Question

The production optimization is evaluated on the basis of discounted revenue to be generated by the projects. The net present value (NPV) for calculating the discounted revenue is defined by
\[ NPV = NPV_R - \text{cost} \] where, \(NPV_R\) = present value of cash flow discounted at a given rate \(i\).
If \(\Delta R_n\) is the annual incremental revenue after optimization for the \(n^{th}\) year, and \(m\) is the remaining life of the project at the end of \(n^{th}\) year, then which ONE of the following options for \(NPV_R\) is CORRECT?

• A. \(\displaystyle NPV_R = \sum_{n=1}^{m} \frac{(1+i)^n}{\Delta R_n}\)
• B. \(\displaystyle NPV_R = \sum_{n=1}^{m} \frac{\Delta R_n}{(1+i)^{n-1}}\)
• C. \(\displaystyle NPV_R = \sum_{n=1}^{m} \left[\frac{\Delta R_n}{(1+i)}\right]^n\)
• D. \(\displaystyle NPV_R = \sum_{n=1}^{m} \frac{\Delta R_n}{(1+i)^n}\)

Hint:

Always discount future revenues using \((1+i)^n\). The farther the cash flow, the higher the discounting applied.

Answer 1

The Correct Option is D

Step 1: Understanding present value of revenue.
The present value (PV) of a future revenue \(\Delta R_n\) occurring at the end of year \(n\) must be discounted using the factor \((1+i)^n\), where \(i\) is the discount rate. This adjusts future cash flows to their equivalent value today.
Step 2: Applying the discounting concept.
The standard formula for the present value of cash flow in year \(n\) is: \[ PV_n = \frac{\Delta R_n}{(1+i)^n} \] This ensures that future revenue is reduced appropriately based on how far in the future it is received.
Step 3: Summing over project life.
For a project with \(m\) remaining years of revenue, the total discounted revenue (i.e., \(NPV_R\)) is the sum of discounted annual revenues: \[ NPV_R = \sum_{n=1}^{m} \frac{\Delta R_n}{(1+i)^n} \]
Step 4: Option analysis.
Options (A), (B), and (C) use incorrect discounting relationships. Only option (D) correctly applies the discount factor to reduce future revenue to its present value.
Step 5: Conclusion.
Thus, the correct formula for discounted incremental revenue is given in option (D).