Present value and discounting

Mathematical statement

Relative to a fixed rate i per period, positive accumulation factor 1+i and period count n, the present value of a future amount F is defined as F/(1+i)^n. This inverts fixed-period-rate compounding. Cash flows, variable rates and fees require a different model.

Representative formula (exact TeX)

P(F)

Assumptions

• Period units, rate conventions and n agree; 1+i>0 and there are no additional cash flows, fees, taxes or rounding during the modeled interval.
• Exercises use nonnegative exact rational amounts. Model computations are not quotes, investment advice or market forecasts.

Symbol scope

• P — present-value operator
• F — future amount

Intuition, definition, and misconceptions

One-period discounting divides F by 1+i; discounting over n periods divides by its nth power.

Compounding and then discounting with the same factor and period count should recover the principal, giving an exact roundtrip check.

The formula does not directly apply when rate periods differ, the factor is nonpositive or several cash flows occur.

Do not multiply by the accumulation factor when discounting or mix annual rates, monthly rates and incompatible period counts.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Compound-interest model — Discounting reverses fixed-accumulation-factor compounding.
• Fraction — Present value uses exact division.
• Equation — Solve the future-value equation backward for present value.

Verification boundary

Only a fixed-period-rate model without cash flows is checked; this is not investment advice or a market forecast.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add cash-flow timing, annuities, nominal/effective rates and sensitivity.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.