Question
Question 4
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 15%?
Day care center
Health Spa
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 15%?
Day care center
Health Spa
Answers
GPT-4o mini
To determine which project has the higher Net Present Value (NPV), we will calculate the NPV for both the daycare center and the health spa using the provided cash flows and a discount rate of 15%.
### NPV Formula
The formula for NPV is:
\[
NPV = \sum \frac{C_t}{(1+r)^t} - C_0
\]
Where:
- \( C_t \) = cash inflow during the period \( t \)
- \( r \) = discount rate
- \( C_0 \) = initial investment (cash outflow)
### Cash Flows for Each Project
1. **Daycare Center:**
- Initial Investment (Year 0): \$5,000
- Year 1: \$2,500
- Year 2: \$2,500
- Year 3: \$2,500
- Year 4 and beyond: \$0
2. **Health Spa:**
- Initial Investment (Year 0): \$5,000
- Year 1: \$0
- Year 2: \$1,000
- Year 3: \$7,100
- Year 4 and beyond: \$0
### NPV Calculation for Daycare Center
\[
\text{NPV} = -5000 + \frac{2500}{(1+0.15)^1} + \frac{2500}{(1+0.15)^2} + \frac{2500}{(1+0.15)^3}
\]
Calculating each term:
1. Year 1: \(\frac{2500}{(1+0.15)^1} = \frac{2500}{1.15} \approx 2173.91\)
2. Year 2: \(\frac{2500}{(1+0.15)^2} = \frac{2500}{1.3225} \approx 1890.40\)
3. Year 3: \(\frac{2500}{(1+0.15)^3} = \frac{2500}{1.520875} \approx 1643.37\)
Now summing these:
\[
\text{NPV}_{\text{daycare}} = -5000 + 2173.91 + 1890.40 + 1643.37 \approx -5000 + 5707.68 \approx -292.32
\]
### NPV Calculation for Health Spa
\[
\text{NPV} = -5000 + \frac{0}{(1+0.15)^1} + \frac{1000}{(1+0.15)^2} + \frac{7100}{(1+0.15)^3}
\]
Calculating each term:
1. Year 1: \(\frac{0}{(1+0.15)^1} = 0\)
2. Year 2: \(\frac{1000}{(1+0.15)^2} = \frac{1000}{1.3225} \approx 755.78\)
3. Year 3: \(\frac{7100}{(1+0.15)^3} = \frac{7100}{1.520875} \approx 4663.18\)
Now summing these:
\[
\text{NPV}_{\text{health spa}} = -5000 + 0 + 755.78 + 4663.18 \approx -5000 + 5418.96 \approx -581.04
\]
### Summary of NPVs
1. **NPV of Daycare Center:** \(-292.32\)
2. **NPV of Health Spa:** \(-581.04\)
### Conclusion
When comparing the NPVs, the daycare center has a higher NPV than the health spa (less negative). Therefore, **you should choose the daycare center** based on the NPV criterion.
### NPV Formula
The formula for NPV is:
\[
NPV = \sum \frac{C_t}{(1+r)^t} - C_0
\]
Where:
- \( C_t \) = cash inflow during the period \( t \)
- \( r \) = discount rate
- \( C_0 \) = initial investment (cash outflow)
### Cash Flows for Each Project
1. **Daycare Center:**
- Initial Investment (Year 0): \$5,000
- Year 1: \$2,500
- Year 2: \$2,500
- Year 3: \$2,500
- Year 4 and beyond: \$0
2. **Health Spa:**
- Initial Investment (Year 0): \$5,000
- Year 1: \$0
- Year 2: \$1,000
- Year 3: \$7,100
- Year 4 and beyond: \$0
### NPV Calculation for Daycare Center
\[
\text{NPV} = -5000 + \frac{2500}{(1+0.15)^1} + \frac{2500}{(1+0.15)^2} + \frac{2500}{(1+0.15)^3}
\]
Calculating each term:
1. Year 1: \(\frac{2500}{(1+0.15)^1} = \frac{2500}{1.15} \approx 2173.91\)
2. Year 2: \(\frac{2500}{(1+0.15)^2} = \frac{2500}{1.3225} \approx 1890.40\)
3. Year 3: \(\frac{2500}{(1+0.15)^3} = \frac{2500}{1.520875} \approx 1643.37\)
Now summing these:
\[
\text{NPV}_{\text{daycare}} = -5000 + 2173.91 + 1890.40 + 1643.37 \approx -5000 + 5707.68 \approx -292.32
\]
### NPV Calculation for Health Spa
\[
\text{NPV} = -5000 + \frac{0}{(1+0.15)^1} + \frac{1000}{(1+0.15)^2} + \frac{7100}{(1+0.15)^3}
\]
Calculating each term:
1. Year 1: \(\frac{0}{(1+0.15)^1} = 0\)
2. Year 2: \(\frac{1000}{(1+0.15)^2} = \frac{1000}{1.3225} \approx 755.78\)
3. Year 3: \(\frac{7100}{(1+0.15)^3} = \frac{7100}{1.520875} \approx 4663.18\)
Now summing these:
\[
\text{NPV}_{\text{health spa}} = -5000 + 0 + 755.78 + 4663.18 \approx -5000 + 5418.96 \approx -581.04
\]
### Summary of NPVs
1. **NPV of Daycare Center:** \(-292.32\)
2. **NPV of Health Spa:** \(-581.04\)
### Conclusion
When comparing the NPVs, the daycare center has a higher NPV than the health spa (less negative). Therefore, **you should choose the daycare center** based on the NPV criterion.