Present worth from a known future amount If S is the amount available after n interest periods from principal P at a discrete compound interest rate i, what is the correct formula for the present worth of S?

Introduction / Context:Discounting a future amount back to present worth is a core step in process economic evaluations such as net present value and profitability analysis. Correctly applying compound interest is essential to avoid under- or over-valuing future cash flows.

Given Data / Assumptions:

• Discrete compounding at interest rate i per period.
• Future amount S is realized at the end of n periods.
• No interim payments between 0 and n.

Concept / Approach:The present worth relation for single-payment compound interest is: S = P * (1 + i)^n. Rearranging for P yields P = S / (1 + i)^n. This is the standard present worth factor applied to discount a single future sum to time zero.

Step-by-Step Solution:

Verification / Alternative check:Finance tables list the present worth factor (P/F, i, n) = 1 / (1 + i)^n; multiplying by S gives P.

Why Other Options Are Wrong:

• (1 + i)^n / S: Inverted; not a monetary amount.
• S / (1 + i n): Uses simple interest, not compound interest.
• S / (1 + n) i: Algebraically incorrect and dimensionally inconsistent.

Common Pitfalls:Confusing simple with compound interest; neglecting that discounting must mirror the compounding convention used to grow the amount.

Final Answer:S / (1 + i)^n