91Ó°ÊÓ

The payback period is a simple measure that helps in determining how long it will take to recover the initial investment from the cash inflows of a project. In this exercise, we calculated the payback period by setting it equal to the project life, denoted by \( N \). The goal is to find the annual payment, \( C \), that balances the total cost at exactly the end of the project's lifetime.
To achieve a payback period equal to the lifetime of the project, or \( N \), the annual payment \( C \) should meet the condition where it covers the initial cost \( I \) over \( N \) years. We calculate it as:
\[C = \frac{I}{N}\]
This formula implies that each year, the cash flow returns an equal slice of the total initial cost, allowing the project to break even by the end of its life.

Net Present Value is a widely used financial metric that helps assess the profitability of a project. It involves calculating all the cash inflows and outflows over the project's life and discounting them back to their present value using a specified required return, \( R \).
The aim here is to make sure the project's NPV is positive, indicating it generates more value than its cost. The equation for NPV is:
\[NPV = \sum_{t=1}^N \frac{C}{(1+R)^t} - I\]
A positive NPV means that the present value of cash inflows \( C \) over \( N \) years, after accounting for the required return, exceeds the initial investment \( I \). To find \( C \) for achieving a profitable NPV, we rearrange the formula:
\[C > \frac{I(1+R)^t}{\sum_{t=1}^N (1+R)^t}\]
By fulfilling this condition, the project ensures that the returns outweigh the costs when adjusted for the time value of money.

The Benefit-Cost Ratio (BCR) is a decision-making tool used to evaluate the relationship between the benefits and costs of a project. For the given scenario, we aim for a BCR of 2, meaning the project's benefits should be twice its costs.
The equation for determining this is:
\[\frac{ \sum_{t=1}^N \frac{C}{(1+R)^t}}{I} = 2\]
To find the annual cash flow \( C \) that fulfills a BCR of 2, indicating a project where benefits are double the initial investment, we rearrange the formula:
\[C = \frac{2I(1+R)^t}{\sum_{t=1}^N (1+R)^t}\]
This configuration underscores that by properly managing and structuring cash inflows, the benefits can significantly outweigh the costs, making the project highly attractive from an economic standpoint.