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Now give this answer completely without skipping even a small step of calculation and try not to give answer in long paragraphs and answer the questions by following these instructions: ?Solution should be in proper format. ?A complete answer starts with the problem's objective OR addition of available data/ information Or the right approach. ?Add all necessary steps/supporting explanations until the answer is complete. ?If there are any formulae, you should mention it. ?Write calculations in a format so it can be easily pasted in chegg equation render tool. ?Do not use Python or any code-based for calculations (Important). ?Double check the solution to accomplish the correct concept, explanation, and calculations before submitting. ✅ The Final Answer should be a simple and clear summary of the solution in lucid language. ✅ The Final Answer Step is to include the crux of the entire solution. ✅ The final answer step contains the result of the solution only in a clear & concise manner. ✅ The Final Answer should be a simple and clear summary of the solution in lucid language. ✅ Final Answer Step is to include the crux of the entire solution. ✅Final answer step contains the result of the solution only in clear & concise manner.Donald has made up his mind—he wants a pool in the family’s backyard! He figures his kids and spouse will be thrilled. However, to cover the cost of the pool, they'll have to pack up and live away from home for a few weeks during the summer to rent their home to vacationers. He crunched the numbers based on the following estimates. 1 Costof pool/installation $52,302 2. Lifeof the pool (no salvage value) 20 years 3. Annual net cash inflows from renting (net of cash expenses for renting and pool maintenance) ~~ $6,900 4. Taxrate 24% 5. Averagerateof return 8% Donald's daughter, Sarah, found the above information written on a sheet of paper in his office, along with the following notes. “This is a no-brainer! We'll recover the cost of this pool in just 7 years, even though we plan to live here until we're old and gray, or at least as long as the pool hangs on. If we can rent our house out for just 3 weeks each year, it'll be almost pure profit that we can put toward paying off the pool. I studied time value of money a little bit in college; the present value of this investment is positive, so therefore we should definitely move ahead with it. My family will think they're in heaven” Click here to view the factor table To generate even more benefit from the pool, Donald is considering a full one-month (4 weeks) rental for the family's home. He realizes there are pros and cons to this scenario but figures it would be worth it to have one more option to discuss when he brings this up with his family. Under this scenario, determine the NPV of the investment. (Round present value factor calculations to 5 decimal places, e.g. 1.25124 and final answer to O decimal places e.g. 125.25. Enter negative amounts using either a negative sign preceding the number e.g. -45.25 or parentheses e.g. (45.25).) NPV §

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Now give this answer completely without skipping even a small step of calculation and try not to give answer in long paragraphs and answer the questions by following these instructions: ?Solution should be in proper format. ?A complete answer starts with the problem's objective OR addition of available data/ information Or the right approach. ?Add all necessary steps/supporting explanations until the answer is complete. ?If there are any formulae, you should mention it. ?Write calculations in a format so it can be easily pasted in chegg equation render tool. ?Do not use Python or any code-based for calculations (Important). ?Double check the solution to accomplish the correct concept, explanation, and calculations before submitting. ✅ The Final Answer should be a simple and clear summary of the solution in lucid language. ✅ The Final Answer Step is to include the crux of the entire solution. ✅ The final answer step contains the result of the solution only in a clear & concise manner. ✅ The Final Answer should be a simple and clear summary of the solution in lucid language. ✅ Final Answer Step is to include the crux of the entire solution. ✅Final answer step contains the result of the solution only in clear & concise manner.Uploaded ImageUploaded ImageDonald has made up his mind—he wants a pool in the family’s backyard! He figures his kids and spouse will be thrilled. However, to cover the cost of the pool, they'll have to pack up and live away from home for a few weeks during the summer to rent their home to vacationers. He crunched the numbers based on the following estimates. 1 Costof pool/installation $52,302 2. Lifeof the pool (no salvage value) 20 years 3. Annual net cash inflows from renting (net of cash expenses for renting and pool maintenance) ~~ $6,900 4. Taxrate 24% 5. Averagerateof return 8% Donald's daughter, Sarah, found the above information written on a sheet of paper in his office, along with the following notes. “This is a no-brainer! We'll recover the cost of this pool in just 7 years, even though we plan to live here until we're old and gray, or at least as long as the pool hangs on. If we can rent our house out for just 3 weeks each year, it'll be almost pure profit that we can put toward paying off the pool. I studied time value of money a little bit in college; the present value of this investment is positive, so therefore we should definitely move ahead with it. My family will think they're in heaven” Click here to view the factor table To generate even more benefit from the pool, Donald is considering a full one-month (4 weeks) rental for the family's home. He realizes there are pros and cons to this scenario but figures it would be worth it to have one more option to discuss when he brings this up with his family. Under this scenario, determine the NPV of the investment. (Round present value factor calculations to 5 decimal places, e.g. 1.25124 and final answer to O decimal places e.g. 125.25. Enter negative amounts using either a negative sign preceding the number e.g. -45.25 or parentheses e.g. (45.25).) NPV §

