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?
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AR * [(1 + i)^n − 1] / i
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BR * [1 − (1 + i)^(−n)] / i
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CR(1 + i)^n
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DR / (1 + i)^n
Answer
Correct Answer: R * [1 − (1 + i)^(−n)] / i
Explanation
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:
Write PV = Σ (R / (1 + i)^k) for k = 1 to n.Recognize geometric series with ratio 1/(1 + i).Sum to obtain PV = R * [1 − (1 + i)^(−n)] / i.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:
R * [(1 + i)^n − 1] / i: this is the future value of an annuity (end-of-year compounding), not present value.R(1 + i)^n: future value of a single payment, not an annuity.R / (1 + i)^n: present value of a single payment at year n, not a series.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