Time value of money (end-of-year payments): If an amount R is paid at the end of every year for n years at an interest rate i, what is the net present value (NPV) of this annuity?

Introduction / Context:Evaluating capital projects requires discounting future cash flows. The present value of a uniform series of end-of-year payments (an ordinary annuity) appears in equipment leases, service contracts, and maintenance reserves in chemical plant economics.

Given Data / Assumptions:

• Annuity with constant payment R at the end of each year.
• Number of payments n; interest (discount) rate i per year.
• Ordinary annuity (end-of-period), not annuity due.

Concept / Approach:The present value of an ordinary annuity is the sum of discounted payments. The closed-form factor is PV = R * [1 − (1 + i)^(−n)] / i. This factor is widely tabulated as the present worth of an annuity (P/A, i, n).

Step-by-Step Solution:

Verification / Alternative check:Check limiting case: as n → ∞ with i > 0, PV → R / i, consistent with a perpetuity. For n = 1, PV = R / (1 + i), matching the formula.

Why Other Options Are Wrong:

Common Pitfalls:Confusing ordinary annuity with annuity due (beginning-of-year payments), which multiplies by (1 + i). Also, mixing up present vs. future value factors.

Final Answer:R * [1 − (1 + i)^(−n)] / i