Question 5

Question

Suppose you have the following two mutually exclusive projects that you can carry out on the corner of 39th Street and Walnut Street: Build a daycare center or a health spa.

Suppose the day care center has the following cash flows: An immediate cash outlay of $5,000 followed by inflows of $2500 in each of the next 3 years and zero thereafter.

Suppose the health spa has the following cash flows: An immediate outlay of $5000 followed by inflows of nothing in year one, $1000 in year 2 and $7100 in year 3 and zero thereafter.

If you base your investment decision on which investment has the highest NPV, which do you choose when the discount rate is 5%?

Day care center

Health Spa

Answer 1

To determine which project to choose based on the Net Present Value (NPV) method, we need to calculate the NPV for both the daycare center and the health spa at a discount rate of 5%.

Cash Flows:

1. Day Care Center

   • Year 0: -$5,000 (outflow)
   • Year 1: +$2,500 (inflow)
   • Year 2: +$2,500 (inflow)
   • Year 3: +$2,500 (inflow)
   • Year 4 and beyond: $0

2. Health Spa

   • Year 0: -$5,000 (outflow)
   • Year 1: $0 (inflow)
   • Year 2: +$1,000 (inflow)
   • Year 3: +$7,100 (inflow)
   • Year 4 and beyond: $0

Step 1: Calculate NPV for each project

Day Care Center NPV Calculation

\[ NPV = \sum_{t=0}^{n} \frac{C_t}{(1+r)^t} \]

Where:

• \( C_t \) = cash flow at time \( t \)
• \( r \) = discount rate (5% = 0.05)
• \( n \) = number of periods

Calculating NPV for Day Care Center:

\[ NPV = -5000 + \frac{2500}{(1+0.05)^1} + \frac{2500}{(1+0.05)^2} + \frac{2500}{(1+0.05)^3} \]

Calculating each term:

1. Year 0: -5000
2. Year 1: \( \frac{2500}{1.05^1} = \frac{2500}{1.05} \approx 2380.95 \)
3. Year 2: \( \frac{2500}{1.05^2} = \frac{2500}{1.1025} \approx 2270.39 \)
4. Year 3: \( \frac{2500}{1.05^3} = \frac{2500}{1.157625} \approx 2165.83 \)

Now sum these values:

\[ NPV_{Day Care} \approx -5000 + 2380.95 + 2270.39 + 2165.83 \] \[ NPV_{Day Care} \approx -5000 + 6817.17 \approx 1817.17 \]

Health Spa NPV Calculation

Calculating NPV for Health Spa:

\[ NPV = -5000 + \frac{0}{(1+0.05)^1} + \frac{1000}{(1+0.05)^2} + \frac{7100}{(1+0.05)^3} \]

Calculating each important term:

1. Year 0: -5000
2. Year 1: 0 (cash flow is 0)
3. Year 2: \( \frac{1000}{1.05^2} = \frac{1000}{1.1025} \approx 907.03 \)
4. Year 3: \( \frac{7100}{1.05^3} = \frac{7100}{1.157625} \approx 6135.11 \)

Now sum these values:

\[ NPV_{Health Spa} \approx -5000 + 0 + 907.03 + 6135.11 \] \[ NPV_{Health Spa} \approx -5000 + 7042.14 \approx 2042.14 \]

Conclusion

Now we will compare the NPVs:

• NPV for Day Care Center: \( 1817.17 \)
• NPV for Health Spa: \( 2042.14 \)

Since \( 2042.14 > 1817.17 \), the project with the highest NPV is the Health Spa.

Therefore, based on NPV, you should choose the Health Spa.