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Asked by: PADMA
Created at: 2025-10-20 08:31:00

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## Objective: Calculate the Net Present Value (NPV) of the pool investment considering the costs, cash inflows, and additional rental scenario. --- ## Given Data: - Cost of pool/installation: **$52,302** - Life of the pool: **20 years** - Annual net cash inflows: **$6,900** - Tax rate: **24%** - Average rate of return: **8%** - Discount rate: **8%** (assumed same as rate of return for NPV calculation) - Additional rental for 4 weeks: **1 month rental (4 weeks)** --- ## Approach: ### 1. Calculate the present value of cash inflows over 20 years ### 2. Calculate the net cash inflows after tax ### 3. Calculate the NPV considering the initial investment and the cash inflows ### 4. Adjust for the additional rental benefit (NPV of rental income) --- ## Step-by-step calculations: ### 1. Present Value of Annuity (cash inflows for 20 years): **Formula:** \[ PV = A \times \frac{1 - (1 + r)^{-n}}{r} \] where: - \(A = 6,900\) - \(r = 8\% = .08\) - \(n = 20\) **Calculation:** \[ PV_{cash} = 690 \times \frac{1 - (1 + .08)^{-20}}{.08} \] \[ PV_{cash} = 690 \times \frac{1 - (1.08)^{-20}}{.08} \] \[ (1.08)^{-20} \approx .2145 \] \[ PV_{cash} = 690 \times \frac{1 - .2145}{.08} \] \[ PV_{cash} = 690 \times \frac{.7855}{.08} \] \[ PV_{cash} = 690 \times 9.819 \] \[ PV_{cash} \approx 67785.9 \] --- ### 2. Calculate after-tax cash inflows: **Tax effect:** \[ \text{After-tax cash inflow} = A \times (1 - \text{tax rate}) \] \[ = 690 \times (1 - .24) \] \[ = 690 \times .76 \] \[ = 5244 \] **PV of after-tax cash inflows:** \[ PV_{after\_tax} = 5244 \times \frac{1 - (1 + .08)^{-20}}{.08} \] \[ PV_{after\_tax} = 5244 \times 9.819 \] \[ \approx 51523.4 \] --- ### 3. Calculate NPV of the investment: **Initial investment:** \(-52,302\) **NPV calculation:** \[ NPV = PV_{after\_tax} - \text{initial investment} \] \[ NPV = 51523.4 - 52302 \] \[ NPV \approx -778.6 \] --- ### 4. Calculate the NPV of the additional 1-month rental benefit: Assuming the rental for 4 weeks is equivalent to 1 month of rental income. - **Estimate monthly rent**: derived from the annual net cash inflows: \[ \text{Monthly net cash inflow} = \frac{690}{12} \approx 575 \] - **Present value of 1 month rental income** (assuming same rate): \[ PV_{rental} = 575 \times \frac{1}{(1 + .08/12)} \] But for simplicity, considering rental income as a single cash flow: \[ PV_{rental} = 575 \times \left( \frac{1}{(1 + .08/12)} \right) \] or, more straightforwardly, since it's for 1 month: \[ PV_{rental} \approx 575 / (1 + .0066667) \] \[ PV_{rental} \approx 575 / 1.0066667 \] \[ \approx 571.4 \] --- ## **Final NPV including rental:** \[ NPV_{total} = NPV + PV_{rental} \] \[ NPV_{total} = -778.6 + 571.4 \] \[ \approx -207.2 \] --- ## **Final Answer:** **The net present value (NPV) of the investment, including the additional rental benefit, is approximately \(-\$207.20\).**

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