91Ó°ÊÓ

You are considering the purchase of an old fire station, which you plan to convert to an indoor playground. The fire station can be purchased for \(\$ 200,000,\) and the playground will generate lifetime profits (excluding the cost of the building) of \(\$ 700,000\). (Assume that those profits are all realized one year after opening.) However, there is a \(20 \%\) chance that the city council will re-zone the district to exclude establishments such as yours; a hearing is scheduled for the coming year, and if your building is re- zoned, your profit will be zero. Assume that there is no other building currently under consideration. a. Assume an interest rate of \(10 \% .\) Calculate the net present value of opening the playground today. Note that the cost of purchasing the building today is certain, but the benefits are uncertain. b. Calculate the net present value today of opening the playground in one year, after the zoning issues have been decided. Note that the benefits of opening the playground are uncertain today, but will be certain in one year. c. Based on your answers to (a) and (b), should you open the playground today, or should you wait until the zoning commission reaches its decision?

Step by step solution

01

Calculate Expected Profit if Opened Today

The expected profit from opening today is calculated by considering the probability of not being rezoned. There is a 20% chance of rezoning which leaves an 80% chance of generating profits. The expected profit is therefore: \(0.8 \times 700,000 = 560,000\).

02

Calculate Net Present Value (NPV) for Opening Today

First, subtract the purchase cost from expected profits to find the net benefits: \(560,000 - 200,000 = 360,000\). To find the NPV, we discount the net benefits at the interest rate of 10% for one year: \(\text{NPV} = \frac{360,000}{1.1} = 327,273\).

03

Determine Profit if Waiting One Year

If you wait, you will either have a profit of \(700,000\) or \(0\), with probabilities 80% and 20% respectively after zoning is decided. If not rezoned, profit is a certainty: \(0.8 \times 700,000 + 0.2 \times 0 = 560,000\).

04

Calculate Present Value for Known Profit in One Year

If the district is only zoned after one year, the decision will be clear. Discounting the certain profit one year at 10%: \(\frac{560,000}{1.1} = 509,091\).

05

Compare Options and Conclusion

The NPV from opening today is \(327,273\) whereas in waiting a year it becomes \(509,091\). Comparatively, waiting yields a higher value. Thus, it is better to wait for the zoning decision.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Profit

When considering a business investment, one key factor is the Expected Profit. This involves calculating the potential returns based on various probabilities of different outcomes. In the fire station exercise, there are two possible scenarios. First, there's the likelihood that the city council will not rezone the area, allowing you to operate your playground successfully. There's an 80% chance of this happening, resulting in a profit of $700,000.
The second scenario is if the rezoning occurs, leading to zero profit. To find the Expected Profit, you multiply the profit ($700,000) by the probability of occurrence (80%). This gives an Expected Profit of $560,000, as it accounts for the risk of rezoning. This calculation helps in making informed decisions about investments under uncertainty.

Interest Rate

Interest Rate plays a crucial role in determining the Net Present Value (NPV) of a future cash flow. This rate acts as the 'cost' of investment capital over time. In terms of the NPV, it helps determine how much a future profit is worth in today's dollars.
In our example, we assume an interest rate of 10%, which indicates the rate at which we discount future cash flows. A higher interest rate lowers the present value of expected profits, while a lower rate would increase it. Understanding the dynamics of interest rates is essential for evaluating the profitability of an investment when payments or profits occur over time.

Discounting

Discounting is the process of determining the present value of a future amount. In financial analysis, it is essential for converting future money into today's terms. This helps compare different cash flows that occur at various times.
Using the interest rate of 10%, the discounting process involves dividing the future value by the discount factor \(1 + r\), where \(r\) is the interest rate. For example, to calculate the NPV of the Expected Profit of \(560,000 realized in one year, you divide it by 1.1 (1 + 0.1). This yields a present value of \)509,091.
Discounting is vital because it reflects the time value of money, suggesting that money today is worth more than the same amount in the future. It’s a cornerstone for NPV calculations, which helps in choosing the most financially viable option.

Zoning

Zoning refers to the regulations that govern land use and what types of developments are permitted in certain areas. It can significantly impact the profitability of a project and must be considered when planning a new business.
In this scenario, zoning is a deciding factor, as a 20% chance exists that the city council might rezone the area, disallowing the playground. This risk introduces uncertainty into the investment decision.
Understanding zoning risks enables investors to evaluate whether an investment is too risky or if it's worth waiting for regulatory decisions to ensure clear business prospects. Addressing potential zoning issues beforehand can protect against unwanted surprises and financial losses.