Created At: [[2024-12-08]] ## Explanation The Net Present Value (NPV) formula is a fundamental concept in finance that helps determine the value today of a series of future cash flows. The basic idea is that money you receive or pay in the future is worth less than money you have today, due to the time value of money. This loss of value over time is typically captured by something called a “discount rate,” which accounts for factors like inflation, risk, and the opportunity cost of not having the money available to invest immediately. ### The Formula The formula you’ve provided: $ \text{NPV} = \sum_{i=1}^n \frac{CF_i}{(1 + d)^i} $ can be broken down into several components: ### 1. Future Cash Flows ($CF_i$) These are the amounts of money you expect to receive (inflows) or pay (outflows) at different points in the future. Each $CF_i$ corresponds to the net cash flow occurring at time period $i$. For example, $CF_1$ might be the cash flow at the end of the first year, $CF_2$ at the end of the second year, and so on. ### 2. Discount Rate ($d$) The discount rate is the rate of return or the required rate of return used to “discount” future cash flows back to the present. It encapsulates: - **Opportunity Cost**: The return you could earn if you invested your money elsewhere. - **Risk**: The uncertainty or variability in receiving the expected cash flows. - **Time Value of Money**: The principle that a dollar today is generally worth more than a dollar received in the future. ### 3. Discounting Factor ($(1+d)^i$) This is the mechanism by which we translate a future value into a present value. The farther into the future the cash flow occurs, the larger the exponent $i$ becomes, and thus the more the cash flow is “discounted.” ### Putting it All Together - For each future time period $i$, you take the expected cash flow $CF_i$ and divide it by $(1+d)^i$, effectively shrinking its value back to what it would be worth in today’s terms. - You then sum all these present values of the future cash flows. The result is a single number: the Net Present Value. ## Why NPV Matters ### Investment Decisions NPV helps determine whether a project or investment is worthwhile. If the NPV is positive, it means the investment, considering the cost of capital and risk, is expected to generate more value than it costs. If the NPV is negative, the investment might destroy value rather than create it. ### Comparing Projects NPV allows you to compare different projects or investments on a level playing field, because each one’s future cash flows are translated into current dollar values. In summary, the NPV formula takes a series of future cash flows and “pulls” them back to today’s value using the chosen discount rate. By summing these present values, you get a measure of how much value (positive or negative) the project or investment is likely to add from today’s perspective.