs as sales, variable costs, fixed costs, cost of capital, and marginal tax rate will affect such outputs as net present value (NPV) of a project, internal rate of return (IRR), and discounted payback period. It also provides a better understanding of the risks associated with a project.

## How to conduct sensitivity analysis

Several steps are involved in sensitivity analysis of a project.

- Determine input and output variables.
- Calculate the baseline value of the output variable using the baseline input variables value.
- Change the value of one of the input variables while others remain constant, and calculate the new value of the output variable.
- Calculate the percentage change of both output and input variable compared to baseline values.
- Calculate the sensitivity of the output variable to the change in the input variable using the formula below.

### Formula

Sensitivity = % change in output / % change in input

Steps three through five should be repeated for the other input variables.

Sensitivity analysis allows identification of input variables that represent the greatest vulnerability for the project.

## Example

A company is considering a project with initial costs of $500,000 involving purchase of new machinery with a useful life of 5 years. The after-tax cost of capital is 16%, and the marginal tax rate is 30%. The other key parameters of a project are presented in the table below.

The depreciation expense of $40,000 per annum is included in fixed costs.

The management of a company will conduct sensitivity analysis of a project.

### Solution

__Step 1__

Management designates the net present value (NPV) and the internal rate of return (IRR) as outputs. The inputs are:

- Fixed costs
- Sales in units
- Sales price per unit
- Variable costs per unit

__Step 2__

Let’s find the baseline NPV and IRR. The calculation of discounted cash flows is shown in the table below.

The calculation of discounted net cash flow for designated Year 1 is shown below.

S = 20,000 × $35 = $700,000

TVC = 20,000 × $22 = $440,000

EBT = $700,000 – $440,000 – $100,000 = $160,000

T = $160,000 × 30% = $48,000

NI = $160,000 – $48,000 = $112,000

NCF = $112,000 + $40,000 = $152,000

DNCF = $152,000 / (1+0.16)^{1} = $131,034

The discounted net cash flows in other designated years are calculated in the same manner.

The baseline net present value of a project is as follows:

Baseline NPV = -$500,000 + $131,034 + $136,891 + $160,164 + $137,686 + $111,030 = $176,805

The baseline IRR is 29.03%. You can read here how to calculate IRR in Excel.

__Step 3__

Let’s assume that fixed costs will be 5% higher than baseline values. Provided other inputs remain constant, the discounted net cash flows will be as follows:

NPV = -$500,000 + $128,017 + $134,238 + $157,810 + $135,579 + $109,113 = $164,757

% change in NPV = ($164,757 – $176,805) / $176,805 = -6.81%

Sensitivity of NPV = -6,81% / 5% = -1.362

This means that with an increase in fixed costs of 1%, the NPV of a project will decrease by 1.362%, and vice versa, if fixed costs are reduced by 1%, the NPV will increase by 1.362%.

The new value of IRR is 28.17%.

% change in IRR = (28.17%-29.03%) / 29.03% = -2.96%

Sensitivity of IRR = -2.96% / 5% = -0.592

__Step 4__

Let’s assume that sales in units will be 5% higher than baseline values. Provided other inputs remain constant, the discounted net cash flows will be as follows:

Please note that the change in the number of units sold affects both sales and total variable costs figures!

NPV = -$500,000 + $138,879 + $144,902 + $169,246 + $145,573 + $117,545 = $216,145

% change in NPV = ($216,145 – $176,805) / $176,805 = 22.25%

Sensitivity of NPV = 22.25% / 5% = 4.450

Thus, an increase in sales by 1% will cause an increase in NPV by 4.450% and vice versa.

The new value of IRR is 31.73%.

% change in IRR = (31.73%-29.03%) / 29.03% = 9.30%

Sensitivity of IRR = 9.30% / 5% = 1.860

__Step 5__

Let’s assume that the sales price of a unit will be 5% higher than baseline values. Provided other inputs remain constant, the discounted net cash flows will be as follows:

Please note that the change in sales price affects sales figures only!

NPV = -$500,000 + $152,155 + $157,491 + $183,170 + $157,896 + $128,277 = $278,989

% change in NPV = ($278,989-$176,805) / $176,805 = 57.79%

Sensitivity of NPV = 57.79% / 5% = 11.558

An increase in the sales price by 1% will result in an increase in the NPV by 11.558%, and if the sales price drops by 1%, the NPV of a project will decrease by 11.558%.

The new value of IRR is 35.95%

% change in IRR = (35.95%-29.03%) / 29.03% = 23.84%

Sensitivity of IRR = 9.30% / 5% = 4.768

__Step 6__

Let’s assume that variable costs per unit will be 5% higher than baseline values. Provided other inputs remain constant, the discounted net cash flows will be as follows:

NPV = -$500,000 + $117,759 + $124,301 + $146,240 + $125,363 + $100,298 = $113,961

% change in NPV = ($113,961 – $176,805) / $176,805 = -35.54%

Sensitivity of NPV = -35,54% / 5% = -7.109

An increase in the variable costs per unit by 1% will result in an decrease in the NPV by 7.109% and vice versa.

The new value of IRR is 24.57%.

% change in IRR = (24.57% – 29.03%) / 29.03% = -15.36%

Sensitivity of IRR = -15.36% / 5% = -3.072

### Results of sensitivity analysis

Sensitivity analysis shows that the NPV and IRR of a project are most vulnerable to the change in the sales price and variable costs per unit and less vulnerable to the change in fixed costs and sales in units.

### Graph

The results of sensitivity analysis in the example above is illustrated in the graph below.

As we can see, management of a company should estimate the sales price and variable costs as accurately as possible because they have the greatest impact on the net present value of a project.