- Use the NPV formula \(\sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}\) to evaluate both projects at a discount rate of \(11\%\).
- Calculate NPV for Project A: \(-320 + \frac{150}{1.11^1} + \frac{150}{1.11^2} + \frac{150}{1.11^3} + \frac{150}{1.11^4} = \boxed{\$145.37}\).
- Calculate NPV for Project B: \(-320 + \frac{220}{1.11^1} + \frac{220}{1.11^2} + \frac{220}{1.11^3} = \boxed{\$217.62}\).
- Compare the results; Project B has the higher NPV, so choose \(\boxed{\text{Project B}}\).
Explanation
- Understanding the NPV Formula and Data
To determine the Net Present Value (NPV) of both projects, we use the formula for the sum of discounted cash flows:
\[NPV = \sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}\]
where \(CF_t\) is the cash flow at the end of year \(t\), \(r\) is the discount rate of \(11\%\) (or \(0.11\)), and \(n\) is the duration of the project in years. We have:
- For Project A: Initial investment of \(\$320\), followed by cash flows of \(\$150\) per year for \(4\) years.
- For Project B: Initial investment of \(\$320\), followed by cash flows of \(\$220\) per year for \(3\) years.
- Calculating NPV for Project A
We calculate the NPV for Project A:
\[NPV_A = -320 + \frac{150}{(1.11)^1} + \frac{150}{(1.11)^2} + \frac{150}{(1.11)^3} + \frac{150}{(1.11)^4}\]
\[NPV_A = -320 + 135.1351 + 121.7434 + 109.6787 + 98.8097\]
\[NPV_A \approx 145.37\]
The NPV for Project A is \(\$145.37\).
- Calculating NPV for Project B
We calculate the NPV for Project B:
\[NPV_B = -320 + \frac{220}{(1.11)^1} + \frac{220}{(1.11)^2} + \frac{220}{(1.11)^3}\]
\[NPV_B = -320 + 198.1982 + 178.5569 + 160.8621\]
\[NPV_B \approx 217.62\]
The NPV for Project B is \(\$217.62\).
- Comparison and Final Conclusion
Comparing the two projects, we see that the NPV for Project B (\(\$217.62\)) is significantly higher than the NPV for Project A (\(\$145.37\)). Since both projects have positive NPVs, they are both technically profitable, but we should choose the one that adds the most value. Therefore, Project B is the better investment. The final results are:
- NPV of Project A: \(\boxed{\$145.37}\)
- NPV of Project B: \(\boxed{\$217.62}\)
- Preferred project: \(\boxed{\text{Project B}}\)
Examples
Net Present Value (NPV) is a fundamental tool for businesses and individuals to evaluate the profitability of an investment. By discounting future cash flows back to their present value, we can account for the time value of money—the principle that a dollar today is worth more than a dollar tomorrow. For instance, if you are deciding whether to invest in a new solar panel system or a high-efficiency HVAC unit, you would calculate the NPV of the savings each system generates over its lifetime, using a specific interest rate, to determine which one offers the best long-term financial return. This ensures that you aren't just looking at total cash back, but also how long you have to wait to receive those funds.