$3-1$ Scenario Analysis Family Security is considering the introduction of a child security product consisting of tiny Global Positioning System $($GPS$)$ trackers that can be inserted in the sole of a child$$s shoe. The trackers allow parents to track the child if he or she were ever lost or abducted. The estimates, plus or minus $10\%,$ associated with this new product are as follows:

Unit price: $$125$

Variable costs per unit: $$75$

Fixed costs: $$250,000$ per year

Expected sales: $10,000$ per year

Because this is a new product line, the firm$$s analysts are not confident in their estimates and would like to know how well the investment would fare if the estimates on the items listed above are $10\%$ higher or $10\%$ lower than expected. Assume that this new product line will require an initial outlay of $$1$ million, with no working capital investment, and will last for ten years, being depreciated down to zero using straight$-$line depreciation. In addition, the firm$$s management uses a discount rate of $10\%$ and a $34\%$ tax rate in its project analyses.

Calculate the project$$s NPV under each of the following sets of assumptions: $(1)$ the best$-$case scenario $($use the high estimates$$unit price $10\%$ above expected, variable costs $10\%$ less than expected, fixed costs $10\%$ less than expected, and unit sales $10\%$ higher than expected$),(2)$ the base case using expected values, and $(3)$ the worst$-$case scenario.

Given your estimates of the range of NPVs for the investment, what is your assessment of the investment$$s potential?

What are the limitations of this type of scenario analysis? $($Hint: What are the chances that all variables will deviate from their expected values in a best$-$ or worst$-$case scenario?$)$